src/HOL/Analysis/Linear_Algebra.thy
author paulson <lp15@cam.ac.uk>
Wed Jul 19 16:41:26 2017 +0100 (22 months ago)
changeset 66287 005a30862ed0
parent 65680 378a2f11bec9
child 66297 d425bdf419f5
permissions -rw-r--r--
new material: Colinearity, convex sets, polytopes
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(*  Title:      HOL/Analysis/Linear_Algebra.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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section \<open>Elementary linear algebra on Euclidean spaces\<close>
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theory Linear_Algebra
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imports
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  Euclidean_Space
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  "~~/src/HOL/Library/Infinite_Set"
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begin
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lemma linear_simps:
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  assumes "bounded_linear f"
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  shows
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    "f (a + b) = f a + f b"
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    "f (a - b) = f a - f b"
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    "f 0 = 0"
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    "f (- a) = - f a"
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    "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
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proof -
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  interpret f: bounded_linear f by fact
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  show "f (a + b) = f a + f b" by (rule f.add)
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  show "f (a - b) = f a - f b" by (rule f.diff)
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  show "f 0 = 0" by (rule f.zero)
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  show "f (- a) = - f a" by (rule f.minus)
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  show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR)
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qed
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lemma bounded_linearI:
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  assumes "\<And>x y. f (x + y) = f x + f y"
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    and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
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    and "\<And>x. norm (f x) \<le> norm x * K"
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  shows "bounded_linear f"
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  using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)
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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_uminus: "linear uminus"
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by (simp add: linear_iff)
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lemma linear_compose_sum:
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  assumes lS: "\<forall>a \<in> S. linear (f a)"
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  shows "linear (\<lambda>x. sum (\<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_diff: "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_sum:
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  assumes f: "linear f"
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  shows "f (sum g S) = sum (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_sum_mul:
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  assumes lin: "linear f"
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  shows "f (sum (\<lambda>i. c i *\<^sub>R v i) S) = sum (\<lambda>i. c i *\<^sub>R f (v i)) S"
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  using linear_sum[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_diff[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_diff: "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_sum:
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  assumes sA: "subspace A"
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    and f: "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A"
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  shows "sum 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 [iff]: "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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lemma subspace_sums: "\<lbrakk>subspace S; subspace T\<rbrakk> \<Longrightarrow> subspace {x + y|x y. x \<in> S \<and> y \<in> T}"
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apply (simp add: subspace_def)
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apply (intro conjI impI allI)
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  using add.right_neutral apply blast
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 apply clarify
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 apply (metis add.assoc add.left_commute)
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using scaleR_add_right by blast
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subsection \<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 [iff]: "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"
huffman@44521
   304
  unfolding span_def by (rule hull_minimal)
huffman@44521
   305
hoelzl@63053
   306
lemma span_UNIV: "span UNIV = UNIV"
hoelzl@63053
   307
  by (intro span_unique) auto
hoelzl@63053
   308
huffman@44521
   309
lemma (in real_vector) span_induct:
wenzelm@49522
   310
  assumes x: "x \<in> span S"
lp15@63469
   311
    and P: "subspace (Collect P)"
lp15@63469
   312
    and SP: "\<And>x. x \<in> S \<Longrightarrow> P x"
lp15@63469
   313
  shows "P x"
wenzelm@49522
   314
proof -
lp15@63469
   315
  from SP have SP': "S \<subseteq> Collect P"
wenzelm@53406
   316
    by (simp add: subset_eq)
huffman@44170
   317
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
lp15@63469
   318
  show ?thesis
lp15@63469
   319
    using subset_eq by force
huffman@44133
   320
qed
huffman@44133
   321
huffman@44133
   322
lemma span_empty[simp]: "span {} = {0}"
huffman@44133
   323
  apply (simp add: span_def)
huffman@44133
   324
  apply (rule hull_unique)
huffman@44170
   325
  apply (auto simp add: subspace_def)
huffman@44133
   326
  done
huffman@44133
   327
lp15@62948
   328
lemma (in real_vector) independent_empty [iff]: "independent {}"
huffman@44133
   329
  by (simp add: dependent_def)
huffman@44133
   330
wenzelm@49522
   331
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
huffman@44133
   332
  unfolding dependent_def by auto
huffman@44133
   333
wenzelm@53406
   334
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
huffman@44133
   335
  apply (clarsimp simp add: dependent_def span_mono)
huffman@44133
   336
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
huffman@44133
   337
  apply force
huffman@44133
   338
  apply (rule span_mono)
huffman@44133
   339
  apply auto
huffman@44133
   340
  done
huffman@44133
   341
huffman@44133
   342
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   343
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   344
wenzelm@49711
   345
lemma (in real_vector) span_induct':
hoelzl@63050
   346
  "\<forall>x \<in> S. P x \<Longrightarrow> subspace {x. P x} \<Longrightarrow> \<forall>x \<in> span S. P x"
hoelzl@63050
   347
  unfolding span_def by (rule hull_induct) auto
huffman@44133
   348
wenzelm@56444
   349
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
wenzelm@53406
   350
where
huffman@44170
   351
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   352
| span_induct_alt_help_S:
wenzelm@53406
   353
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   354
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   355
huffman@44133
   356
lemma span_induct_alt':
wenzelm@53406
   357
  assumes h0: "h 0"
wenzelm@53406
   358
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   359
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   360
proof -
wenzelm@53406
   361
  {
wenzelm@53406
   362
    fix x :: 'a
wenzelm@53406
   363
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   364
    have "h x"
huffman@44133
   365
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   366
      apply (rule h0)
wenzelm@53406
   367
      apply (rule hS)
wenzelm@53406
   368
      apply assumption
wenzelm@53406
   369
      apply assumption
wenzelm@53406
   370
      done
wenzelm@53406
   371
  }
huffman@44133
   372
  note th0 = this
wenzelm@53406
   373
  {
wenzelm@53406
   374
    fix x
wenzelm@53406
   375
    assume x: "x \<in> span S"
huffman@44170
   376
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   377
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   378
      show "x \<in> span S" by (rule x)
wenzelm@49522
   379
    next
wenzelm@53406
   380
      fix x
wenzelm@53406
   381
      assume xS: "x \<in> S"
wenzelm@53406
   382
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   383
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   384
        by simp
wenzelm@49522
   385
    next
wenzelm@49522
   386
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   387
      moreover
wenzelm@53406
   388
      {
wenzelm@53406
   389
        fix x y
wenzelm@49522
   390
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   391
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   392
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   393
          apply simp
haftmann@57512
   394
          unfolding add.assoc
wenzelm@49522
   395
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   396
          apply assumption
wenzelm@49522
   397
          apply simp
wenzelm@53406
   398
          done
wenzelm@53406
   399
      }
wenzelm@49522
   400
      moreover
wenzelm@53406
   401
      {
wenzelm@53406
   402
        fix c x
wenzelm@49522
   403
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   404
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   405
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   406
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   407
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   408
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   409
          apply assumption
wenzelm@49522
   410
          apply simp
wenzelm@49522
   411
          done }
lp15@63469
   412
      ultimately show "subspace {a. a \<in> span_induct_alt_help S}"
wenzelm@49522
   413
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   414
    qed
wenzelm@53406
   415
  }
huffman@44133
   416
  with th0 show ?thesis by blast
huffman@44133
   417
qed
huffman@44133
   418
huffman@44133
   419
lemma span_induct_alt:
wenzelm@53406
   420
  assumes h0: "h 0"
wenzelm@53406
   421
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   422
    and x: "x \<in> span S"
huffman@44133
   423
  shows "h x"
wenzelm@49522
   424
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   425
wenzelm@60420
   426
text \<open>Individual closure properties.\<close>
huffman@44133
   427
huffman@44133
   428
lemma span_span: "span (span A) = span A"
huffman@44133
   429
  unfolding span_def hull_hull ..
huffman@44133
   430
wenzelm@53406
   431
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   432
  by (metis span_clauses(1))
wenzelm@53406
   433
lp15@63075
   434
lemma (in real_vector) span_0 [simp]: "0 \<in> span S"
wenzelm@53406
   435
  by (metis subspace_span subspace_0)
huffman@44133
   436
huffman@44133
   437
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   438
  by (metis subset_eq span_superset)
huffman@44133
   439
hoelzl@63053
   440
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
hoelzl@63053
   441
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
hoelzl@63053
   442
  by (auto simp add: span_span)
hoelzl@63053
   443
wenzelm@53406
   444
lemma (in real_vector) dependent_0:
wenzelm@53406
   445
  assumes "0 \<in> A"
wenzelm@53406
   446
  shows "dependent A"
wenzelm@53406
   447
  unfolding dependent_def
wenzelm@53406
   448
  using assms span_0
lp15@63075
   449
  by blast
wenzelm@53406
   450
wenzelm@53406
   451
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   452
  by (metis subspace_add subspace_span)
huffman@44133
   453
wenzelm@53406
   454
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   455
  by (metis subspace_span subspace_mul)
huffman@44133
   456
wenzelm@53406
   457
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   458
  by (metis subspace_neg subspace_span)
huffman@44133
   459
lp15@63938
   460
lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
lp15@63114
   461
  by (metis subspace_span subspace_diff)
huffman@44133
   462
nipkow@64267
   463
lemma (in real_vector) span_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> sum f A \<in> span S"
nipkow@64267
   464
  by (rule subspace_sum [OF subspace_span])
huffman@44133
   465
huffman@44133
   466
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   467
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   468
hoelzl@63050
   469
text \<open>The key breakdown property.\<close>
hoelzl@63050
   470
hoelzl@63050
   471
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
hoelzl@63050
   472
proof (rule span_unique)
hoelzl@63050
   473
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
hoelzl@63050
   474
    by (fast intro: scaleR_one [symmetric])
hoelzl@63050
   475
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
hoelzl@63050
   476
    unfolding subspace_def
hoelzl@63050
   477
    by (auto intro: scaleR_add_left [symmetric])
hoelzl@63050
   478
next
hoelzl@63050
   479
  fix T
hoelzl@63050
   480
  assume "{x} \<subseteq> T" and "subspace T"
hoelzl@63050
   481
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
hoelzl@63050
   482
    unfolding subspace_def by auto
hoelzl@63050
   483
qed
hoelzl@63050
   484
wenzelm@60420
   485
text \<open>Mapping under linear image.\<close>
huffman@44133
   486
hoelzl@63050
   487
lemma subspace_linear_image:
hoelzl@63050
   488
  assumes lf: "linear f"
hoelzl@63050
   489
    and sS: "subspace S"
hoelzl@63050
   490
  shows "subspace (f ` S)"
hoelzl@63050
   491
  using lf sS linear_0[OF lf]
hoelzl@63050
   492
  unfolding linear_iff subspace_def
hoelzl@63050
   493
  apply (auto simp add: image_iff)
hoelzl@63050
   494
  apply (rule_tac x="x + y" in bexI)
hoelzl@63050
   495
  apply auto
hoelzl@63050
   496
  apply (rule_tac x="c *\<^sub>R x" in bexI)
hoelzl@63050
   497
  apply auto
hoelzl@63050
   498
  done
hoelzl@63050
   499
hoelzl@63050
   500
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
hoelzl@63050
   501
  by (auto simp add: subspace_def linear_iff linear_0[of f])
hoelzl@63050
   502
hoelzl@63050
   503
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
hoelzl@63050
   504
  by (auto simp add: subspace_def linear_iff linear_0[of f])
hoelzl@63050
   505
huffman@44521
   506
lemma span_linear_image:
huffman@44521
   507
  assumes lf: "linear f"
wenzelm@56444
   508
  shows "span (f ` S) = f ` span S"
huffman@44521
   509
proof (rule span_unique)
huffman@44521
   510
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   511
    by (intro image_mono span_inc)
huffman@44521
   512
  show "subspace (f ` span S)"
huffman@44521
   513
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   514
next
wenzelm@53406
   515
  fix T
wenzelm@53406
   516
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   517
  then show "f ` span S \<subseteq> T"
huffman@44521
   518
    unfolding image_subset_iff_subset_vimage
huffman@44521
   519
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   520
qed
huffman@44521
   521
hoelzl@63053
   522
lemma spans_image:
hoelzl@63053
   523
  assumes lf: "linear f"
hoelzl@63053
   524
    and VB: "V \<subseteq> span B"
hoelzl@63053
   525
  shows "f ` V \<subseteq> span (f ` B)"
hoelzl@63053
   526
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
hoelzl@63053
   527
lp15@63469
   528
lemma span_Un: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   529
proof (rule span_unique)
huffman@44521
   530
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   531
    by safe (force intro: span_clauses)+
huffman@44521
   532
next
huffman@44521
   533
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   534
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   535
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   536
    by (intro subspace_Times subspace_span)
huffman@44521
   537
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   538
    by (rule subspace_linear_image)
huffman@44521
   539
next
wenzelm@49711
   540
  fix T
wenzelm@49711
   541
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   542
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   543
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   544
qed
huffman@44133
   545
wenzelm@49522
   546
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   547
proof -
huffman@44521
   548
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
lp15@63469
   549
    unfolding span_Un span_singleton
huffman@44521
   550
    apply safe
huffman@44521
   551
    apply (rule_tac x=k in exI, simp)
huffman@44521
   552
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   553
    apply auto
huffman@44521
   554
    done
wenzelm@49522
   555
  then show ?thesis by simp
huffman@44521
   556
qed
huffman@44521
   557
huffman@44133
   558
lemma span_breakdown:
wenzelm@53406
   559
  assumes bS: "b \<in> S"
wenzelm@53406
   560
    and aS: "a \<in> span S"
huffman@44521
   561
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   562
  using assms span_insert [of b "S - {b}"]
huffman@44521
   563
  by (simp add: insert_absorb)
huffman@44133
   564
wenzelm@53406
   565
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
   566
  by (simp add: span_insert)
huffman@44133
   567
wenzelm@60420
   568
text \<open>Hence some "reversal" results.\<close>
huffman@44133
   569
huffman@44133
   570
lemma in_span_insert:
wenzelm@49711
   571
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
   572
    and na: "a \<notin> span S"
huffman@44133
   573
  shows "b \<in> span (insert a S)"
wenzelm@49663
   574
proof -
huffman@55910
   575
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
   576
    unfolding span_insert by fast
wenzelm@53406
   577
  show ?thesis
wenzelm@53406
   578
  proof (cases "k = 0")
wenzelm@53406
   579
    case True
huffman@55910
   580
    with k have "a \<in> span S" by simp
huffman@55910
   581
    with na show ?thesis by simp
wenzelm@53406
   582
  next
wenzelm@53406
   583
    case False
huffman@55910
   584
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
   585
      by (rule span_mul)
huffman@55910
   586
    then have "b - inverse k *\<^sub>R a \<in> span S"
wenzelm@60420
   587
      using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
huffman@55910
   588
    then show ?thesis
huffman@55910
   589
      unfolding span_insert by fast
wenzelm@53406
   590
  qed
huffman@44133
   591
qed
huffman@44133
   592
huffman@44133
   593
lemma in_span_delete:
huffman@44133
   594
  assumes a: "a \<in> span S"
wenzelm@53716
   595
    and na: "a \<notin> span (S - {b})"
huffman@44133
   596
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
   597
  apply (rule in_span_insert)
huffman@44133
   598
  apply (rule set_rev_mp)
huffman@44133
   599
  apply (rule a)
huffman@44133
   600
  apply (rule span_mono)
huffman@44133
   601
  apply blast
huffman@44133
   602
  apply (rule na)
huffman@44133
   603
  done
huffman@44133
   604
wenzelm@60420
   605
text \<open>Transitivity property.\<close>
huffman@44133
   606
huffman@44521
   607
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
   608
  unfolding span_def by (rule hull_redundant)
huffman@44521
   609
huffman@44133
   610
lemma span_trans:
wenzelm@53406
   611
  assumes x: "x \<in> span S"
wenzelm@53406
   612
    and y: "y \<in> span (insert x S)"
huffman@44133
   613
  shows "y \<in> span S"
huffman@44521
   614
  using assms by (simp only: span_redundant)
huffman@44133
   615
huffman@44133
   616
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
   617
  by (simp only: span_redundant span_0)
huffman@44133
   618
wenzelm@60420
   619
text \<open>An explicit expansion is sometimes needed.\<close>
huffman@44133
   620
huffman@44133
   621
lemma span_explicit:
nipkow@64267
   622
  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
   623
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
   624
proof -
wenzelm@53406
   625
  {
wenzelm@53406
   626
    fix x
huffman@55910
   627
    assume "?h x"
nipkow@64267
   628
    then obtain S u where "finite S" and "S \<subseteq> P" and "sum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
   629
      by blast
huffman@55910
   630
    then have "x \<in> span P"
nipkow@64267
   631
      by (auto intro: span_sum span_mul span_superset)
wenzelm@53406
   632
  }
huffman@44133
   633
  moreover
huffman@55910
   634
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
   635
  proof (rule span_induct_alt')
huffman@55910
   636
    show "?h 0"
huffman@55910
   637
      by (rule exI[where x="{}"], simp)
huffman@44133
   638
  next
huffman@44133
   639
    fix c x y
wenzelm@53406
   640
    assume x: "x \<in> P"
huffman@55910
   641
    assume hy: "?h y"
huffman@44133
   642
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
nipkow@64267
   643
      and u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
   644
    let ?S = "insert x S"
wenzelm@49522
   645
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
   646
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
   647
      by blast+
wenzelm@53406
   648
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
   649
    proof cases
wenzelm@53406
   650
      assume xS: "x \<in> S"
nipkow@64267
   651
      have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
nipkow@64267
   652
        using xS by (simp add: sum.remove [OF fS xS] insert_absorb)
huffman@44133
   653
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
nipkow@64267
   654
        by (simp add: sum.remove [OF fS xS] algebra_simps)
huffman@44133
   655
      also have "\<dots> = c*\<^sub>R x + y"
haftmann@57512
   656
        by (simp add: add.commute u)
nipkow@64267
   657
      finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
   658
      then show ?thesis using th0 by blast
wenzelm@53406
   659
    next
wenzelm@53406
   660
      assume xS: "x \<notin> S"
wenzelm@49522
   661
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
   662
        unfolding u[symmetric]
nipkow@64267
   663
        apply (rule sum.cong)
wenzelm@53406
   664
        using xS
wenzelm@53406
   665
        apply auto
wenzelm@49522
   666
        done
wenzelm@53406
   667
      show ?thesis using fS xS th0
haftmann@57512
   668
        by (simp add: th00 add.commute cong del: if_weak_cong)
wenzelm@53406
   669
    qed
huffman@55910
   670
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
   671
      by fast
huffman@44133
   672
  qed
huffman@44133
   673
  ultimately show ?thesis by blast
huffman@44133
   674
qed
huffman@44133
   675
huffman@44133
   676
lemma dependent_explicit:
nipkow@64267
   677
  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0))"
wenzelm@49522
   678
  (is "?lhs = ?rhs")
wenzelm@49522
   679
proof -
wenzelm@53406
   680
  {
wenzelm@53406
   681
    assume dP: "dependent P"
huffman@44133
   682
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
nipkow@64267
   683
      and SP: "S \<subseteq> P - {a}" and ua: "sum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
   684
      unfolding dependent_def span_explicit by blast
huffman@44133
   685
    let ?S = "insert a S"
huffman@44133
   686
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
   687
    let ?v = a
wenzelm@53406
   688
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
   689
      by blast
wenzelm@53406
   690
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
   691
      by auto
nipkow@64267
   692
    have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
   693
      using fS aS
huffman@55910
   694
      apply simp
huffman@44133
   695
      apply (subst (2) ua[symmetric])
nipkow@64267
   696
      apply (rule sum.cong)
wenzelm@49522
   697
      apply auto
wenzelm@49522
   698
      done
huffman@55910
   699
    with th0 have ?rhs by fast
wenzelm@49522
   700
  }
huffman@44133
   701
  moreover
wenzelm@53406
   702
  {
wenzelm@53406
   703
    fix S u v
wenzelm@49522
   704
    assume fS: "finite S"
wenzelm@53406
   705
      and SP: "S \<subseteq> P"
wenzelm@53406
   706
      and vS: "v \<in> S"
wenzelm@53406
   707
      and uv: "u v \<noteq> 0"
nipkow@64267
   708
      and u: "sum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
   709
    let ?a = v
huffman@44133
   710
    let ?S = "S - {v}"
huffman@44133
   711
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
   712
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
   713
      using fS SP vS by auto
nipkow@64267
   714
    have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S =
nipkow@64267
   715
      sum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
nipkow@64267
   716
      using fS vS uv by (simp add: sum_diff1 field_simps)
wenzelm@53406
   717
    also have "\<dots> = ?a"
nipkow@64267
   718
      unfolding scaleR_right.sum [symmetric] u using uv by simp
nipkow@64267
   719
    finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
   720
    with th0 have ?lhs
huffman@44133
   721
      unfolding dependent_def span_explicit
huffman@44133
   722
      apply -
huffman@44133
   723
      apply (rule bexI[where x= "?a"])
huffman@44133
   724
      apply (simp_all del: scaleR_minus_left)
huffman@44133
   725
      apply (rule exI[where x= "?S"])
wenzelm@49522
   726
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
   727
      done
wenzelm@49522
   728
  }
huffman@44133
   729
  ultimately show ?thesis by blast
huffman@44133
   730
qed
huffman@44133
   731
lp15@63075
   732
lemma dependent_finite:
lp15@63075
   733
  assumes "finite S"
lp15@63075
   734
    shows "dependent S \<longleftrightarrow> (\<exists>u. (\<exists>v \<in> S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = 0)"
lp15@63075
   735
           (is "?lhs = ?rhs")
lp15@63075
   736
proof
lp15@63075
   737
  assume ?lhs
lp15@63075
   738
  then obtain T u v
lp15@63075
   739
         where "finite T" "T \<subseteq> S" "v\<in>T" "u v \<noteq> 0" "(\<Sum>v\<in>T. u v *\<^sub>R v) = 0"
lp15@63075
   740
    by (force simp: dependent_explicit)
lp15@63075
   741
  with assms show ?rhs
lp15@63075
   742
    apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
nipkow@64267
   743
    apply (auto simp: sum.mono_neutral_right)
lp15@63075
   744
    done
lp15@63075
   745
next
lp15@63075
   746
  assume ?rhs  with assms show ?lhs
lp15@63075
   747
    by (fastforce simp add: dependent_explicit)
lp15@63075
   748
qed
lp15@63075
   749
hoelzl@63051
   750
lemma span_alt:
hoelzl@63051
   751
  "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
   752
  unfolding span_explicit
hoelzl@63051
   753
  apply safe
hoelzl@63051
   754
  subgoal for x S u
hoelzl@63051
   755
    by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
nipkow@64267
   756
        (auto intro!: sum.mono_neutral_cong_right)
hoelzl@63051
   757
  apply auto
hoelzl@63051
   758
  done
hoelzl@63051
   759
hoelzl@63051
   760
lemma dependent_alt:
hoelzl@63051
   761
  "dependent B \<longleftrightarrow>
hoelzl@63051
   762
    (\<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
   763
  unfolding dependent_explicit
hoelzl@63051
   764
  apply safe
hoelzl@63051
   765
  subgoal for S u v
hoelzl@63051
   766
    apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
nipkow@64267
   767
    apply (subst sum.mono_neutral_cong_left[where T=S])
nipkow@64267
   768
    apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong)
hoelzl@63051
   769
    done
hoelzl@63051
   770
  apply auto
hoelzl@63051
   771
  done
hoelzl@63051
   772
hoelzl@63051
   773
lemma independent_alt:
hoelzl@63051
   774
  "independent B \<longleftrightarrow>
hoelzl@63051
   775
    (\<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
   776
  unfolding dependent_alt by auto
hoelzl@63051
   777
hoelzl@63051
   778
lemma independentD_alt:
hoelzl@63051
   779
  "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
   780
  unfolding independent_alt by blast
hoelzl@63051
   781
hoelzl@63051
   782
lemma independentD_unique:
hoelzl@63051
   783
  assumes B: "independent B"
hoelzl@63051
   784
    and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   785
    and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   786
    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
   787
  shows "X = Y"
wenzelm@49522
   788
proof -
hoelzl@63051
   789
  have "X x - Y x = 0" for x
hoelzl@63051
   790
    using B
hoelzl@63051
   791
  proof (rule independentD_alt)
hoelzl@63051
   792
    have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
hoelzl@63051
   793
      by auto
hoelzl@63051
   794
    then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   795
      using X Y by (auto dest: finite_subset)
hoelzl@63051
   796
    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)"
nipkow@64267
   797
      using X Y by (intro sum.mono_neutral_cong_left) auto
hoelzl@63051
   798
    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)"
nipkow@64267
   799
      by (simp add: scaleR_diff_left sum_subtractf assms)
hoelzl@63051
   800
    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)"
nipkow@64267
   801
      using X Y by (intro sum.mono_neutral_cong_right) auto
hoelzl@63051
   802
    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)"
nipkow@64267
   803
      using X Y by (intro sum.mono_neutral_cong_right) auto
hoelzl@63051
   804
    finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
hoelzl@63051
   805
      using assms by simp
hoelzl@63051
   806
  qed
hoelzl@63051
   807
  then show ?thesis
hoelzl@63051
   808
    by auto
huffman@44133
   809
qed
huffman@44133
   810
wenzelm@60420
   811
text \<open>This is useful for building a basis step-by-step.\<close>
huffman@44133
   812
huffman@44133
   813
lemma independent_insert:
wenzelm@53406
   814
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
   815
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
   816
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
   817
proof (cases "a \<in> S")
wenzelm@53406
   818
  case True
wenzelm@53406
   819
  then show ?thesis
wenzelm@53406
   820
    using insert_absorb[OF True] by simp
wenzelm@53406
   821
next
wenzelm@53406
   822
  case False
wenzelm@53406
   823
  show ?thesis
wenzelm@53406
   824
  proof
wenzelm@53406
   825
    assume i: ?lhs
wenzelm@53406
   826
    then show ?rhs
wenzelm@53406
   827
      using False
wenzelm@53406
   828
      apply simp
wenzelm@53406
   829
      apply (rule conjI)
wenzelm@53406
   830
      apply (rule independent_mono)
wenzelm@53406
   831
      apply assumption
wenzelm@53406
   832
      apply blast
wenzelm@53406
   833
      apply (simp add: dependent_def)
wenzelm@53406
   834
      done
wenzelm@53406
   835
  next
wenzelm@53406
   836
    assume i: ?rhs
wenzelm@53406
   837
    show ?lhs
wenzelm@53406
   838
      using i False
wenzelm@53406
   839
      apply (auto simp add: dependent_def)
lp15@60810
   840
      by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
wenzelm@53406
   841
  qed
huffman@44133
   842
qed
huffman@44133
   843
hoelzl@63051
   844
lemma independent_Union_directed:
hoelzl@63051
   845
  assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
hoelzl@63051
   846
  assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
hoelzl@63051
   847
  shows "independent (\<Union>C)"
hoelzl@63051
   848
proof
hoelzl@63051
   849
  assume "dependent (\<Union>C)"
hoelzl@63051
   850
  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
   851
    by (auto simp: dependent_explicit)
hoelzl@63051
   852
hoelzl@63051
   853
  have "S \<noteq> {}"
hoelzl@63051
   854
    using \<open>v \<in> S\<close> by auto
hoelzl@63051
   855
  have "\<exists>c\<in>C. S \<subseteq> c"
hoelzl@63051
   856
    using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
hoelzl@63051
   857
  proof (induction rule: finite_ne_induct)
hoelzl@63051
   858
    case (insert i I)
hoelzl@63051
   859
    then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
hoelzl@63051
   860
      by blast
hoelzl@63051
   861
    from directed[OF cd] cd have "c \<union> d \<in> C"
hoelzl@63051
   862
      by (auto simp: sup.absorb1 sup.absorb2)
hoelzl@63051
   863
    with iI show ?case
hoelzl@63051
   864
      by (intro bexI[of _ "c \<union> d"]) auto
hoelzl@63051
   865
  qed auto
hoelzl@63051
   866
  then obtain c where "c \<in> C" "S \<subseteq> c"
hoelzl@63051
   867
    by auto
hoelzl@63051
   868
  have "dependent c"
hoelzl@63051
   869
    unfolding dependent_explicit
hoelzl@63051
   870
    by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
hoelzl@63051
   871
  with indep[OF \<open>c \<in> C\<close>] show False
hoelzl@63051
   872
    by auto
hoelzl@63051
   873
qed
hoelzl@63051
   874
hoelzl@63051
   875
text \<open>Hence we can create a maximal independent subset.\<close>
hoelzl@63051
   876
hoelzl@63051
   877
lemma maximal_independent_subset_extend:
hoelzl@63051
   878
  assumes "S \<subseteq> V" "independent S"
hoelzl@63051
   879
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
hoelzl@63051
   880
proof -
hoelzl@63051
   881
  let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}"
hoelzl@63051
   882
  have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M"
hoelzl@63051
   883
  proof (rule subset_Zorn)
hoelzl@63051
   884
    fix C :: "'a set set" assume "subset.chain ?C C"
hoelzl@63051
   885
    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
   886
      "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
hoelzl@63051
   887
      unfolding subset.chain_def by blast+
hoelzl@63051
   888
hoelzl@63051
   889
    show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U"
hoelzl@63051
   890
    proof cases
hoelzl@63051
   891
      assume "C = {}" with assms show ?thesis
hoelzl@63051
   892
        by (auto intro!: exI[of _ S])
hoelzl@63051
   893
    next
hoelzl@63051
   894
      assume "C \<noteq> {}"
hoelzl@63051
   895
      with C(2) have "S \<subseteq> \<Union>C"
hoelzl@63051
   896
        by auto
hoelzl@63051
   897
      moreover have "independent (\<Union>C)"
hoelzl@63051
   898
        by (intro independent_Union_directed C)
hoelzl@63051
   899
      moreover have "\<Union>C \<subseteq> V"
hoelzl@63051
   900
        using C by auto
hoelzl@63051
   901
      ultimately show ?thesis
hoelzl@63051
   902
        by auto
hoelzl@63051
   903
    qed
hoelzl@63051
   904
  qed
hoelzl@63051
   905
  then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B"
hoelzl@63051
   906
    and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B"
hoelzl@63051
   907
    by auto
hoelzl@63051
   908
  moreover
hoelzl@63051
   909
  { assume "\<not> V \<subseteq> span B"
hoelzl@63051
   910
    then obtain v where "v \<in> V" "v \<notin> span B"
hoelzl@63051
   911
      by auto
hoelzl@63051
   912
    with B have "independent (insert v B)"
hoelzl@63051
   913
      unfolding independent_insert by auto
hoelzl@63051
   914
    from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close>
hoelzl@63051
   915
    have "v \<in> B"
hoelzl@63051
   916
      by auto
hoelzl@63051
   917
    with \<open>v \<notin> span B\<close> have False
hoelzl@63051
   918
      by (auto intro: span_superset) }
hoelzl@63051
   919
  ultimately show ?thesis
hoelzl@63051
   920
    by (auto intro!: exI[of _ B])
hoelzl@63051
   921
qed
hoelzl@63051
   922
hoelzl@63051
   923
hoelzl@63051
   924
lemma maximal_independent_subset:
hoelzl@63051
   925
  "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
hoelzl@63051
   926
  by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
hoelzl@63051
   927
hoelzl@63051
   928
lemma span_finite:
hoelzl@63051
   929
  assumes fS: "finite S"
nipkow@64267
   930
  shows "span S = {y. \<exists>u. sum (\<lambda>v. u v *\<^sub>R v) S = y}"
hoelzl@63051
   931
  (is "_ = ?rhs")
hoelzl@63051
   932
proof -
hoelzl@63051
   933
  {
hoelzl@63051
   934
    fix y
hoelzl@63051
   935
    assume y: "y \<in> span S"
hoelzl@63051
   936
    from y obtain S' u where fS': "finite S'"
hoelzl@63051
   937
      and SS': "S' \<subseteq> S"
nipkow@64267
   938
      and u: "sum (\<lambda>v. u v *\<^sub>R v) S' = y"
hoelzl@63051
   939
      unfolding span_explicit by blast
hoelzl@63051
   940
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
nipkow@64267
   941
    have "sum (\<lambda>v. ?u v *\<^sub>R v) S = sum (\<lambda>v. u v *\<^sub>R v) S'"
nipkow@64267
   942
      using SS' fS by (auto intro!: sum.mono_neutral_cong_right)
nipkow@64267
   943
    then have "sum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
hoelzl@63051
   944
    then have "y \<in> ?rhs" by auto
hoelzl@63051
   945
  }
hoelzl@63051
   946
  moreover
hoelzl@63051
   947
  {
hoelzl@63051
   948
    fix y u
nipkow@64267
   949
    assume u: "sum (\<lambda>v. u v *\<^sub>R v) S = y"
hoelzl@63051
   950
    then have "y \<in> span S" using fS unfolding span_explicit by auto
hoelzl@63051
   951
  }
hoelzl@63051
   952
  ultimately show ?thesis by blast
hoelzl@63051
   953
qed
hoelzl@63051
   954
hoelzl@63052
   955
lemma linear_independent_extend_subspace:
hoelzl@63052
   956
  assumes "independent B"
hoelzl@63052
   957
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)"
hoelzl@63052
   958
proof -
hoelzl@63052
   959
  from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV]
hoelzl@63052
   960
  obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV"
hoelzl@63052
   961
    by (auto simp: top_unique)
hoelzl@63052
   962
  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
   963
    using \<open>span B' = UNIV\<close> unfolding span_alt by auto
hoelzl@63052
   964
  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
   965
    "\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)"
hoelzl@63052
   966
    unfolding choice_iff by auto
hoelzl@63052
   967
hoelzl@63052
   968
  have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y
hoelzl@63052
   969
    using \<open>independent B'\<close>
hoelzl@63052
   970
  proof (rule independentD_unique)
hoelzl@63052
   971
    have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
hoelzl@63052
   972
      = (\<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)"
nipkow@64267
   973
      by (intro sum.mono_neutral_cong_left) (auto intro: X)
hoelzl@63052
   974
    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)"
nipkow@64267
   975
      by (auto simp add: scaleR_add_left sum.distrib
nipkow@64267
   976
               intro!: arg_cong2[where f="op +"]  sum.mono_neutral_cong_right X)
hoelzl@63052
   977
    also have "\<dots> = x + y"
hoelzl@63052
   978
      by (simp add: X(3)[symmetric])
hoelzl@63052
   979
    also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
hoelzl@63052
   980
      by (rule X(3))
hoelzl@63052
   981
    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
   982
      ..
hoelzl@63052
   983
    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
   984
      by auto
hoelzl@63052
   985
    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
   986
        "finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   987
      using X(1) by (auto dest: finite_subset intro: X)
hoelzl@63052
   988
  qed
hoelzl@63052
   989
hoelzl@63052
   990
  have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c
hoelzl@63052
   991
    using \<open>independent B'\<close>
hoelzl@63052
   992
  proof (rule independentD_unique)
hoelzl@63052
   993
    show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   994
      "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
hoelzl@63052
   995
      using X(1,2) by auto
hoelzl@63052
   996
    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)"
nipkow@64267
   997
      unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
hoelzl@63052
   998
      by (cases "c = 0") (auto simp: X(3)[symmetric])
hoelzl@63052
   999
  qed
hoelzl@63052
  1000
hoelzl@63052
  1001
  have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x
hoelzl@63052
  1002
    using \<open>independent B'\<close>
hoelzl@63052
  1003
    by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric])
hoelzl@63052
  1004
wenzelm@63148
  1005
  define f' where "f' y = (if y \<in> B then f y else 0)" for y
wenzelm@63148
  1006
  define g where "g y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x)" for y
hoelzl@63052
  1007
hoelzl@63052
  1008
  have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x
hoelzl@63052
  1009
    by (auto simp: g_def X_B')
hoelzl@63052
  1010
hoelzl@63052
  1011
  have "linear g"
hoelzl@63052
  1012
  proof
hoelzl@63052
  1013
    fix x y
hoelzl@63052
  1014
    have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
hoelzl@63052
  1015
      = (\<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)"
nipkow@64267
  1016
      by (intro sum.mono_neutral_cong_left) (auto intro: X)
hoelzl@63052
  1017
    show "g (x + y) = g x + g y"
hoelzl@63052
  1018
      unfolding g_def X_add *
nipkow@64267
  1019
      by (auto simp add: scaleR_add_left sum.distrib
nipkow@64267
  1020
               intro!: arg_cong2[where f="op +"]  sum.mono_neutral_cong_right X)
hoelzl@63052
  1021
  next
hoelzl@63052
  1022
    show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
nipkow@64267
  1023
      by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X)
hoelzl@63052
  1024
  qed
hoelzl@63052
  1025
  moreover have "\<forall>x\<in>B. g x = f x"
hoelzl@63052
  1026
    using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
hoelzl@63052
  1027
  moreover have "range g = span (f`B)"
hoelzl@63052
  1028
    unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric]
hoelzl@63052
  1029
  proof (rule span_subspace)
hoelzl@63052
  1030
    have "g ` B' \<subseteq> f`B \<union> {0}"
hoelzl@63052
  1031
      by (auto simp: g_f' f'_def)
hoelzl@63052
  1032
    also have "\<dots> \<subseteq> span (f`B)"
hoelzl@63052
  1033
      by (auto intro: span_superset span_0)
hoelzl@63052
  1034
    finally show "g ` B' \<subseteq> span (f`B)"
hoelzl@63052
  1035
      by auto
hoelzl@63052
  1036
    have "x \<in> B \<Longrightarrow> f x = g x" for x
hoelzl@63052
  1037
      using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def)
hoelzl@63052
  1038
    then show "span (f ` B) \<subseteq> span (g ` B')"
hoelzl@63052
  1039
      using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto
hoelzl@63052
  1040
  qed (rule subspace_span)
hoelzl@63052
  1041
  ultimately show ?thesis
hoelzl@63052
  1042
    by auto
hoelzl@63052
  1043
qed
hoelzl@63052
  1044
hoelzl@63052
  1045
lemma linear_independent_extend:
hoelzl@63052
  1046
  "independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
hoelzl@63052
  1047
  using linear_independent_extend_subspace[of B f] by auto
hoelzl@63052
  1048
hoelzl@63053
  1049
text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
hoelzl@63053
  1050
hoelzl@63053
  1051
lemma subspace_kernel:
hoelzl@63053
  1052
  assumes lf: "linear f"
hoelzl@63053
  1053
  shows "subspace {x. f x = 0}"
hoelzl@63053
  1054
  apply (simp add: subspace_def)
hoelzl@63053
  1055
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
hoelzl@63053
  1056
  done
hoelzl@63053
  1057
hoelzl@63053
  1058
lemma linear_eq_0_span:
hoelzl@63053
  1059
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
hoelzl@63053
  1060
  shows "\<forall>x \<in> span B. f x = 0"
hoelzl@63053
  1061
  using f0 subspace_kernel[OF lf]
hoelzl@63053
  1062
  by (rule span_induct')
hoelzl@63053
  1063
hoelzl@63053
  1064
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
  1065
  using linear_eq_0_span[of f B] by auto
hoelzl@63053
  1066
hoelzl@63053
  1067
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
  1068
  using linear_eq_0_span[of "\<lambda>x. f x - g x" B] by (auto simp: linear_compose_sub)
hoelzl@63053
  1069
hoelzl@63053
  1070
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
  1071
  using linear_eq_span[of f g B] by auto
hoelzl@63053
  1072
wenzelm@60420
  1073
text \<open>The degenerate case of the Exchange Lemma.\<close>
huffman@44133
  1074
huffman@44133
  1075
lemma spanning_subset_independent:
wenzelm@49711
  1076
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1077
    and iA: "independent A"
wenzelm@49522
  1078
    and AsB: "A \<subseteq> span B"
huffman@44133
  1079
  shows "A = B"
huffman@44133
  1080
proof
wenzelm@49663
  1081
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1082
huffman@44133
  1083
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1084
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1085
wenzelm@53406
  1086
  {
wenzelm@53406
  1087
    fix x
wenzelm@53406
  1088
    assume x: "x \<in> A"
huffman@44133
  1089
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1090
      unfolding dependent_def using x by blast
wenzelm@53406
  1091
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1092
      by (blast intro: span_superset)
huffman@44133
  1093
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1094
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1095
      by (metis span_mono)
wenzelm@53406
  1096
    {
wenzelm@53406
  1097
      assume xB: "x \<notin> B"
wenzelm@53406
  1098
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1099
        by blast
wenzelm@53406
  1100
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1101
        by (metis span_mono)
wenzelm@53406
  1102
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1103
        by blast
wenzelm@53406
  1104
      with x have False
wenzelm@53406
  1105
        by (metis span_superset)
wenzelm@53406
  1106
    }
wenzelm@53406
  1107
    then have "x \<in> B" by blast
wenzelm@53406
  1108
  }
huffman@44133
  1109
  then show "A \<subseteq> B" by blast
huffman@44133
  1110
qed
huffman@44133
  1111
hoelzl@63053
  1112
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
hoelzl@63053
  1113
hoelzl@63053
  1114
lemma spanning_surjective_image:
hoelzl@63053
  1115
  assumes us: "UNIV \<subseteq> span S"
hoelzl@63053
  1116
    and lf: "linear f"
hoelzl@63053
  1117
    and sf: "surj f"
hoelzl@63053
  1118
  shows "UNIV \<subseteq> span (f ` S)"
hoelzl@63053
  1119
proof -
hoelzl@63053
  1120
  have "UNIV \<subseteq> f ` UNIV"
hoelzl@63053
  1121
    using sf by (auto simp add: surj_def)
hoelzl@63053
  1122
  also have " \<dots> \<subseteq> span (f ` S)"
hoelzl@63053
  1123
    using spans_image[OF lf us] .
hoelzl@63053
  1124
  finally show ?thesis .
hoelzl@63053
  1125
qed
hoelzl@63053
  1126
hoelzl@63053
  1127
lemma independent_inj_on_image:
hoelzl@63053
  1128
  assumes iS: "independent S"
hoelzl@63053
  1129
    and lf: "linear f"
hoelzl@63053
  1130
    and fi: "inj_on f (span S)"
hoelzl@63053
  1131
  shows "independent (f ` S)"
hoelzl@63053
  1132
proof -
hoelzl@63053
  1133
  {
hoelzl@63053
  1134
    fix a
hoelzl@63053
  1135
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
hoelzl@63053
  1136
    have eq: "f ` S - {f a} = f ` (S - {a})"
hoelzl@63053
  1137
      using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset)
hoelzl@63053
  1138
    from a have "f a \<in> f ` span (S - {a})"
hoelzl@63053
  1139
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
hoelzl@63053
  1140
    then have "a \<in> span (S - {a})"
hoelzl@63053
  1141
      by (rule inj_on_image_mem_iff_alt[OF fi, rotated])
hoelzl@63053
  1142
         (insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>)
hoelzl@63053
  1143
    with a(1) iS have False
hoelzl@63053
  1144
      by (simp add: dependent_def)
hoelzl@63053
  1145
  }
hoelzl@63053
  1146
  then show ?thesis
hoelzl@63053
  1147
    unfolding dependent_def by blast
hoelzl@63053
  1148
qed
hoelzl@63053
  1149
hoelzl@63053
  1150
lemma independent_injective_image:
hoelzl@63053
  1151
  "independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)"
hoelzl@63053
  1152
  using independent_inj_on_image[of S f] by (auto simp: subset_inj_on)
hoelzl@63053
  1153
hoelzl@63053
  1154
text \<open>Detailed theorems about left and right invertibility in general case.\<close>
hoelzl@63053
  1155
hoelzl@63053
  1156
lemma linear_inj_on_left_inverse:
hoelzl@63053
  1157
  assumes lf: "linear f" and fi: "inj_on f (span S)"
hoelzl@63053
  1158
  shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)"
hoelzl@63053
  1159
proof -
hoelzl@63053
  1160
  obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B"
hoelzl@63053
  1161
    using maximal_independent_subset[of S] by auto
hoelzl@63053
  1162
  then have "span S = span B"
hoelzl@63053
  1163
    unfolding span_eq by (auto simp: span_superset)
hoelzl@63053
  1164
  with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi
hoelzl@63053
  1165
  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
  1166
    by fastforce
hoelzl@63053
  1167
  have fB: "inj_on f B"
hoelzl@63053
  1168
    using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
hoelzl@63053
  1169
hoelzl@63053
  1170
  have "\<forall>x\<in>span B. g (f x) = x"
hoelzl@63053
  1171
  proof (intro linear_eq_span)
hoelzl@63053
  1172
    show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
hoelzl@63053
  1173
      using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
hoelzl@63053
  1174
    show "\<forall>x \<in> B. g (f x) = x"
hoelzl@63053
  1175
      using g fi \<open>span S = span B\<close> by (auto simp: fB)
hoelzl@63053
  1176
  qed
hoelzl@63053
  1177
  moreover
hoelzl@63053
  1178
  have "inv_into B f ` f ` B \<subseteq> B"
hoelzl@63053
  1179
    by (auto simp: fB)
hoelzl@63053
  1180
  then have "range g \<subseteq> span S"
hoelzl@63053
  1181
    unfolding g \<open>span S = span B\<close> by (intro span_mono)
hoelzl@63053
  1182
  ultimately show ?thesis
hoelzl@63053
  1183
    using \<open>span S = span B\<close> \<open>linear g\<close> by auto
hoelzl@63053
  1184
qed
hoelzl@63053
  1185
hoelzl@63053
  1186
lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id"
hoelzl@63053
  1187
  using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV)
hoelzl@63053
  1188
hoelzl@63053
  1189
lemma linear_surj_right_inverse:
hoelzl@63053
  1190
  assumes lf: "linear f" and sf: "span T \<subseteq> f`span S"
hoelzl@63053
  1191
  shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)"
hoelzl@63053
  1192
proof -
hoelzl@63053
  1193
  obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B"
hoelzl@63053
  1194
    using maximal_independent_subset[of T] by auto
hoelzl@63053
  1195
  then have "span T = span B"
hoelzl@63053
  1196
    unfolding span_eq by (auto simp: span_superset)
hoelzl@63053
  1197
hoelzl@63053
  1198
  from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"]
hoelzl@63053
  1199
  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
  1200
    by auto
hoelzl@63053
  1201
  moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x
hoelzl@63053
  1202
    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
  1203
  ultimately have "\<forall>x\<in>B. f (g x) = x"
hoelzl@63053
  1204
    by auto
hoelzl@63053
  1205
  then have "\<forall>x\<in>span B. f (g x) = x"
hoelzl@63053
  1206
    using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>]
hoelzl@63053
  1207
    by (intro linear_eq_span) (auto simp: id_def comp_def)
hoelzl@63053
  1208
  moreover have "inv_into (span S) f ` B \<subseteq> span S"
hoelzl@63053
  1209
    using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
hoelzl@63053
  1210
  then have "range g \<subseteq> span S"
hoelzl@63053
  1211
    unfolding g by (intro span_minimal subspace_span) auto
hoelzl@63053
  1212
  ultimately show ?thesis
hoelzl@63053
  1213
    using \<open>linear g\<close> \<open>span T = span B\<close> by auto
hoelzl@63053
  1214
qed
hoelzl@63053
  1215
hoelzl@63053
  1216
lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id"
hoelzl@63053
  1217
  using linear_surj_right_inverse[of f UNIV UNIV]
hoelzl@63053
  1218
  by (auto simp: span_UNIV fun_eq_iff)
hoelzl@63053
  1219
wenzelm@60420
  1220
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
huffman@44133
  1221
huffman@44133
  1222
lemma exchange_lemma:
wenzelm@49711
  1223
  assumes f:"finite t"
wenzelm@49711
  1224
    and i: "independent s"
wenzelm@49711
  1225
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1226
  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
  1227
  using f i sp
wenzelm@49522
  1228
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1229
  case less
wenzelm@60420
  1230
  note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
wenzelm@53406
  1231
  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
  1232
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1233
  {
lp15@55775
  1234
    assume "s \<subseteq> t"
lp15@55775
  1235
    then have ?ths
lp15@55775
  1236
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1237
  }
huffman@44133
  1238
  moreover
wenzelm@53406
  1239
  {
wenzelm@53406
  1240
    assume st: "t \<subseteq> s"
wenzelm@53406
  1241
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1242
    have ?ths
lp15@55775
  1243
      by (metis Un_absorb sp)
wenzelm@53406
  1244
  }
huffman@44133
  1245
  moreover
wenzelm@53406
  1246
  {
wenzelm@53406
  1247
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1248
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1249
      by blast
wenzelm@53406
  1250
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1251
      by blast
wenzelm@53406
  1252
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1253
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1254
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1255
      by auto
wenzelm@53406
  1256
    have ?ths
wenzelm@53406
  1257
    proof cases
wenzelm@53716
  1258
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1259
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1260
        by auto
huffman@44133
  1261
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1262
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1263
        and fu: "finite u" by blast
huffman@44133
  1264
      let ?w = "insert b u"
wenzelm@53406
  1265
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1266
        using u by blast
wenzelm@53406
  1267
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1268
        by blast
wenzelm@53406
  1269
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1270
        using u b by blast
wenzelm@53406
  1271
      have bu: "b \<notin> u"
wenzelm@53406
  1272
        using b u by blast
wenzelm@53406
  1273
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1274
        by auto
wenzelm@49522
  1275
      then have th2: "card (insert b u) = card t"
huffman@44133
  1276
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1277
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1278
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1279
        by (rule span_mono) blast
huffman@44133
  1280
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1281
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1282
        by blast
wenzelm@53406
  1283
      from th show ?thesis by blast
wenzelm@53406
  1284
    next
wenzelm@53716
  1285
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1286
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1287
        by blast
wenzelm@53406
  1288
      have ab: "a \<noteq> b"
wenzelm@53406
  1289
        using a b by blast
wenzelm@53406
  1290
      have at: "a \<notin> t"
wenzelm@53406
  1291
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1292
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1293
        using cardlt ft a b by auto
wenzelm@53406
  1294
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1295
        using ft by auto
wenzelm@53406
  1296
      {
wenzelm@53406
  1297
        fix x
wenzelm@53406
  1298
        assume xs: "x \<in> s"
wenzelm@53406
  1299
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1300
          using b by auto
wenzelm@53406
  1301
        from b(1) have "b \<in> span t"
wenzelm@53406
  1302
          by (simp add: span_superset)
wenzelm@53406
  1303
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1304
          apply (rule in_span_delete)
wenzelm@53406
  1305
          using a sp unfolding subset_eq
wenzelm@53406
  1306
          apply auto
wenzelm@53406
  1307
          done
wenzelm@53406
  1308
        from xs sp have "x \<in> span t"
wenzelm@53406
  1309
          by blast
wenzelm@53406
  1310
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1311
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1312
      }
wenzelm@53406
  1313
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1314
        by blast
wenzelm@53406
  1315
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1316
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1317
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1318
        "s \<subseteq> span u" by blast
wenzelm@53406
  1319
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1320
        by auto
wenzelm@53406
  1321
      then show ?thesis by blast
wenzelm@53406
  1322
    qed
huffman@44133
  1323
  }
wenzelm@49522
  1324
  ultimately show ?ths by blast
huffman@44133
  1325
qed
huffman@44133
  1326
wenzelm@60420
  1327
text \<open>This implies corresponding size bounds.\<close>
huffman@44133
  1328
huffman@44133
  1329
lemma independent_span_bound:
wenzelm@53406
  1330
  assumes f: "finite t"
wenzelm@53406
  1331
    and i: "independent s"
wenzelm@53406
  1332
    and sp: "s \<subseteq> span t"
huffman@44133
  1333
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1334
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1335
huffman@44133
  1336
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1337
proof -
wenzelm@53406
  1338
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1339
    by auto
huffman@44133
  1340
  show ?thesis unfolding eq
huffman@44133
  1341
    apply (rule finite_imageI)
huffman@44133
  1342
    apply (rule finite)
huffman@44133
  1343
    done
huffman@44133
  1344
qed
huffman@44133
  1345
wenzelm@53406
  1346
hoelzl@63050
  1347
subsection \<open>More interesting properties of the norm.\<close>
hoelzl@63050
  1348
hoelzl@63050
  1349
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
hoelzl@63050
  1350
  by auto
hoelzl@63050
  1351
hoelzl@63050
  1352
notation inner (infix "\<bullet>" 70)
hoelzl@63050
  1353
hoelzl@63050
  1354
lemma square_bound_lemma:
hoelzl@63050
  1355
  fixes x :: real
hoelzl@63050
  1356
  shows "x < (1 + x) * (1 + x)"
hoelzl@63050
  1357
proof -
hoelzl@63050
  1358
  have "(x + 1/2)\<^sup>2 + 3/4 > 0"
hoelzl@63050
  1359
    using zero_le_power2[of "x+1/2"] by arith
hoelzl@63050
  1360
  then show ?thesis
hoelzl@63050
  1361
    by (simp add: field_simps power2_eq_square)
hoelzl@63050
  1362
qed
hoelzl@63050
  1363
hoelzl@63050
  1364
lemma square_continuous:
hoelzl@63050
  1365
  fixes e :: real
hoelzl@63050
  1366
  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
  1367
  using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
hoelzl@63050
  1368
  by (force simp add: power2_eq_square)
hoelzl@63050
  1369
hoelzl@63050
  1370
hoelzl@63050
  1371
lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
hoelzl@63050
  1372
  by simp (* TODO: delete *)
hoelzl@63050
  1373
hoelzl@63050
  1374
lemma norm_triangle_sub:
hoelzl@63050
  1375
  fixes x y :: "'a::real_normed_vector"
hoelzl@63050
  1376
  shows "norm x \<le> norm y + norm (x - y)"
hoelzl@63050
  1377
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
hoelzl@63050
  1378
hoelzl@63050
  1379
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
hoelzl@63050
  1380
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1381
hoelzl@63050
  1382
lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
hoelzl@63050
  1383
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1384
hoelzl@63050
  1385
lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
hoelzl@63050
  1386
  apply (subst order_eq_iff)
hoelzl@63050
  1387
  apply (auto simp: norm_le)
hoelzl@63050
  1388
  done
hoelzl@63050
  1389
hoelzl@63050
  1390
lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
hoelzl@63050
  1391
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1392
hoelzl@63050
  1393
text\<open>Squaring equations and inequalities involving norms.\<close>
hoelzl@63050
  1394
hoelzl@63050
  1395
lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
hoelzl@63050
  1396
  by (simp only: power2_norm_eq_inner) (* TODO: move? *)
hoelzl@63050
  1397
hoelzl@63050
  1398
lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
hoelzl@63050
  1399
  by (auto simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1400
hoelzl@63050
  1401
lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
hoelzl@63050
  1402
  apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
hoelzl@63050
  1403
  using norm_ge_zero[of x]
hoelzl@63050
  1404
  apply arith
hoelzl@63050
  1405
  done
hoelzl@63050
  1406
hoelzl@63050
  1407
lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
hoelzl@63050
  1408
  apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
hoelzl@63050
  1409
  using norm_ge_zero[of x]
hoelzl@63050
  1410
  apply arith
hoelzl@63050
  1411
  done
hoelzl@63050
  1412
hoelzl@63050
  1413
lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
hoelzl@63050
  1414
  by (metis not_le norm_ge_square)
hoelzl@63050
  1415
hoelzl@63050
  1416
lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
hoelzl@63050
  1417
  by (metis norm_le_square not_less)
hoelzl@63050
  1418
hoelzl@63050
  1419
text\<open>Dot product in terms of the norm rather than conversely.\<close>
hoelzl@63050
  1420
hoelzl@63050
  1421
lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
hoelzl@63050
  1422
  inner_scaleR_left inner_scaleR_right
hoelzl@63050
  1423
hoelzl@63050
  1424
lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
wenzelm@63170
  1425
  by (simp only: power2_norm_eq_inner inner_simps inner_commute) auto
hoelzl@63050
  1426
hoelzl@63050
  1427
lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
wenzelm@63170
  1428
  by (simp only: power2_norm_eq_inner inner_simps inner_commute)
wenzelm@63170
  1429
    (auto simp add: algebra_simps)
hoelzl@63050
  1430
hoelzl@63050
  1431
text\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close>
hoelzl@63050
  1432
hoelzl@63050
  1433
lemma linear_componentwise:
hoelzl@63050
  1434
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
hoelzl@63050
  1435
  assumes lf: "linear f"
hoelzl@63050
  1436
  shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
hoelzl@63050
  1437
proof -
hoelzl@63050
  1438
  have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
nipkow@64267
  1439
    by (simp add: inner_sum_left)
hoelzl@63050
  1440
  then show ?thesis
nipkow@64267
  1441
    unfolding linear_sum_mul[OF lf, symmetric]
hoelzl@63050
  1442
    unfolding euclidean_representation ..
hoelzl@63050
  1443
qed
hoelzl@63050
  1444
hoelzl@63050
  1445
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
hoelzl@63050
  1446
  (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@63050
  1447
proof
hoelzl@63050
  1448
  assume ?lhs
hoelzl@63050
  1449
  then show ?rhs by simp
hoelzl@63050
  1450
next
hoelzl@63050
  1451
  assume ?rhs
hoelzl@63050
  1452
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
hoelzl@63050
  1453
    by simp
hoelzl@63050
  1454
  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
hoelzl@63050
  1455
    by (simp add: inner_diff inner_commute)
hoelzl@63050
  1456
  then have "(x - y) \<bullet> (x - y) = 0"
hoelzl@63050
  1457
    by (simp add: field_simps inner_diff inner_commute)
hoelzl@63050
  1458
  then show "x = y" by simp
hoelzl@63050
  1459
qed
hoelzl@63050
  1460
hoelzl@63050
  1461
lemma norm_triangle_half_r:
hoelzl@63050
  1462
  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
hoelzl@63050
  1463
  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
hoelzl@63050
  1464
hoelzl@63050
  1465
lemma norm_triangle_half_l:
hoelzl@63050
  1466
  assumes "norm (x - y) < e / 2"
hoelzl@63050
  1467
    and "norm (x' - y) < e / 2"
hoelzl@63050
  1468
  shows "norm (x - x') < e"
hoelzl@63050
  1469
  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
hoelzl@63050
  1470
  unfolding dist_norm[symmetric] .
hoelzl@63050
  1471
hoelzl@63050
  1472
lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
hoelzl@63050
  1473
  by (rule norm_triangle_ineq [THEN order_trans])
hoelzl@63050
  1474
hoelzl@63050
  1475
lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
hoelzl@63050
  1476
  by (rule norm_triangle_ineq [THEN le_less_trans])
hoelzl@63050
  1477
nipkow@64267
  1478
lemma sum_clauses:
nipkow@64267
  1479
  shows "sum f {} = 0"
nipkow@64267
  1480
    and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
hoelzl@63050
  1481
  by (auto simp add: insert_absorb)
hoelzl@63050
  1482
nipkow@64267
  1483
lemma sum_norm_bound:
hoelzl@63050
  1484
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
lp15@64773
  1485
  assumes K: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K"
lp15@64773
  1486
  shows "norm (sum f S) \<le> of_nat (card S)*K"
nipkow@64267
  1487
  using sum_norm_le[OF K] sum_constant[symmetric]
hoelzl@63050
  1488
  by simp
hoelzl@63050
  1489
nipkow@64267
  1490
lemma sum_group:
hoelzl@63050
  1491
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
nipkow@64267
  1492
  shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
nipkow@64267
  1493
  apply (subst sum_image_gen[OF fS, of g f])
nipkow@64267
  1494
  apply (rule sum.mono_neutral_right[OF fT fST])
nipkow@64267
  1495
  apply (auto intro: sum.neutral)
hoelzl@63050
  1496
  done
hoelzl@63050
  1497
hoelzl@63050
  1498
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
hoelzl@63050
  1499
proof
hoelzl@63050
  1500
  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
hoelzl@63050
  1501
  then have "\<forall>x. x \<bullet> (y - z) = 0"
hoelzl@63050
  1502
    by (simp add: inner_diff)
hoelzl@63050
  1503
  then have "(y - z) \<bullet> (y - z) = 0" ..
hoelzl@63050
  1504
  then show "y = z" by simp
hoelzl@63050
  1505
qed simp
hoelzl@63050
  1506
hoelzl@63050
  1507
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
hoelzl@63050
  1508
proof
hoelzl@63050
  1509
  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
hoelzl@63050
  1510
  then have "\<forall>z. (x - y) \<bullet> z = 0"
hoelzl@63050
  1511
    by (simp add: inner_diff)
hoelzl@63050
  1512
  then have "(x - y) \<bullet> (x - y) = 0" ..
hoelzl@63050
  1513
  then show "x = y" by simp
hoelzl@63050
  1514
qed simp
hoelzl@63050
  1515
hoelzl@63050
  1516
hoelzl@63050
  1517
subsection \<open>Orthogonality.\<close>
hoelzl@63050
  1518
hoelzl@63050
  1519
context real_inner
hoelzl@63050
  1520
begin
hoelzl@63050
  1521
hoelzl@63050
  1522
definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
hoelzl@63050
  1523
lp15@63072
  1524
lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
lp15@63072
  1525
  by (simp add: orthogonal_def)
lp15@63072
  1526
hoelzl@63050
  1527
lemma orthogonal_clauses:
hoelzl@63050
  1528
  "orthogonal a 0"
hoelzl@63050
  1529
  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
hoelzl@63050
  1530
  "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
hoelzl@63050
  1531
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
hoelzl@63050
  1532
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
hoelzl@63050
  1533
  "orthogonal 0 a"
hoelzl@63050
  1534
  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
hoelzl@63050
  1535
  "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
hoelzl@63050
  1536
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
hoelzl@63050
  1537
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
hoelzl@63050
  1538
  unfolding orthogonal_def inner_add inner_diff by auto
hoelzl@63050
  1539
hoelzl@63050
  1540
end
hoelzl@63050
  1541
hoelzl@63050
  1542
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
hoelzl@63050
  1543
  by (simp add: orthogonal_def inner_commute)
hoelzl@63050
  1544
lp15@63114
  1545
lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
lp15@63114
  1546
  by (rule ext) (simp add: orthogonal_def)
lp15@63114
  1547
lp15@63114
  1548
lemma pairwise_ortho_scaleR:
lp15@63114
  1549
    "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
lp15@63114
  1550
    \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
lp15@63114
  1551
  by (auto simp: pairwise_def orthogonal_clauses)
lp15@63114
  1552
lp15@63114
  1553
lemma orthogonal_rvsum:
nipkow@64267
  1554
    "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
lp15@63114
  1555
  by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
lp15@63114
  1556
lp15@63114
  1557
lemma orthogonal_lvsum:
nipkow@64267
  1558
    "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
lp15@63114
  1559
  by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
lp15@63114
  1560
lp15@63114
  1561
lemma norm_add_Pythagorean:
lp15@63114
  1562
  assumes "orthogonal a b"
lp15@63114
  1563
    shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
lp15@63114
  1564
proof -
lp15@63114
  1565
  from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
lp15@63114
  1566
    by (simp add: algebra_simps orthogonal_def inner_commute)
lp15@63114
  1567
  then show ?thesis
lp15@63114
  1568
    by (simp add: power2_norm_eq_inner)
lp15@63114
  1569
qed
lp15@63114
  1570
nipkow@64267
  1571
lemma norm_sum_Pythagorean:
lp15@63114
  1572
  assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
nipkow@64267
  1573
    shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
lp15@63114
  1574
using assms
lp15@63114
  1575
proof (induction I rule: finite_induct)
lp15@63114
  1576
  case empty then show ?case by simp
lp15@63114
  1577
next
lp15@63114
  1578
  case (insert x I)
nipkow@64267
  1579
  then have "orthogonal (f x) (sum f I)"
lp15@63114
  1580
    by (metis pairwise_insert orthogonal_rvsum)
lp15@63114
  1581
  with insert show ?case
lp15@63114
  1582
    by (simp add: pairwise_insert norm_add_Pythagorean)
lp15@63114
  1583
qed
lp15@63114
  1584
hoelzl@63050
  1585
hoelzl@63050
  1586
subsection \<open>Bilinear functions.\<close>
hoelzl@63050
  1587
hoelzl@63050
  1588
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
hoelzl@63050
  1589
hoelzl@63050
  1590
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
hoelzl@63050
  1591
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1592
hoelzl@63050
  1593
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
hoelzl@63050
  1594
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1595
hoelzl@63050
  1596
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
hoelzl@63050
  1597
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1598
hoelzl@63050
  1599
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
hoelzl@63050
  1600
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1601
hoelzl@63050
  1602
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
hoelzl@63050
  1603
  by (drule bilinear_lmul [of _ "- 1"]) simp
hoelzl@63050
  1604
hoelzl@63050
  1605
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
hoelzl@63050
  1606
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
hoelzl@63050
  1607
hoelzl@63050
  1608
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
hoelzl@63050
  1609
  using add_left_imp_eq[of x y 0] by auto
hoelzl@63050
  1610
hoelzl@63050
  1611
lemma bilinear_lzero:
hoelzl@63050
  1612
  assumes "bilinear h"
hoelzl@63050
  1613
  shows "h 0 x = 0"
hoelzl@63050
  1614
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
hoelzl@63050
  1615
hoelzl@63050
  1616
lemma bilinear_rzero:
hoelzl@63050
  1617
  assumes "bilinear h"
hoelzl@63050
  1618
  shows "h x 0 = 0"
hoelzl@63050
  1619
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
hoelzl@63050
  1620
hoelzl@63050
  1621
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
hoelzl@63050
  1622
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
hoelzl@63050
  1623
hoelzl@63050
  1624
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
hoelzl@63050
  1625
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
hoelzl@63050
  1626
nipkow@64267
  1627
lemma bilinear_sum:
hoelzl@63050
  1628
  assumes bh: "bilinear h"
hoelzl@63050
  1629
    and fS: "finite S"
hoelzl@63050
  1630
    and fT: "finite T"
nipkow@64267
  1631
  shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
hoelzl@63050
  1632
proof -
nipkow@64267
  1633
  have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
nipkow@64267
  1634
    apply (rule linear_sum[unfolded o_def])
hoelzl@63050
  1635
    using bh fS
hoelzl@63050
  1636
    apply (auto simp add: bilinear_def)
hoelzl@63050
  1637
    done
nipkow@64267
  1638
  also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
nipkow@64267
  1639
    apply (rule sum.cong, simp)
nipkow@64267
  1640
    apply (rule linear_sum[unfolded o_def])
hoelzl@63050
  1641
    using bh fT
hoelzl@63050
  1642
    apply (auto simp add: bilinear_def)
hoelzl@63050
  1643
    done
hoelzl@63050
  1644
  finally show ?thesis
nipkow@64267
  1645
    unfolding sum.cartesian_product .
hoelzl@63050
  1646
qed
hoelzl@63050
  1647
hoelzl@63050
  1648
hoelzl@63050
  1649
subsection \<open>Adjoints.\<close>
hoelzl@63050
  1650
hoelzl@63050
  1651
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
hoelzl@63050
  1652
hoelzl@63050
  1653
lemma adjoint_unique:
hoelzl@63050
  1654
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
hoelzl@63050
  1655
  shows "adjoint f = g"
hoelzl@63050
  1656
  unfolding adjoint_def
hoelzl@63050
  1657
proof (rule some_equality)
hoelzl@63050
  1658
  show "\<forall>x y. inner (f x) y = inner x (g y)"
hoelzl@63050
  1659
    by (rule assms)
hoelzl@63050
  1660
next
hoelzl@63050
  1661
  fix h
hoelzl@63050
  1662
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
hoelzl@63050
  1663
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
hoelzl@63050
  1664
    using assms by simp
hoelzl@63050
  1665
  then have "\<forall>x y. inner x (g y - h y) = 0"
hoelzl@63050
  1666
    by (simp add: inner_diff_right)
hoelzl@63050
  1667
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
hoelzl@63050
  1668
    by simp
hoelzl@63050
  1669
  then have "\<forall>y. h y = g y"
hoelzl@63050
  1670
    by simp
hoelzl@63050
  1671
  then show "h = g" by (simp add: ext)
hoelzl@63050
  1672
qed
hoelzl@63050
  1673
hoelzl@63050
  1674
text \<open>TODO: The following lemmas about adjoints should hold for any
wenzelm@63680
  1675
  Hilbert space (i.e. complete inner product space).
wenzelm@63680
  1676
  (see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
hoelzl@63050
  1677
\<close>
hoelzl@63050
  1678
hoelzl@63050
  1679
lemma adjoint_works:
hoelzl@63050
  1680
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1681
  assumes lf: "linear f"
hoelzl@63050
  1682
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@63050
  1683
proof -
hoelzl@63050
  1684
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
hoelzl@63050
  1685
  proof (intro allI exI)
hoelzl@63050
  1686
    fix y :: "'m" and x
hoelzl@63050
  1687
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
hoelzl@63050
  1688
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
hoelzl@63050
  1689
      by (simp add: euclidean_representation)
hoelzl@63050
  1690
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
nipkow@64267
  1691
      unfolding linear_sum[OF lf]
hoelzl@63050
  1692
      by (simp add: linear_cmul[OF lf])
hoelzl@63050
  1693
    finally show "f x \<bullet> y = x \<bullet> ?w"
nipkow@64267
  1694
      by (simp add: inner_sum_left inner_sum_right mult.commute)
hoelzl@63050
  1695
  qed
hoelzl@63050
  1696
  then show ?thesis
hoelzl@63050
  1697
    unfolding adjoint_def choice_iff
hoelzl@63050
  1698
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
hoelzl@63050
  1699
qed
hoelzl@63050
  1700
hoelzl@63050
  1701
lemma adjoint_clauses:
hoelzl@63050
  1702
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1703
  assumes lf: "linear f"
hoelzl@63050
  1704
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@63050
  1705
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@63050
  1706
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@63050
  1707
hoelzl@63050
  1708
lemma adjoint_linear:
hoelzl@63050
  1709
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1710
  assumes lf: "linear f"
hoelzl@63050
  1711
  shows "linear (adjoint f)"
hoelzl@63050
  1712
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
hoelzl@63050
  1713
    adjoint_clauses[OF lf] inner_distrib)
hoelzl@63050
  1714
hoelzl@63050
  1715
lemma adjoint_adjoint:
hoelzl@63050
  1716
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1717
  assumes lf: "linear f"
hoelzl@63050
  1718
  shows "adjoint (adjoint f) = f"
hoelzl@63050
  1719
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@63050
  1720
hoelzl@63050
  1721
hoelzl@63050
  1722
subsection \<open>Interlude: Some properties of real sets\<close>
hoelzl@63050
  1723
hoelzl@63050
  1724
lemma seq_mono_lemma:
hoelzl@63050
  1725
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
hoelzl@63050
  1726
    and "\<forall>n \<ge> m. e n \<le> e m"
hoelzl@63050
  1727
  shows "\<forall>n \<ge> m. d n < e m"
hoelzl@63050
  1728
  using assms
hoelzl@63050
  1729
  apply auto
hoelzl@63050
  1730
  apply (erule_tac x="n" in allE)
hoelzl@63050
  1731
  apply (erule_tac x="n" in allE)
hoelzl@63050
  1732
  apply auto
hoelzl@63050
  1733
  done
hoelzl@63050
  1734
hoelzl@63050
  1735
lemma infinite_enumerate:
hoelzl@63050
  1736
  assumes fS: "infinite S"
hoelzl@63050
  1737
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
hoelzl@63050
  1738
  unfolding subseq_def
hoelzl@63050
  1739
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
hoelzl@63050
  1740
hoelzl@63050
  1741
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
  1742
  apply auto
hoelzl@63050
  1743
  apply (rule_tac x="d/2" in exI)
hoelzl@63050
  1744
  apply auto
hoelzl@63050
  1745
  done
hoelzl@63050
  1746
hoelzl@63050
  1747
lemma approachable_lt_le2:  \<comment>\<open>like the above, but pushes aside an extra formula\<close>
hoelzl@63050
  1748
    "(\<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
  1749
  apply auto
hoelzl@63050
  1750
  apply (rule_tac x="d/2" in exI, auto)
hoelzl@63050
  1751
  done
hoelzl@63050
  1752
hoelzl@63050
  1753
lemma triangle_lemma:
hoelzl@63050
  1754
  fixes x y z :: real
hoelzl@63050
  1755
  assumes x: "0 \<le> x"
hoelzl@63050
  1756
    and y: "0 \<le> y"
hoelzl@63050
  1757
    and z: "0 \<le> z"
hoelzl@63050
  1758
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
hoelzl@63050
  1759
  shows "x \<le> y + z"
hoelzl@63050
  1760
proof -
hoelzl@63050
  1761
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
hoelzl@63050
  1762
    using z y by simp
hoelzl@63050
  1763
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
hoelzl@63050
  1764
    by (simp add: power2_eq_square field_simps)
hoelzl@63050
  1765
  from y z have yz: "y + z \<ge> 0"
hoelzl@63050
  1766
    by arith
hoelzl@63050
  1767
  from power2_le_imp_le[OF th yz] show ?thesis .
hoelzl@63050
  1768
qed
hoelzl@63050
  1769
hoelzl@63050
  1770
hoelzl@63050
  1771
hoelzl@63050
  1772
subsection \<open>Archimedean properties and useful consequences\<close>
hoelzl@63050
  1773
hoelzl@63050
  1774
text\<open>Bernoulli's inequality\<close>
hoelzl@63050
  1775
proposition Bernoulli_inequality:
hoelzl@63050
  1776
  fixes x :: real
hoelzl@63050
  1777
  assumes "-1 \<le> x"
hoelzl@63050
  1778
    shows "1 + n * x \<le> (1 + x) ^ n"
hoelzl@63050
  1779
proof (induct n)
hoelzl@63050
  1780
  case 0
hoelzl@63050
  1781
  then show ?case by simp
hoelzl@63050
  1782
next
hoelzl@63050
  1783
  case (Suc n)
hoelzl@63050
  1784
  have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
hoelzl@63050
  1785
    by (simp add: algebra_simps)
hoelzl@63050
  1786
  also have "... = (1 + x) * (1 + n*x)"
hoelzl@63050
  1787
    by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
hoelzl@63050
  1788
  also have "... \<le> (1 + x) ^ Suc n"
hoelzl@63050
  1789
    using Suc.hyps assms mult_left_mono by fastforce
hoelzl@63050
  1790
  finally show ?case .
hoelzl@63050
  1791
qed
hoelzl@63050
  1792
hoelzl@63050
  1793
corollary Bernoulli_inequality_even:
hoelzl@63050
  1794
  fixes x :: real
hoelzl@63050
  1795
  assumes "even n"
hoelzl@63050
  1796
    shows "1 + n * x \<le> (1 + x) ^ n"
hoelzl@63050
  1797
proof (cases "-1 \<le> x \<or> n=0")
hoelzl@63050
  1798
  case True
hoelzl@63050
  1799
  then show ?thesis
hoelzl@63050
  1800
    by (auto simp: Bernoulli_inequality)
hoelzl@63050
  1801
next
hoelzl@63050
  1802
  case False
hoelzl@63050
  1803
  then have "real n \<ge> 1"
hoelzl@63050
  1804
    by simp
hoelzl@63050
  1805
  with False have "n * x \<le> -1"
hoelzl@63050
  1806
    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
  1807
  then have "1 + n * x \<le> 0"
hoelzl@63050
  1808
    by auto
hoelzl@63050
  1809
  also have "... \<le> (1 + x) ^ n"
hoelzl@63050
  1810
    using assms
hoelzl@63050
  1811
    using zero_le_even_power by blast
hoelzl@63050
  1812
  finally show ?thesis .
hoelzl@63050
  1813
qed
hoelzl@63050
  1814
hoelzl@63050
  1815
corollary real_arch_pow:
hoelzl@63050
  1816
  fixes x :: real
hoelzl@63050
  1817
  assumes x: "1 < x"
hoelzl@63050
  1818
  shows "\<exists>n. y < x^n"
hoelzl@63050
  1819
proof -
hoelzl@63050
  1820
  from x have x0: "x - 1 > 0"
hoelzl@63050
  1821
    by arith
hoelzl@63050
  1822
  from reals_Archimedean3[OF x0, rule_format, of y]
hoelzl@63050
  1823
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
hoelzl@63050
  1824
  from x0 have x00: "x- 1 \<ge> -1" by arith
hoelzl@63050
  1825
  from Bernoulli_inequality[OF x00, of n] n
hoelzl@63050
  1826
  have "y < x^n" by auto
hoelzl@63050
  1827
  then show ?thesis by metis
hoelzl@63050
  1828
qed
hoelzl@63050
  1829
hoelzl@63050
  1830
corollary real_arch_pow_inv:
hoelzl@63050
  1831
  fixes x y :: real
hoelzl@63050
  1832
  assumes y: "y > 0"
hoelzl@63050
  1833
    and x1: "x < 1"
hoelzl@63050
  1834
  shows "\<exists>n. x^n < y"
hoelzl@63050
  1835
proof (cases "x > 0")
hoelzl@63050
  1836
  case True
hoelzl@63050
  1837
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
hoelzl@63050
  1838
  from real_arch_pow[OF ix, of "1/y"]
hoelzl@63050
  1839
  obtain n where n: "1/y < (1/x)^n" by blast
hoelzl@63050
  1840
  then show ?thesis using y \<open>x > 0\<close>
hoelzl@63050
  1841
    by (auto simp add: field_simps)
hoelzl@63050
  1842
next
hoelzl@63050
  1843
  case False
hoelzl@63050
  1844
  with y x1 show ?thesis
hoelzl@63050
  1845
    apply auto
hoelzl@63050
  1846
    apply (rule exI[where x=1])
hoelzl@63050
  1847
    apply auto
hoelzl@63050
  1848
    done
hoelzl@63050
  1849
qed
hoelzl@63050
  1850
hoelzl@63050
  1851
lemma forall_pos_mono:
hoelzl@63050
  1852
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
hoelzl@63050
  1853
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
hoelzl@63050
  1854
  by (metis real_arch_inverse)
hoelzl@63050
  1855
hoelzl@63050
  1856
lemma forall_pos_mono_1:
hoelzl@63050
  1857
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
hoelzl@63050
  1858
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
hoelzl@63050
  1859
  apply (rule forall_pos_mono)
hoelzl@63050
  1860
  apply auto
hoelzl@63050
  1861
  apply (metis Suc_pred of_nat_Suc)
hoelzl@63050
  1862
  done
hoelzl@63050
  1863
hoelzl@63050
  1864
wenzelm@60420
  1865
subsection \<open>Euclidean Spaces as Typeclass\<close>
huffman@44133
  1866
hoelzl@50526
  1867
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1868
  unfolding dependent_def
hoelzl@50526
  1869
  apply (subst span_finite)
hoelzl@50526
  1870
  apply simp
huffman@44133
  1871
  apply clarify
hoelzl@50526
  1872
  apply (drule_tac f="inner a" in arg_cong)
nipkow@64267
  1873
  apply (simp add: inner_Basis inner_sum_right eq_commute)
hoelzl@50526
  1874
  done
hoelzl@50526
  1875
huffman@53939
  1876
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1877
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1878
  by (fast intro: euclidean_representation)
huffman@44133
  1879
hoelzl@50526
  1880
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1881
  unfolding span_Basis ..
hoelzl@50526
  1882
hoelzl@50526
  1883
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1884
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1885
hoelzl@50526
  1886
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1887
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1888
hoelzl@50526
  1889
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1890
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1891
hoelzl@50526
  1892
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1893
  apply (subst euclidean_representation[of x, symmetric])
nipkow@64267
  1894
  apply (rule order_trans[OF norm_sum])
nipkow@64267
  1895
  apply (auto intro!: sum_mono)
wenzelm@49522
  1896
  done
huffman@44133
  1897
nipkow@64267
  1898
lemma sum_norm_allsubsets_bound:
wenzelm@56444
  1899
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1900
  assumes fP: "finite P"
nipkow@64267
  1901
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
hoelzl@50526
  1902
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1903
proof -
hoelzl@50526
  1904
  have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
nipkow@64267
  1905
    by (rule sum_mono) (rule norm_le_l1)
hoelzl@50526
  1906
  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>)"
nipkow@64267
  1907
    by (rule sum.commute)
hoelzl@50526
  1908
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
nipkow@64267
  1909
  proof (rule sum_bounded_above)
wenzelm@53406
  1910
    fix i :: 'n
wenzelm@53406
  1911
    assume i: "i \<in> Basis"
wenzelm@53406
  1912
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1913
      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)"
nipkow@64267
  1914
      by (simp add: abs_real_def sum.If_cases[OF fP] sum_negf norm_triangle_ineq4 inner_sum_left
wenzelm@56444
  1915
        del: real_norm_def)
wenzelm@53406
  1916
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1917
      unfolding real_norm_def
hoelzl@50526
  1918
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1919
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1920
  qed
lp15@61609
  1921
  also have "\<dots> = 2 * real DIM('n) * e" by simp
huffman@44133
  1922
  finally show ?thesis .
huffman@44133
  1923
qed
huffman@44133
  1924
wenzelm@53406
  1925
wenzelm@60420
  1926
subsection \<open>Linearity and Bilinearity continued\<close>
huffman@44133
  1927
huffman@44133
  1928
lemma linear_bounded:
wenzelm@56444
  1929
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1930
  assumes lf: "linear f"
huffman@44133
  1931
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1932
proof
hoelzl@50526
  1933
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1934
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1935
  proof
wenzelm@53406
  1936
    fix x :: 'a
hoelzl@50526
  1937
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1938
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1939
      unfolding euclidean_representation ..
nipkow@64267
  1940
    also have "\<dots> = norm (sum ?g Basis)"
nipkow@64267
  1941
      by (simp add: linear_sum [OF lf] linear_cmul [OF lf])
nipkow@64267
  1942
    finally have th0: "norm (f x) = norm (sum ?g Basis)" .
lp15@64773
  1943
    have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
lp15@64773
  1944
    proof -
lp15@64773
  1945
      from Basis_le_norm[OF that, of x]
huffman@53939
  1946
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1947
        unfolding norm_scaleR
haftmann@57512
  1948
        apply (subst mult.commute)
wenzelm@49663
  1949
        apply (rule mult_mono)
wenzelm@49663
  1950
        apply (auto simp add: field_simps)
wenzelm@53406
  1951
        done
huffman@53939
  1952
    qed
nipkow@64267
  1953
    from sum_norm_le[of _ ?g, OF th]
huffman@53939
  1954
    show "norm (f x) \<le> ?B * norm x"
nipkow@64267
  1955
      unfolding th0 sum_distrib_right by metis
huffman@53939
  1956
  qed
huffman@44133
  1957
qed
huffman@44133
  1958
huffman@44133
  1959
lemma linear_conv_bounded_linear:
huffman@44133
  1960
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1961
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1962
proof
huffman@44133
  1963
  assume "linear f"
huffman@53939
  1964
  then interpret f: linear f .
huffman@44133
  1965
  show "bounded_linear f"
huffman@44133
  1966
  proof
huffman@44133
  1967
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
wenzelm@60420
  1968
      using \<open>linear f\<close> by (rule linear_bounded)
wenzelm@49522
  1969
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
haftmann@57512
  1970
      by (simp add: mult.commute)
huffman@44133
  1971
  qed
huffman@44133
  1972
next
huffman@44133
  1973
  assume "bounded_linear f"
huffman@44133
  1974
  then interpret f: bounded_linear f .
huffman@53939
  1975
  show "linear f" ..
huffman@53939
  1976
qed
huffman@53939
  1977
paulson@61518
  1978
lemmas linear_linear = linear_conv_bounded_linear[symmetric]
paulson@61518
  1979
huffman@53939
  1980
lemma linear_bounded_pos:
wenzelm@56444
  1981
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1982
  assumes lf: "linear f"
huffman@53939
  1983
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1984
proof -
huffman@53939
  1985
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1986
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1987
    by (rule bounded_linear.pos_bounded)
huffman@53939
  1988
  then show ?thesis
haftmann@57512
  1989
    by (simp only: mult.commute)
huffman@44133
  1990
qed
huffman@44133
  1991
wenzelm@49522
  1992
lemma bounded_linearI':
wenzelm@56444
  1993
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1994
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1995
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1996
  shows "bounded_linear f"
wenzelm@53406
  1997
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1998
  by (rule linearI[OF assms])
huffman@44133
  1999
huffman@44133
  2000
lemma bilinear_bounded:
wenzelm@56444
  2001
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  2002
  assumes bh: "bilinear h"
huffman@44133
  2003
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  2004
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  2005
  fix x :: 'm
wenzelm@53406
  2006
  fix y :: 'n
nipkow@64267
  2007
  have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
wenzelm@53406
  2008
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  2009
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  2010
    apply rule
hoelzl@50526
  2011
    done
nipkow@64267
  2012
  also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
nipkow@64267
  2013
    unfolding bilinear_sum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  2014
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  2015
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
nipkow@64267
  2016
    apply (auto simp add: sum_distrib_right th sum.cartesian_product)
nipkow@64267
  2017
    apply (rule sum_norm_le)
wenzelm@53406
  2018
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  2019
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  2020
    apply (rule mult_mono)
wenzelm@53406
  2021
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  2022
    apply (rule mult_mono)
wenzelm@53406
  2023
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  2024
    done
huffman@44133
  2025
qed
huffman@44133
  2026
huffman@44133
  2027
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  2028
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  2029
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  2030
proof
huffman@44133
  2031
  assume "bilinear h"
huffman@44133
  2032
  show "bounded_bilinear h"
huffman@44133
  2033
  proof
wenzelm@53406
  2034
    fix x y z
wenzelm@53406
  2035
    show "h (x + y) z = h x z + h y z"
wenzelm@60420
  2036
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  2037
  next
wenzelm@53406
  2038
    fix x y z
wenzelm@53406
  2039
    show "h x (y + z) = h x y + h x z"
wenzelm@60420
  2040
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  2041
  next
wenzelm@53406
  2042
    fix r x y
wenzelm@53406
  2043
    show "h (scaleR r x) y = scaleR r (h x y)"
wenzelm@60420
  2044
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  2045
      by simp
huffman@44133
  2046
  next
wenzelm@53406
  2047
    fix r x y
wenzelm@53406
  2048
    show "h x (scaleR r y) = scaleR r (h x y)"
wenzelm@60420
  2049
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  2050
      by simp
huffman@44133
  2051
  next
huffman@44133
  2052
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
wenzelm@60420
  2053
      using \<open>bilinear h\<close> by (rule bilinear_bounded)
wenzelm@49522
  2054
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
haftmann@57514
  2055
      by (simp add: ac_simps)
huffman@44133
  2056
  qed
huffman@44133
  2057
next
huffman@44133
  2058
  assume "bounded_bilinear h"
huffman@44133
  2059
  then interpret h: bounded_bilinear h .
huffman@44133
  2060
  show "bilinear h"
huffman@44133
  2061
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  2062
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  2063
qed
huffman@44133
  2064
huffman@53939
  2065
lemma bilinear_bounded_pos:
wenzelm@56444
  2066
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  2067
  assumes bh: "bilinear h"
huffman@53939
  2068
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  2069
proof -
huffman@53939
  2070
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  2071
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  2072
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  2073
  then show ?thesis
haftmann@57514
  2074
    by (simp only: ac_simps)
huffman@53939
  2075
qed
huffman@53939
  2076
lp15@63469
  2077
lemma bounded_linear_imp_has_derivative:
lp15@63469
  2078
     "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
lp15@63469
  2079
  by (simp add: has_derivative_def bounded_linear.linear linear_diff)
lp15@63469
  2080
lp15@63469
  2081
lemma linear_imp_has_derivative:
lp15@63469
  2082
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
lp15@63469
  2083
  shows "linear f \<Longrightarrow> (f has_derivative f) net"
lp15@63469
  2084
by (simp add: has_derivative_def linear_conv_bounded_linear linear_diff)
lp15@63469
  2085
lp15@63469
  2086
lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
lp15@63469
  2087
  using bounded_linear_imp_has_derivative differentiable_def by blast
lp15@63469
  2088
lp15@63469
  2089
lemma linear_imp_differentiable:
lp15@63469
  2090
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
lp15@63469
  2091
  shows "linear f \<Longrightarrow> f differentiable net"
lp15@63469
  2092
by (metis linear_imp_has_derivative differentiable_def)
lp15@63469
  2093
wenzelm@49522
  2094
wenzelm@60420
  2095
subsection \<open>We continue.\<close>
huffman@44133
  2096
huffman@44133
  2097
lemma independent_bound:
wenzelm@53716
  2098
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  2099
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  2100
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  2101
lp15@61609
  2102
corollary
paulson@60303
  2103
  fixes S :: "'a::euclidean_space set"
paulson@60303
  2104
  assumes "independent S"
paulson@60303
  2105
  shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
paulson@60303
  2106
using assms independent_bound by auto
lp15@61609
  2107
lp15@63075
  2108
lemma independent_explicit:
lp15@63075
  2109
  fixes B :: "'a::euclidean_space set"
lp15@63075
  2110
  shows "independent B \<longleftrightarrow>
lp15@63075
  2111
         finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *\<^sub>R v) = 0 \<longrightarrow> (\<forall>v \<in> B. c v = 0))"
lp15@63075
  2112
apply (cases "finite B")
lp15@63075
  2113
 apply (force simp: dependent_finite)
lp15@63075
  2114
using independent_bound
lp15@63075
  2115
apply auto
lp15@63075
  2116
done
lp15@63075
  2117
wenzelm@49663
  2118
lemma dependent_biggerset:
wenzelm@56444
  2119
  fixes S :: "'a::euclidean_space set"
wenzelm@56444
  2120
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  2121
  by (metis independent_bound not_less)
huffman@44133
  2122
wenzelm@60420
  2123
text \<open>Notion of dimension.\<close>
huffman@44133
  2124
wenzelm@53406
  2125
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  2126
wenzelm@49522
  2127
lemma basis_exists:
wenzelm@49522
  2128
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  2129
  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
  2130
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  2131
  by auto
huffman@44133
  2132
lp15@60307
  2133
corollary dim_le_card:
lp15@60307
  2134
  fixes s :: "'a::euclidean_space set"
lp15@60307
  2135
  shows "finite s \<Longrightarrow> dim s \<le> card s"
lp15@60307
  2136
by (metis basis_exists card_mono)
lp15@60307
  2137
wenzelm@60420
  2138
text \<open>Consequences of independence or spanning for cardinality.\<close>
huffman@44133
  2139
wenzelm@53406
  2140
lemma independent_card_le_dim:
wenzelm@53406
  2141
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2142
  assumes "B \<subseteq> V"
wenzelm@53406
  2143
    and "independent B"
wenzelm@49522
  2144
  shows "card B \<le> dim V"
huffman@44133
  2145
proof -
wenzelm@60420
  2146
  from basis_exists[of V] \<open>B \<subseteq> V\<close>
wenzelm@53406
  2147
  obtain B' where "independent B'"
wenzelm@53406
  2148
    and "B \<subseteq> span B'"
wenzelm@53406
  2149
    and "card B' = dim V"
wenzelm@53406
  2150
    by blast
wenzelm@60420
  2151
  with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
huffman@44133
  2152
  show ?thesis by auto
huffman@44133
  2153
qed
huffman@44133
  2154
wenzelm@49522
  2155
lemma span_card_ge_dim:
wenzelm@53406
  2156
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2157
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  2158
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  2159
huffman@44133
  2160
lemma basis_card_eq_dim:
wenzelm@53406
  2161
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  2162
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  2163
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  2164
wenzelm@53406
  2165
lemma dim_unique:
wenzelm@53406
  2166
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2167
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  2168
  by (metis basis_card_eq_dim)
huffman@44133
  2169
wenzelm@60420
  2170
text \<open>More lemmas about dimension.\<close>
huffman@44133
  2171
lp15@64122
  2172
lemma dim_UNIV [simp]: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  2173
  using independent_Basis
hoelzl@50526
  2174
  by (intro dim_unique[of Basis]) auto
huffman@44133
  2175
huffman@44133
  2176
lemma dim_subset:
wenzelm@53406
  2177
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2178
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  2179
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  2180
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  2181
wenzelm@53406
  2182
lemma dim_subset_UNIV:
wenzelm@53406
  2183
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2184
  shows "dim S \<le> DIM('a)"
huffman@44133
  2185
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  2186
wenzelm@60420
  2187
text \<open>Converses to those.\<close>
huffman@44133
  2188
huffman@44133
  2189
lemma card_ge_dim_independent:
wenzelm@53406
  2190
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2191
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  2192
    and iB: "independent B"
wenzelm@53406
  2193
    and dVB: "dim V \<le> card B"
huffman@44133
  2194
  shows "V \<subseteq> span B"
wenzelm@53406
  2195
proof
wenzelm@53406
  2196
  fix a
wenzelm@53406
  2197
  assume aV: "a \<in> V"
wenzelm@53406
  2198
  {
wenzelm@53406
  2199
    assume aB: "a \<notin> span B"
wenzelm@53406
  2200
    then have iaB: "independent (insert a B)"
wenzelm@53406
  2201
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  2202
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  2203
      by blast
wenzelm@53406
  2204
    from aB have "a \<notin>B"
wenzelm@53406
  2205
      by (auto simp add: span_superset)
wenzelm@53406
  2206
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  2207
    have False by auto
wenzelm@53406
  2208
  }
wenzelm@53406
  2209
  then show "a \<in> span B" by blast
huffman@44133
  2210
qed
huffman@44133
  2211
huffman@44133
  2212
lemma card_le_dim_spanning:
wenzelm@49663
  2213
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  2214
    and VB: "V \<subseteq> span B"
wenzelm@49663
  2215
    and fB: "finite B"
wenzelm@49663
  2216
    and dVB: "dim V \<ge> card B"
huffman@44133
  2217
  shows "independent B"
wenzelm@49522
  2218
proof -
wenzelm@53406
  2219
  {
wenzelm@53406
  2220
    fix a
wenzelm@53716
  2221
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  2222
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  2223
      by auto
wenzelm@53716
  2224
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  2225
      by auto
wenzelm@53716
  2226
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  2227
      by blast
wenzelm@53406
  2228
    {
wenzelm@53406
  2229
      fix x
wenzelm@53406
  2230
      assume x: "x \<in> V"
wenzelm@53716
  2231
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  2232
        by blast
wenzelm@53406
  2233
      from x VB have x': "x \<in> span B"
wenzelm@53406
  2234
        by blast
huffman@44133
  2235
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  2236
      have "x \<in> span (B - {a})" .
wenzelm@53406
  2237
    }
wenzelm@53716
  2238
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  2239
      by blast
wenzelm@53716
  2240
    have th2: "finite (B - {a})"
wenzelm@53406
  2241
      using fB by auto
huffman@44133
  2242
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  2243
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  2244
    from c c0 dVB cb have False by simp
wenzelm@53406
  2245
  }
wenzelm@53406
  2246
  then show ?thesis
wenzelm@53406
  2247
    unfolding dependent_def by blast
huffman@44133
  2248
qed
huffman@44133
  2249
wenzelm@53406
  2250
lemma card_eq_dim:
wenzelm@53406
  2251
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2252
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  2253
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  2254
wenzelm@60420
  2255
text \<open>More general size bound lemmas.\<close>
huffman@44133
  2256
huffman@44133
  2257
lemma independent_bound_general:
wenzelm@53406
  2258
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2259
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  2260
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  2261
wenzelm@49522
  2262
lemma dependent_biggerset_general:
wenzelm@53406
  2263
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2264
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  2265
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  2266
paulson@60303
  2267
lemma dim_span [simp]:
wenzelm@53406
  2268
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2269
  shows "dim (span S) = dim S"
wenzelm@49522
  2270
proof -
huffman@44133
  2271
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  2272
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  2273
  from basis_exists[of S]
wenzelm@53406
  2274
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  2275
    by blast
wenzelm@53406
  2276
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2277
    using independent_bound by blast+
wenzelm@53406
  2278
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  2279
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  2280
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  2281
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  2282
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  2283
    using fB(2) by arith
huffman@44133
  2284
qed
huffman@44133
  2285
wenzelm@53406
  2286
lemma subset_le_dim:
wenzelm@53406
  2287
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2288
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  2289
  by (metis dim_span dim_subset)
huffman@44133
  2290
wenzelm@53406
  2291
lemma span_eq_dim:
wenzelm@56444
  2292
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2293
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  2294
  by (metis dim_span)
huffman@44133
  2295
huffman@44133
  2296
lemma dim_image_le:
huffman@44133
  2297
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  2298
  assumes lf: "linear f"
wenzelm@49663
  2299
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  2300
proof -
huffman@44133
  2301
  from basis_exists[of S] obtain B where
huffman@44133
  2302
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  2303
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2304
    using independent_bound by blast+
huffman@44133
  2305
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  2306
    apply (rule span_card_ge_dim)
wenzelm@53406
  2307
    using lf B fB
wenzelm@53406
  2308
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  2309
    done
wenzelm@53406
  2310
  also have "\<dots> \<le> dim S"
wenzelm@53406
  2311
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  2312
  finally show ?thesis .
huffman@44133
  2313
qed
huffman@44133
  2314
wenzelm@60420
  2315
text \<open>Picking an orthogonal replacement for a spanning set.\<close>
huffman@44133
  2316
wenzelm@53406
  2317
lemma vector_sub_project_orthogonal:
wenzelm@53406
  2318
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  2319
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  2320
  unfolding inner_simps by auto
huffman@44133
  2321
huffman@44528
  2322
lemma pairwise_orthogonal_insert:
huffman@44528
  2323
  assumes "pairwise orthogonal S"
wenzelm@49522
  2324
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  2325
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  2326
  using assms unfolding pairwise_def
huffman@44528
  2327
  by (auto simp add: orthogonal_commute)
huffman@44528
  2328
huffman@44133
  2329
lemma basis_orthogonal:
wenzelm@53406
  2330
  fixes B :: "'a::real_inner set"
huffman@44133
  2331
  assumes fB: "finite B"
huffman@44133
  2332
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  2333
  (is " \<exists>C. ?P B C")
wenzelm@49522
  2334
  using fB
wenzelm@49522
  2335
proof (induct rule: finite_induct)
wenzelm@49522
  2336
  case empty
wenzelm@53406
  2337
  then show ?case
wenzelm@53406
  2338
    apply (rule exI[where x="{}"])
wenzelm@53406
  2339
    apply (auto simp add: pairwise_def)
wenzelm@53406
  2340
    done
huffman@44133
  2341
next
wenzelm@49522
  2342
  case (insert a B)
wenzelm@60420
  2343
  note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
wenzelm@60420
  2344
  from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
huffman@44133
  2345
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  2346
    "span C = span B" "pairwise orthogonal C" by blast
nipkow@64267
  2347
  let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  2348
  let ?C = "insert ?a C"
wenzelm@53406
  2349
  from C(1) have fC: "finite ?C"
wenzelm@53406
  2350
    by simp
wenzelm@49522
  2351
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  2352
    by (simp add: card_insert_if)
wenzelm@53406
  2353
  {
wenzelm@53406
  2354
    fix x k
wenzelm@49522
  2355
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  2356
      by (simp add: field_simps)
huffman@44133
  2357
    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
  2358
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  2359
      apply (rule span_add_eq)
huffman@44133
  2360
      apply (rule span_mul)
nipkow@64267
  2361
      apply (rule span_sum)
huffman@44133
  2362
      apply (rule span_mul)
wenzelm@49522
  2363
      apply (rule span_superset)
wenzelm@49522
  2364
      apply assumption
wenzelm@53406
  2365
      done
wenzelm@53406
  2366
  }
huffman@44133
  2367
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  2368
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  2369
  {
wenzelm@53406
  2370
    fix y
wenzelm@53406
  2371
    assume yC: "y \<in> C"
wenzelm@53406
  2372
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  2373
      by blast
wenzelm@53406
  2374
    have fth: "finite (C - {y})"
wenzelm@53406
  2375
      using C by simp
huffman@44528
  2376
    have "orthogonal ?a y"
huffman@44528
  2377
      unfolding orthogonal_def
nipkow@64267
  2378
      unfolding inner_diff inner_sum_left right_minus_eq
nipkow@64267
  2379
      unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
huffman@44528
  2380
      apply (clarsimp simp add: inner_commute[of y a])
nipkow@64267
  2381
      apply (rule sum.neutral)
huffman@44528
  2382
      apply clarsimp
huffman@44528
  2383
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@60420
  2384
      using \<open>y \<in> C\<close> by auto
wenzelm@53406
  2385
  }
wenzelm@60420
  2386
  with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
huffman@44528
  2387
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  2388
  from fC cC SC CPO have "?P (insert a B) ?C"