src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author hoelzl
Fri Apr 22 17:22:29 2016 +0200 (2016-04-22)
changeset 63053 4a108f280dc2
parent 63052 c968bce3921e
child 63072 eb5d493a9e03
permissions -rw-r--r--
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
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(*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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section \<open>Elementary linear algebra on Euclidean spaces\<close>
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theory Linear_Algebra
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imports
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  Euclidean_Space
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  "~~/src/HOL/Library/Infinite_Set"
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begin
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subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
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definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
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  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
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lemma hull_same: "S s \<Longrightarrow> S hull s = s"
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  unfolding hull_def by auto
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lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
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  unfolding hull_def Ball_def by auto
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lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
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  using hull_same[of S s] hull_in[of S s] by metis
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lemma hull_hull [simp]: "S hull (S hull s) = S hull s"
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  unfolding hull_def by blast
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lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
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  unfolding hull_def by blast
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lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
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  unfolding hull_def by blast
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lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
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  unfolding hull_def by blast
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lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
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  unfolding hull_def by blast
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lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
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  unfolding hull_def by blast
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lemma hull_UNIV [simp]: "S hull UNIV = UNIV"
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  unfolding hull_def by auto
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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)"
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  unfolding hull_def by auto
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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"
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  using hull_minimal[of S "{x. P x}" Q]
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  by (auto simp add: subset_eq)
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lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
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  by (metis hull_subset subset_eq)
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lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
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  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
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lemma hull_union:
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  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
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  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
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  apply rule
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  apply (rule hull_mono)
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  unfolding Un_subset_iff
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  apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
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  apply (rule hull_minimal)
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  apply (metis hull_union_subset)
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  apply (metis hull_in T)
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  done
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lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
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  unfolding hull_def by blast
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lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
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  by (metis hull_redundant_eq)
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subsection \<open>Linear functions.\<close>
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lemma linear_iff:
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  "linear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x)"
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  (is "linear f \<longleftrightarrow> ?rhs")
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proof
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  assume "linear f"
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  then interpret f: linear f .
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  show "?rhs" by (simp add: f.add f.scaleR)
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next
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  assume "?rhs"
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  then show "linear f" by unfold_locales simp_all
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qed
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lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
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  by (simp add: linear_iff algebra_simps)
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lemma linear_compose_scaleR: "linear f \<Longrightarrow> linear (\<lambda>x. f x *\<^sub>R c)"
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  by (simp add: linear_iff scaleR_add_left)
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lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
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  by (simp add: linear_iff)
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lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
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  by (simp add: linear_iff algebra_simps)
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lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
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  by (simp add: linear_iff algebra_simps)
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lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
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  by (simp add: linear_iff)
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lemma linear_id: "linear id"
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  by (simp add: linear_iff id_def)
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lemma linear_zero: "linear (\<lambda>x. 0)"
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  by (simp add: linear_iff)
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lemma linear_compose_setsum:
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  assumes lS: "\<forall>a \<in> S. linear (f a)"
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  shows "linear (\<lambda>x. setsum (\<lambda>a. f a x) S)"
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proof (cases "finite S")
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  case True
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  then show ?thesis
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    using lS by induct (simp_all add: linear_zero linear_compose_add)
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next
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  case False
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  then show ?thesis
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    by (simp add: linear_zero)
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qed
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lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
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  unfolding linear_iff
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  apply clarsimp
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  apply (erule allE[where x="0::'a"])
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  apply simp
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  done
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lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
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  by (rule linear.scaleR)
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lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
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  using linear_cmul [where c="-1"] by simp
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lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
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  by (metis linear_iff)
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lemma linear_sub: "linear f \<Longrightarrow> f (x - y) = f x - f y"
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  using linear_add [of f x "- y"] by (simp add: linear_neg)
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lemma linear_setsum:
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  assumes f: "linear f"
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  shows "f (setsum g S) = setsum (f \<circ> g) S"
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proof (cases "finite S")
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  case True
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  then show ?thesis
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    by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
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next
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  case False
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  then show ?thesis
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    by (simp add: linear_0 [OF f])
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qed
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lemma linear_setsum_mul:
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  assumes lin: "linear f"
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  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
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  using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
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  by simp
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lemma linear_injective_0:
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  assumes lin: "linear f"
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  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
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proof -
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  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
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    by (simp add: inj_on_def)
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  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
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    by simp
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  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
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    by (simp add: linear_sub[OF lin])
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  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
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    by auto
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  finally show ?thesis .
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qed
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lemma linear_scaleR  [simp]: "linear (\<lambda>x. scaleR c x)"
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  by (simp add: linear_iff scaleR_add_right)
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lemma linear_scaleR_left [simp]: "linear (\<lambda>r. scaleR r x)"
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  by (simp add: linear_iff scaleR_add_left)
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lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
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  by (simp add: inj_on_def)
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lemma linear_add_cmul:
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  assumes "linear f"
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  shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
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  using linear_add[of f] linear_cmul[of f] assms by simp
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subsection \<open>Subspaces of vector spaces\<close>
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definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
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  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)"
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definition (in real_vector) "span S = (subspace hull S)"
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definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
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abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
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text \<open>Closure properties of subspaces.\<close>
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lemma subspace_UNIV[simp]: "subspace UNIV"
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  by (simp add: subspace_def)
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lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
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  by (metis subspace_def)
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lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
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  by (metis subspace_def)
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lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
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  by (metis subspace_def)
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lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
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  by (metis scaleR_minus1_left subspace_mul)
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lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
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  using subspace_add [of S x "- y"] by (simp add: subspace_neg)
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lemma (in real_vector) subspace_setsum:
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  assumes sA: "subspace A"
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    and f: "\<forall>x\<in>B. f x \<in> A"
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  shows "setsum f B \<in> A"
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proof (cases "finite B")
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  case True
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  then show ?thesis
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    using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
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qed (simp add: subspace_0 [OF sA])
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lemma subspace_trivial: "subspace {0}"
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  by (simp add: subspace_def)
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lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
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  by (simp add: subspace_def)
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lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
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  unfolding subspace_def zero_prod_def by simp
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text \<open>Properties of span.\<close>
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lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
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  by (metis span_def hull_mono)
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lemma (in real_vector) subspace_span: "subspace (span S)"
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  unfolding span_def
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  apply (rule hull_in)
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  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
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  apply auto
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  done
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lemma (in real_vector) span_clauses:
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  "a \<in> S \<Longrightarrow> a \<in> span S"
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  "0 \<in> span S"
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  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
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  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
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  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
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lemma span_unique:
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  "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
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  unfolding span_def by (rule hull_unique)
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lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
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  unfolding span_def by (rule hull_minimal)
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lemma span_UNIV: "span UNIV = UNIV"
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  by (intro span_unique) auto
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lemma (in real_vector) span_induct:
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  assumes x: "x \<in> span S"
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    and P: "subspace P"
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    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
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  shows "x \<in> P"
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proof -
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  from SP have SP': "S \<subseteq> P"
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    by (simp add: subset_eq)
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  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
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  show "x \<in> P"
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    by (metis subset_eq)
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qed
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lemma span_empty[simp]: "span {} = {0}"
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  apply (simp add: span_def)
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  apply (rule hull_unique)
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  apply (auto simp add: subspace_def)
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  done
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lemma (in real_vector) independent_empty [iff]: "independent {}"
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  by (simp add: dependent_def)
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lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
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  unfolding dependent_def by auto
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lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
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  apply (clarsimp simp add: dependent_def span_mono)
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  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
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  apply force
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  apply (rule span_mono)
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  apply auto
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  done
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huffman@44133
   307
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   308
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   309
wenzelm@49711
   310
lemma (in real_vector) span_induct':
hoelzl@63050
   311
  "\<forall>x \<in> S. P x \<Longrightarrow> subspace {x. P x} \<Longrightarrow> \<forall>x \<in> span S. P x"
hoelzl@63050
   312
  unfolding span_def by (rule hull_induct) auto
huffman@44133
   313
wenzelm@56444
   314
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
wenzelm@53406
   315
where
huffman@44170
   316
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   317
| span_induct_alt_help_S:
wenzelm@53406
   318
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   319
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   320
huffman@44133
   321
lemma span_induct_alt':
wenzelm@53406
   322
  assumes h0: "h 0"
wenzelm@53406
   323
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   324
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   325
proof -
wenzelm@53406
   326
  {
wenzelm@53406
   327
    fix x :: 'a
wenzelm@53406
   328
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   329
    have "h x"
huffman@44133
   330
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   331
      apply (rule h0)
wenzelm@53406
   332
      apply (rule hS)
wenzelm@53406
   333
      apply assumption
wenzelm@53406
   334
      apply assumption
wenzelm@53406
   335
      done
wenzelm@53406
   336
  }
huffman@44133
   337
  note th0 = this
wenzelm@53406
   338
  {
wenzelm@53406
   339
    fix x
wenzelm@53406
   340
    assume x: "x \<in> span S"
huffman@44170
   341
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   342
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   343
      show "x \<in> span S" by (rule x)
wenzelm@49522
   344
    next
wenzelm@53406
   345
      fix x
wenzelm@53406
   346
      assume xS: "x \<in> S"
wenzelm@53406
   347
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   348
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   349
        by simp
wenzelm@49522
   350
    next
wenzelm@49522
   351
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   352
      moreover
wenzelm@53406
   353
      {
wenzelm@53406
   354
        fix x y
wenzelm@49522
   355
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   356
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   357
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   358
          apply simp
haftmann@57512
   359
          unfolding add.assoc
wenzelm@49522
   360
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   361
          apply assumption
wenzelm@49522
   362
          apply simp
wenzelm@53406
   363
          done
wenzelm@53406
   364
      }
wenzelm@49522
   365
      moreover
wenzelm@53406
   366
      {
wenzelm@53406
   367
        fix c x
wenzelm@49522
   368
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   369
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   370
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   371
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   372
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   373
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   374
          apply assumption
wenzelm@49522
   375
          apply simp
wenzelm@49522
   376
          done }
wenzelm@53406
   377
      ultimately show "subspace (span_induct_alt_help S)"
wenzelm@49522
   378
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   379
    qed
wenzelm@53406
   380
  }
huffman@44133
   381
  with th0 show ?thesis by blast
huffman@44133
   382
qed
huffman@44133
   383
huffman@44133
   384
lemma span_induct_alt:
wenzelm@53406
   385
  assumes h0: "h 0"
wenzelm@53406
   386
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   387
    and x: "x \<in> span S"
huffman@44133
   388
  shows "h x"
wenzelm@49522
   389
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   390
wenzelm@60420
   391
text \<open>Individual closure properties.\<close>
huffman@44133
   392
huffman@44133
   393
lemma span_span: "span (span A) = span A"
huffman@44133
   394
  unfolding span_def hull_hull ..
huffman@44133
   395
wenzelm@53406
   396
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   397
  by (metis span_clauses(1))
wenzelm@53406
   398
wenzelm@53406
   399
lemma (in real_vector) span_0: "0 \<in> span S"
wenzelm@53406
   400
  by (metis subspace_span subspace_0)
huffman@44133
   401
huffman@44133
   402
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   403
  by (metis subset_eq span_superset)
huffman@44133
   404
hoelzl@63053
   405
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
hoelzl@63053
   406
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
hoelzl@63053
   407
  by (auto simp add: span_span)
hoelzl@63053
   408
wenzelm@53406
   409
lemma (in real_vector) dependent_0:
wenzelm@53406
   410
  assumes "0 \<in> A"
wenzelm@53406
   411
  shows "dependent A"
wenzelm@53406
   412
  unfolding dependent_def
wenzelm@53406
   413
  using assms span_0
lp15@60162
   414
  by auto
wenzelm@53406
   415
wenzelm@53406
   416
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   417
  by (metis subspace_add subspace_span)
huffman@44133
   418
wenzelm@53406
   419
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   420
  by (metis subspace_span subspace_mul)
huffman@44133
   421
wenzelm@53406
   422
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   423
  by (metis subspace_neg subspace_span)
huffman@44133
   424
wenzelm@53406
   425
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
huffman@44133
   426
  by (metis subspace_span subspace_sub)
huffman@44133
   427
huffman@56196
   428
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
huffman@56196
   429
  by (rule subspace_setsum [OF subspace_span])
huffman@44133
   430
huffman@44133
   431
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   432
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   433
hoelzl@63050
   434
text \<open>The key breakdown property.\<close>
hoelzl@63050
   435
hoelzl@63050
   436
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
hoelzl@63050
   437
proof (rule span_unique)
hoelzl@63050
   438
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
hoelzl@63050
   439
    by (fast intro: scaleR_one [symmetric])
hoelzl@63050
   440
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
hoelzl@63050
   441
    unfolding subspace_def
hoelzl@63050
   442
    by (auto intro: scaleR_add_left [symmetric])
hoelzl@63050
   443
next
hoelzl@63050
   444
  fix T
hoelzl@63050
   445
  assume "{x} \<subseteq> T" and "subspace T"
hoelzl@63050
   446
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
hoelzl@63050
   447
    unfolding subspace_def by auto
hoelzl@63050
   448
qed
hoelzl@63050
   449
wenzelm@60420
   450
text \<open>Mapping under linear image.\<close>
huffman@44133
   451
hoelzl@63050
   452
lemma subspace_linear_image:
hoelzl@63050
   453
  assumes lf: "linear f"
hoelzl@63050
   454
    and sS: "subspace S"
hoelzl@63050
   455
  shows "subspace (f ` S)"
hoelzl@63050
   456
  using lf sS linear_0[OF lf]
hoelzl@63050
   457
  unfolding linear_iff subspace_def
hoelzl@63050
   458
  apply (auto simp add: image_iff)
hoelzl@63050
   459
  apply (rule_tac x="x + y" in bexI)
hoelzl@63050
   460
  apply auto
hoelzl@63050
   461
  apply (rule_tac x="c *\<^sub>R x" in bexI)
hoelzl@63050
   462
  apply auto
hoelzl@63050
   463
  done
hoelzl@63050
   464
hoelzl@63050
   465
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
hoelzl@63050
   466
  by (auto simp add: subspace_def linear_iff linear_0[of f])
hoelzl@63050
   467
hoelzl@63050
   468
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
hoelzl@63050
   469
  by (auto simp add: subspace_def linear_iff linear_0[of f])
hoelzl@63050
   470
huffman@44521
   471
lemma span_linear_image:
huffman@44521
   472
  assumes lf: "linear f"
wenzelm@56444
   473
  shows "span (f ` S) = f ` span S"
huffman@44521
   474
proof (rule span_unique)
huffman@44521
   475
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   476
    by (intro image_mono span_inc)
huffman@44521
   477
  show "subspace (f ` span S)"
huffman@44521
   478
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   479
next
wenzelm@53406
   480
  fix T
wenzelm@53406
   481
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   482
  then show "f ` span S \<subseteq> T"
huffman@44521
   483
    unfolding image_subset_iff_subset_vimage
huffman@44521
   484
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   485
qed
huffman@44521
   486
hoelzl@63053
   487
lemma spans_image:
hoelzl@63053
   488
  assumes lf: "linear f"
hoelzl@63053
   489
    and VB: "V \<subseteq> span B"
hoelzl@63053
   490
  shows "f ` V \<subseteq> span (f ` B)"
hoelzl@63053
   491
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
hoelzl@63053
   492
huffman@44521
   493
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   494
proof (rule span_unique)
huffman@44521
   495
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   496
    by safe (force intro: span_clauses)+
huffman@44521
   497
next
huffman@44521
   498
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   499
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   500
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   501
    by (intro subspace_Times subspace_span)
huffman@44521
   502
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   503
    by (rule subspace_linear_image)
huffman@44521
   504
next
wenzelm@49711
   505
  fix T
wenzelm@49711
   506
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   507
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   508
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   509
qed
huffman@44133
   510
wenzelm@49522
   511
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   512
proof -
huffman@44521
   513
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   514
    unfolding span_union span_singleton
huffman@44521
   515
    apply safe
huffman@44521
   516
    apply (rule_tac x=k in exI, simp)
huffman@44521
   517
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   518
    apply auto
huffman@44521
   519
    done
wenzelm@49522
   520
  then show ?thesis by simp
huffman@44521
   521
qed
huffman@44521
   522
huffman@44133
   523
lemma span_breakdown:
wenzelm@53406
   524
  assumes bS: "b \<in> S"
wenzelm@53406
   525
    and aS: "a \<in> span S"
huffman@44521
   526
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   527
  using assms span_insert [of b "S - {b}"]
huffman@44521
   528
  by (simp add: insert_absorb)
huffman@44133
   529
wenzelm@53406
   530
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
   531
  by (simp add: span_insert)
huffman@44133
   532
wenzelm@60420
   533
text \<open>Hence some "reversal" results.\<close>
huffman@44133
   534
huffman@44133
   535
lemma in_span_insert:
wenzelm@49711
   536
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
   537
    and na: "a \<notin> span S"
huffman@44133
   538
  shows "b \<in> span (insert a S)"
wenzelm@49663
   539
proof -
huffman@55910
   540
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
   541
    unfolding span_insert by fast
wenzelm@53406
   542
  show ?thesis
wenzelm@53406
   543
  proof (cases "k = 0")
wenzelm@53406
   544
    case True
huffman@55910
   545
    with k have "a \<in> span S" by simp
huffman@55910
   546
    with na show ?thesis by simp
wenzelm@53406
   547
  next
wenzelm@53406
   548
    case False
huffman@55910
   549
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
   550
      by (rule span_mul)
huffman@55910
   551
    then have "b - inverse k *\<^sub>R a \<in> span S"
wenzelm@60420
   552
      using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
huffman@55910
   553
    then show ?thesis
huffman@55910
   554
      unfolding span_insert by fast
wenzelm@53406
   555
  qed
huffman@44133
   556
qed
huffman@44133
   557
huffman@44133
   558
lemma in_span_delete:
huffman@44133
   559
  assumes a: "a \<in> span S"
wenzelm@53716
   560
    and na: "a \<notin> span (S - {b})"
huffman@44133
   561
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
   562
  apply (rule in_span_insert)
huffman@44133
   563
  apply (rule set_rev_mp)
huffman@44133
   564
  apply (rule a)
huffman@44133
   565
  apply (rule span_mono)
huffman@44133
   566
  apply blast
huffman@44133
   567
  apply (rule na)
huffman@44133
   568
  done
huffman@44133
   569
wenzelm@60420
   570
text \<open>Transitivity property.\<close>
huffman@44133
   571
huffman@44521
   572
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
   573
  unfolding span_def by (rule hull_redundant)
huffman@44521
   574
huffman@44133
   575
lemma span_trans:
wenzelm@53406
   576
  assumes x: "x \<in> span S"
wenzelm@53406
   577
    and y: "y \<in> span (insert x S)"
huffman@44133
   578
  shows "y \<in> span S"
huffman@44521
   579
  using assms by (simp only: span_redundant)
huffman@44133
   580
huffman@44133
   581
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
   582
  by (simp only: span_redundant span_0)
huffman@44133
   583
wenzelm@60420
   584
text \<open>An explicit expansion is sometimes needed.\<close>
huffman@44133
   585
huffman@44133
   586
lemma span_explicit:
huffman@44133
   587
  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
   588
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
   589
proof -
wenzelm@53406
   590
  {
wenzelm@53406
   591
    fix x
huffman@55910
   592
    assume "?h x"
huffman@55910
   593
    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
   594
      by blast
huffman@55910
   595
    then have "x \<in> span P"
huffman@55910
   596
      by (auto intro: span_setsum span_mul span_superset)
wenzelm@53406
   597
  }
huffman@44133
   598
  moreover
huffman@55910
   599
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
   600
  proof (rule span_induct_alt')
huffman@55910
   601
    show "?h 0"
huffman@55910
   602
      by (rule exI[where x="{}"], simp)
huffman@44133
   603
  next
huffman@44133
   604
    fix c x y
wenzelm@53406
   605
    assume x: "x \<in> P"
huffman@55910
   606
    assume hy: "?h y"
huffman@44133
   607
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
huffman@44133
   608
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
   609
    let ?S = "insert x S"
wenzelm@49522
   610
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
   611
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
   612
      by blast+
wenzelm@53406
   613
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
   614
    proof cases
wenzelm@53406
   615
      assume xS: "x \<in> S"
huffman@55910
   616
      have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
huffman@55910
   617
        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
huffman@44133
   618
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
huffman@55910
   619
        by (simp add: setsum.remove [OF fS xS] algebra_simps)
huffman@44133
   620
      also have "\<dots> = c*\<^sub>R x + y"
haftmann@57512
   621
        by (simp add: add.commute u)
huffman@44133
   622
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
   623
      then show ?thesis using th0 by blast
wenzelm@53406
   624
    next
wenzelm@53406
   625
      assume xS: "x \<notin> S"
wenzelm@49522
   626
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
   627
        unfolding u[symmetric]
haftmann@57418
   628
        apply (rule setsum.cong)
wenzelm@53406
   629
        using xS
wenzelm@53406
   630
        apply auto
wenzelm@49522
   631
        done
wenzelm@53406
   632
      show ?thesis using fS xS th0
haftmann@57512
   633
        by (simp add: th00 add.commute cong del: if_weak_cong)
wenzelm@53406
   634
    qed
huffman@55910
   635
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
   636
      by fast
huffman@44133
   637
  qed
huffman@44133
   638
  ultimately show ?thesis by blast
huffman@44133
   639
qed
huffman@44133
   640
huffman@44133
   641
lemma dependent_explicit:
wenzelm@49522
   642
  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
wenzelm@49522
   643
  (is "?lhs = ?rhs")
wenzelm@49522
   644
proof -
wenzelm@53406
   645
  {
wenzelm@53406
   646
    assume dP: "dependent P"
huffman@44133
   647
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
huffman@44133
   648
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
   649
      unfolding dependent_def span_explicit by blast
huffman@44133
   650
    let ?S = "insert a S"
huffman@44133
   651
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
   652
    let ?v = a
wenzelm@53406
   653
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
   654
      by blast
wenzelm@53406
   655
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
   656
      by auto
huffman@44133
   657
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
   658
      using fS aS
huffman@55910
   659
      apply simp
huffman@44133
   660
      apply (subst (2) ua[symmetric])
haftmann@57418
   661
      apply (rule setsum.cong)
wenzelm@49522
   662
      apply auto
wenzelm@49522
   663
      done
huffman@55910
   664
    with th0 have ?rhs by fast
wenzelm@49522
   665
  }
huffman@44133
   666
  moreover
wenzelm@53406
   667
  {
wenzelm@53406
   668
    fix S u v
wenzelm@49522
   669
    assume fS: "finite S"
wenzelm@53406
   670
      and SP: "S \<subseteq> P"
wenzelm@53406
   671
      and vS: "v \<in> S"
wenzelm@53406
   672
      and uv: "u v \<noteq> 0"
wenzelm@49522
   673
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
   674
    let ?a = v
huffman@44133
   675
    let ?S = "S - {v}"
huffman@44133
   676
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
   677
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
   678
      using fS SP vS by auto
wenzelm@53406
   679
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
wenzelm@53406
   680
      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
hoelzl@56480
   681
      using fS vS uv by (simp add: setsum_diff1 field_simps)
wenzelm@53406
   682
    also have "\<dots> = ?a"
hoelzl@56479
   683
      unfolding scaleR_right.setsum [symmetric] u using uv by simp
wenzelm@53406
   684
    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
   685
    with th0 have ?lhs
huffman@44133
   686
      unfolding dependent_def span_explicit
huffman@44133
   687
      apply -
huffman@44133
   688
      apply (rule bexI[where x= "?a"])
huffman@44133
   689
      apply (simp_all del: scaleR_minus_left)
huffman@44133
   690
      apply (rule exI[where x= "?S"])
wenzelm@49522
   691
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
   692
      done
wenzelm@49522
   693
  }
huffman@44133
   694
  ultimately show ?thesis by blast
huffman@44133
   695
qed
huffman@44133
   696
hoelzl@63051
   697
lemma span_alt:
hoelzl@63051
   698
  "span B = {(\<Sum>x | f x \<noteq> 0. f x *\<^sub>R x) | f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}"
hoelzl@63051
   699
  unfolding span_explicit
hoelzl@63051
   700
  apply safe
hoelzl@63051
   701
  subgoal for x S u
hoelzl@63051
   702
    by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
hoelzl@63051
   703
        (auto intro!: setsum.mono_neutral_cong_right)
hoelzl@63051
   704
  apply auto
hoelzl@63051
   705
  done
hoelzl@63051
   706
hoelzl@63051
   707
lemma dependent_alt:
hoelzl@63051
   708
  "dependent B \<longleftrightarrow>
hoelzl@63051
   709
    (\<exists>X. finite {x. X x \<noteq> 0} \<and> {x. X x \<noteq> 0} \<subseteq> B \<and> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<and> (\<exists>x. X x \<noteq> 0))"
hoelzl@63051
   710
  unfolding dependent_explicit
hoelzl@63051
   711
  apply safe
hoelzl@63051
   712
  subgoal for S u v
hoelzl@63051
   713
    apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
hoelzl@63051
   714
    apply (subst setsum.mono_neutral_cong_left[where T=S])
hoelzl@63051
   715
    apply (auto intro!: setsum.mono_neutral_cong_right cong: rev_conj_cong)
hoelzl@63051
   716
    done
hoelzl@63051
   717
  apply auto
hoelzl@63051
   718
  done
hoelzl@63051
   719
hoelzl@63051
   720
lemma independent_alt:
hoelzl@63051
   721
  "independent B \<longleftrightarrow>
hoelzl@63051
   722
    (\<forall>X. finite {x. X x \<noteq> 0} \<longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<longrightarrow> (\<forall>x. X x = 0))"
hoelzl@63051
   723
  unfolding dependent_alt by auto
hoelzl@63051
   724
hoelzl@63051
   725
lemma independentD_alt:
hoelzl@63051
   726
  "independent B \<Longrightarrow> finite {x. X x \<noteq> 0} \<Longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<Longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<Longrightarrow> X x = 0"
hoelzl@63051
   727
  unfolding independent_alt by blast
hoelzl@63051
   728
hoelzl@63051
   729
lemma independentD_unique:
hoelzl@63051
   730
  assumes B: "independent B"
hoelzl@63051
   731
    and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   732
    and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   733
    and "(\<Sum>x | X x \<noteq> 0. X x *\<^sub>R x) = (\<Sum>x| Y x \<noteq> 0. Y x *\<^sub>R x)"
hoelzl@63051
   734
  shows "X = Y"
wenzelm@49522
   735
proof -
hoelzl@63051
   736
  have "X x - Y x = 0" for x
hoelzl@63051
   737
    using B
hoelzl@63051
   738
  proof (rule independentD_alt)
hoelzl@63051
   739
    have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
hoelzl@63051
   740
      by auto
hoelzl@63051
   741
    then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   742
      using X Y by (auto dest: finite_subset)
hoelzl@63051
   743
    then have "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. (X v - Y v) *\<^sub>R v)"
hoelzl@63051
   744
      using X Y by (intro setsum.mono_neutral_cong_left) auto
hoelzl@63051
   745
    also have "\<dots> = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) - (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v)"
hoelzl@63051
   746
      by (simp add: scaleR_diff_left setsum_subtractf assms)
hoelzl@63051
   747
    also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) = (\<Sum>v\<in>{S. X S \<noteq> 0}. X v *\<^sub>R v)"
hoelzl@63051
   748
      using X Y by (intro setsum.mono_neutral_cong_right) auto
hoelzl@63051
   749
    also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v) = (\<Sum>v\<in>{S. Y S \<noteq> 0}. Y v *\<^sub>R v)"
hoelzl@63051
   750
      using X Y by (intro setsum.mono_neutral_cong_right) auto
hoelzl@63051
   751
    finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
hoelzl@63051
   752
      using assms by simp
hoelzl@63051
   753
  qed
hoelzl@63051
   754
  then show ?thesis
hoelzl@63051
   755
    by auto
huffman@44133
   756
qed
huffman@44133
   757
wenzelm@60420
   758
text \<open>This is useful for building a basis step-by-step.\<close>
huffman@44133
   759
huffman@44133
   760
lemma independent_insert:
wenzelm@53406
   761
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
   762
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
   763
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
   764
proof (cases "a \<in> S")
wenzelm@53406
   765
  case True
wenzelm@53406
   766
  then show ?thesis
wenzelm@53406
   767
    using insert_absorb[OF True] by simp
wenzelm@53406
   768
next
wenzelm@53406
   769
  case False
wenzelm@53406
   770
  show ?thesis
wenzelm@53406
   771
  proof
wenzelm@53406
   772
    assume i: ?lhs
wenzelm@53406
   773
    then show ?rhs
wenzelm@53406
   774
      using False
wenzelm@53406
   775
      apply simp
wenzelm@53406
   776
      apply (rule conjI)
wenzelm@53406
   777
      apply (rule independent_mono)
wenzelm@53406
   778
      apply assumption
wenzelm@53406
   779
      apply blast
wenzelm@53406
   780
      apply (simp add: dependent_def)
wenzelm@53406
   781
      done
wenzelm@53406
   782
  next
wenzelm@53406
   783
    assume i: ?rhs
wenzelm@53406
   784
    show ?lhs
wenzelm@53406
   785
      using i False
wenzelm@53406
   786
      apply (auto simp add: dependent_def)
lp15@60810
   787
      by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
wenzelm@53406
   788
  qed
huffman@44133
   789
qed
huffman@44133
   790
hoelzl@63051
   791
lemma independent_Union_directed:
hoelzl@63051
   792
  assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
hoelzl@63051
   793
  assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
hoelzl@63051
   794
  shows "independent (\<Union>C)"
hoelzl@63051
   795
proof
hoelzl@63051
   796
  assume "dependent (\<Union>C)"
hoelzl@63051
   797
  then obtain u v S where S: "finite S" "S \<subseteq> \<Union>C" "v \<in> S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
hoelzl@63051
   798
    by (auto simp: dependent_explicit)
hoelzl@63051
   799
hoelzl@63051
   800
  have "S \<noteq> {}"
hoelzl@63051
   801
    using \<open>v \<in> S\<close> by auto
hoelzl@63051
   802
  have "\<exists>c\<in>C. S \<subseteq> c"
hoelzl@63051
   803
    using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
hoelzl@63051
   804
  proof (induction rule: finite_ne_induct)
hoelzl@63051
   805
    case (insert i I)
hoelzl@63051
   806
    then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
hoelzl@63051
   807
      by blast
hoelzl@63051
   808
    from directed[OF cd] cd have "c \<union> d \<in> C"
hoelzl@63051
   809
      by (auto simp: sup.absorb1 sup.absorb2)
hoelzl@63051
   810
    with iI show ?case
hoelzl@63051
   811
      by (intro bexI[of _ "c \<union> d"]) auto
hoelzl@63051
   812
  qed auto
hoelzl@63051
   813
  then obtain c where "c \<in> C" "S \<subseteq> c"
hoelzl@63051
   814
    by auto
hoelzl@63051
   815
  have "dependent c"
hoelzl@63051
   816
    unfolding dependent_explicit
hoelzl@63051
   817
    by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
hoelzl@63051
   818
  with indep[OF \<open>c \<in> C\<close>] show False
hoelzl@63051
   819
    by auto
hoelzl@63051
   820
qed
hoelzl@63051
   821
hoelzl@63051
   822
text \<open>Hence we can create a maximal independent subset.\<close>
hoelzl@63051
   823
hoelzl@63051
   824
lemma maximal_independent_subset_extend:
hoelzl@63051
   825
  assumes "S \<subseteq> V" "independent S"
hoelzl@63051
   826
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
hoelzl@63051
   827
proof -
hoelzl@63051
   828
  let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}"
hoelzl@63051
   829
  have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M"
hoelzl@63051
   830
  proof (rule subset_Zorn)
hoelzl@63051
   831
    fix C :: "'a set set" assume "subset.chain ?C C"
hoelzl@63051
   832
    then have C: "\<And>c. c \<in> C \<Longrightarrow> c \<subseteq> V" "\<And>c. c \<in> C \<Longrightarrow> S \<subseteq> c" "\<And>c. c \<in> C \<Longrightarrow> independent c"
hoelzl@63051
   833
      "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
hoelzl@63051
   834
      unfolding subset.chain_def by blast+
hoelzl@63051
   835
hoelzl@63051
   836
    show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U"
hoelzl@63051
   837
    proof cases
hoelzl@63051
   838
      assume "C = {}" with assms show ?thesis
hoelzl@63051
   839
        by (auto intro!: exI[of _ S])
hoelzl@63051
   840
    next
hoelzl@63051
   841
      assume "C \<noteq> {}"
hoelzl@63051
   842
      with C(2) have "S \<subseteq> \<Union>C"
hoelzl@63051
   843
        by auto
hoelzl@63051
   844
      moreover have "independent (\<Union>C)"
hoelzl@63051
   845
        by (intro independent_Union_directed C)
hoelzl@63051
   846
      moreover have "\<Union>C \<subseteq> V"
hoelzl@63051
   847
        using C by auto
hoelzl@63051
   848
      ultimately show ?thesis
hoelzl@63051
   849
        by auto
hoelzl@63051
   850
    qed
hoelzl@63051
   851
  qed
hoelzl@63051
   852
  then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B"
hoelzl@63051
   853
    and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B"
hoelzl@63051
   854
    by auto
hoelzl@63051
   855
  moreover
hoelzl@63051
   856
  { assume "\<not> V \<subseteq> span B"
hoelzl@63051
   857
    then obtain v where "v \<in> V" "v \<notin> span B"
hoelzl@63051
   858
      by auto
hoelzl@63051
   859
    with B have "independent (insert v B)"
hoelzl@63051
   860
      unfolding independent_insert by auto
hoelzl@63051
   861
    from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close>
hoelzl@63051
   862
    have "v \<in> B"
hoelzl@63051
   863
      by auto
hoelzl@63051
   864
    with \<open>v \<notin> span B\<close> have False
hoelzl@63051
   865
      by (auto intro: span_superset) }
hoelzl@63051
   866
  ultimately show ?thesis
hoelzl@63051
   867
    by (auto intro!: exI[of _ B])
hoelzl@63051
   868
qed
hoelzl@63051
   869
hoelzl@63051
   870
hoelzl@63051
   871
lemma maximal_independent_subset:
hoelzl@63051
   872
  "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
hoelzl@63051
   873
  by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
hoelzl@63051
   874
hoelzl@63051
   875
lemma span_finite:
hoelzl@63051
   876
  assumes fS: "finite S"
hoelzl@63051
   877
  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
hoelzl@63051
   878
  (is "_ = ?rhs")
hoelzl@63051
   879
proof -
hoelzl@63051
   880
  {
hoelzl@63051
   881
    fix y
hoelzl@63051
   882
    assume y: "y \<in> span S"
hoelzl@63051
   883
    from y obtain S' u where fS': "finite S'"
hoelzl@63051
   884
      and SS': "S' \<subseteq> S"
hoelzl@63051
   885
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
hoelzl@63051
   886
      unfolding span_explicit by blast
hoelzl@63051
   887
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
hoelzl@63051
   888
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
hoelzl@63051
   889
      using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
hoelzl@63051
   890
    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
hoelzl@63051
   891
    then have "y \<in> ?rhs" by auto
hoelzl@63051
   892
  }
hoelzl@63051
   893
  moreover
hoelzl@63051
   894
  {
hoelzl@63051
   895
    fix y u
hoelzl@63051
   896
    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
hoelzl@63051
   897
    then have "y \<in> span S" using fS unfolding span_explicit by auto
hoelzl@63051
   898
  }
hoelzl@63051
   899
  ultimately show ?thesis by blast
hoelzl@63051
   900
qed
hoelzl@63051
   901
hoelzl@63052
   902
lemma linear_independent_extend_subspace:
hoelzl@63052
   903
  assumes "independent B"
hoelzl@63052
   904
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)"
hoelzl@63052
   905
proof -
hoelzl@63052
   906
  from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV]
hoelzl@63052
   907
  obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV"
hoelzl@63052
   908
    by (auto simp: top_unique)
hoelzl@63052
   909
  have "\<forall>y. \<exists>X. {x. X x \<noteq> 0} \<subseteq> B' \<and> finite {x. X x \<noteq> 0} \<and> y = (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x)"
hoelzl@63052
   910
    using \<open>span B' = UNIV\<close> unfolding span_alt by auto
hoelzl@63052
   911
  then obtain X where X: "\<And>y. {x. X y x \<noteq> 0} \<subseteq> B'" "\<And>y. finite {x. X y x \<noteq> 0}"
hoelzl@63052
   912
    "\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)"
hoelzl@63052
   913
    unfolding choice_iff by auto
hoelzl@63052
   914
hoelzl@63052
   915
  have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y
hoelzl@63052
   916
    using \<open>independent B'\<close>
hoelzl@63052
   917
  proof (rule independentD_unique)
hoelzl@63052
   918
    have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
hoelzl@63052
   919
      = (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R z)"
hoelzl@63052
   920
      by (intro setsum.mono_neutral_cong_left) (auto intro: X)
hoelzl@63052
   921
    also have "\<dots> = (\<Sum>z\<in>{z. X x z \<noteq> 0}. X x z *\<^sub>R z) + (\<Sum>z\<in>{z. X y z \<noteq> 0}. X y z *\<^sub>R z)"
hoelzl@63052
   922
      by (auto simp add: scaleR_add_left setsum.distrib
hoelzl@63052
   923
               intro!: arg_cong2[where f="op +"]  setsum.mono_neutral_cong_right X)
hoelzl@63052
   924
    also have "\<dots> = x + y"
hoelzl@63052
   925
      by (simp add: X(3)[symmetric])
hoelzl@63052
   926
    also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
hoelzl@63052
   927
      by (rule X(3))
hoelzl@63052
   928
    finally show "(\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z) = (\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)"
hoelzl@63052
   929
      ..
hoelzl@63052
   930
    have "{z. X x z + X y z \<noteq> 0} \<subseteq> {z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}"
hoelzl@63052
   931
      by auto
hoelzl@63052
   932
    then show "finite {z. X x z + X y z \<noteq> 0}" "{xa. X x xa + X y xa \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   933
        "finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   934
      using X(1) by (auto dest: finite_subset intro: X)
hoelzl@63052
   935
  qed
hoelzl@63052
   936
hoelzl@63052
   937
  have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c
hoelzl@63052
   938
    using \<open>independent B'\<close>
hoelzl@63052
   939
  proof (rule independentD_unique)
hoelzl@63052
   940
    show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   941
      "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
hoelzl@63052
   942
      using X(1,2) by auto
hoelzl@63052
   943
    show "(\<Sum>z | X (c *\<^sub>R x) z \<noteq> 0. X (c *\<^sub>R x) z *\<^sub>R z) = (\<Sum>z | c * X x z \<noteq> 0. (c * X x z) *\<^sub>R z)"
hoelzl@63052
   944
      unfolding scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric]
hoelzl@63052
   945
      by (cases "c = 0") (auto simp: X(3)[symmetric])
hoelzl@63052
   946
  qed
hoelzl@63052
   947
hoelzl@63052
   948
  have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x
hoelzl@63052
   949
    using \<open>independent B'\<close>
hoelzl@63052
   950
    by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric])
hoelzl@63052
   951
hoelzl@63052
   952
  def f' \<equiv> "\<lambda>y. if y \<in> B then f y else 0"
hoelzl@63052
   953
  def g \<equiv> "\<lambda>y. \<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x"
hoelzl@63052
   954
hoelzl@63052
   955
  have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x
hoelzl@63052
   956
    by (auto simp: g_def X_B')
hoelzl@63052
   957
hoelzl@63052
   958
  have "linear g"
hoelzl@63052
   959
  proof
hoelzl@63052
   960
    fix x y
hoelzl@63052
   961
    have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
hoelzl@63052
   962
      = (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R f' z)"
hoelzl@63052
   963
      by (intro setsum.mono_neutral_cong_left) (auto intro: X)
hoelzl@63052
   964
    show "g (x + y) = g x + g y"
hoelzl@63052
   965
      unfolding g_def X_add *
hoelzl@63052
   966
      by (auto simp add: scaleR_add_left setsum.distrib
hoelzl@63052
   967
               intro!: arg_cong2[where f="op +"]  setsum.mono_neutral_cong_right X)
hoelzl@63052
   968
  next
hoelzl@63052
   969
    show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
hoelzl@63052
   970
      by (auto simp add: g_def X_cmult scaleR_setsum_right intro!: setsum.mono_neutral_cong_left X)
hoelzl@63052
   971
  qed
hoelzl@63052
   972
  moreover have "\<forall>x\<in>B. g x = f x"
hoelzl@63052
   973
    using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
hoelzl@63052
   974
  moreover have "range g = span (f`B)"
hoelzl@63052
   975
    unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric]
hoelzl@63052
   976
  proof (rule span_subspace)
hoelzl@63052
   977
    have "g ` B' \<subseteq> f`B \<union> {0}"
hoelzl@63052
   978
      by (auto simp: g_f' f'_def)
hoelzl@63052
   979
    also have "\<dots> \<subseteq> span (f`B)"
hoelzl@63052
   980
      by (auto intro: span_superset span_0)
hoelzl@63052
   981
    finally show "g ` B' \<subseteq> span (f`B)"
hoelzl@63052
   982
      by auto
hoelzl@63052
   983
    have "x \<in> B \<Longrightarrow> f x = g x" for x
hoelzl@63052
   984
      using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def)
hoelzl@63052
   985
    then show "span (f ` B) \<subseteq> span (g ` B')"
hoelzl@63052
   986
      using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto
hoelzl@63052
   987
  qed (rule subspace_span)
hoelzl@63052
   988
  ultimately show ?thesis
hoelzl@63052
   989
    by auto
hoelzl@63052
   990
qed
hoelzl@63052
   991
hoelzl@63052
   992
lemma linear_independent_extend:
hoelzl@63052
   993
  "independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
hoelzl@63052
   994
  using linear_independent_extend_subspace[of B f] by auto
hoelzl@63052
   995
hoelzl@63053
   996
text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
hoelzl@63053
   997
hoelzl@63053
   998
lemma subspace_kernel:
hoelzl@63053
   999
  assumes lf: "linear f"
hoelzl@63053
  1000
  shows "subspace {x. f x = 0}"
hoelzl@63053
  1001
  apply (simp add: subspace_def)
hoelzl@63053
  1002
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
hoelzl@63053
  1003
  done
hoelzl@63053
  1004
hoelzl@63053
  1005
lemma linear_eq_0_span:
hoelzl@63053
  1006
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
hoelzl@63053
  1007
  shows "\<forall>x \<in> span B. f x = 0"
hoelzl@63053
  1008
  using f0 subspace_kernel[OF lf]
hoelzl@63053
  1009
  by (rule span_induct')
hoelzl@63053
  1010
hoelzl@63053
  1011
lemma linear_eq_0: "linear f \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = 0 \<Longrightarrow> \<forall>x\<in>S. f x = 0"
hoelzl@63053
  1012
  using linear_eq_0_span[of f B] by auto
hoelzl@63053
  1013
hoelzl@63053
  1014
lemma linear_eq_span:  "linear f \<Longrightarrow> linear g \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x \<in> span B. f x = g x"
hoelzl@63053
  1015
  using linear_eq_0_span[of "\<lambda>x. f x - g x" B] by (auto simp: linear_compose_sub)
hoelzl@63053
  1016
hoelzl@63053
  1017
lemma linear_eq: "linear f \<Longrightarrow> linear g \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x\<in>S. f x = g x"
hoelzl@63053
  1018
  using linear_eq_span[of f g B] by auto
hoelzl@63053
  1019
wenzelm@60420
  1020
text \<open>The degenerate case of the Exchange Lemma.\<close>
huffman@44133
  1021
huffman@44133
  1022
lemma spanning_subset_independent:
wenzelm@49711
  1023
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1024
    and iA: "independent A"
wenzelm@49522
  1025
    and AsB: "A \<subseteq> span B"
huffman@44133
  1026
  shows "A = B"
huffman@44133
  1027
proof
wenzelm@49663
  1028
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1029
huffman@44133
  1030
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1031
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1032
wenzelm@53406
  1033
  {
wenzelm@53406
  1034
    fix x
wenzelm@53406
  1035
    assume x: "x \<in> A"
huffman@44133
  1036
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1037
      unfolding dependent_def using x by blast
wenzelm@53406
  1038
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1039
      by (blast intro: span_superset)
huffman@44133
  1040
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1041
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1042
      by (metis span_mono)
wenzelm@53406
  1043
    {
wenzelm@53406
  1044
      assume xB: "x \<notin> B"
wenzelm@53406
  1045
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1046
        by blast
wenzelm@53406
  1047
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1048
        by (metis span_mono)
wenzelm@53406
  1049
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1050
        by blast
wenzelm@53406
  1051
      with x have False
wenzelm@53406
  1052
        by (metis span_superset)
wenzelm@53406
  1053
    }
wenzelm@53406
  1054
    then have "x \<in> B" by blast
wenzelm@53406
  1055
  }
huffman@44133
  1056
  then show "A \<subseteq> B" by blast
huffman@44133
  1057
qed
huffman@44133
  1058
hoelzl@63053
  1059
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
hoelzl@63053
  1060
hoelzl@63053
  1061
lemma spanning_surjective_image:
hoelzl@63053
  1062
  assumes us: "UNIV \<subseteq> span S"
hoelzl@63053
  1063
    and lf: "linear f"
hoelzl@63053
  1064
    and sf: "surj f"
hoelzl@63053
  1065
  shows "UNIV \<subseteq> span (f ` S)"
hoelzl@63053
  1066
proof -
hoelzl@63053
  1067
  have "UNIV \<subseteq> f ` UNIV"
hoelzl@63053
  1068
    using sf by (auto simp add: surj_def)
hoelzl@63053
  1069
  also have " \<dots> \<subseteq> span (f ` S)"
hoelzl@63053
  1070
    using spans_image[OF lf us] .
hoelzl@63053
  1071
  finally show ?thesis .
hoelzl@63053
  1072
qed
hoelzl@63053
  1073
hoelzl@63053
  1074
lemma independent_inj_on_image:
hoelzl@63053
  1075
  assumes iS: "independent S"
hoelzl@63053
  1076
    and lf: "linear f"
hoelzl@63053
  1077
    and fi: "inj_on f (span S)"
hoelzl@63053
  1078
  shows "independent (f ` S)"
hoelzl@63053
  1079
proof -
hoelzl@63053
  1080
  {
hoelzl@63053
  1081
    fix a
hoelzl@63053
  1082
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
hoelzl@63053
  1083
    have eq: "f ` S - {f a} = f ` (S - {a})"
hoelzl@63053
  1084
      using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset)
hoelzl@63053
  1085
    from a have "f a \<in> f ` span (S - {a})"
hoelzl@63053
  1086
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
hoelzl@63053
  1087
    then have "a \<in> span (S - {a})"
hoelzl@63053
  1088
      by (rule inj_on_image_mem_iff_alt[OF fi, rotated])
hoelzl@63053
  1089
         (insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>)
hoelzl@63053
  1090
    with a(1) iS have False
hoelzl@63053
  1091
      by (simp add: dependent_def)
hoelzl@63053
  1092
  }
hoelzl@63053
  1093
  then show ?thesis
hoelzl@63053
  1094
    unfolding dependent_def by blast
hoelzl@63053
  1095
qed
hoelzl@63053
  1096
hoelzl@63053
  1097
lemma independent_injective_image:
hoelzl@63053
  1098
  "independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)"
hoelzl@63053
  1099
  using independent_inj_on_image[of S f] by (auto simp: subset_inj_on)
hoelzl@63053
  1100
hoelzl@63053
  1101
text \<open>Detailed theorems about left and right invertibility in general case.\<close>
hoelzl@63053
  1102
hoelzl@63053
  1103
lemma linear_inj_on_left_inverse:
hoelzl@63053
  1104
  assumes lf: "linear f" and fi: "inj_on f (span S)"
hoelzl@63053
  1105
  shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)"
hoelzl@63053
  1106
proof -
hoelzl@63053
  1107
  obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B"
hoelzl@63053
  1108
    using maximal_independent_subset[of S] by auto
hoelzl@63053
  1109
  then have "span S = span B"
hoelzl@63053
  1110
    unfolding span_eq by (auto simp: span_superset)
hoelzl@63053
  1111
  with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi
hoelzl@63053
  1112
  obtain g where g: "linear g" "\<forall>x\<in>f ` B. g x = inv_into B f x" "range g = span (inv_into B f ` f ` B)"
hoelzl@63053
  1113
    by fastforce
hoelzl@63053
  1114
  have fB: "inj_on f B"
hoelzl@63053
  1115
    using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
hoelzl@63053
  1116
hoelzl@63053
  1117
  have "\<forall>x\<in>span B. g (f x) = x"
hoelzl@63053
  1118
  proof (intro linear_eq_span)
hoelzl@63053
  1119
    show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
hoelzl@63053
  1120
      using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
hoelzl@63053
  1121
    show "\<forall>x \<in> B. g (f x) = x"
hoelzl@63053
  1122
      using g fi \<open>span S = span B\<close> by (auto simp: fB)
hoelzl@63053
  1123
  qed
hoelzl@63053
  1124
  moreover
hoelzl@63053
  1125
  have "inv_into B f ` f ` B \<subseteq> B"
hoelzl@63053
  1126
    by (auto simp: fB)
hoelzl@63053
  1127
  then have "range g \<subseteq> span S"
hoelzl@63053
  1128
    unfolding g \<open>span S = span B\<close> by (intro span_mono)
hoelzl@63053
  1129
  ultimately show ?thesis
hoelzl@63053
  1130
    using \<open>span S = span B\<close> \<open>linear g\<close> by auto
hoelzl@63053
  1131
qed
hoelzl@63053
  1132
hoelzl@63053
  1133
lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id"
hoelzl@63053
  1134
  using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV)
hoelzl@63053
  1135
hoelzl@63053
  1136
lemma linear_surj_right_inverse:
hoelzl@63053
  1137
  assumes lf: "linear f" and sf: "span T \<subseteq> f`span S"
hoelzl@63053
  1138
  shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)"
hoelzl@63053
  1139
proof -
hoelzl@63053
  1140
  obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B"
hoelzl@63053
  1141
    using maximal_independent_subset[of T] by auto
hoelzl@63053
  1142
  then have "span T = span B"
hoelzl@63053
  1143
    unfolding span_eq by (auto simp: span_superset)
hoelzl@63053
  1144
hoelzl@63053
  1145
  from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"]
hoelzl@63053
  1146
  obtain g where g: "linear g" "\<forall>x\<in>B. g x = inv_into (span S) f x" "range g = span (inv_into (span S) f`B)"
hoelzl@63053
  1147
    by auto
hoelzl@63053
  1148
  moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x
hoelzl@63053
  1149
    using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (intro f_inv_into_f) (auto intro: span_superset)
hoelzl@63053
  1150
  ultimately have "\<forall>x\<in>B. f (g x) = x"
hoelzl@63053
  1151
    by auto
hoelzl@63053
  1152
  then have "\<forall>x\<in>span B. f (g x) = x"
hoelzl@63053
  1153
    using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>]
hoelzl@63053
  1154
    by (intro linear_eq_span) (auto simp: id_def comp_def)
hoelzl@63053
  1155
  moreover have "inv_into (span S) f ` B \<subseteq> span S"
hoelzl@63053
  1156
    using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
hoelzl@63053
  1157
  then have "range g \<subseteq> span S"
hoelzl@63053
  1158
    unfolding g by (intro span_minimal subspace_span) auto
hoelzl@63053
  1159
  ultimately show ?thesis
hoelzl@63053
  1160
    using \<open>linear g\<close> \<open>span T = span B\<close> by auto
hoelzl@63053
  1161
qed
hoelzl@63053
  1162
hoelzl@63053
  1163
lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id"
hoelzl@63053
  1164
  using linear_surj_right_inverse[of f UNIV UNIV]
hoelzl@63053
  1165
  by (auto simp: span_UNIV fun_eq_iff)
hoelzl@63053
  1166
wenzelm@60420
  1167
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
huffman@44133
  1168
huffman@44133
  1169
lemma exchange_lemma:
wenzelm@49711
  1170
  assumes f:"finite t"
wenzelm@49711
  1171
    and i: "independent s"
wenzelm@49711
  1172
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1173
  shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
wenzelm@49663
  1174
  using f i sp
wenzelm@49522
  1175
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1176
  case less
wenzelm@60420
  1177
  note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
wenzelm@53406
  1178
  let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
huffman@44133
  1179
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1180
  {
lp15@55775
  1181
    assume "s \<subseteq> t"
lp15@55775
  1182
    then have ?ths
lp15@55775
  1183
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1184
  }
huffman@44133
  1185
  moreover
wenzelm@53406
  1186
  {
wenzelm@53406
  1187
    assume st: "t \<subseteq> s"
wenzelm@53406
  1188
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1189
    have ?ths
lp15@55775
  1190
      by (metis Un_absorb sp)
wenzelm@53406
  1191
  }
huffman@44133
  1192
  moreover
wenzelm@53406
  1193
  {
wenzelm@53406
  1194
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1195
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1196
      by blast
wenzelm@53406
  1197
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1198
      by blast
wenzelm@53406
  1199
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1200
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1201
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1202
      by auto
wenzelm@53406
  1203
    have ?ths
wenzelm@53406
  1204
    proof cases
wenzelm@53716
  1205
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1206
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1207
        by auto
huffman@44133
  1208
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1209
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1210
        and fu: "finite u" by blast
huffman@44133
  1211
      let ?w = "insert b u"
wenzelm@53406
  1212
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1213
        using u by blast
wenzelm@53406
  1214
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1215
        by blast
wenzelm@53406
  1216
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1217
        using u b by blast
wenzelm@53406
  1218
      have bu: "b \<notin> u"
wenzelm@53406
  1219
        using b u by blast
wenzelm@53406
  1220
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1221
        by auto
wenzelm@49522
  1222
      then have th2: "card (insert b u) = card t"
huffman@44133
  1223
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1224
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1225
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1226
        by (rule span_mono) blast
huffman@44133
  1227
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1228
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1229
        by blast
wenzelm@53406
  1230
      from th show ?thesis by blast
wenzelm@53406
  1231
    next
wenzelm@53716
  1232
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1233
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1234
        by blast
wenzelm@53406
  1235
      have ab: "a \<noteq> b"
wenzelm@53406
  1236
        using a b by blast
wenzelm@53406
  1237
      have at: "a \<notin> t"
wenzelm@53406
  1238
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1239
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1240
        using cardlt ft a b by auto
wenzelm@53406
  1241
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1242
        using ft by auto
wenzelm@53406
  1243
      {
wenzelm@53406
  1244
        fix x
wenzelm@53406
  1245
        assume xs: "x \<in> s"
wenzelm@53406
  1246
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1247
          using b by auto
wenzelm@53406
  1248
        from b(1) have "b \<in> span t"
wenzelm@53406
  1249
          by (simp add: span_superset)
wenzelm@53406
  1250
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1251
          apply (rule in_span_delete)
wenzelm@53406
  1252
          using a sp unfolding subset_eq
wenzelm@53406
  1253
          apply auto
wenzelm@53406
  1254
          done
wenzelm@53406
  1255
        from xs sp have "x \<in> span t"
wenzelm@53406
  1256
          by blast
wenzelm@53406
  1257
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1258
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1259
      }
wenzelm@53406
  1260
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1261
        by blast
wenzelm@53406
  1262
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1263
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1264
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1265
        "s \<subseteq> span u" by blast
wenzelm@53406
  1266
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1267
        by auto
wenzelm@53406
  1268
      then show ?thesis by blast
wenzelm@53406
  1269
    qed
huffman@44133
  1270
  }
wenzelm@49522
  1271
  ultimately show ?ths by blast
huffman@44133
  1272
qed
huffman@44133
  1273
wenzelm@60420
  1274
text \<open>This implies corresponding size bounds.\<close>
huffman@44133
  1275
huffman@44133
  1276
lemma independent_span_bound:
wenzelm@53406
  1277
  assumes f: "finite t"
wenzelm@53406
  1278
    and i: "independent s"
wenzelm@53406
  1279
    and sp: "s \<subseteq> span t"
huffman@44133
  1280
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1281
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1282
huffman@44133
  1283
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1284
proof -
wenzelm@53406
  1285
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1286
    by auto
huffman@44133
  1287
  show ?thesis unfolding eq
huffman@44133
  1288
    apply (rule finite_imageI)
huffman@44133
  1289
    apply (rule finite)
huffman@44133
  1290
    done
huffman@44133
  1291
qed
huffman@44133
  1292
wenzelm@53406
  1293
hoelzl@63050
  1294
subsection \<open>More interesting properties of the norm.\<close>
hoelzl@63050
  1295
hoelzl@63050
  1296
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
hoelzl@63050
  1297
  by auto
hoelzl@63050
  1298
hoelzl@63050
  1299
notation inner (infix "\<bullet>" 70)
hoelzl@63050
  1300
hoelzl@63050
  1301
lemma square_bound_lemma:
hoelzl@63050
  1302
  fixes x :: real
hoelzl@63050
  1303
  shows "x < (1 + x) * (1 + x)"
hoelzl@63050
  1304
proof -
hoelzl@63050
  1305
  have "(x + 1/2)\<^sup>2 + 3/4 > 0"
hoelzl@63050
  1306
    using zero_le_power2[of "x+1/2"] by arith
hoelzl@63050
  1307
  then show ?thesis
hoelzl@63050
  1308
    by (simp add: field_simps power2_eq_square)
hoelzl@63050
  1309
qed
hoelzl@63050
  1310
hoelzl@63050
  1311
lemma square_continuous:
hoelzl@63050
  1312
  fixes e :: real
hoelzl@63050
  1313
  shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
hoelzl@63050
  1314
  using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
hoelzl@63050
  1315
  by (force simp add: power2_eq_square)
hoelzl@63050
  1316
hoelzl@63050
  1317
hoelzl@63050
  1318
lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
hoelzl@63050
  1319
  by simp (* TODO: delete *)
hoelzl@63050
  1320
hoelzl@63050
  1321
lemma norm_triangle_sub:
hoelzl@63050
  1322
  fixes x y :: "'a::real_normed_vector"
hoelzl@63050
  1323
  shows "norm x \<le> norm y + norm (x - y)"
hoelzl@63050
  1324
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
hoelzl@63050
  1325
hoelzl@63050
  1326
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
hoelzl@63050
  1327
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1328
hoelzl@63050
  1329
lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
hoelzl@63050
  1330
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1331
hoelzl@63050
  1332
lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
hoelzl@63050
  1333
  apply (subst order_eq_iff)
hoelzl@63050
  1334
  apply (auto simp: norm_le)
hoelzl@63050
  1335
  done
hoelzl@63050
  1336
hoelzl@63050
  1337
lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
hoelzl@63050
  1338
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1339
hoelzl@63050
  1340
text\<open>Squaring equations and inequalities involving norms.\<close>
hoelzl@63050
  1341
hoelzl@63050
  1342
lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
hoelzl@63050
  1343
  by (simp only: power2_norm_eq_inner) (* TODO: move? *)
hoelzl@63050
  1344
hoelzl@63050
  1345
lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
hoelzl@63050
  1346
  by (auto simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1347
hoelzl@63050
  1348
lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
hoelzl@63050
  1349
  apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
hoelzl@63050
  1350
  using norm_ge_zero[of x]
hoelzl@63050
  1351
  apply arith
hoelzl@63050
  1352
  done
hoelzl@63050
  1353
hoelzl@63050
  1354
lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
hoelzl@63050
  1355
  apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
hoelzl@63050
  1356
  using norm_ge_zero[of x]
hoelzl@63050
  1357
  apply arith
hoelzl@63050
  1358
  done
hoelzl@63050
  1359
hoelzl@63050
  1360
lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
hoelzl@63050
  1361
  by (metis not_le norm_ge_square)
hoelzl@63050
  1362
hoelzl@63050
  1363
lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
hoelzl@63050
  1364
  by (metis norm_le_square not_less)
hoelzl@63050
  1365
hoelzl@63050
  1366
text\<open>Dot product in terms of the norm rather than conversely.\<close>
hoelzl@63050
  1367
hoelzl@63050
  1368
lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
hoelzl@63050
  1369
  inner_scaleR_left inner_scaleR_right
hoelzl@63050
  1370
hoelzl@63050
  1371
lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
hoelzl@63050
  1372
  unfolding power2_norm_eq_inner inner_simps inner_commute by auto
hoelzl@63050
  1373
hoelzl@63050
  1374
lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
hoelzl@63050
  1375
  unfolding power2_norm_eq_inner inner_simps inner_commute
hoelzl@63050
  1376
  by (auto simp add: algebra_simps)
hoelzl@63050
  1377
hoelzl@63050
  1378
text\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close>
hoelzl@63050
  1379
hoelzl@63050
  1380
lemma linear_componentwise:
hoelzl@63050
  1381
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
hoelzl@63050
  1382
  assumes lf: "linear f"
hoelzl@63050
  1383
  shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
hoelzl@63050
  1384
proof -
hoelzl@63050
  1385
  have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
hoelzl@63050
  1386
    by (simp add: inner_setsum_left)
hoelzl@63050
  1387
  then show ?thesis
hoelzl@63050
  1388
    unfolding linear_setsum_mul[OF lf, symmetric]
hoelzl@63050
  1389
    unfolding euclidean_representation ..
hoelzl@63050
  1390
qed
hoelzl@63050
  1391
hoelzl@63050
  1392
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
hoelzl@63050
  1393
  (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@63050
  1394
proof
hoelzl@63050
  1395
  assume ?lhs
hoelzl@63050
  1396
  then show ?rhs by simp
hoelzl@63050
  1397
next
hoelzl@63050
  1398
  assume ?rhs
hoelzl@63050
  1399
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
hoelzl@63050
  1400
    by simp
hoelzl@63050
  1401
  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
hoelzl@63050
  1402
    by (simp add: inner_diff inner_commute)
hoelzl@63050
  1403
  then have "(x - y) \<bullet> (x - y) = 0"
hoelzl@63050
  1404
    by (simp add: field_simps inner_diff inner_commute)
hoelzl@63050
  1405
  then show "x = y" by simp
hoelzl@63050
  1406
qed
hoelzl@63050
  1407
hoelzl@63050
  1408
lemma norm_triangle_half_r:
hoelzl@63050
  1409
  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
hoelzl@63050
  1410
  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
hoelzl@63050
  1411
hoelzl@63050
  1412
lemma norm_triangle_half_l:
hoelzl@63050
  1413
  assumes "norm (x - y) < e / 2"
hoelzl@63050
  1414
    and "norm (x' - y) < e / 2"
hoelzl@63050
  1415
  shows "norm (x - x') < e"
hoelzl@63050
  1416
  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
hoelzl@63050
  1417
  unfolding dist_norm[symmetric] .
hoelzl@63050
  1418
hoelzl@63050
  1419
lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
hoelzl@63050
  1420
  by (rule norm_triangle_ineq [THEN order_trans])
hoelzl@63050
  1421
hoelzl@63050
  1422
lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
hoelzl@63050
  1423
  by (rule norm_triangle_ineq [THEN le_less_trans])
hoelzl@63050
  1424
hoelzl@63050
  1425
lemma setsum_clauses:
hoelzl@63050
  1426
  shows "setsum f {} = 0"
hoelzl@63050
  1427
    and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
hoelzl@63050
  1428
  by (auto simp add: insert_absorb)
hoelzl@63050
  1429
hoelzl@63050
  1430
lemma setsum_norm_le:
hoelzl@63050
  1431
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
hoelzl@63050
  1432
  assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
hoelzl@63050
  1433
  shows "norm (setsum f S) \<le> setsum g S"
hoelzl@63050
  1434
  by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
hoelzl@63050
  1435
hoelzl@63050
  1436
lemma setsum_norm_bound:
hoelzl@63050
  1437
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
hoelzl@63050
  1438
  assumes K: "\<forall>x \<in> S. norm (f x) \<le> K"
hoelzl@63050
  1439
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
hoelzl@63050
  1440
  using setsum_norm_le[OF K] setsum_constant[symmetric]
hoelzl@63050
  1441
  by simp
hoelzl@63050
  1442
hoelzl@63050
  1443
lemma setsum_group:
hoelzl@63050
  1444
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
hoelzl@63050
  1445
  shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S"
hoelzl@63050
  1446
  apply (subst setsum_image_gen[OF fS, of g f])
hoelzl@63050
  1447
  apply (rule setsum.mono_neutral_right[OF fT fST])
hoelzl@63050
  1448
  apply (auto intro: setsum.neutral)
hoelzl@63050
  1449
  done
hoelzl@63050
  1450
hoelzl@63050
  1451
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
hoelzl@63050
  1452
proof
hoelzl@63050
  1453
  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
hoelzl@63050
  1454
  then have "\<forall>x. x \<bullet> (y - z) = 0"
hoelzl@63050
  1455
    by (simp add: inner_diff)
hoelzl@63050
  1456
  then have "(y - z) \<bullet> (y - z) = 0" ..
hoelzl@63050
  1457
  then show "y = z" by simp
hoelzl@63050
  1458
qed simp
hoelzl@63050
  1459
hoelzl@63050
  1460
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
hoelzl@63050
  1461
proof
hoelzl@63050
  1462
  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
hoelzl@63050
  1463
  then have "\<forall>z. (x - y) \<bullet> z = 0"
hoelzl@63050
  1464
    by (simp add: inner_diff)
hoelzl@63050
  1465
  then have "(x - y) \<bullet> (x - y) = 0" ..
hoelzl@63050
  1466
  then show "x = y" by simp
hoelzl@63050
  1467
qed simp
hoelzl@63050
  1468
hoelzl@63050
  1469
hoelzl@63050
  1470
subsection \<open>Orthogonality.\<close>
hoelzl@63050
  1471
hoelzl@63050
  1472
context real_inner
hoelzl@63050
  1473
begin
hoelzl@63050
  1474
hoelzl@63050
  1475
definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
hoelzl@63050
  1476
hoelzl@63050
  1477
lemma orthogonal_clauses:
hoelzl@63050
  1478
  "orthogonal a 0"
hoelzl@63050
  1479
  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
hoelzl@63050
  1480
  "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
hoelzl@63050
  1481
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
hoelzl@63050
  1482
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
hoelzl@63050
  1483
  "orthogonal 0 a"
hoelzl@63050
  1484
  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
hoelzl@63050
  1485
  "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
hoelzl@63050
  1486
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
hoelzl@63050
  1487
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
hoelzl@63050
  1488
  unfolding orthogonal_def inner_add inner_diff by auto
hoelzl@63050
  1489
hoelzl@63050
  1490
end
hoelzl@63050
  1491
hoelzl@63050
  1492
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
hoelzl@63050
  1493
  by (simp add: orthogonal_def inner_commute)
hoelzl@63050
  1494
hoelzl@63050
  1495
hoelzl@63050
  1496
subsection \<open>Bilinear functions.\<close>
hoelzl@63050
  1497
hoelzl@63050
  1498
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
hoelzl@63050
  1499
hoelzl@63050
  1500
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
hoelzl@63050
  1501
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1502
hoelzl@63050
  1503
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
hoelzl@63050
  1504
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1505
hoelzl@63050
  1506
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
hoelzl@63050
  1507
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1508
hoelzl@63050
  1509
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
hoelzl@63050
  1510
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1511
hoelzl@63050
  1512
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
hoelzl@63050
  1513
  by (drule bilinear_lmul [of _ "- 1"]) simp
hoelzl@63050
  1514
hoelzl@63050
  1515
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
hoelzl@63050
  1516
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
hoelzl@63050
  1517
hoelzl@63050
  1518
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
hoelzl@63050
  1519
  using add_left_imp_eq[of x y 0] by auto
hoelzl@63050
  1520
hoelzl@63050
  1521
lemma bilinear_lzero:
hoelzl@63050
  1522
  assumes "bilinear h"
hoelzl@63050
  1523
  shows "h 0 x = 0"
hoelzl@63050
  1524
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
hoelzl@63050
  1525
hoelzl@63050
  1526
lemma bilinear_rzero:
hoelzl@63050
  1527
  assumes "bilinear h"
hoelzl@63050
  1528
  shows "h x 0 = 0"
hoelzl@63050
  1529
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
hoelzl@63050
  1530
hoelzl@63050
  1531
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
hoelzl@63050
  1532
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
hoelzl@63050
  1533
hoelzl@63050
  1534
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
hoelzl@63050
  1535
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
hoelzl@63050
  1536
hoelzl@63050
  1537
lemma bilinear_setsum:
hoelzl@63050
  1538
  assumes bh: "bilinear h"
hoelzl@63050
  1539
    and fS: "finite S"
hoelzl@63050
  1540
    and fT: "finite T"
hoelzl@63050
  1541
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
hoelzl@63050
  1542
proof -
hoelzl@63050
  1543
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
hoelzl@63050
  1544
    apply (rule linear_setsum[unfolded o_def])
hoelzl@63050
  1545
    using bh fS
hoelzl@63050
  1546
    apply (auto simp add: bilinear_def)
hoelzl@63050
  1547
    done
hoelzl@63050
  1548
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
hoelzl@63050
  1549
    apply (rule setsum.cong, simp)
hoelzl@63050
  1550
    apply (rule linear_setsum[unfolded o_def])
hoelzl@63050
  1551
    using bh fT
hoelzl@63050
  1552
    apply (auto simp add: bilinear_def)
hoelzl@63050
  1553
    done
hoelzl@63050
  1554
  finally show ?thesis
hoelzl@63050
  1555
    unfolding setsum.cartesian_product .
hoelzl@63050
  1556
qed
hoelzl@63050
  1557
hoelzl@63050
  1558
hoelzl@63050
  1559
subsection \<open>Adjoints.\<close>
hoelzl@63050
  1560
hoelzl@63050
  1561
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
hoelzl@63050
  1562
hoelzl@63050
  1563
lemma adjoint_unique:
hoelzl@63050
  1564
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
hoelzl@63050
  1565
  shows "adjoint f = g"
hoelzl@63050
  1566
  unfolding adjoint_def
hoelzl@63050
  1567
proof (rule some_equality)
hoelzl@63050
  1568
  show "\<forall>x y. inner (f x) y = inner x (g y)"
hoelzl@63050
  1569
    by (rule assms)
hoelzl@63050
  1570
next
hoelzl@63050
  1571
  fix h
hoelzl@63050
  1572
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
hoelzl@63050
  1573
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
hoelzl@63050
  1574
    using assms by simp
hoelzl@63050
  1575
  then have "\<forall>x y. inner x (g y - h y) = 0"
hoelzl@63050
  1576
    by (simp add: inner_diff_right)
hoelzl@63050
  1577
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
hoelzl@63050
  1578
    by simp
hoelzl@63050
  1579
  then have "\<forall>y. h y = g y"
hoelzl@63050
  1580
    by simp
hoelzl@63050
  1581
  then show "h = g" by (simp add: ext)
hoelzl@63050
  1582
qed
hoelzl@63050
  1583
hoelzl@63050
  1584
text \<open>TODO: The following lemmas about adjoints should hold for any
hoelzl@63050
  1585
Hilbert space (i.e. complete inner product space).
hoelzl@63050
  1586
(see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
hoelzl@63050
  1587
\<close>
hoelzl@63050
  1588
hoelzl@63050
  1589
lemma adjoint_works:
hoelzl@63050
  1590
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1591
  assumes lf: "linear f"
hoelzl@63050
  1592
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@63050
  1593
proof -
hoelzl@63050
  1594
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
hoelzl@63050
  1595
  proof (intro allI exI)
hoelzl@63050
  1596
    fix y :: "'m" and x
hoelzl@63050
  1597
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
hoelzl@63050
  1598
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
hoelzl@63050
  1599
      by (simp add: euclidean_representation)
hoelzl@63050
  1600
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
hoelzl@63050
  1601
      unfolding linear_setsum[OF lf]
hoelzl@63050
  1602
      by (simp add: linear_cmul[OF lf])
hoelzl@63050
  1603
    finally show "f x \<bullet> y = x \<bullet> ?w"
hoelzl@63050
  1604
      by (simp add: inner_setsum_left inner_setsum_right mult.commute)
hoelzl@63050
  1605
  qed
hoelzl@63050
  1606
  then show ?thesis
hoelzl@63050
  1607
    unfolding adjoint_def choice_iff
hoelzl@63050
  1608
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
hoelzl@63050
  1609
qed
hoelzl@63050
  1610
hoelzl@63050
  1611
lemma adjoint_clauses:
hoelzl@63050
  1612
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1613
  assumes lf: "linear f"
hoelzl@63050
  1614
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@63050
  1615
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@63050
  1616
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@63050
  1617
hoelzl@63050
  1618
lemma adjoint_linear:
hoelzl@63050
  1619
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1620
  assumes lf: "linear f"
hoelzl@63050
  1621
  shows "linear (adjoint f)"
hoelzl@63050
  1622
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
hoelzl@63050
  1623
    adjoint_clauses[OF lf] inner_distrib)
hoelzl@63050
  1624
hoelzl@63050
  1625
lemma adjoint_adjoint:
hoelzl@63050
  1626
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1627
  assumes lf: "linear f"
hoelzl@63050
  1628
  shows "adjoint (adjoint f) = f"
hoelzl@63050
  1629
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@63050
  1630
hoelzl@63050
  1631
hoelzl@63050
  1632
subsection \<open>Interlude: Some properties of real sets\<close>
hoelzl@63050
  1633
hoelzl@63050
  1634
lemma seq_mono_lemma:
hoelzl@63050
  1635
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
hoelzl@63050
  1636
    and "\<forall>n \<ge> m. e n \<le> e m"
hoelzl@63050
  1637
  shows "\<forall>n \<ge> m. d n < e m"
hoelzl@63050
  1638
  using assms
hoelzl@63050
  1639
  apply auto
hoelzl@63050
  1640
  apply (erule_tac x="n" in allE)
hoelzl@63050
  1641
  apply (erule_tac x="n" in allE)
hoelzl@63050
  1642
  apply auto
hoelzl@63050
  1643
  done
hoelzl@63050
  1644
hoelzl@63050
  1645
lemma infinite_enumerate:
hoelzl@63050
  1646
  assumes fS: "infinite S"
hoelzl@63050
  1647
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
hoelzl@63050
  1648
  unfolding subseq_def
hoelzl@63050
  1649
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
hoelzl@63050
  1650
hoelzl@63050
  1651
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)"
hoelzl@63050
  1652
  apply auto
hoelzl@63050
  1653
  apply (rule_tac x="d/2" in exI)
hoelzl@63050
  1654
  apply auto
hoelzl@63050
  1655
  done
hoelzl@63050
  1656
hoelzl@63050
  1657
lemma approachable_lt_le2:  \<comment>\<open>like the above, but pushes aside an extra formula\<close>
hoelzl@63050
  1658
    "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
hoelzl@63050
  1659
  apply auto
hoelzl@63050
  1660
  apply (rule_tac x="d/2" in exI, auto)
hoelzl@63050
  1661
  done
hoelzl@63050
  1662
hoelzl@63050
  1663
lemma triangle_lemma:
hoelzl@63050
  1664
  fixes x y z :: real
hoelzl@63050
  1665
  assumes x: "0 \<le> x"
hoelzl@63050
  1666
    and y: "0 \<le> y"
hoelzl@63050
  1667
    and z: "0 \<le> z"
hoelzl@63050
  1668
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
hoelzl@63050
  1669
  shows "x \<le> y + z"
hoelzl@63050
  1670
proof -
hoelzl@63050
  1671
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
hoelzl@63050
  1672
    using z y by simp
hoelzl@63050
  1673
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
hoelzl@63050
  1674
    by (simp add: power2_eq_square field_simps)
hoelzl@63050
  1675
  from y z have yz: "y + z \<ge> 0"
hoelzl@63050
  1676
    by arith
hoelzl@63050
  1677
  from power2_le_imp_le[OF th yz] show ?thesis .
hoelzl@63050
  1678
qed
hoelzl@63050
  1679
hoelzl@63050
  1680
hoelzl@63050
  1681
hoelzl@63050
  1682
subsection \<open>Archimedean properties and useful consequences\<close>
hoelzl@63050
  1683
hoelzl@63050
  1684
text\<open>Bernoulli's inequality\<close>
hoelzl@63050
  1685
proposition Bernoulli_inequality:
hoelzl@63050
  1686
  fixes x :: real
hoelzl@63050
  1687
  assumes "-1 \<le> x"
hoelzl@63050
  1688
    shows "1 + n * x \<le> (1 + x) ^ n"
hoelzl@63050
  1689
proof (induct n)
hoelzl@63050
  1690
  case 0
hoelzl@63050
  1691
  then show ?case by simp
hoelzl@63050
  1692
next
hoelzl@63050
  1693
  case (Suc n)
hoelzl@63050
  1694
  have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
hoelzl@63050
  1695
    by (simp add: algebra_simps)
hoelzl@63050
  1696
  also have "... = (1 + x) * (1 + n*x)"
hoelzl@63050
  1697
    by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
hoelzl@63050
  1698
  also have "... \<le> (1 + x) ^ Suc n"
hoelzl@63050
  1699
    using Suc.hyps assms mult_left_mono by fastforce
hoelzl@63050
  1700
  finally show ?case .
hoelzl@63050
  1701
qed
hoelzl@63050
  1702
hoelzl@63050
  1703
corollary Bernoulli_inequality_even:
hoelzl@63050
  1704
  fixes x :: real
hoelzl@63050
  1705
  assumes "even n"
hoelzl@63050
  1706
    shows "1 + n * x \<le> (1 + x) ^ n"
hoelzl@63050
  1707
proof (cases "-1 \<le> x \<or> n=0")
hoelzl@63050
  1708
  case True
hoelzl@63050
  1709
  then show ?thesis
hoelzl@63050
  1710
    by (auto simp: Bernoulli_inequality)
hoelzl@63050
  1711
next
hoelzl@63050
  1712
  case False
hoelzl@63050
  1713
  then have "real n \<ge> 1"
hoelzl@63050
  1714
    by simp
hoelzl@63050
  1715
  with False have "n * x \<le> -1"
hoelzl@63050
  1716
    by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
hoelzl@63050
  1717
  then have "1 + n * x \<le> 0"
hoelzl@63050
  1718
    by auto
hoelzl@63050
  1719
  also have "... \<le> (1 + x) ^ n"
hoelzl@63050
  1720
    using assms
hoelzl@63050
  1721
    using zero_le_even_power by blast
hoelzl@63050
  1722
  finally show ?thesis .
hoelzl@63050
  1723
qed
hoelzl@63050
  1724
hoelzl@63050
  1725
corollary real_arch_pow:
hoelzl@63050
  1726
  fixes x :: real
hoelzl@63050
  1727
  assumes x: "1 < x"
hoelzl@63050
  1728
  shows "\<exists>n. y < x^n"
hoelzl@63050
  1729
proof -
hoelzl@63050
  1730
  from x have x0: "x - 1 > 0"
hoelzl@63050
  1731
    by arith
hoelzl@63050
  1732
  from reals_Archimedean3[OF x0, rule_format, of y]
hoelzl@63050
  1733
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
hoelzl@63050
  1734
  from x0 have x00: "x- 1 \<ge> -1" by arith
hoelzl@63050
  1735
  from Bernoulli_inequality[OF x00, of n] n
hoelzl@63050
  1736
  have "y < x^n" by auto
hoelzl@63050
  1737
  then show ?thesis by metis
hoelzl@63050
  1738
qed
hoelzl@63050
  1739
hoelzl@63050
  1740
corollary real_arch_pow_inv:
hoelzl@63050
  1741
  fixes x y :: real
hoelzl@63050
  1742
  assumes y: "y > 0"
hoelzl@63050
  1743
    and x1: "x < 1"
hoelzl@63050
  1744
  shows "\<exists>n. x^n < y"
hoelzl@63050
  1745
proof (cases "x > 0")
hoelzl@63050
  1746
  case True
hoelzl@63050
  1747
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
hoelzl@63050
  1748
  from real_arch_pow[OF ix, of "1/y"]
hoelzl@63050
  1749
  obtain n where n: "1/y < (1/x)^n" by blast
hoelzl@63050
  1750
  then show ?thesis using y \<open>x > 0\<close>
hoelzl@63050
  1751
    by (auto simp add: field_simps)
hoelzl@63050
  1752
next
hoelzl@63050
  1753
  case False
hoelzl@63050
  1754
  with y x1 show ?thesis
hoelzl@63050
  1755
    apply auto
hoelzl@63050
  1756
    apply (rule exI[where x=1])
hoelzl@63050
  1757
    apply auto
hoelzl@63050
  1758
    done
hoelzl@63050
  1759
qed
hoelzl@63050
  1760
hoelzl@63050
  1761
lemma forall_pos_mono:
hoelzl@63050
  1762
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
hoelzl@63050
  1763
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
hoelzl@63050
  1764
  by (metis real_arch_inverse)
hoelzl@63050
  1765
hoelzl@63050
  1766
lemma forall_pos_mono_1:
hoelzl@63050
  1767
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
hoelzl@63050
  1768
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
hoelzl@63050
  1769
  apply (rule forall_pos_mono)
hoelzl@63050
  1770
  apply auto
hoelzl@63050
  1771
  apply (metis Suc_pred of_nat_Suc)
hoelzl@63050
  1772
  done
hoelzl@63050
  1773
hoelzl@63050
  1774
wenzelm@60420
  1775
subsection \<open>Euclidean Spaces as Typeclass\<close>
huffman@44133
  1776
hoelzl@50526
  1777
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1778
  unfolding dependent_def
hoelzl@50526
  1779
  apply (subst span_finite)
hoelzl@50526
  1780
  apply simp
huffman@44133
  1781
  apply clarify
hoelzl@50526
  1782
  apply (drule_tac f="inner a" in arg_cong)
hoelzl@50526
  1783
  apply (simp add: inner_Basis inner_setsum_right eq_commute)
hoelzl@50526
  1784
  done
hoelzl@50526
  1785
huffman@53939
  1786
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1787
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1788
  by (fast intro: euclidean_representation)
huffman@44133
  1789
hoelzl@50526
  1790
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1791
  unfolding span_Basis ..
hoelzl@50526
  1792
hoelzl@50526
  1793
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1794
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1795
hoelzl@50526
  1796
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1797
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1798
hoelzl@50526
  1799
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1800
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1801
hoelzl@50526
  1802
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1803
  apply (subst euclidean_representation[of x, symmetric])
huffman@44176
  1804
  apply (rule order_trans[OF norm_setsum])
wenzelm@49522
  1805
  apply (auto intro!: setsum_mono)
wenzelm@49522
  1806
  done
huffman@44133
  1807
huffman@44133
  1808
lemma setsum_norm_allsubsets_bound:
wenzelm@56444
  1809
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1810
  assumes fP: "finite P"
wenzelm@53406
  1811
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@50526
  1812
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1813
proof -
hoelzl@50526
  1814
  have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
hoelzl@50526
  1815
    by (rule setsum_mono) (rule norm_le_l1)
hoelzl@50526
  1816
  also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
haftmann@57418
  1817
    by (rule setsum.commute)
hoelzl@50526
  1818
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
lp15@60974
  1819
  proof (rule setsum_bounded_above)
wenzelm@53406
  1820
    fix i :: 'n
wenzelm@53406
  1821
    assume i: "i \<in> Basis"
wenzelm@53406
  1822
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1823
      norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
haftmann@57418
  1824
      by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
wenzelm@56444
  1825
        del: real_norm_def)
wenzelm@53406
  1826
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1827
      unfolding real_norm_def
hoelzl@50526
  1828
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1829
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1830
  qed
lp15@61609
  1831
  also have "\<dots> = 2 * real DIM('n) * e" by simp
huffman@44133
  1832
  finally show ?thesis .
huffman@44133
  1833
qed
huffman@44133
  1834
wenzelm@53406
  1835
wenzelm@60420
  1836
subsection \<open>Linearity and Bilinearity continued\<close>
huffman@44133
  1837
huffman@44133
  1838
lemma linear_bounded:
wenzelm@56444
  1839
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1840
  assumes lf: "linear f"
huffman@44133
  1841
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1842
proof
hoelzl@50526
  1843
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1844
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1845
  proof
wenzelm@53406
  1846
    fix x :: 'a
hoelzl@50526
  1847
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1848
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1849
      unfolding euclidean_representation ..
hoelzl@50526
  1850
    also have "\<dots> = norm (setsum ?g Basis)"
huffman@53939
  1851
      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
hoelzl@50526
  1852
    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
huffman@53939
  1853
    have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
huffman@53939
  1854
    proof
wenzelm@53406
  1855
      fix i :: 'a
wenzelm@53406
  1856
      assume i: "i \<in> Basis"
hoelzl@50526
  1857
      from Basis_le_norm[OF i, of x]
huffman@53939
  1858
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1859
        unfolding norm_scaleR
haftmann@57512
  1860
        apply (subst mult.commute)
wenzelm@49663
  1861
        apply (rule mult_mono)
wenzelm@49663
  1862
        apply (auto simp add: field_simps)
wenzelm@53406
  1863
        done
huffman@53939
  1864
    qed
hoelzl@50526
  1865
    from setsum_norm_le[of _ ?g, OF th]
huffman@53939
  1866
    show "norm (f x) \<le> ?B * norm x"
wenzelm@53406
  1867
      unfolding th0 setsum_left_distrib by metis
huffman@53939
  1868
  qed
huffman@44133
  1869
qed
huffman@44133
  1870
huffman@44133
  1871
lemma linear_conv_bounded_linear:
huffman@44133
  1872
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1873
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1874
proof
huffman@44133
  1875
  assume "linear f"
huffman@53939
  1876
  then interpret f: linear f .
huffman@44133
  1877
  show "bounded_linear f"
huffman@44133
  1878
  proof
huffman@44133
  1879
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
wenzelm@60420
  1880
      using \<open>linear f\<close> by (rule linear_bounded)
wenzelm@49522
  1881
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
haftmann@57512
  1882
      by (simp add: mult.commute)
huffman@44133
  1883
  qed
huffman@44133
  1884
next
huffman@44133
  1885
  assume "bounded_linear f"
huffman@44133
  1886
  then interpret f: bounded_linear f .
huffman@53939
  1887
  show "linear f" ..
huffman@53939
  1888
qed
huffman@53939
  1889
paulson@61518
  1890
lemmas linear_linear = linear_conv_bounded_linear[symmetric]
paulson@61518
  1891
huffman@53939
  1892
lemma linear_bounded_pos:
wenzelm@56444
  1893
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1894
  assumes lf: "linear f"
huffman@53939
  1895
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1896
proof -
huffman@53939
  1897
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1898
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1899
    by (rule bounded_linear.pos_bounded)
huffman@53939
  1900
  then show ?thesis
haftmann@57512
  1901
    by (simp only: mult.commute)
huffman@44133
  1902
qed
huffman@44133
  1903
wenzelm@49522
  1904
lemma bounded_linearI':
wenzelm@56444
  1905
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1906
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1907
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1908
  shows "bounded_linear f"
wenzelm@53406
  1909
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1910
  by (rule linearI[OF assms])
huffman@44133
  1911
huffman@44133
  1912
lemma bilinear_bounded:
wenzelm@56444
  1913
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  1914
  assumes bh: "bilinear h"
huffman@44133
  1915
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  1916
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  1917
  fix x :: 'm
wenzelm@53406
  1918
  fix y :: 'n
wenzelm@53406
  1919
  have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
wenzelm@53406
  1920
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  1921
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  1922
    apply rule
hoelzl@50526
  1923
    done
wenzelm@53406
  1924
  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
hoelzl@50526
  1925
    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  1926
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  1927
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
haftmann@57418
  1928
    apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
wenzelm@53406
  1929
    apply (rule setsum_norm_le)
wenzelm@53406
  1930
    apply simp
wenzelm@53406
  1931
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  1932
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  1933
    apply (rule mult_mono)
wenzelm@53406
  1934
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1935
    apply (rule mult_mono)
wenzelm@53406
  1936
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1937
    done
huffman@44133
  1938
qed
huffman@44133
  1939
huffman@44133
  1940
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  1941
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  1942
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  1943
proof
huffman@44133
  1944
  assume "bilinear h"
huffman@44133
  1945
  show "bounded_bilinear h"
huffman@44133
  1946
  proof
wenzelm@53406
  1947
    fix x y z
wenzelm@53406
  1948
    show "h (x + y) z = h x z + h y z"
wenzelm@60420
  1949
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  1950
  next
wenzelm@53406
  1951
    fix x y z
wenzelm@53406
  1952
    show "h x (y + z) = h x y + h x z"
wenzelm@60420
  1953
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  1954
  next
wenzelm@53406
  1955
    fix r x y
wenzelm@53406
  1956
    show "h (scaleR r x) y = scaleR r (h x y)"
wenzelm@60420
  1957
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  1958
      by simp
huffman@44133
  1959
  next
wenzelm@53406
  1960
    fix r x y
wenzelm@53406
  1961
    show "h x (scaleR r y) = scaleR r (h x y)"
wenzelm@60420
  1962
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  1963
      by simp
huffman@44133
  1964
  next
huffman@44133
  1965
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
wenzelm@60420
  1966
      using \<open>bilinear h\<close> by (rule bilinear_bounded)
wenzelm@49522
  1967
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
haftmann@57514
  1968
      by (simp add: ac_simps)
huffman@44133
  1969
  qed
huffman@44133
  1970
next
huffman@44133
  1971
  assume "bounded_bilinear h"
huffman@44133
  1972
  then interpret h: bounded_bilinear h .
huffman@44133
  1973
  show "bilinear h"
huffman@44133
  1974
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  1975
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  1976
qed
huffman@44133
  1977
huffman@53939
  1978
lemma bilinear_bounded_pos:
wenzelm@56444
  1979
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  1980
  assumes bh: "bilinear h"
huffman@53939
  1981
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  1982
proof -
huffman@53939
  1983
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  1984
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  1985
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  1986
  then show ?thesis
haftmann@57514
  1987
    by (simp only: ac_simps)
huffman@53939
  1988
qed
huffman@53939
  1989
wenzelm@49522
  1990
wenzelm@60420
  1991
subsection \<open>We continue.\<close>
huffman@44133
  1992
huffman@44133
  1993
lemma independent_bound:
wenzelm@53716
  1994
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  1995
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  1996
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  1997
lp15@61609
  1998
corollary
paulson@60303
  1999
  fixes S :: "'a::euclidean_space set"
paulson@60303
  2000
  assumes "independent S"
paulson@60303
  2001
  shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
paulson@60303
  2002
using assms independent_bound by auto
lp15@61609
  2003
wenzelm@49663
  2004
lemma dependent_biggerset:
wenzelm@56444
  2005
  fixes S :: "'a::euclidean_space set"
wenzelm@56444
  2006
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  2007
  by (metis independent_bound not_less)
huffman@44133
  2008
wenzelm@60420
  2009
text \<open>Notion of dimension.\<close>
huffman@44133
  2010
wenzelm@53406
  2011
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  2012
wenzelm@49522
  2013
lemma basis_exists:
wenzelm@49522
  2014
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  2015
  unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
wenzelm@49522
  2016
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  2017
  by auto
huffman@44133
  2018
lp15@60307
  2019
corollary dim_le_card:
lp15@60307
  2020
  fixes s :: "'a::euclidean_space set"
lp15@60307
  2021
  shows "finite s \<Longrightarrow> dim s \<le> card s"
lp15@60307
  2022
by (metis basis_exists card_mono)
lp15@60307
  2023
wenzelm@60420
  2024
text \<open>Consequences of independence or spanning for cardinality.\<close>
huffman@44133
  2025
wenzelm@53406
  2026
lemma independent_card_le_dim:
wenzelm@53406
  2027
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2028
  assumes "B \<subseteq> V"
wenzelm@53406
  2029
    and "independent B"
wenzelm@49522
  2030
  shows "card B \<le> dim V"
huffman@44133
  2031
proof -
wenzelm@60420
  2032
  from basis_exists[of V] \<open>B \<subseteq> V\<close>
wenzelm@53406
  2033
  obtain B' where "independent B'"
wenzelm@53406
  2034
    and "B \<subseteq> span B'"
wenzelm@53406
  2035
    and "card B' = dim V"
wenzelm@53406
  2036
    by blast
wenzelm@60420
  2037
  with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
huffman@44133
  2038
  show ?thesis by auto
huffman@44133
  2039
qed
huffman@44133
  2040
wenzelm@49522
  2041
lemma span_card_ge_dim:
wenzelm@53406
  2042
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2043
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  2044
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  2045
huffman@44133
  2046
lemma basis_card_eq_dim:
wenzelm@53406
  2047
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  2048
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  2049
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  2050
wenzelm@53406
  2051
lemma dim_unique:
wenzelm@53406
  2052
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2053
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  2054
  by (metis basis_card_eq_dim)
huffman@44133
  2055
wenzelm@60420
  2056
text \<open>More lemmas about dimension.\<close>
huffman@44133
  2057
wenzelm@53406
  2058
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  2059
  using independent_Basis
hoelzl@50526
  2060
  by (intro dim_unique[of Basis]) auto
huffman@44133
  2061
huffman@44133
  2062
lemma dim_subset:
wenzelm@53406
  2063
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2064
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  2065
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  2066
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  2067
wenzelm@53406
  2068
lemma dim_subset_UNIV:
wenzelm@53406
  2069
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2070
  shows "dim S \<le> DIM('a)"
huffman@44133
  2071
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  2072
wenzelm@60420
  2073
text \<open>Converses to those.\<close>
huffman@44133
  2074
huffman@44133
  2075
lemma card_ge_dim_independent:
wenzelm@53406
  2076
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2077
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  2078
    and iB: "independent B"
wenzelm@53406
  2079
    and dVB: "dim V \<le> card B"
huffman@44133
  2080
  shows "V \<subseteq> span B"
wenzelm@53406
  2081
proof
wenzelm@53406
  2082
  fix a
wenzelm@53406
  2083
  assume aV: "a \<in> V"
wenzelm@53406
  2084
  {
wenzelm@53406
  2085
    assume aB: "a \<notin> span B"
wenzelm@53406
  2086
    then have iaB: "independent (insert a B)"
wenzelm@53406
  2087
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  2088
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  2089
      by blast
wenzelm@53406
  2090
    from aB have "a \<notin>B"
wenzelm@53406
  2091
      by (auto simp add: span_superset)
wenzelm@53406
  2092
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  2093
    have False by auto
wenzelm@53406
  2094
  }
wenzelm@53406
  2095
  then show "a \<in> span B" by blast
huffman@44133
  2096
qed
huffman@44133
  2097
huffman@44133
  2098
lemma card_le_dim_spanning:
wenzelm@49663
  2099
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  2100
    and VB: "V \<subseteq> span B"
wenzelm@49663
  2101
    and fB: "finite B"
wenzelm@49663
  2102
    and dVB: "dim V \<ge> card B"
huffman@44133
  2103
  shows "independent B"
wenzelm@49522
  2104
proof -
wenzelm@53406
  2105
  {
wenzelm@53406
  2106
    fix a
wenzelm@53716
  2107
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  2108
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  2109
      by auto
wenzelm@53716
  2110
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  2111
      by auto
wenzelm@53716
  2112
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  2113
      by blast
wenzelm@53406
  2114
    {
wenzelm@53406
  2115
      fix x
wenzelm@53406
  2116
      assume x: "x \<in> V"
wenzelm@53716
  2117
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  2118
        by blast
wenzelm@53406
  2119
      from x VB have x': "x \<in> span B"
wenzelm@53406
  2120
        by blast
huffman@44133
  2121
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  2122
      have "x \<in> span (B - {a})" .
wenzelm@53406
  2123
    }
wenzelm@53716
  2124
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  2125
      by blast
wenzelm@53716
  2126
    have th2: "finite (B - {a})"
wenzelm@53406
  2127
      using fB by auto
huffman@44133
  2128
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  2129
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  2130
    from c c0 dVB cb have False by simp
wenzelm@53406
  2131
  }
wenzelm@53406
  2132
  then show ?thesis
wenzelm@53406
  2133
    unfolding dependent_def by blast
huffman@44133
  2134
qed
huffman@44133
  2135
wenzelm@53406
  2136
lemma card_eq_dim:
wenzelm@53406
  2137
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2138
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  2139
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  2140
wenzelm@60420
  2141
text \<open>More general size bound lemmas.\<close>
huffman@44133
  2142
huffman@44133
  2143
lemma independent_bound_general:
wenzelm@53406
  2144
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2145
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  2146
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  2147
wenzelm@49522
  2148
lemma dependent_biggerset_general:
wenzelm@53406
  2149
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2150
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  2151
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  2152
paulson@60303
  2153
lemma dim_span [simp]:
wenzelm@53406
  2154
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2155
  shows "dim (span S) = dim S"
wenzelm@49522
  2156
proof -
huffman@44133
  2157
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  2158
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  2159
  from basis_exists[of S]
wenzelm@53406
  2160
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  2161
    by blast
wenzelm@53406
  2162
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2163
    using independent_bound by blast+
wenzelm@53406
  2164
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  2165
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  2166
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  2167
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  2168
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  2169
    using fB(2) by arith
huffman@44133
  2170
qed
huffman@44133
  2171
wenzelm@53406
  2172
lemma subset_le_dim:
wenzelm@53406
  2173
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2174
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  2175
  by (metis dim_span dim_subset)
huffman@44133
  2176
wenzelm@53406
  2177
lemma span_eq_dim:
wenzelm@56444
  2178
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2179
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  2180
  by (metis dim_span)
huffman@44133
  2181
huffman@44133
  2182
lemma dim_image_le:
huffman@44133
  2183
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  2184
  assumes lf: "linear f"
wenzelm@49663
  2185
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  2186
proof -
huffman@44133
  2187
  from basis_exists[of S] obtain B where
huffman@44133
  2188
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  2189
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2190
    using independent_bound by blast+
huffman@44133
  2191
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  2192
    apply (rule span_card_ge_dim)
wenzelm@53406
  2193
    using lf B fB
wenzelm@53406
  2194
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  2195
    done
wenzelm@53406
  2196
  also have "\<dots> \<le> dim S"
wenzelm@53406
  2197
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  2198
  finally show ?thesis .
huffman@44133
  2199
qed
huffman@44133
  2200
wenzelm@60420
  2201
text \<open>Picking an orthogonal replacement for a spanning set.\<close>
huffman@44133
  2202
wenzelm@53406
  2203
lemma vector_sub_project_orthogonal:
wenzelm@53406
  2204
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  2205
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  2206
  unfolding inner_simps by auto
huffman@44133
  2207
huffman@44528
  2208
lemma pairwise_orthogonal_insert:
huffman@44528
  2209
  assumes "pairwise orthogonal S"
wenzelm@49522
  2210
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  2211
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  2212
  using assms unfolding pairwise_def
huffman@44528
  2213
  by (auto simp add: orthogonal_commute)
huffman@44528
  2214
huffman@44133
  2215
lemma basis_orthogonal:
wenzelm@53406
  2216
  fixes B :: "'a::real_inner set"
huffman@44133
  2217
  assumes fB: "finite B"
huffman@44133
  2218
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  2219
  (is " \<exists>C. ?P B C")
wenzelm@49522
  2220
  using fB
wenzelm@49522
  2221
proof (induct rule: finite_induct)
wenzelm@49522
  2222
  case empty
wenzelm@53406
  2223
  then show ?case
wenzelm@53406
  2224
    apply (rule exI[where x="{}"])
wenzelm@53406
  2225
    apply (auto simp add: pairwise_def)
wenzelm@53406
  2226
    done
huffman@44133
  2227
next
wenzelm@49522
  2228
  case (insert a B)
wenzelm@60420
  2229
  note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
wenzelm@60420
  2230
  from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
huffman@44133
  2231
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  2232
    "span C = span B" "pairwise orthogonal C" by blast
huffman@44133
  2233
  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  2234
  let ?C = "insert ?a C"
wenzelm@53406
  2235
  from C(1) have fC: "finite ?C"
wenzelm@53406
  2236
    by simp
wenzelm@49522
  2237
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  2238
    by (simp add: card_insert_if)
wenzelm@53406
  2239
  {
wenzelm@53406
  2240
    fix x k
wenzelm@49522
  2241
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  2242
      by (simp add: field_simps)
huffman@44133
  2243
    have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
huffman@44133
  2244
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  2245
      apply (rule span_add_eq)
huffman@44133
  2246
      apply (rule span_mul)
huffman@56196
  2247
      apply (rule span_setsum)
huffman@44133
  2248
      apply clarify
huffman@44133
  2249
      apply (rule span_mul)
wenzelm@49522
  2250
      apply (rule span_superset)
wenzelm@49522
  2251
      apply assumption
wenzelm@53406
  2252
      done
wenzelm@53406
  2253
  }
huffman@44133
  2254
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  2255
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  2256
  {
wenzelm@53406
  2257
    fix y
wenzelm@53406
  2258
    assume yC: "y \<in> C"
wenzelm@53406
  2259
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  2260
      by blast
wenzelm@53406
  2261
    have fth: "finite (C - {y})"
wenzelm@53406
  2262
      using C by simp
huffman@44528
  2263
    have "orthogonal ?a y"
huffman@44528
  2264
      unfolding orthogonal_def
haftmann@54230
  2265
      unfolding inner_diff inner_setsum_left right_minus_eq
wenzelm@60420
  2266
      unfolding setsum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
huffman@44528
  2267
      apply (clarsimp simp add: inner_commute[of y a])
haftmann@57418
  2268
      apply (rule setsum.neutral)
huffman@44528
  2269
      apply clarsimp
huffman@44528
  2270
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@60420
  2271
      using \<open>y \<in> C\<close> by auto
wenzelm@53406
  2272
  }
wenzelm@60420
  2273
  with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
huffman@44528
  2274
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  2275
  from fC cC SC CPO have "?P (insert a B) ?C"
wenzelm@53406
  2276
    by blast
huffman@44133
  2277
  then show ?case by blast
huffman@44133
  2278
qed
huffman@44133
  2279
huffman@44133
  2280
lemma orthogonal_basis_exists:
huffman@44133
  2281
  fixes V :: "('a::euclidean_space) set"
huffman@44133
  2282
  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
wenzelm@49663
  2283
proof -
wenzelm@49522
  2284
  from basis_exists[of V] obtain B where
wenzelm@53406
  2285
    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
wenzelm@53406
  2286
    by blast
wenzelm@53406
  2287
  from B have fB: "finite B" "card B = dim V"
wenzelm@53406
  2288
    using independent_bound by auto
huffman@44133
  2289
  from basis_orthogonal[OF fB(1)] obtain C where
wenzelm@53406
  2290
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
wenzelm@53406
  2291
    by blast
wenzelm@53406
  2292
  from C B have CSV: "C \<subseteq> span V"
wenzelm@53406
  2293
    by (metis span_inc span_mono subset_trans)
wenzelm@53406
  2294
  from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
wenzelm@53406
  2295
    by (simp add: span_span)
huffman@44133
  2296
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
wenzelm@53406
  2297
  have iC: "independent C"
huffman@44133
  2298
    by (simp add: dim_span)
wenzelm@53406
  2299
  from C fB have "card C \<le> dim V"
wenzelm@53406
  2300
    by simp
wenzelm@53406
  2301
  moreover have "dim V \<le> card C"
wenzelm@53406
  2302
    using span_card_ge_dim[OF CSV SVC C(1)]
wenzelm@53406
  2303
    by (simp add: dim_span)
wenzelm@53406
  2304
  ultimately have CdV: "card C = dim V"
wenzelm@53406
  2305
    using C(1) by simp
wenzelm@53406
  2306
  from C B CSV CdV iC show ?thesis
wenzelm@53406
  2307
    by auto
huffman@44133
  2308
qed
huffman@44133
  2309
wenzelm@60420
  2310
text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
huffman@44133
  2311
wenzelm@49522
  2312
lemma span_not_univ_orthogonal:
wenzelm@53406
  2313
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2314
  assumes sU: "span S \<noteq> UNIV"
wenzelm@56444
  2315
  shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
wenzelm@49522
  2316
proof -
wenzelm@53406
  2317
  from sU obtain a where a: "a \<notin> span S"
wenzelm@53406
  2318
    by blast
huffman@44133
  2319
  from orthogonal_basis_exists obtain B where
huffman@44133
  2320
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
huffman@44133
  2321
    by blast
wenzelm@53406
  2322
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2323
    using independent_bound by auto
huffman@44133
  2324
  from span_mono[OF B(2)] span_mono[OF B(3)]
wenzelm@53406
  2325
  have sSB: "span S = span B"
wenzelm@53406
  2326
    by (simp add: span_span)
huffman@44133
  2327
  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
huffman@44133
  2328
  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
huffman@44133
  2329
    unfolding sSB
huffman@56196
  2330
    apply (rule span_setsum)
huffman@44133
  2331
    apply clarsimp
huffman@44133
  2332
    apply (rule span_mul)
wenzelm@49522
  2333
    apply (rule span_superset)
wenzelm@49522
  2334
    apply assumption
wenzelm@49522
  2335
    done
wenzelm@53406
  2336
  with a have a0:"?a  \<noteq> 0"
wenzelm@53406
  2337
    by auto
huffman@44133
  2338
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
wenzelm@49522
  2339
  proof (rule span_induct')
wenzelm@49522
  2340
    show "subspace {x. ?a \<bullet> x = 0}"
wenzelm@49522
  2341
      by (auto simp add: subspace_def inner_add)
wenzelm@49522
  2342
  next
wenzelm@53406
  2343
    {
wenzelm@53406
  2344
      fix x
wenzelm@53406
  2345
      assume x: "x \<in> B"
wenzelm@53406
  2346
      from x have B': "B = insert x (B - {x})"
wenzelm@53406
  2347
        by blast
wenzelm@53406
  2348
      have fth: "finite (B - {x})"
wenzelm@53406
  2349
        using fB by simp
huffman@44133
  2350
      have "?a \<bullet> x = 0"
wenzelm@53406
  2351
        apply (subst B')
wenzelm@53406
  2352
        using fB fth
huffman@44133
  2353
        unfolding setsum_clauses(2)[OF fth]
huffman@44133
  2354
        apply simp unfolding inner_simps
huffman@44527
  2355
        apply (clarsimp simp add: inner_add inner_setsum_left)
haftmann@57418
  2356
        apply (rule setsum.neutral, rule ballI)
huffman@44133
  2357
        unfolding inner_commute
wenzelm@49711
  2358
        apply (auto simp add: x field_simps
wenzelm@49711
  2359
          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  2360
        done
wenzelm@53406
  2361
    }
wenzelm@53406
  2362
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
wenzelm@53406
  2363
      by blast
huffman@44133
  2364
  qed
wenzelm@53406
  2365
  with a0 show ?thesis
wenzelm@53406
  2366
    unfolding sSB by (auto intro: exI[where x="?a"])
huffman@44133
  2367
qed
huffman@44133
  2368
huffman@44133
  2369
lemma span_not_univ_subset_hyperplane:
wenzelm@53406
  2370
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2371
  assumes SU: "span S \<noteq> UNIV"
huffman@44133
  2372
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
huffman@44133
  2373
  using span_not_univ_orthogonal[OF SU] by auto
huffman@44133
  2374
wenzelm@49663
  2375
lemma lowdim_subset_hyperplane:
wenzelm@53406
  2376
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2377
  assumes d: "dim S < DIM('a)"
wenzelm@56444
  2378
  shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
wenzelm@49522
  2379
proof -
wenzelm@53406
  2380
  {
wenzelm@53406
  2381
    assume "span S = UNIV"
wenzelm@53406
  2382
    then have "dim (span S) = dim (UNIV :: ('a) set)"
wenzelm@53406
  2383
      by simp
wenzelm@53406
  2384
    then have "dim S = DIM('a)"
wenzelm@53406
  2385
      by (simp add: dim_span dim_UNIV)