src/HOL/Analysis/Linear_Algebra.thy
author paulson <lp15@cam.ac.uk>
Mon Apr 30 22:13:04 2018 +0100 (12 months ago)
changeset 68058 69715dfdc286
parent 67982 7643b005b29a
child 68062 ee88c0fccbae
permissions -rw-r--r--
more general tidying up
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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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  "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%important 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: "\<lbrakk>a \<in> Q hull S; \<And>x. x\<in> S \<Longrightarrow> P x; Q {x. P x}\<rbrakk> \<Longrightarrow> P a"
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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_Un_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_Un:
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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 equalityI)
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  apply (meson hull_mono hull_subset sup.mono)
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  by (metis hull_Un_subset hull_hull hull_mono)
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lemma hull_Un_left: "P hull (S \<union> T) = P hull (P hull S \<union> T)"
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  apply (rule equalityI)
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   apply (simp add: Un_commute hull_mono hull_subset sup.coboundedI2)
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  by (metis Un_subset_iff hull_hull hull_mono hull_subset)
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lemma hull_Un_right: "P hull (S \<union> T) = P hull (S \<union> P hull T)"
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  by (metis hull_Un_left sup.commute)
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lemma hull_insert:
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   "P hull (insert a S) = P hull (insert a (P hull S))"
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  by (metis hull_Un_right insert_is_Un)
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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%important 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%unimportant
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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%important (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%important (in real_vector) "span S = (subspace hull S)"
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definition%important (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%unimportant \<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"
wenzelm@53406
   304
  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
huffman@44133
   305
huffman@44521
   306
lemma span_unique:
wenzelm@49522
   307
  "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
huffman@44521
   308
  unfolding span_def by (rule hull_unique)
huffman@44521
   309
huffman@44521
   310
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
huffman@44521
   311
  unfolding span_def by (rule hull_minimal)
huffman@44521
   312
lp15@66641
   313
lemma span_UNIV [simp]: "span UNIV = UNIV"
hoelzl@63053
   314
  by (intro span_unique) auto
hoelzl@63053
   315
huffman@44521
   316
lemma (in real_vector) span_induct:
wenzelm@49522
   317
  assumes x: "x \<in> span S"
lp15@63469
   318
    and P: "subspace (Collect P)"
lp15@63469
   319
    and SP: "\<And>x. x \<in> S \<Longrightarrow> P x"
lp15@63469
   320
  shows "P x"
wenzelm@49522
   321
proof -
lp15@63469
   322
  from SP have SP': "S \<subseteq> Collect P"
wenzelm@53406
   323
    by (simp add: subset_eq)
huffman@44170
   324
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
lp15@63469
   325
  show ?thesis
lp15@63469
   326
    using subset_eq by force
huffman@44133
   327
qed
huffman@44133
   328
huffman@44133
   329
lemma span_empty[simp]: "span {} = {0}"
huffman@44133
   330
  apply (simp add: span_def)
huffman@44133
   331
  apply (rule hull_unique)
huffman@44170
   332
  apply (auto simp add: subspace_def)
huffman@44133
   333
  done
huffman@44133
   334
lp15@62948
   335
lemma (in real_vector) independent_empty [iff]: "independent {}"
huffman@44133
   336
  by (simp add: dependent_def)
huffman@44133
   337
wenzelm@49522
   338
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
huffman@44133
   339
  unfolding dependent_def by auto
huffman@44133
   340
wenzelm@53406
   341
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
lp15@68058
   342
  unfolding dependent_def span_mono
lp15@68058
   343
  by (metis insert_Diff local.span_mono subsetCE subset_insert_iff) 
huffman@44133
   344
huffman@44133
   345
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   346
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   347
wenzelm@56444
   348
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
wenzelm@53406
   349
where
huffman@44170
   350
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   351
| span_induct_alt_help_S:
wenzelm@53406
   352
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   353
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   354
huffman@44133
   355
lemma span_induct_alt':
wenzelm@53406
   356
  assumes h0: "h 0"
wenzelm@53406
   357
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   358
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   359
proof -
wenzelm@53406
   360
  {
wenzelm@53406
   361
    fix x :: 'a
wenzelm@53406
   362
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   363
    have "h x"
huffman@44133
   364
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   365
      apply (rule h0)
wenzelm@53406
   366
      apply (rule hS)
wenzelm@53406
   367
      apply assumption
wenzelm@53406
   368
      apply assumption
wenzelm@53406
   369
      done
wenzelm@53406
   370
  }
huffman@44133
   371
  note th0 = this
wenzelm@53406
   372
  {
wenzelm@53406
   373
    fix x
wenzelm@53406
   374
    assume x: "x \<in> span S"
huffman@44170
   375
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   376
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   377
      show "x \<in> span S" by (rule x)
wenzelm@49522
   378
    next
wenzelm@53406
   379
      fix x
wenzelm@53406
   380
      assume xS: "x \<in> S"
wenzelm@53406
   381
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   382
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   383
        by simp
wenzelm@49522
   384
    next
wenzelm@49522
   385
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   386
      moreover
wenzelm@53406
   387
      {
wenzelm@53406
   388
        fix x y
wenzelm@49522
   389
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   390
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   391
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   392
          apply simp
haftmann@57512
   393
          unfolding add.assoc
wenzelm@49522
   394
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   395
          apply assumption
wenzelm@49522
   396
          apply simp
wenzelm@53406
   397
          done
wenzelm@53406
   398
      }
wenzelm@49522
   399
      moreover
wenzelm@53406
   400
      {
wenzelm@53406
   401
        fix c x
wenzelm@49522
   402
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   403
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   404
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   405
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   406
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   407
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   408
          apply assumption
wenzelm@49522
   409
          apply simp
wenzelm@49522
   410
          done }
lp15@63469
   411
      ultimately show "subspace {a. a \<in> span_induct_alt_help S}"
wenzelm@49522
   412
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   413
    qed
wenzelm@53406
   414
  }
huffman@44133
   415
  with th0 show ?thesis by blast
huffman@44133
   416
qed
huffman@44133
   417
huffman@44133
   418
lemma span_induct_alt:
wenzelm@53406
   419
  assumes h0: "h 0"
wenzelm@53406
   420
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   421
    and x: "x \<in> span S"
huffman@44133
   422
  shows "h x"
wenzelm@49522
   423
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   424
wenzelm@60420
   425
text \<open>Individual closure properties.\<close>
huffman@44133
   426
huffman@44133
   427
lemma span_span: "span (span A) = span A"
huffman@44133
   428
  unfolding span_def hull_hull ..
huffman@44133
   429
wenzelm@53406
   430
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   431
  by (metis span_clauses(1))
wenzelm@53406
   432
lp15@63075
   433
lemma (in real_vector) span_0 [simp]: "0 \<in> span S"
wenzelm@53406
   434
  by (metis subspace_span subspace_0)
huffman@44133
   435
huffman@44133
   436
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   437
  by (metis subset_eq span_superset)
huffman@44133
   438
hoelzl@63053
   439
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
hoelzl@63053
   440
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
hoelzl@63053
   441
  by (auto simp add: span_span)
hoelzl@63053
   442
wenzelm@53406
   443
lemma (in real_vector) dependent_0:
wenzelm@53406
   444
  assumes "0 \<in> A"
wenzelm@53406
   445
  shows "dependent A"
wenzelm@53406
   446
  unfolding dependent_def
wenzelm@53406
   447
  using assms span_0
lp15@63075
   448
  by blast
wenzelm@53406
   449
wenzelm@53406
   450
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   451
  by (metis subspace_add subspace_span)
huffman@44133
   452
wenzelm@53406
   453
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   454
  by (metis subspace_span subspace_mul)
huffman@44133
   455
wenzelm@53406
   456
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   457
  by (metis subspace_neg subspace_span)
huffman@44133
   458
lp15@63938
   459
lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
lp15@63114
   460
  by (metis subspace_span subspace_diff)
huffman@44133
   461
nipkow@64267
   462
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
   463
  by (rule subspace_sum [OF subspace_span])
huffman@44133
   464
huffman@44133
   465
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   466
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   467
hoelzl@63050
   468
text \<open>The key breakdown property.\<close>
hoelzl@63050
   469
hoelzl@63050
   470
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
hoelzl@63050
   471
proof (rule span_unique)
hoelzl@63050
   472
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
hoelzl@63050
   473
    by (fast intro: scaleR_one [symmetric])
hoelzl@63050
   474
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
hoelzl@63050
   475
    unfolding subspace_def
hoelzl@63050
   476
    by (auto intro: scaleR_add_left [symmetric])
hoelzl@63050
   477
next
hoelzl@63050
   478
  fix T
hoelzl@63050
   479
  assume "{x} \<subseteq> T" and "subspace T"
hoelzl@63050
   480
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
hoelzl@63050
   481
    unfolding subspace_def by auto
hoelzl@63050
   482
qed
hoelzl@63050
   483
wenzelm@60420
   484
text \<open>Mapping under linear image.\<close>
huffman@44133
   485
hoelzl@63050
   486
lemma subspace_linear_image:
hoelzl@63050
   487
  assumes lf: "linear f"
hoelzl@63050
   488
    and sS: "subspace S"
hoelzl@63050
   489
  shows "subspace (f ` S)"
hoelzl@63050
   490
  using lf sS linear_0[OF lf]
hoelzl@63050
   491
  unfolding linear_iff subspace_def
hoelzl@63050
   492
  apply (auto simp add: image_iff)
hoelzl@63050
   493
  apply (rule_tac x="x + y" in bexI)
hoelzl@63050
   494
  apply auto
hoelzl@63050
   495
  apply (rule_tac x="c *\<^sub>R x" in bexI)
hoelzl@63050
   496
  apply auto
hoelzl@63050
   497
  done
hoelzl@63050
   498
hoelzl@63050
   499
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
hoelzl@63050
   500
  by (auto simp add: subspace_def linear_iff linear_0[of f])
hoelzl@63050
   501
hoelzl@63050
   502
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
hoelzl@63050
   503
  by (auto simp add: subspace_def linear_iff linear_0[of f])
hoelzl@63050
   504
huffman@44521
   505
lemma span_linear_image:
huffman@44521
   506
  assumes lf: "linear f"
wenzelm@56444
   507
  shows "span (f ` S) = f ` span S"
huffman@44521
   508
proof (rule span_unique)
huffman@44521
   509
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   510
    by (intro image_mono span_inc)
huffman@44521
   511
  show "subspace (f ` span S)"
huffman@44521
   512
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   513
next
wenzelm@53406
   514
  fix T
wenzelm@53406
   515
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   516
  then show "f ` span S \<subseteq> T"
huffman@44521
   517
    unfolding image_subset_iff_subset_vimage
huffman@44521
   518
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   519
qed
huffman@44521
   520
hoelzl@63053
   521
lemma spans_image:
hoelzl@63053
   522
  assumes lf: "linear f"
hoelzl@63053
   523
    and VB: "V \<subseteq> span B"
hoelzl@63053
   524
  shows "f ` V \<subseteq> span (f ` B)"
hoelzl@63053
   525
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
hoelzl@63053
   526
lp15@63469
   527
lemma span_Un: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   528
proof (rule span_unique)
huffman@44521
   529
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   530
    by safe (force intro: span_clauses)+
huffman@44521
   531
next
huffman@44521
   532
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   533
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   534
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   535
    by (intro subspace_Times subspace_span)
huffman@44521
   536
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   537
    by (rule subspace_linear_image)
huffman@44521
   538
next
wenzelm@49711
   539
  fix T
wenzelm@49711
   540
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   541
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   542
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   543
qed
huffman@44133
   544
wenzelm@49522
   545
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   546
proof -
huffman@44521
   547
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
lp15@63469
   548
    unfolding span_Un span_singleton
huffman@44521
   549
    apply safe
huffman@44521
   550
    apply (rule_tac x=k in exI, simp)
huffman@44521
   551
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   552
    apply auto
huffman@44521
   553
    done
wenzelm@49522
   554
  then show ?thesis by simp
huffman@44521
   555
qed
huffman@44521
   556
huffman@44133
   557
lemma span_breakdown:
wenzelm@53406
   558
  assumes bS: "b \<in> S"
wenzelm@53406
   559
    and aS: "a \<in> span S"
huffman@44521
   560
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   561
  using assms span_insert [of b "S - {b}"]
huffman@44521
   562
  by (simp add: insert_absorb)
huffman@44133
   563
wenzelm@53406
   564
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
   565
  by (simp add: span_insert)
huffman@44133
   566
wenzelm@60420
   567
text \<open>Hence some "reversal" results.\<close>
huffman@44133
   568
huffman@44133
   569
lemma in_span_insert:
wenzelm@49711
   570
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
   571
    and na: "a \<notin> span S"
huffman@44133
   572
  shows "b \<in> span (insert a S)"
wenzelm@49663
   573
proof -
huffman@55910
   574
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
   575
    unfolding span_insert by fast
wenzelm@53406
   576
  show ?thesis
wenzelm@53406
   577
  proof (cases "k = 0")
wenzelm@53406
   578
    case True
huffman@55910
   579
    with k have "a \<in> span S" by simp
huffman@55910
   580
    with na show ?thesis by simp
wenzelm@53406
   581
  next
wenzelm@53406
   582
    case False
huffman@55910
   583
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
   584
      by (rule span_mul)
huffman@55910
   585
    then have "b - inverse k *\<^sub>R a \<in> span S"
wenzelm@60420
   586
      using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
huffman@55910
   587
    then show ?thesis
huffman@55910
   588
      unfolding span_insert by fast
wenzelm@53406
   589
  qed
huffman@44133
   590
qed
huffman@44133
   591
huffman@44133
   592
lemma in_span_delete:
huffman@44133
   593
  assumes a: "a \<in> span S"
wenzelm@53716
   594
    and na: "a \<notin> span (S - {b})"
huffman@44133
   595
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
   596
  apply (rule in_span_insert)
huffman@44133
   597
  apply (rule set_rev_mp)
huffman@44133
   598
  apply (rule a)
huffman@44133
   599
  apply (rule span_mono)
huffman@44133
   600
  apply blast
huffman@44133
   601
  apply (rule na)
huffman@44133
   602
  done
huffman@44133
   603
wenzelm@60420
   604
text \<open>Transitivity property.\<close>
huffman@44133
   605
huffman@44521
   606
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
   607
  unfolding span_def by (rule hull_redundant)
huffman@44521
   608
huffman@44133
   609
lemma span_trans:
wenzelm@53406
   610
  assumes x: "x \<in> span S"
wenzelm@53406
   611
    and y: "y \<in> span (insert x S)"
huffman@44133
   612
  shows "y \<in> span S"
huffman@44521
   613
  using assms by (simp only: span_redundant)
huffman@44133
   614
huffman@44133
   615
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
   616
  by (simp only: span_redundant span_0)
huffman@44133
   617
wenzelm@60420
   618
text \<open>An explicit expansion is sometimes needed.\<close>
huffman@44133
   619
huffman@44133
   620
lemma span_explicit:
nipkow@64267
   621
  "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
   622
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
   623
proof -
wenzelm@53406
   624
  {
wenzelm@53406
   625
    fix x
huffman@55910
   626
    assume "?h x"
nipkow@64267
   627
    then obtain S u where "finite S" and "S \<subseteq> P" and "sum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
   628
      by blast
huffman@55910
   629
    then have "x \<in> span P"
nipkow@64267
   630
      by (auto intro: span_sum span_mul span_superset)
wenzelm@53406
   631
  }
huffman@44133
   632
  moreover
huffman@55910
   633
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
   634
  proof (rule span_induct_alt')
huffman@55910
   635
    show "?h 0"
huffman@55910
   636
      by (rule exI[where x="{}"], simp)
huffman@44133
   637
  next
huffman@44133
   638
    fix c x y
wenzelm@53406
   639
    assume x: "x \<in> P"
huffman@55910
   640
    assume hy: "?h y"
huffman@44133
   641
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
nipkow@64267
   642
      and u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
   643
    let ?S = "insert x S"
wenzelm@49522
   644
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
   645
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
   646
      by blast+
wenzelm@53406
   647
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
   648
    proof cases
wenzelm@53406
   649
      assume xS: "x \<in> S"
nipkow@64267
   650
      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
   651
        using xS by (simp add: sum.remove [OF fS xS] insert_absorb)
huffman@44133
   652
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
nipkow@64267
   653
        by (simp add: sum.remove [OF fS xS] algebra_simps)
huffman@44133
   654
      also have "\<dots> = c*\<^sub>R x + y"
haftmann@57512
   655
        by (simp add: add.commute u)
nipkow@64267
   656
      finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
   657
      then show ?thesis using th0 by blast
wenzelm@53406
   658
    next
wenzelm@53406
   659
      assume xS: "x \<notin> S"
wenzelm@49522
   660
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
   661
        unfolding u[symmetric]
nipkow@64267
   662
        apply (rule sum.cong)
wenzelm@53406
   663
        using xS
wenzelm@53406
   664
        apply auto
wenzelm@49522
   665
        done
wenzelm@53406
   666
      show ?thesis using fS xS th0
haftmann@57512
   667
        by (simp add: th00 add.commute cong del: if_weak_cong)
wenzelm@53406
   668
    qed
huffman@55910
   669
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
   670
      by fast
huffman@44133
   671
  qed
huffman@44133
   672
  ultimately show ?thesis by blast
huffman@44133
   673
qed
huffman@44133
   674
huffman@44133
   675
lemma dependent_explicit:
nipkow@64267
   676
  "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
   677
  (is "?lhs = ?rhs")
wenzelm@49522
   678
proof -
wenzelm@53406
   679
  {
wenzelm@53406
   680
    assume dP: "dependent P"
huffman@44133
   681
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
nipkow@64267
   682
      and SP: "S \<subseteq> P - {a}" and ua: "sum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
   683
      unfolding dependent_def span_explicit by blast
huffman@44133
   684
    let ?S = "insert a S"
huffman@44133
   685
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
   686
    let ?v = a
wenzelm@53406
   687
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
   688
      by blast
wenzelm@53406
   689
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
   690
      by auto
nipkow@64267
   691
    have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
   692
      using fS aS
huffman@55910
   693
      apply simp
huffman@44133
   694
      apply (subst (2) ua[symmetric])
nipkow@64267
   695
      apply (rule sum.cong)
wenzelm@49522
   696
      apply auto
wenzelm@49522
   697
      done
huffman@55910
   698
    with th0 have ?rhs by fast
wenzelm@49522
   699
  }
huffman@44133
   700
  moreover
wenzelm@53406
   701
  {
wenzelm@53406
   702
    fix S u v
wenzelm@49522
   703
    assume fS: "finite S"
wenzelm@53406
   704
      and SP: "S \<subseteq> P"
wenzelm@53406
   705
      and vS: "v \<in> S"
wenzelm@53406
   706
      and uv: "u v \<noteq> 0"
nipkow@64267
   707
      and u: "sum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
   708
    let ?a = v
huffman@44133
   709
    let ?S = "S - {v}"
huffman@44133
   710
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
   711
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
   712
      using fS SP vS by auto
nipkow@64267
   713
    have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S =
nipkow@64267
   714
      sum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
nipkow@64267
   715
      using fS vS uv by (simp add: sum_diff1 field_simps)
wenzelm@53406
   716
    also have "\<dots> = ?a"
nipkow@64267
   717
      unfolding scaleR_right.sum [symmetric] u using uv by simp
nipkow@64267
   718
    finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
   719
    with th0 have ?lhs
huffman@44133
   720
      unfolding dependent_def span_explicit
huffman@44133
   721
      apply -
huffman@44133
   722
      apply (rule bexI[where x= "?a"])
huffman@44133
   723
      apply (simp_all del: scaleR_minus_left)
huffman@44133
   724
      apply (rule exI[where x= "?S"])
wenzelm@49522
   725
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
   726
      done
wenzelm@49522
   727
  }
huffman@44133
   728
  ultimately show ?thesis by blast
huffman@44133
   729
qed
huffman@44133
   730
lp15@63075
   731
lemma dependent_finite:
lp15@63075
   732
  assumes "finite S"
lp15@63075
   733
    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
   734
           (is "?lhs = ?rhs")
lp15@63075
   735
proof
lp15@63075
   736
  assume ?lhs
lp15@63075
   737
  then obtain T u v
lp15@63075
   738
         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
   739
    by (force simp: dependent_explicit)
lp15@63075
   740
  with assms show ?rhs
lp15@63075
   741
    apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
nipkow@64267
   742
    apply (auto simp: sum.mono_neutral_right)
lp15@63075
   743
    done
lp15@63075
   744
next
lp15@63075
   745
  assume ?rhs  with assms show ?lhs
lp15@63075
   746
    by (fastforce simp add: dependent_explicit)
lp15@63075
   747
qed
lp15@63075
   748
hoelzl@63051
   749
lemma span_alt:
hoelzl@63051
   750
  "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
   751
  unfolding span_explicit
hoelzl@63051
   752
  apply safe
hoelzl@63051
   753
  subgoal for x S u
hoelzl@63051
   754
    by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
nipkow@64267
   755
        (auto intro!: sum.mono_neutral_cong_right)
hoelzl@63051
   756
  apply auto
hoelzl@63051
   757
  done
hoelzl@63051
   758
hoelzl@63051
   759
lemma dependent_alt:
hoelzl@63051
   760
  "dependent B \<longleftrightarrow>
hoelzl@63051
   761
    (\<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
   762
  unfolding dependent_explicit
hoelzl@63051
   763
  apply safe
hoelzl@63051
   764
  subgoal for S u v
hoelzl@63051
   765
    apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
nipkow@64267
   766
    apply (subst sum.mono_neutral_cong_left[where T=S])
nipkow@64267
   767
    apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong)
hoelzl@63051
   768
    done
hoelzl@63051
   769
  apply auto
hoelzl@63051
   770
  done
hoelzl@63051
   771
hoelzl@63051
   772
lemma independent_alt:
hoelzl@63051
   773
  "independent B \<longleftrightarrow>
hoelzl@63051
   774
    (\<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
   775
  unfolding dependent_alt by auto
hoelzl@63051
   776
hoelzl@63051
   777
lemma independentD_alt:
hoelzl@63051
   778
  "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
   779
  unfolding independent_alt by blast
hoelzl@63051
   780
hoelzl@63051
   781
lemma independentD_unique:
hoelzl@63051
   782
  assumes B: "independent B"
hoelzl@63051
   783
    and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   784
    and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   785
    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
   786
  shows "X = Y"
wenzelm@49522
   787
proof -
hoelzl@63051
   788
  have "X x - Y x = 0" for x
hoelzl@63051
   789
    using B
hoelzl@63051
   790
  proof (rule independentD_alt)
hoelzl@63051
   791
    have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
hoelzl@63051
   792
      by auto
hoelzl@63051
   793
    then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
hoelzl@63051
   794
      using X Y by (auto dest: finite_subset)
hoelzl@63051
   795
    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
   796
      using X Y by (intro sum.mono_neutral_cong_left) auto
hoelzl@63051
   797
    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
   798
      by (simp add: scaleR_diff_left sum_subtractf assms)
hoelzl@63051
   799
    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
   800
      using X Y by (intro sum.mono_neutral_cong_right) auto
hoelzl@63051
   801
    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
   802
      using X Y by (intro sum.mono_neutral_cong_right) auto
hoelzl@63051
   803
    finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
hoelzl@63051
   804
      using assms by simp
hoelzl@63051
   805
  qed
hoelzl@63051
   806
  then show ?thesis
hoelzl@63051
   807
    by auto
huffman@44133
   808
qed
huffman@44133
   809
wenzelm@60420
   810
text \<open>This is useful for building a basis step-by-step.\<close>
huffman@44133
   811
huffman@44133
   812
lemma independent_insert:
wenzelm@53406
   813
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
   814
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
   815
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
   816
proof (cases "a \<in> S")
wenzelm@53406
   817
  case True
wenzelm@53406
   818
  then show ?thesis
wenzelm@53406
   819
    using insert_absorb[OF True] by simp
wenzelm@53406
   820
next
wenzelm@53406
   821
  case False
wenzelm@53406
   822
  show ?thesis
wenzelm@53406
   823
  proof
wenzelm@53406
   824
    assume i: ?lhs
wenzelm@53406
   825
    then show ?rhs
wenzelm@53406
   826
      using False
wenzelm@53406
   827
      apply simp
wenzelm@53406
   828
      apply (rule conjI)
wenzelm@53406
   829
      apply (rule independent_mono)
wenzelm@53406
   830
      apply assumption
wenzelm@53406
   831
      apply blast
wenzelm@53406
   832
      apply (simp add: dependent_def)
wenzelm@53406
   833
      done
wenzelm@53406
   834
  next
wenzelm@53406
   835
    assume i: ?rhs
wenzelm@53406
   836
    show ?lhs
wenzelm@53406
   837
      using i False
wenzelm@53406
   838
      apply (auto simp add: dependent_def)
lp15@60810
   839
      by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
wenzelm@53406
   840
  qed
huffman@44133
   841
qed
huffman@44133
   842
hoelzl@63051
   843
lemma independent_Union_directed:
hoelzl@63051
   844
  assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
hoelzl@63051
   845
  assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
hoelzl@63051
   846
  shows "independent (\<Union>C)"
hoelzl@63051
   847
proof
hoelzl@63051
   848
  assume "dependent (\<Union>C)"
hoelzl@63051
   849
  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
   850
    by (auto simp: dependent_explicit)
hoelzl@63051
   851
hoelzl@63051
   852
  have "S \<noteq> {}"
hoelzl@63051
   853
    using \<open>v \<in> S\<close> by auto
hoelzl@63051
   854
  have "\<exists>c\<in>C. S \<subseteq> c"
hoelzl@63051
   855
    using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
hoelzl@63051
   856
  proof (induction rule: finite_ne_induct)
hoelzl@63051
   857
    case (insert i I)
hoelzl@63051
   858
    then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
hoelzl@63051
   859
      by blast
hoelzl@63051
   860
    from directed[OF cd] cd have "c \<union> d \<in> C"
hoelzl@63051
   861
      by (auto simp: sup.absorb1 sup.absorb2)
hoelzl@63051
   862
    with iI show ?case
hoelzl@63051
   863
      by (intro bexI[of _ "c \<union> d"]) auto
hoelzl@63051
   864
  qed auto
hoelzl@63051
   865
  then obtain c where "c \<in> C" "S \<subseteq> c"
hoelzl@63051
   866
    by auto
hoelzl@63051
   867
  have "dependent c"
hoelzl@63051
   868
    unfolding dependent_explicit
hoelzl@63051
   869
    by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
hoelzl@63051
   870
  with indep[OF \<open>c \<in> C\<close>] show False
hoelzl@63051
   871
    by auto
hoelzl@63051
   872
qed
hoelzl@63051
   873
hoelzl@63051
   874
text \<open>Hence we can create a maximal independent subset.\<close>
hoelzl@63051
   875
hoelzl@63051
   876
lemma maximal_independent_subset_extend:
hoelzl@63051
   877
  assumes "S \<subseteq> V" "independent S"
hoelzl@63051
   878
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
hoelzl@63051
   879
proof -
hoelzl@63051
   880
  let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}"
hoelzl@63051
   881
  have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M"
hoelzl@63051
   882
  proof (rule subset_Zorn)
hoelzl@63051
   883
    fix C :: "'a set set" assume "subset.chain ?C C"
hoelzl@63051
   884
    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
   885
      "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
hoelzl@63051
   886
      unfolding subset.chain_def by blast+
hoelzl@63051
   887
hoelzl@63051
   888
    show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U"
hoelzl@63051
   889
    proof cases
hoelzl@63051
   890
      assume "C = {}" with assms show ?thesis
hoelzl@63051
   891
        by (auto intro!: exI[of _ S])
hoelzl@63051
   892
    next
hoelzl@63051
   893
      assume "C \<noteq> {}"
hoelzl@63051
   894
      with C(2) have "S \<subseteq> \<Union>C"
hoelzl@63051
   895
        by auto
hoelzl@63051
   896
      moreover have "independent (\<Union>C)"
hoelzl@63051
   897
        by (intro independent_Union_directed C)
hoelzl@63051
   898
      moreover have "\<Union>C \<subseteq> V"
hoelzl@63051
   899
        using C by auto
hoelzl@63051
   900
      ultimately show ?thesis
hoelzl@63051
   901
        by auto
hoelzl@63051
   902
    qed
hoelzl@63051
   903
  qed
hoelzl@63051
   904
  then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B"
hoelzl@63051
   905
    and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B"
hoelzl@63051
   906
    by auto
hoelzl@63051
   907
  moreover
hoelzl@63051
   908
  { assume "\<not> V \<subseteq> span B"
hoelzl@63051
   909
    then obtain v where "v \<in> V" "v \<notin> span B"
hoelzl@63051
   910
      by auto
hoelzl@63051
   911
    with B have "independent (insert v B)"
hoelzl@63051
   912
      unfolding independent_insert by auto
hoelzl@63051
   913
    from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close>
hoelzl@63051
   914
    have "v \<in> B"
hoelzl@63051
   915
      by auto
hoelzl@63051
   916
    with \<open>v \<notin> span B\<close> have False
hoelzl@63051
   917
      by (auto intro: span_superset) }
hoelzl@63051
   918
  ultimately show ?thesis
hoelzl@63051
   919
    by (auto intro!: exI[of _ B])
hoelzl@63051
   920
qed
hoelzl@63051
   921
hoelzl@63051
   922
hoelzl@63051
   923
lemma maximal_independent_subset:
hoelzl@63051
   924
  "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
hoelzl@63051
   925
  by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
hoelzl@63051
   926
hoelzl@63051
   927
lemma span_finite:
hoelzl@63051
   928
  assumes fS: "finite S"
nipkow@64267
   929
  shows "span S = {y. \<exists>u. sum (\<lambda>v. u v *\<^sub>R v) S = y}"
hoelzl@63051
   930
  (is "_ = ?rhs")
hoelzl@63051
   931
proof -
hoelzl@63051
   932
  {
hoelzl@63051
   933
    fix y
hoelzl@63051
   934
    assume y: "y \<in> span S"
hoelzl@63051
   935
    from y obtain S' u where fS': "finite S'"
hoelzl@63051
   936
      and SS': "S' \<subseteq> S"
nipkow@64267
   937
      and u: "sum (\<lambda>v. u v *\<^sub>R v) S' = y"
hoelzl@63051
   938
      unfolding span_explicit by blast
hoelzl@63051
   939
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
nipkow@64267
   940
    have "sum (\<lambda>v. ?u v *\<^sub>R v) S = sum (\<lambda>v. u v *\<^sub>R v) S'"
nipkow@64267
   941
      using SS' fS by (auto intro!: sum.mono_neutral_cong_right)
nipkow@64267
   942
    then have "sum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
hoelzl@63051
   943
    then have "y \<in> ?rhs" by auto
hoelzl@63051
   944
  }
hoelzl@63051
   945
  moreover
hoelzl@63051
   946
  {
hoelzl@63051
   947
    fix y u
nipkow@64267
   948
    assume u: "sum (\<lambda>v. u v *\<^sub>R v) S = y"
hoelzl@63051
   949
    then have "y \<in> span S" using fS unfolding span_explicit by auto
hoelzl@63051
   950
  }
hoelzl@63051
   951
  ultimately show ?thesis by blast
hoelzl@63051
   952
qed
hoelzl@63051
   953
hoelzl@63052
   954
lemma linear_independent_extend_subspace:
hoelzl@63052
   955
  assumes "independent B"
hoelzl@63052
   956
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)"
hoelzl@63052
   957
proof -
hoelzl@63052
   958
  from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV]
hoelzl@63052
   959
  obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV"
hoelzl@63052
   960
    by (auto simp: top_unique)
hoelzl@63052
   961
  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
   962
    using \<open>span B' = UNIV\<close> unfolding span_alt by auto
hoelzl@63052
   963
  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
   964
    "\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)"
hoelzl@63052
   965
    unfolding choice_iff by auto
hoelzl@63052
   966
hoelzl@63052
   967
  have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y
hoelzl@63052
   968
    using \<open>independent B'\<close>
hoelzl@63052
   969
  proof (rule independentD_unique)
hoelzl@63052
   970
    have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
hoelzl@63052
   971
      = (\<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
   972
      by (intro sum.mono_neutral_cong_left) (auto intro: X)
hoelzl@63052
   973
    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
   974
      by (auto simp add: scaleR_add_left sum.distrib
nipkow@67399
   975
               intro!: arg_cong2[where f="(+)"]  sum.mono_neutral_cong_right X)
hoelzl@63052
   976
    also have "\<dots> = x + y"
hoelzl@63052
   977
      by (simp add: X(3)[symmetric])
hoelzl@63052
   978
    also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
hoelzl@63052
   979
      by (rule X(3))
hoelzl@63052
   980
    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
   981
      ..
hoelzl@63052
   982
    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
   983
      by auto
hoelzl@63052
   984
    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
   985
        "finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   986
      using X(1) by (auto dest: finite_subset intro: X)
hoelzl@63052
   987
  qed
hoelzl@63052
   988
hoelzl@63052
   989
  have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c
hoelzl@63052
   990
    using \<open>independent B'\<close>
hoelzl@63052
   991
  proof (rule independentD_unique)
hoelzl@63052
   992
    show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'"
hoelzl@63052
   993
      "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
hoelzl@63052
   994
      using X(1,2) by auto
hoelzl@63052
   995
    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
   996
      unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
hoelzl@63052
   997
      by (cases "c = 0") (auto simp: X(3)[symmetric])
hoelzl@63052
   998
  qed
hoelzl@63052
   999
hoelzl@63052
  1000
  have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x
hoelzl@63052
  1001
    using \<open>independent B'\<close>
hoelzl@63052
  1002
    by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric])
hoelzl@63052
  1003
wenzelm@63148
  1004
  define f' where "f' y = (if y \<in> B then f y else 0)" for y
wenzelm@63148
  1005
  define g where "g y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x)" for y
hoelzl@63052
  1006
hoelzl@63052
  1007
  have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x
hoelzl@63052
  1008
    by (auto simp: g_def X_B')
hoelzl@63052
  1009
hoelzl@63052
  1010
  have "linear g"
hoelzl@63052
  1011
  proof
hoelzl@63052
  1012
    fix x y
hoelzl@63052
  1013
    have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
hoelzl@63052
  1014
      = (\<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
  1015
      by (intro sum.mono_neutral_cong_left) (auto intro: X)
hoelzl@63052
  1016
    show "g (x + y) = g x + g y"
hoelzl@63052
  1017
      unfolding g_def X_add *
nipkow@64267
  1018
      by (auto simp add: scaleR_add_left sum.distrib
nipkow@67399
  1019
               intro!: arg_cong2[where f="(+)"]  sum.mono_neutral_cong_right X)
hoelzl@63052
  1020
  next
hoelzl@63052
  1021
    show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
nipkow@64267
  1022
      by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X)
hoelzl@63052
  1023
  qed
hoelzl@63052
  1024
  moreover have "\<forall>x\<in>B. g x = f x"
hoelzl@63052
  1025
    using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
hoelzl@63052
  1026
  moreover have "range g = span (f`B)"
hoelzl@63052
  1027
    unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric]
hoelzl@63052
  1028
  proof (rule span_subspace)
hoelzl@63052
  1029
    have "g ` B' \<subseteq> f`B \<union> {0}"
hoelzl@63052
  1030
      by (auto simp: g_f' f'_def)
hoelzl@63052
  1031
    also have "\<dots> \<subseteq> span (f`B)"
hoelzl@63052
  1032
      by (auto intro: span_superset span_0)
hoelzl@63052
  1033
    finally show "g ` B' \<subseteq> span (f`B)"
hoelzl@63052
  1034
      by auto
hoelzl@63052
  1035
    have "x \<in> B \<Longrightarrow> f x = g x" for x
hoelzl@63052
  1036
      using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def)
hoelzl@63052
  1037
    then show "span (f ` B) \<subseteq> span (g ` B')"
hoelzl@63052
  1038
      using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto
hoelzl@63052
  1039
  qed (rule subspace_span)
hoelzl@63052
  1040
  ultimately show ?thesis
hoelzl@63052
  1041
    by auto
hoelzl@63052
  1042
qed
hoelzl@63052
  1043
hoelzl@63052
  1044
lemma linear_independent_extend:
hoelzl@63052
  1045
  "independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
hoelzl@63052
  1046
  using linear_independent_extend_subspace[of B f] by auto
hoelzl@63052
  1047
hoelzl@63053
  1048
text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
hoelzl@63053
  1049
hoelzl@63053
  1050
lemma subspace_kernel:
hoelzl@63053
  1051
  assumes lf: "linear f"
hoelzl@63053
  1052
  shows "subspace {x. f x = 0}"
hoelzl@63053
  1053
  apply (simp add: subspace_def)
hoelzl@63053
  1054
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
hoelzl@63053
  1055
  done
hoelzl@63053
  1056
hoelzl@63053
  1057
lemma linear_eq_0_span:
lp15@68058
  1058
  assumes x: "x \<in> span B" and lf: "linear f" and f0: "\<And>x. x\<in>B \<Longrightarrow> f x = 0"
lp15@68058
  1059
  shows "f x = 0"
lp15@68058
  1060
  using x f0 subspace_kernel[OF lf] span_induct
lp15@68058
  1061
  by blast
lp15@68058
  1062
lp15@68058
  1063
lemma linear_eq_0: "\<lbrakk>x \<in> S; linear f; S \<subseteq> span B; \<And>x. x\<in>B \<Longrightarrow> f x = 0\<rbrakk> \<Longrightarrow> f x = 0"
lp15@68058
  1064
  using linear_eq_0_span[of x B f] by auto
lp15@68058
  1065
lp15@68058
  1066
lemma linear_eq_span: "\<lbrakk>x \<in> span B; linear f; linear g; \<And>x. x\<in>B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> f x = g x"
lp15@68058
  1067
  using linear_eq_0_span[of x B "\<lambda>x. f x - g x"]  by (auto simp: linear_compose_sub)
lp15@68058
  1068
lp15@68058
  1069
lemma linear_eq: "\<lbrakk>x \<in> S; linear f; linear g; S \<subseteq> span B; \<And>x. x\<in>B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> f x = g x"
lp15@68058
  1070
  using linear_eq_span[of _ B f g] by auto
hoelzl@63053
  1071
wenzelm@60420
  1072
text \<open>The degenerate case of the Exchange Lemma.\<close>
huffman@44133
  1073
huffman@44133
  1074
lemma spanning_subset_independent:
wenzelm@49711
  1075
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1076
    and iA: "independent A"
wenzelm@49522
  1077
    and AsB: "A \<subseteq> span B"
huffman@44133
  1078
  shows "A = B"
huffman@44133
  1079
proof
wenzelm@49663
  1080
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1081
huffman@44133
  1082
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1083
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1084
wenzelm@53406
  1085
  {
wenzelm@53406
  1086
    fix x
wenzelm@53406
  1087
    assume x: "x \<in> A"
huffman@44133
  1088
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1089
      unfolding dependent_def using x by blast
wenzelm@53406
  1090
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1091
      by (blast intro: span_superset)
huffman@44133
  1092
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1093
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1094
      by (metis span_mono)
wenzelm@53406
  1095
    {
wenzelm@53406
  1096
      assume xB: "x \<notin> B"
wenzelm@53406
  1097
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1098
        by blast
wenzelm@53406
  1099
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1100
        by (metis span_mono)
wenzelm@53406
  1101
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1102
        by blast
wenzelm@53406
  1103
      with x have False
wenzelm@53406
  1104
        by (metis span_superset)
wenzelm@53406
  1105
    }
wenzelm@53406
  1106
    then have "x \<in> B" by blast
wenzelm@53406
  1107
  }
huffman@44133
  1108
  then show "A \<subseteq> B" by blast
huffman@44133
  1109
qed
huffman@44133
  1110
hoelzl@63053
  1111
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
hoelzl@63053
  1112
hoelzl@63053
  1113
lemma spanning_surjective_image:
hoelzl@63053
  1114
  assumes us: "UNIV \<subseteq> span S"
hoelzl@63053
  1115
    and lf: "linear f"
hoelzl@63053
  1116
    and sf: "surj f"
hoelzl@63053
  1117
  shows "UNIV \<subseteq> span (f ` S)"
hoelzl@63053
  1118
proof -
hoelzl@63053
  1119
  have "UNIV \<subseteq> f ` UNIV"
hoelzl@63053
  1120
    using sf by (auto simp add: surj_def)
hoelzl@63053
  1121
  also have " \<dots> \<subseteq> span (f ` S)"
hoelzl@63053
  1122
    using spans_image[OF lf us] .
hoelzl@63053
  1123
  finally show ?thesis .
hoelzl@63053
  1124
qed
hoelzl@63053
  1125
hoelzl@63053
  1126
lemma independent_inj_on_image:
hoelzl@63053
  1127
  assumes iS: "independent S"
hoelzl@63053
  1128
    and lf: "linear f"
hoelzl@63053
  1129
    and fi: "inj_on f (span S)"
hoelzl@63053
  1130
  shows "independent (f ` S)"
hoelzl@63053
  1131
proof -
hoelzl@63053
  1132
  {
hoelzl@63053
  1133
    fix a
hoelzl@63053
  1134
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
hoelzl@63053
  1135
    have eq: "f ` S - {f a} = f ` (S - {a})"
hoelzl@63053
  1136
      using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset)
hoelzl@63053
  1137
    from a have "f a \<in> f ` span (S - {a})"
hoelzl@63053
  1138
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
hoelzl@63053
  1139
    then have "a \<in> span (S - {a})"
hoelzl@63053
  1140
      by (rule inj_on_image_mem_iff_alt[OF fi, rotated])
hoelzl@63053
  1141
         (insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>)
hoelzl@63053
  1142
    with a(1) iS have False
hoelzl@63053
  1143
      by (simp add: dependent_def)
hoelzl@63053
  1144
  }
hoelzl@63053
  1145
  then show ?thesis
hoelzl@63053
  1146
    unfolding dependent_def by blast
hoelzl@63053
  1147
qed
hoelzl@63053
  1148
hoelzl@63053
  1149
lemma independent_injective_image:
hoelzl@63053
  1150
  "independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)"
hoelzl@63053
  1151
  using independent_inj_on_image[of S f] by (auto simp: subset_inj_on)
hoelzl@63053
  1152
hoelzl@63053
  1153
text \<open>Detailed theorems about left and right invertibility in general case.\<close>
hoelzl@63053
  1154
hoelzl@63053
  1155
lemma linear_inj_on_left_inverse:
hoelzl@63053
  1156
  assumes lf: "linear f" and fi: "inj_on f (span S)"
hoelzl@63053
  1157
  shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)"
hoelzl@63053
  1158
proof -
hoelzl@63053
  1159
  obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B"
hoelzl@63053
  1160
    using maximal_independent_subset[of S] by auto
hoelzl@63053
  1161
  then have "span S = span B"
hoelzl@63053
  1162
    unfolding span_eq by (auto simp: span_superset)
hoelzl@63053
  1163
  with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi
hoelzl@63053
  1164
  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
  1165
    by fastforce
hoelzl@63053
  1166
  have fB: "inj_on f B"
hoelzl@63053
  1167
    using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
hoelzl@63053
  1168
lp15@68058
  1169
  have "g (f x) = x" if "x \<in> span B" for x
lp15@68058
  1170
  proof (rule linear_eq_span)
hoelzl@63053
  1171
    show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
hoelzl@63053
  1172
      using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
lp15@68058
  1173
    show "g (f x) = x" if "x \<in> B" for x
lp15@68058
  1174
      using g fi \<open>span S = span B\<close>   by (simp add: fB that)
lp15@68058
  1175
  qed (rule that)
hoelzl@63053
  1176
  moreover
hoelzl@63053
  1177
  have "inv_into B f ` f ` B \<subseteq> B"
hoelzl@63053
  1178
    by (auto simp: fB)
hoelzl@63053
  1179
  then have "range g \<subseteq> span S"
hoelzl@63053
  1180
    unfolding g \<open>span S = span B\<close> by (intro span_mono)
hoelzl@63053
  1181
  ultimately show ?thesis
hoelzl@63053
  1182
    using \<open>span S = span B\<close> \<open>linear g\<close> by auto
hoelzl@63053
  1183
qed
hoelzl@63053
  1184
hoelzl@63053
  1185
lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id"
hoelzl@63053
  1186
  using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV)
hoelzl@63053
  1187
hoelzl@63053
  1188
lemma linear_surj_right_inverse:
hoelzl@63053
  1189
  assumes lf: "linear f" and sf: "span T \<subseteq> f`span S"
hoelzl@63053
  1190
  shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)"
hoelzl@63053
  1191
proof -
hoelzl@63053
  1192
  obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B"
hoelzl@63053
  1193
    using maximal_independent_subset[of T] by auto
hoelzl@63053
  1194
  then have "span T = span B"
hoelzl@63053
  1195
    unfolding span_eq by (auto simp: span_superset)
hoelzl@63053
  1196
hoelzl@63053
  1197
  from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"]
hoelzl@63053
  1198
  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
  1199
    by auto
hoelzl@63053
  1200
  moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x
hoelzl@63053
  1201
    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
  1202
  ultimately have "\<forall>x\<in>B. f (g x) = x"
hoelzl@63053
  1203
    by auto
hoelzl@63053
  1204
  then have "\<forall>x\<in>span B. f (g x) = x"
lp15@68058
  1205
    using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>] linear_eq_span by fastforce
hoelzl@63053
  1206
  moreover have "inv_into (span S) f ` B \<subseteq> span S"
hoelzl@63053
  1207
    using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
hoelzl@63053
  1208
  then have "range g \<subseteq> span S"
hoelzl@63053
  1209
    unfolding g by (intro span_minimal subspace_span) auto
hoelzl@63053
  1210
  ultimately show ?thesis
hoelzl@63053
  1211
    using \<open>linear g\<close> \<open>span T = span B\<close> by auto
hoelzl@63053
  1212
qed
hoelzl@63053
  1213
hoelzl@63053
  1214
lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id"
hoelzl@63053
  1215
  using linear_surj_right_inverse[of f UNIV UNIV]
hoelzl@63053
  1216
  by (auto simp: span_UNIV fun_eq_iff)
hoelzl@63053
  1217
wenzelm@60420
  1218
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
huffman@44133
  1219
huffman@44133
  1220
lemma exchange_lemma:
wenzelm@49711
  1221
  assumes f:"finite t"
wenzelm@49711
  1222
    and i: "independent s"
wenzelm@49711
  1223
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1224
  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
  1225
  using f i sp
wenzelm@49522
  1226
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1227
  case less
wenzelm@60420
  1228
  note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
wenzelm@53406
  1229
  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
  1230
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1231
  {
lp15@55775
  1232
    assume "s \<subseteq> t"
lp15@55775
  1233
    then have ?ths
lp15@55775
  1234
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1235
  }
huffman@44133
  1236
  moreover
wenzelm@53406
  1237
  {
wenzelm@53406
  1238
    assume st: "t \<subseteq> s"
wenzelm@53406
  1239
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1240
    have ?ths
lp15@55775
  1241
      by (metis Un_absorb sp)
wenzelm@53406
  1242
  }
huffman@44133
  1243
  moreover
wenzelm@53406
  1244
  {
wenzelm@53406
  1245
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1246
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1247
      by blast
wenzelm@53406
  1248
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1249
      by blast
wenzelm@53406
  1250
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1251
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1252
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1253
      by auto
wenzelm@53406
  1254
    have ?ths
wenzelm@53406
  1255
    proof cases
wenzelm@53716
  1256
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1257
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1258
        by auto
huffman@44133
  1259
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1260
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1261
        and fu: "finite u" by blast
huffman@44133
  1262
      let ?w = "insert b u"
wenzelm@53406
  1263
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1264
        using u by blast
wenzelm@53406
  1265
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1266
        by blast
wenzelm@53406
  1267
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1268
        using u b by blast
wenzelm@53406
  1269
      have bu: "b \<notin> u"
wenzelm@53406
  1270
        using b u by blast
wenzelm@53406
  1271
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1272
        by auto
wenzelm@49522
  1273
      then have th2: "card (insert b u) = card t"
huffman@44133
  1274
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1275
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1276
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1277
        by (rule span_mono) blast
huffman@44133
  1278
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1279
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1280
        by blast
wenzelm@53406
  1281
      from th show ?thesis by blast
wenzelm@53406
  1282
    next
wenzelm@53716
  1283
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1284
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1285
        by blast
wenzelm@53406
  1286
      have ab: "a \<noteq> b"
wenzelm@53406
  1287
        using a b by blast
wenzelm@53406
  1288
      have at: "a \<notin> t"
wenzelm@53406
  1289
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1290
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1291
        using cardlt ft a b by auto
wenzelm@53406
  1292
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1293
        using ft by auto
wenzelm@53406
  1294
      {
wenzelm@53406
  1295
        fix x
wenzelm@53406
  1296
        assume xs: "x \<in> s"
wenzelm@53406
  1297
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1298
          using b by auto
wenzelm@53406
  1299
        from b(1) have "b \<in> span t"
wenzelm@53406
  1300
          by (simp add: span_superset)
wenzelm@53406
  1301
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1302
          apply (rule in_span_delete)
wenzelm@53406
  1303
          using a sp unfolding subset_eq
wenzelm@53406
  1304
          apply auto
wenzelm@53406
  1305
          done
wenzelm@53406
  1306
        from xs sp have "x \<in> span t"
wenzelm@53406
  1307
          by blast
wenzelm@53406
  1308
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1309
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1310
      }
wenzelm@53406
  1311
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1312
        by blast
wenzelm@53406
  1313
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1314
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1315
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1316
        "s \<subseteq> span u" by blast
wenzelm@53406
  1317
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1318
        by auto
wenzelm@53406
  1319
      then show ?thesis by blast
wenzelm@53406
  1320
    qed
huffman@44133
  1321
  }
wenzelm@49522
  1322
  ultimately show ?ths by blast
huffman@44133
  1323
qed
huffman@44133
  1324
wenzelm@60420
  1325
text \<open>This implies corresponding size bounds.\<close>
huffman@44133
  1326
huffman@44133
  1327
lemma independent_span_bound:
wenzelm@53406
  1328
  assumes f: "finite t"
wenzelm@53406
  1329
    and i: "independent s"
wenzelm@53406
  1330
    and sp: "s \<subseteq> span t"
huffman@44133
  1331
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1332
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1333
huffman@44133
  1334
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1335
proof -
wenzelm@53406
  1336
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1337
    by auto
huffman@44133
  1338
  show ?thesis unfolding eq
huffman@44133
  1339
    apply (rule finite_imageI)
huffman@44133
  1340
    apply (rule finite)
huffman@44133
  1341
    done
huffman@44133
  1342
qed
huffman@44133
  1343
wenzelm@53406
  1344
immler@67962
  1345
subsection%unimportant \<open>More interesting properties of the norm.\<close>
hoelzl@63050
  1346
hoelzl@63050
  1347
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
hoelzl@63050
  1348
  by auto
hoelzl@63050
  1349
hoelzl@63050
  1350
notation inner (infix "\<bullet>" 70)
hoelzl@63050
  1351
hoelzl@63050
  1352
lemma square_bound_lemma:
hoelzl@63050
  1353
  fixes x :: real
hoelzl@63050
  1354
  shows "x < (1 + x) * (1 + x)"
hoelzl@63050
  1355
proof -
hoelzl@63050
  1356
  have "(x + 1/2)\<^sup>2 + 3/4 > 0"
hoelzl@63050
  1357
    using zero_le_power2[of "x+1/2"] by arith
hoelzl@63050
  1358
  then show ?thesis
hoelzl@63050
  1359
    by (simp add: field_simps power2_eq_square)
hoelzl@63050
  1360
qed
hoelzl@63050
  1361
hoelzl@63050
  1362
lemma square_continuous:
hoelzl@63050
  1363
  fixes e :: real
hoelzl@63050
  1364
  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
  1365
  using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
hoelzl@63050
  1366
  by (force simp add: power2_eq_square)
hoelzl@63050
  1367
hoelzl@63050
  1368
hoelzl@63050
  1369
lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
hoelzl@63050
  1370
  by simp (* TODO: delete *)
hoelzl@63050
  1371
hoelzl@63050
  1372
lemma norm_triangle_sub:
hoelzl@63050
  1373
  fixes x y :: "'a::real_normed_vector"
hoelzl@63050
  1374
  shows "norm x \<le> norm y + norm (x - y)"
hoelzl@63050
  1375
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
hoelzl@63050
  1376
hoelzl@63050
  1377
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
hoelzl@63050
  1378
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1379
hoelzl@63050
  1380
lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
hoelzl@63050
  1381
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1382
hoelzl@63050
  1383
lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
hoelzl@63050
  1384
  apply (subst order_eq_iff)
hoelzl@63050
  1385
  apply (auto simp: norm_le)
hoelzl@63050
  1386
  done
hoelzl@63050
  1387
hoelzl@63050
  1388
lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
hoelzl@63050
  1389
  by (simp add: norm_eq_sqrt_inner)
hoelzl@63050
  1390
hoelzl@63050
  1391
nipkow@67399
  1392
text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
hoelzl@63050
  1393
hoelzl@63050
  1394
lemma linear_componentwise:
hoelzl@63050
  1395
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
hoelzl@63050
  1396
  assumes lf: "linear f"
hoelzl@63050
  1397
  shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
hoelzl@63050
  1398
proof -
hoelzl@63050
  1399
  have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
nipkow@64267
  1400
    by (simp add: inner_sum_left)
hoelzl@63050
  1401
  then show ?thesis
nipkow@64267
  1402
    unfolding linear_sum_mul[OF lf, symmetric]
hoelzl@63050
  1403
    unfolding euclidean_representation ..
hoelzl@63050
  1404
qed
hoelzl@63050
  1405
hoelzl@63050
  1406
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
hoelzl@63050
  1407
  (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@63050
  1408
proof
hoelzl@63050
  1409
  assume ?lhs
hoelzl@63050
  1410
  then show ?rhs by simp
hoelzl@63050
  1411
next
hoelzl@63050
  1412
  assume ?rhs
hoelzl@63050
  1413
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
hoelzl@63050
  1414
    by simp
hoelzl@63050
  1415
  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
hoelzl@63050
  1416
    by (simp add: inner_diff inner_commute)
hoelzl@63050
  1417
  then have "(x - y) \<bullet> (x - y) = 0"
hoelzl@63050
  1418
    by (simp add: field_simps inner_diff inner_commute)
hoelzl@63050
  1419
  then show "x = y" by simp
hoelzl@63050
  1420
qed
hoelzl@63050
  1421
hoelzl@63050
  1422
lemma norm_triangle_half_r:
hoelzl@63050
  1423
  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
hoelzl@63050
  1424
  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
hoelzl@63050
  1425
hoelzl@63050
  1426
lemma norm_triangle_half_l:
hoelzl@63050
  1427
  assumes "norm (x - y) < e / 2"
hoelzl@63050
  1428
    and "norm (x' - y) < e / 2"
hoelzl@63050
  1429
  shows "norm (x - x') < e"
hoelzl@63050
  1430
  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
hoelzl@63050
  1431
  unfolding dist_norm[symmetric] .
hoelzl@63050
  1432
hoelzl@63050
  1433
lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
hoelzl@63050
  1434
  by (rule norm_triangle_ineq [THEN order_trans])
hoelzl@63050
  1435
hoelzl@63050
  1436
lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
hoelzl@63050
  1437
  by (rule norm_triangle_ineq [THEN le_less_trans])
hoelzl@63050
  1438
lp15@66420
  1439
lemma abs_triangle_half_r:
lp15@66420
  1440
  fixes y :: "'a::linordered_field"
lp15@66420
  1441
  shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
lp15@66420
  1442
  by linarith
lp15@66420
  1443
lp15@66420
  1444
lemma abs_triangle_half_l:
lp15@66420
  1445
  fixes y :: "'a::linordered_field"
lp15@66420
  1446
  assumes "abs (x - y) < e / 2"
lp15@66420
  1447
    and "abs (x' - y) < e / 2"
lp15@66420
  1448
  shows "abs (x - x') < e"
lp15@66420
  1449
  using assms by linarith
lp15@66420
  1450
nipkow@64267
  1451
lemma sum_clauses:
nipkow@64267
  1452
  shows "sum f {} = 0"
nipkow@64267
  1453
    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
  1454
  by (auto simp add: insert_absorb)
hoelzl@63050
  1455
nipkow@64267
  1456
lemma sum_norm_bound:
hoelzl@63050
  1457
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
lp15@64773
  1458
  assumes K: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K"
lp15@64773
  1459
  shows "norm (sum f S) \<le> of_nat (card S)*K"
nipkow@64267
  1460
  using sum_norm_le[OF K] sum_constant[symmetric]
hoelzl@63050
  1461
  by simp
hoelzl@63050
  1462
nipkow@64267
  1463
lemma sum_group:
hoelzl@63050
  1464
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
nipkow@64267
  1465
  shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
nipkow@64267
  1466
  apply (subst sum_image_gen[OF fS, of g f])
nipkow@64267
  1467
  apply (rule sum.mono_neutral_right[OF fT fST])
nipkow@64267
  1468
  apply (auto intro: sum.neutral)
hoelzl@63050
  1469
  done
hoelzl@63050
  1470
hoelzl@63050
  1471
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
hoelzl@63050
  1472
proof
hoelzl@63050
  1473
  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
hoelzl@63050
  1474
  then have "\<forall>x. x \<bullet> (y - z) = 0"
hoelzl@63050
  1475
    by (simp add: inner_diff)
hoelzl@63050
  1476
  then have "(y - z) \<bullet> (y - z) = 0" ..
hoelzl@63050
  1477
  then show "y = z" by simp
hoelzl@63050
  1478
qed simp
hoelzl@63050
  1479
hoelzl@63050
  1480
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
hoelzl@63050
  1481
proof
hoelzl@63050
  1482
  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
hoelzl@63050
  1483
  then have "\<forall>z. (x - y) \<bullet> z = 0"
hoelzl@63050
  1484
    by (simp add: inner_diff)
hoelzl@63050
  1485
  then have "(x - y) \<bullet> (x - y) = 0" ..
hoelzl@63050
  1486
  then show "x = y" by simp
hoelzl@63050
  1487
qed simp
hoelzl@63050
  1488
hoelzl@63050
  1489
hoelzl@63050
  1490
subsection \<open>Orthogonality.\<close>
hoelzl@63050
  1491
immler@67962
  1492
definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
immler@67962
  1493
hoelzl@63050
  1494
context real_inner
hoelzl@63050
  1495
begin
hoelzl@63050
  1496
lp15@63072
  1497
lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
lp15@63072
  1498
  by (simp add: orthogonal_def)
lp15@63072
  1499
hoelzl@63050
  1500
lemma orthogonal_clauses:
hoelzl@63050
  1501
  "orthogonal a 0"
hoelzl@63050
  1502
  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
hoelzl@63050
  1503
  "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
hoelzl@63050
  1504
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
hoelzl@63050
  1505
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
hoelzl@63050
  1506
  "orthogonal 0 a"
hoelzl@63050
  1507
  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
hoelzl@63050
  1508
  "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
hoelzl@63050
  1509
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
hoelzl@63050
  1510
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
hoelzl@63050
  1511
  unfolding orthogonal_def inner_add inner_diff by auto
hoelzl@63050
  1512
hoelzl@63050
  1513
end
hoelzl@63050
  1514
hoelzl@63050
  1515
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
hoelzl@63050
  1516
  by (simp add: orthogonal_def inner_commute)
hoelzl@63050
  1517
lp15@63114
  1518
lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
lp15@63114
  1519
  by (rule ext) (simp add: orthogonal_def)
lp15@63114
  1520
lp15@63114
  1521
lemma pairwise_ortho_scaleR:
lp15@63114
  1522
    "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
lp15@63114
  1523
    \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
lp15@63114
  1524
  by (auto simp: pairwise_def orthogonal_clauses)
lp15@63114
  1525
lp15@63114
  1526
lemma orthogonal_rvsum:
nipkow@64267
  1527
    "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
lp15@63114
  1528
  by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
lp15@63114
  1529
lp15@63114
  1530
lemma orthogonal_lvsum:
nipkow@64267
  1531
    "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
lp15@63114
  1532
  by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
lp15@63114
  1533
lp15@63114
  1534
lemma norm_add_Pythagorean:
lp15@63114
  1535
  assumes "orthogonal a b"
lp15@63114
  1536
    shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
lp15@63114
  1537
proof -
lp15@63114
  1538
  from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
lp15@63114
  1539
    by (simp add: algebra_simps orthogonal_def inner_commute)
lp15@63114
  1540
  then show ?thesis
lp15@63114
  1541
    by (simp add: power2_norm_eq_inner)
lp15@63114
  1542
qed
lp15@63114
  1543
nipkow@64267
  1544
lemma norm_sum_Pythagorean:
lp15@63114
  1545
  assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
nipkow@64267
  1546
    shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
lp15@63114
  1547
using assms
lp15@63114
  1548
proof (induction I rule: finite_induct)
lp15@63114
  1549
  case empty then show ?case by simp
lp15@63114
  1550
next
lp15@63114
  1551
  case (insert x I)
nipkow@64267
  1552
  then have "orthogonal (f x) (sum f I)"
lp15@63114
  1553
    by (metis pairwise_insert orthogonal_rvsum)
lp15@63114
  1554
  with insert show ?case
lp15@63114
  1555
    by (simp add: pairwise_insert norm_add_Pythagorean)
lp15@63114
  1556
qed
lp15@63114
  1557
hoelzl@63050
  1558
hoelzl@63050
  1559
subsection \<open>Bilinear functions.\<close>
hoelzl@63050
  1560
immler@67962
  1561
definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
hoelzl@63050
  1562
hoelzl@63050
  1563
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
hoelzl@63050
  1564
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1565
hoelzl@63050
  1566
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
hoelzl@63050
  1567
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1568
hoelzl@63050
  1569
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
hoelzl@63050
  1570
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1571
hoelzl@63050
  1572
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
hoelzl@63050
  1573
  by (simp add: bilinear_def linear_iff)
hoelzl@63050
  1574
hoelzl@63050
  1575
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
hoelzl@63050
  1576
  by (drule bilinear_lmul [of _ "- 1"]) simp
hoelzl@63050
  1577
hoelzl@63050
  1578
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
hoelzl@63050
  1579
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
hoelzl@63050
  1580
hoelzl@63050
  1581
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
hoelzl@63050
  1582
  using add_left_imp_eq[of x y 0] by auto
hoelzl@63050
  1583
hoelzl@63050
  1584
lemma bilinear_lzero:
hoelzl@63050
  1585
  assumes "bilinear h"
hoelzl@63050
  1586
  shows "h 0 x = 0"
hoelzl@63050
  1587
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
hoelzl@63050
  1588
hoelzl@63050
  1589
lemma bilinear_rzero:
hoelzl@63050
  1590
  assumes "bilinear h"
hoelzl@63050
  1591
  shows "h x 0 = 0"
hoelzl@63050
  1592
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
hoelzl@63050
  1593
hoelzl@63050
  1594
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
hoelzl@63050
  1595
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
hoelzl@63050
  1596
hoelzl@63050
  1597
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
hoelzl@63050
  1598
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
hoelzl@63050
  1599
nipkow@64267
  1600
lemma bilinear_sum:
hoelzl@63050
  1601
  assumes bh: "bilinear h"
hoelzl@63050
  1602
    and fS: "finite S"
hoelzl@63050
  1603
    and fT: "finite T"
nipkow@64267
  1604
  shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
hoelzl@63050
  1605
proof -
nipkow@64267
  1606
  have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
nipkow@64267
  1607
    apply (rule linear_sum[unfolded o_def])
hoelzl@63050
  1608
    using bh fS
hoelzl@63050
  1609
    apply (auto simp add: bilinear_def)
hoelzl@63050
  1610
    done
nipkow@64267
  1611
  also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
nipkow@64267
  1612
    apply (rule sum.cong, simp)
nipkow@64267
  1613
    apply (rule linear_sum[unfolded o_def])
hoelzl@63050
  1614
    using bh fT
hoelzl@63050
  1615
    apply (auto simp add: bilinear_def)
hoelzl@63050
  1616
    done
hoelzl@63050
  1617
  finally show ?thesis
nipkow@64267
  1618
    unfolding sum.cartesian_product .
hoelzl@63050
  1619
qed
hoelzl@63050
  1620
hoelzl@63050
  1621
hoelzl@63050
  1622
subsection \<open>Adjoints.\<close>
hoelzl@63050
  1623
immler@67962
  1624
definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
hoelzl@63050
  1625
hoelzl@63050
  1626
lemma adjoint_unique:
hoelzl@63050
  1627
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
hoelzl@63050
  1628
  shows "adjoint f = g"
hoelzl@63050
  1629
  unfolding adjoint_def
hoelzl@63050
  1630
proof (rule some_equality)
hoelzl@63050
  1631
  show "\<forall>x y. inner (f x) y = inner x (g y)"
hoelzl@63050
  1632
    by (rule assms)
hoelzl@63050
  1633
next
hoelzl@63050
  1634
  fix h
hoelzl@63050
  1635
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
hoelzl@63050
  1636
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
hoelzl@63050
  1637
    using assms by simp
hoelzl@63050
  1638
  then have "\<forall>x y. inner x (g y - h y) = 0"
hoelzl@63050
  1639
    by (simp add: inner_diff_right)
hoelzl@63050
  1640
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
hoelzl@63050
  1641
    by simp
hoelzl@63050
  1642
  then have "\<forall>y. h y = g y"
hoelzl@63050
  1643
    by simp
hoelzl@63050
  1644
  then show "h = g" by (simp add: ext)
hoelzl@63050
  1645
qed
hoelzl@63050
  1646
hoelzl@63050
  1647
text \<open>TODO: The following lemmas about adjoints should hold for any
wenzelm@63680
  1648
  Hilbert space (i.e. complete inner product space).
wenzelm@63680
  1649
  (see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
hoelzl@63050
  1650
\<close>
hoelzl@63050
  1651
hoelzl@63050
  1652
lemma adjoint_works:
hoelzl@63050
  1653
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1654
  assumes lf: "linear f"
hoelzl@63050
  1655
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@63050
  1656
proof -
hoelzl@63050
  1657
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
hoelzl@63050
  1658
  proof (intro allI exI)
hoelzl@63050
  1659
    fix y :: "'m" and x
hoelzl@63050
  1660
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
hoelzl@63050
  1661
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
hoelzl@63050
  1662
      by (simp add: euclidean_representation)
hoelzl@63050
  1663
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
nipkow@64267
  1664
      unfolding linear_sum[OF lf]
hoelzl@63050
  1665
      by (simp add: linear_cmul[OF lf])
hoelzl@63050
  1666
    finally show "f x \<bullet> y = x \<bullet> ?w"
nipkow@64267
  1667
      by (simp add: inner_sum_left inner_sum_right mult.commute)
hoelzl@63050
  1668
  qed
hoelzl@63050
  1669
  then show ?thesis
hoelzl@63050
  1670
    unfolding adjoint_def choice_iff
hoelzl@63050
  1671
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
hoelzl@63050
  1672
qed
hoelzl@63050
  1673
hoelzl@63050
  1674
lemma adjoint_clauses:
hoelzl@63050
  1675
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1676
  assumes lf: "linear f"
hoelzl@63050
  1677
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@63050
  1678
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@63050
  1679
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@63050
  1680
hoelzl@63050
  1681
lemma adjoint_linear:
hoelzl@63050
  1682
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1683
  assumes lf: "linear f"
hoelzl@63050
  1684
  shows "linear (adjoint f)"
hoelzl@63050
  1685
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
hoelzl@63050
  1686
    adjoint_clauses[OF lf] inner_distrib)
hoelzl@63050
  1687
hoelzl@63050
  1688
lemma adjoint_adjoint:
hoelzl@63050
  1689
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@63050
  1690
  assumes lf: "linear f"
hoelzl@63050
  1691
  shows "adjoint (adjoint f) = f"
hoelzl@63050
  1692
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@63050
  1693
hoelzl@63050
  1694
immler@67962
  1695
subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
hoelzl@63050
  1696
hoelzl@63050
  1697
lemma seq_mono_lemma:
hoelzl@63050
  1698
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
hoelzl@63050
  1699
    and "\<forall>n \<ge> m. e n \<le> e m"
hoelzl@63050
  1700
  shows "\<forall>n \<ge> m. d n < e m"
hoelzl@63050
  1701
  using assms
hoelzl@63050
  1702
  apply auto
hoelzl@63050
  1703
  apply (erule_tac x="n" in allE)
hoelzl@63050
  1704
  apply (erule_tac x="n" in allE)
hoelzl@63050
  1705
  apply auto
hoelzl@63050
  1706
  done
hoelzl@63050
  1707
hoelzl@63050
  1708
lemma infinite_enumerate:
hoelzl@63050
  1709
  assumes fS: "infinite S"
eberlm@66447
  1710
  shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
eberlm@66447
  1711
  unfolding strict_mono_def
hoelzl@63050
  1712
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
hoelzl@63050
  1713
hoelzl@63050
  1714
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
  1715
  apply auto
hoelzl@63050
  1716
  apply (rule_tac x="d/2" in exI)
hoelzl@63050
  1717
  apply auto
hoelzl@63050
  1718
  done
hoelzl@63050
  1719
wenzelm@67443
  1720
lemma approachable_lt_le2:  \<comment> \<open>like the above, but pushes aside an extra formula\<close>
hoelzl@63050
  1721
    "(\<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
  1722
  apply auto
hoelzl@63050
  1723
  apply (rule_tac x="d/2" in exI, auto)
hoelzl@63050
  1724
  done
hoelzl@63050
  1725
hoelzl@63050
  1726
lemma triangle_lemma:
hoelzl@63050
  1727
  fixes x y z :: real
hoelzl@63050
  1728
  assumes x: "0 \<le> x"
hoelzl@63050
  1729
    and y: "0 \<le> y"
hoelzl@63050
  1730
    and z: "0 \<le> z"
hoelzl@63050
  1731
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
hoelzl@63050
  1732
  shows "x \<le> y + z"
hoelzl@63050
  1733
proof -
hoelzl@63050
  1734
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
hoelzl@63050
  1735
    using z y by simp
hoelzl@63050
  1736
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
hoelzl@63050
  1737
    by (simp add: power2_eq_square field_simps)
hoelzl@63050
  1738
  from y z have yz: "y + z \<ge> 0"
hoelzl@63050
  1739
    by arith
hoelzl@63050
  1740
  from power2_le_imp_le[OF th yz] show ?thesis .
hoelzl@63050
  1741
qed
hoelzl@63050
  1742
hoelzl@63050
  1743
hoelzl@63050
  1744
hoelzl@63050
  1745
subsection \<open>Archimedean properties and useful consequences\<close>
hoelzl@63050
  1746
hoelzl@63050
  1747
text\<open>Bernoulli's inequality\<close>
immler@67962
  1748
proposition%important Bernoulli_inequality:
hoelzl@63050
  1749
  fixes x :: real
hoelzl@63050
  1750
  assumes "-1 \<le> x"
hoelzl@63050
  1751
    shows "1 + n * x \<le> (1 + x) ^ n"
immler@67962
  1752
proof%unimportant (induct n)
hoelzl@63050
  1753
  case 0
hoelzl@63050
  1754
  then show ?case by simp
hoelzl@63050
  1755
next
hoelzl@63050
  1756
  case (Suc n)
hoelzl@63050
  1757
  have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
hoelzl@63050
  1758
    by (simp add: algebra_simps)
hoelzl@63050
  1759
  also have "... = (1 + x) * (1 + n*x)"
hoelzl@63050
  1760
    by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
hoelzl@63050
  1761
  also have "... \<le> (1 + x) ^ Suc n"
hoelzl@63050
  1762
    using Suc.hyps assms mult_left_mono by fastforce
hoelzl@63050
  1763
  finally show ?case .
hoelzl@63050
  1764
qed
hoelzl@63050
  1765
hoelzl@63050
  1766
corollary Bernoulli_inequality_even:
hoelzl@63050
  1767
  fixes x :: real
hoelzl@63050
  1768
  assumes "even n"
hoelzl@63050
  1769
    shows "1 + n * x \<le> (1 + x) ^ n"
hoelzl@63050
  1770
proof (cases "-1 \<le> x \<or> n=0")
hoelzl@63050
  1771
  case True
hoelzl@63050
  1772
  then show ?thesis
hoelzl@63050
  1773
    by (auto simp: Bernoulli_inequality)
hoelzl@63050
  1774
next
hoelzl@63050
  1775
  case False
hoelzl@63050
  1776
  then have "real n \<ge> 1"
hoelzl@63050
  1777
    by simp
hoelzl@63050
  1778
  with False have "n * x \<le> -1"
hoelzl@63050
  1779
    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
  1780
  then have "1 + n * x \<le> 0"
hoelzl@63050
  1781
    by auto
hoelzl@63050
  1782
  also have "... \<le> (1 + x) ^ n"
hoelzl@63050
  1783
    using assms
hoelzl@63050
  1784
    using zero_le_even_power by blast
hoelzl@63050
  1785
  finally show ?thesis .
hoelzl@63050
  1786
qed
hoelzl@63050
  1787
hoelzl@63050
  1788
corollary real_arch_pow:
hoelzl@63050
  1789
  fixes x :: real
hoelzl@63050
  1790
  assumes x: "1 < x"
hoelzl@63050
  1791
  shows "\<exists>n. y < x^n"
hoelzl@63050
  1792
proof -
hoelzl@63050
  1793
  from x have x0: "x - 1 > 0"
hoelzl@63050
  1794
    by arith
hoelzl@63050
  1795
  from reals_Archimedean3[OF x0, rule_format, of y]
hoelzl@63050
  1796
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
hoelzl@63050
  1797
  from x0 have x00: "x- 1 \<ge> -1" by arith
hoelzl@63050
  1798
  from Bernoulli_inequality[OF x00, of n] n
hoelzl@63050
  1799
  have "y < x^n" by auto
hoelzl@63050
  1800
  then show ?thesis by metis
hoelzl@63050
  1801
qed
hoelzl@63050
  1802
hoelzl@63050
  1803
corollary real_arch_pow_inv:
hoelzl@63050
  1804
  fixes x y :: real
hoelzl@63050
  1805
  assumes y: "y > 0"
hoelzl@63050
  1806
    and x1: "x < 1"
hoelzl@63050
  1807
  shows "\<exists>n. x^n < y"
hoelzl@63050
  1808
proof (cases "x > 0")
hoelzl@63050
  1809
  case True
hoelzl@63050
  1810
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
hoelzl@63050
  1811
  from real_arch_pow[OF ix, of "1/y"]
hoelzl@63050
  1812
  obtain n where n: "1/y < (1/x)^n" by blast
hoelzl@63050
  1813
  then show ?thesis using y \<open>x > 0\<close>
hoelzl@63050
  1814
    by (auto simp add: field_simps)
hoelzl@63050
  1815
next
hoelzl@63050
  1816
  case False
hoelzl@63050
  1817
  with y x1 show ?thesis
hoelzl@63050
  1818
    apply auto
hoelzl@63050
  1819
    apply (rule exI[where x=1])
hoelzl@63050
  1820
    apply auto
hoelzl@63050
  1821
    done
hoelzl@63050
  1822
qed
hoelzl@63050
  1823
hoelzl@63050
  1824
lemma forall_pos_mono:
hoelzl@63050
  1825
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
hoelzl@63050
  1826
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
hoelzl@63050
  1827
  by (metis real_arch_inverse)
hoelzl@63050
  1828
hoelzl@63050
  1829
lemma forall_pos_mono_1:
hoelzl@63050
  1830
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
hoelzl@63050
  1831
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
hoelzl@63050
  1832
  apply (rule forall_pos_mono)
hoelzl@63050
  1833
  apply auto
hoelzl@63050
  1834
  apply (metis Suc_pred of_nat_Suc)
hoelzl@63050
  1835
  done
hoelzl@63050
  1836
hoelzl@63050
  1837
immler@67962
  1838
subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
huffman@44133
  1839
hoelzl@50526
  1840
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1841
  unfolding dependent_def
hoelzl@50526
  1842
  apply (subst span_finite)
hoelzl@50526
  1843
  apply simp
huffman@44133
  1844
  apply clarify
hoelzl@50526
  1845
  apply (drule_tac f="inner a" in arg_cong)
nipkow@64267
  1846
  apply (simp add: inner_Basis inner_sum_right eq_commute)
hoelzl@50526
  1847
  done
hoelzl@50526
  1848
huffman@53939
  1849
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1850
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1851
  by (fast intro: euclidean_representation)
huffman@44133
  1852
hoelzl@50526
  1853
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1854
  unfolding span_Basis ..
hoelzl@50526
  1855
hoelzl@50526
  1856
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1857
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1858
hoelzl@50526
  1859
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1860
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1861
hoelzl@50526
  1862
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1863
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1864
hoelzl@50526
  1865
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1866
  apply (subst euclidean_representation[of x, symmetric])
nipkow@64267
  1867
  apply (rule order_trans[OF norm_sum])
nipkow@64267
  1868
  apply (auto intro!: sum_mono)
wenzelm@49522
  1869
  done
huffman@44133
  1870
nipkow@64267
  1871
lemma sum_norm_allsubsets_bound:
wenzelm@56444
  1872
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1873
  assumes fP: "finite P"
nipkow@64267
  1874
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
hoelzl@50526
  1875
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1876
proof -
hoelzl@50526
  1877
  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
  1878
    by (rule sum_mono) (rule norm_le_l1)
hoelzl@50526
  1879
  also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
haftmann@66804
  1880
    by (rule sum.swap)
hoelzl@50526
  1881
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
nipkow@64267
  1882
  proof (rule sum_bounded_above)
wenzelm@53406
  1883
    fix i :: 'n
wenzelm@53406
  1884
    assume i: "i \<in> Basis"
wenzelm@53406
  1885
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1886
      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
  1887
      by (simp add: abs_real_def sum.If_cases[OF fP] sum_negf norm_triangle_ineq4 inner_sum_left
wenzelm@56444
  1888
        del: real_norm_def)
wenzelm@53406
  1889
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1890
      unfolding real_norm_def
hoelzl@50526
  1891
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1892
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1893
  qed
lp15@61609
  1894
  also have "\<dots> = 2 * real DIM('n) * e" by simp
huffman@44133
  1895
  finally show ?thesis .
huffman@44133
  1896
qed
huffman@44133
  1897
wenzelm@53406
  1898
immler@67962
  1899
subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
huffman@44133
  1900
huffman@44133
  1901
lemma linear_bounded:
wenzelm@56444
  1902
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1903
  assumes lf: "linear f"
huffman@44133
  1904
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1905
proof
hoelzl@50526
  1906
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1907
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1908
  proof
wenzelm@53406
  1909
    fix x :: 'a
hoelzl@50526
  1910
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1911
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1912
      unfolding euclidean_representation ..
nipkow@64267
  1913
    also have "\<dots> = norm (sum ?g Basis)"
nipkow@64267
  1914
      by (simp add: linear_sum [OF lf] linear_cmul [OF lf])
nipkow@64267
  1915
    finally have th0: "norm (f x) = norm (sum ?g Basis)" .
lp15@64773
  1916
    have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
lp15@64773
  1917
    proof -
lp15@64773
  1918
      from Basis_le_norm[OF that, of x]
huffman@53939
  1919
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1920
        unfolding norm_scaleR
haftmann@57512
  1921
        apply (subst mult.commute)
wenzelm@49663
  1922
        apply (rule mult_mono)
wenzelm@49663
  1923
        apply (auto simp add: field_simps)
wenzelm@53406
  1924
        done
huffman@53939
  1925
    qed
nipkow@64267
  1926
    from sum_norm_le[of _ ?g, OF th]
huffman@53939
  1927
    show "norm (f x) \<le> ?B * norm x"
nipkow@64267
  1928
      unfolding th0 sum_distrib_right by metis
huffman@53939
  1929
  qed
huffman@44133
  1930
qed
huffman@44133
  1931
huffman@44133
  1932
lemma linear_conv_bounded_linear:
huffman@44133
  1933
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1934
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1935
proof
huffman@44133
  1936
  assume "linear f"
huffman@53939
  1937
  then interpret f: linear f .
huffman@44133
  1938
  show "bounded_linear f"
huffman@44133
  1939
  proof
huffman@44133
  1940
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
wenzelm@60420
  1941
      using \<open>linear f\<close> by (rule linear_bounded)
wenzelm@49522
  1942
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
haftmann@57512
  1943
      by (simp add: mult.commute)
huffman@44133
  1944
  qed
huffman@44133
  1945
next
huffman@44133
  1946
  assume "bounded_linear f"
huffman@44133
  1947
  then interpret f: bounded_linear f .
huffman@53939
  1948
  show "linear f" ..
huffman@53939
  1949
qed
huffman@53939
  1950
paulson@61518
  1951
lemmas linear_linear = linear_conv_bounded_linear[symmetric]
paulson@61518
  1952
huffman@53939
  1953
lemma linear_bounded_pos:
wenzelm@56444
  1954
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1955
  assumes lf: "linear f"
lp15@67982
  1956
 obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
huffman@53939
  1957
proof -
huffman@53939
  1958
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1959
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1960
    by (rule bounded_linear.pos_bounded)
lp15@67982
  1961
  with that show ?thesis
lp15@67982
  1962
    by (auto simp: mult.commute)
huffman@44133
  1963
qed
huffman@44133
  1964
lp15@67982
  1965
lemma linear_invertible_bounded_below_pos:
lp15@67982
  1966
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
lp15@67982
  1967
  assumes "linear f" "linear g" "g \<circ> f = id"
lp15@67982
  1968
  obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
lp15@67982
  1969
proof -
lp15@67982
  1970
  obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
lp15@67982
  1971
    using linear_bounded_pos [OF \<open>linear g\<close>] by blast
lp15@67982
  1972
  show thesis
lp15@67982
  1973
  proof
lp15@67982
  1974
    show "0 < 1/B"
lp15@67982
  1975
      by (simp add: \<open>B > 0\<close>)
lp15@67982
  1976
    show "1/B * norm x \<le> norm (f x)" for x
lp15@67982
  1977
    proof -
lp15@67982
  1978
      have "1/B * norm x = 1/B * norm (g (f x))"
lp15@67982
  1979
        using assms by (simp add: pointfree_idE)
lp15@67982
  1980
      also have "\<dots> \<le> norm (f x)"
lp15@67982
  1981
        using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
lp15@67982
  1982
      finally show ?thesis .
lp15@67982
  1983
    qed
lp15@67982
  1984
  qed
lp15@67982
  1985
qed
lp15@67982
  1986
lp15@67982
  1987
lemma linear_inj_bounded_below_pos:
lp15@67982
  1988
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
lp15@67982
  1989
  assumes "linear f" "inj f"
lp15@67982
  1990
  obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
lp15@67982
  1991
  using linear_injective_left_inverse [OF assms] linear_invertible_bounded_below_pos assms by blast
lp15@67982
  1992
wenzelm@49522
  1993
lemma bounded_linearI':
wenzelm@56444
  1994
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1995
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1996
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1997
  shows "bounded_linear f"
wenzelm@53406
  1998
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1999
  by (rule linearI[OF assms])
huffman@44133
  2000
huffman@44133
  2001
lemma bilinear_bounded:
wenzelm@56444
  2002
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  2003
  assumes bh: "bilinear h"
huffman@44133
  2004
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  2005
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  2006
  fix x :: 'm
wenzelm@53406
  2007
  fix y :: 'n
nipkow@64267
  2008
  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
  2009
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  2010
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  2011
    apply rule
hoelzl@50526
  2012
    done
nipkow@64267
  2013
  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
  2014
    unfolding bilinear_sum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  2015
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  2016
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
nipkow@64267
  2017
    apply (auto simp add: sum_distrib_right th sum.cartesian_product)
nipkow@64267
  2018
    apply (rule sum_norm_le)
wenzelm@53406
  2019
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  2020
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  2021
    apply (rule mult_mono)
wenzelm@53406
  2022
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  2023
    apply (rule mult_mono)
wenzelm@53406
  2024
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  2025
    done
huffman@44133
  2026
qed
huffman@44133
  2027
huffman@44133
  2028
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  2029
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  2030
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  2031
proof
huffman@44133
  2032
  assume "bilinear h"
huffman@44133
  2033
  show "bounded_bilinear h"
huffman@44133
  2034
  proof
wenzelm@53406
  2035
    fix x y z
wenzelm@53406
  2036
    show "h (x + y) z = h x z + h y z"
wenzelm@60420
  2037
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  2038
  next
wenzelm@53406
  2039
    fix x y z
wenzelm@53406
  2040
    show "h x (y + z) = h x y + h x z"
wenzelm@60420
  2041
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  2042
  next
wenzelm@53406
  2043
    fix r x y
wenzelm@53406
  2044
    show "h (scaleR r x) y = scaleR r (h x y)"
wenzelm@60420
  2045
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  2046
      by simp
huffman@44133
  2047
  next
wenzelm@53406
  2048
    fix r x y
wenzelm@53406
  2049
    show "h x (scaleR r y) = scaleR r (h x y)"
wenzelm@60420
  2050
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  2051
      by simp
huffman@44133
  2052
  next
huffman@44133
  2053
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
wenzelm@60420
  2054
      using \<open>bilinear h\<close> by (rule bilinear_bounded)
wenzelm@49522
  2055
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
haftmann@57514
  2056
      by (simp add: ac_simps)
huffman@44133
  2057
  qed
huffman@44133
  2058
next
huffman@44133
  2059
  assume "bounded_bilinear h"
huffman@44133
  2060
  then interpret h: bounded_bilinear h .
huffman@44133
  2061
  show "bilinear h"
huffman@44133
  2062
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  2063
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  2064
qed
huffman@44133
  2065
huffman@53939
  2066
lemma bilinear_bounded_pos:
wenzelm@56444
  2067
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  2068
  assumes bh: "bilinear h"
huffman@53939
  2069
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  2070
proof -
huffman@53939
  2071
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  2072
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  2073
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  2074
  then show ?thesis
haftmann@57514
  2075
    by (simp only: ac_simps)
huffman@53939
  2076
qed
huffman@53939
  2077
lp15@63469
  2078
lemma bounded_linear_imp_has_derivative:
lp15@63469
  2079
     "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
lp15@63469
  2080
  by (simp add: has_derivative_def bounded_linear.linear linear_diff)
lp15@63469
  2081
lp15@63469
  2082
lemma linear_imp_has_derivative:
lp15@63469
  2083
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
lp15@63469
  2084
  shows "linear f \<Longrightarrow> (f has_derivative f) net"
lp15@63469
  2085
by (simp add: has_derivative_def linear_conv_bounded_linear linear_diff)
lp15@63469
  2086
lp15@63469
  2087
lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
lp15@63469
  2088
  using bounded_linear_imp_has_derivative differentiable_def by blast
lp15@63469
  2089
lp15@63469
  2090
lemma linear_imp_differentiable:
lp15@63469
  2091
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
lp15@63469
  2092
  shows "linear f \<Longrightarrow> f differentiable net"
lp15@63469
  2093
by (metis linear_imp_has_derivative differentiable_def)
lp15@63469
  2094
wenzelm@49522
  2095
immler@67962
  2096
subsection%unimportant \<open>We continue.\<close>
huffman@44133
  2097
huffman@44133
  2098
lemma independent_bound:
wenzelm@53716
  2099
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  2100
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  2101
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  2102
lp15@61609
  2103
corollary
paulson@60303
  2104
  fixes S :: "'a::euclidean_space set"
paulson@60303
  2105
  assumes "independent S"
paulson@60303
  2106
  shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
paulson@60303
  2107
using assms independent_bound by auto
lp15@61609
  2108
lp15@63075
  2109
lemma independent_explicit:
lp15@63075
  2110
  fixes B :: "'a::euclidean_space set"
lp15@63075
  2111
  shows "independent B \<longleftrightarrow>
lp15@63075
  2112
         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
  2113
apply (cases "finite B")
lp15@63075
  2114
 apply (force simp: dependent_finite)
lp15@63075
  2115
using independent_bound
lp15@63075
  2116
apply auto
lp15@63075
  2117
done
lp15@63075
  2118
wenzelm@49663
  2119
lemma dependent_biggerset:
wenzelm@56444
  2120
  fixes S :: "'a::euclidean_space set"
wenzelm@56444
  2121
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  2122
  by (metis independent_bound not_less)
huffman@44133
  2123
wenzelm@60420
  2124
text \<open>Notion of dimension.\<close>
huffman@44133
  2125
wenzelm@53406
  2126
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  2127
wenzelm@49522
  2128
lemma basis_exists:
wenzelm@49522
  2129
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  2130
  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
  2131
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  2132
  by auto
huffman@44133
  2133
lp15@60307
  2134
corollary dim_le_card:
lp15@60307
  2135
  fixes s :: "'a::euclidean_space set"
lp15@60307
  2136
  shows "finite s \<Longrightarrow> dim s \<le> card s"
lp15@60307
  2137
by (metis basis_exists card_mono)
lp15@60307
  2138
wenzelm@60420
  2139
text \<open>Consequences of independence or spanning for cardinality.\<close>
huffman@44133
  2140
wenzelm@53406
  2141
lemma independent_card_le_dim:
wenzelm@53406
  2142
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2143
  assumes "B \<subseteq> V"
wenzelm@53406
  2144
    and "independent B"
wenzelm@49522
  2145
  shows "card B \<le> dim V"
huffman@44133
  2146
proof -
wenzelm@60420
  2147
  from basis_exists[of V] \<open>B \<subseteq> V\<close>
wenzelm@53406
  2148
  obtain B' where "independent B'"
wenzelm@53406
  2149
    and "B \<subseteq> span B'"
wenzelm@53406
  2150
    and "card B' = dim V"
wenzelm@53406
  2151
    by blast
wenzelm@60420
  2152
  with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
huffman@44133
  2153
  show ?thesis by auto
huffman@44133
  2154
qed
huffman@44133
  2155
wenzelm@49522
  2156
lemma span_card_ge_dim:
wenzelm@53406
  2157
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2158
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  2159
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  2160
huffman@44133
  2161
lemma basis_card_eq_dim:
wenzelm@53406
  2162
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  2163
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  2164
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  2165
wenzelm@53406
  2166
lemma dim_unique:
wenzelm@53406
  2167
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2168
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  2169
  by (metis basis_card_eq_dim)
huffman@44133
  2170
wenzelm@60420
  2171
text \<open>More lemmas about dimension.\<close>
huffman@44133
  2172
lp15@64122
  2173
lemma dim_UNIV [simp]: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  2174
  using independent_Basis
hoelzl@50526
  2175
  by (intro dim_unique[of Basis]) auto
huffman@44133
  2176
huffman@44133
  2177
lemma dim_subset:
wenzelm@53406
  2178
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2179
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  2180
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  2181
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  2182
wenzelm@53406
  2183
lemma dim_subset_UNIV:
wenzelm@53406
  2184
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2185
  shows "dim S \<le> DIM('a)"
huffman@44133
  2186
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  2187
wenzelm@60420
  2188
text \<open>Converses to those.\<close>
huffman@44133
  2189
huffman@44133
  2190
lemma card_ge_dim_independent:
wenzelm@53406
  2191
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2192
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  2193
    and iB: "independent B"
wenzelm@53406
  2194
    and dVB: "dim V \<le> card B"
huffman@44133
  2195
  shows "V \<subseteq> span B"
wenzelm@53406
  2196
proof
wenzelm@53406
  2197
  fix a
wenzelm@53406
  2198
  assume aV: "a \<in> V"
wenzelm@53406
  2199
  {
wenzelm@53406
  2200
    assume aB: "a \<notin> span B"
wenzelm@53406
  2201
    then have iaB: "independent (insert a B)"
wenzelm@53406
  2202
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  2203
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  2204
      by blast
wenzelm@53406
  2205
    from aB have "a \<notin>B"
wenzelm@53406
  2206
      by (auto simp add: span_superset)
wenzelm@53406
  2207
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  2208
    have False by auto
wenzelm@53406
  2209
  }
wenzelm@53406
  2210
  then show "a \<in> span B" by blast
huffman@44133
  2211
qed
huffman@44133
  2212
huffman@44133
  2213
lemma card_le_dim_spanning:
wenzelm@49663
  2214
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  2215
    and VB: "V \<subseteq> span B"
wenzelm@49663
  2216
    and fB: "finite B"
wenzelm@49663
  2217
    and dVB: "dim V \<ge> card B"
huffman@44133
  2218
  shows "independent B"
wenzelm@49522
  2219
proof -
wenzelm@53406
  2220
  {
wenzelm@53406
  2221
    fix a
wenzelm@53716
  2222
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  2223
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  2224
      by auto
wenzelm@53716
  2225
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  2226
      by auto
wenzelm@53716
  2227
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  2228
      by blast
wenzelm@53406
  2229
    {
wenzelm@53406
  2230
      fix x
wenzelm@53406
  2231
      assume x: "x \<in> V"
wenzelm@53716
  2232
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  2233
        by blast
wenzelm@53406
  2234
      from x VB have x': "x \<in> span B"
wenzelm@53406
  2235
        by blast
huffman@44133
  2236
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  2237
      have "x \<in> span (B - {a})" .
wenzelm@53406
  2238
    }
wenzelm@53716
  2239
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  2240
      by blast
wenzelm@53716
  2241
    have th2: "finite (B - {a})"
wenzelm@53406
  2242
      using fB by auto
huffman@44133
  2243
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  2244
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  2245
    from c c0 dVB cb have False by simp
wenzelm@53406
  2246
  }
wenzelm@53406
  2247
  then show ?thesis
wenzelm@53406
  2248
    unfolding dependent_def by blast
huffman@44133
  2249
qed
huffman@44133
  2250
wenzelm@53406
  2251
lemma card_eq_dim:
wenzelm@53406
  2252
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2253
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  2254
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  2255
wenzelm@60420
  2256
text \<open>More general size bound lemmas.\<close>
huffman@44133
  2257
huffman@44133
  2258
lemma independent_bound_general:
wenzelm@53406
  2259
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2260
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  2261
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  2262
wenzelm@49522
  2263
lemma dependent_biggerset_general:
wenzelm@53406
  2264
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2265
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  2266
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  2267
paulson@60303
  2268
lemma dim_span [simp]:
wenzelm@53406
  2269
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2270
  shows "dim (span S) = dim S"
wenzelm@49522
  2271
proof -
huffman@44133
  2272
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  2273
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  2274
  from basis_exists[of S]
wenzelm@53406
  2275
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  2276
    by blast
wenzelm@53406
  2277
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2278
    using independent_bound by blast+
wenzelm@53406
  2279
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  2280
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  2281
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  2282
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  2283
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  2284
    using fB(2) by arith
huffman@44133
  2285
qed
huffman@44133
  2286
wenzelm@53406
  2287
lemma subset_le_dim:
wenzelm@53406
  2288
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2289
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  2290
  by (metis dim_span dim_subset)
huffman@44133
  2291
wenzelm@53406
  2292
lemma span_eq_dim:
wenzelm@56444
  2293
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2294
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  2295
  by (metis dim_span)
huffman@44133
  2296
huffman@44133
  2297
lemma dim_image_le:
huffman@44133
  2298
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  2299
  assumes lf: "linear f"
wenzelm@49663
  2300
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  2301
proof -
huffman@44133
  2302
  from basis_exists[of S] obtain B where
huffman@44133
  2303
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  2304
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2305
    using independent_bound by blast+
huffman@44133
  2306
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  2307
    apply (rule span_card_ge_dim)
wenzelm@53406
  2308
    using lf B fB
wenzelm@53406
  2309
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  2310
    done
wenzelm@53406
  2311
  also have "\<dots> \<le> dim S"
wenzelm@53406
  2312
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  2313
  finally show ?thesis .
huffman@44133
  2314
qed
huffman@44133
  2315
wenzelm@60420
  2316
text \<open>Picking an orthogonal replacement for a spanning set.\<close>
huffman@44133
  2317
wenzelm@53406
  2318
lemma vector_sub_project_orthogonal:
wenzelm@53406
  2319
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  2320
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  2321
  unfolding inner_simps by auto
huffman@44133
  2322
huffman@44528
  2323
lemma pairwise_orthogonal_insert:
huffman@44528
  2324
  assumes "pairwise orthogonal S"
wenzelm@49522
  2325
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  2326
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  2327
  using assms unfolding pairwise_def
huffman@44528
  2328
  by (auto simp add: orthogonal_commute)
huffman@44528
  2329
huffman@44133
  2330
lemma basis_orthogonal:
wenzelm@53406
  2331
  fixes B :: "'a::real_inner set"
huffman@44133
  2332
  assumes fB: "finite B"
huffman@44133
  2333
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  2334
  (is " \<exists>C. ?P B C")
wenzelm@49522
  2335
  using fB
wenzelm@49522
  2336
proof (induct rule: finite_induct)
wenzelm@49522
  2337
  case empty
wenzelm@53406
  2338
  then show ?case
wenzelm@53406
  2339
    apply (rule exI[where x="{}"])
wenzelm@53406
  2340
    apply (auto simp add: pairwise_def)
wenzelm@53406
  2341
    done
huffman@44133
  2342
next
wenzelm@49522
  2343
  case (insert a B)
wenzelm@60420
  2344
  note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
wenzelm@60420
  2345
  from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
huffman@44133
  2346
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  2347
    "span C = span B" "pairwise orthogonal C" by blast
nipkow@64267
  2348
  let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  2349
  let ?C = "insert ?a C"
wenzelm@53406
  2350
  from C(1) have fC: "finite ?C"
wenzelm@53406
  2351
    by simp
wenzelm@49522
  2352
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  2353
    by (simp add: card_insert_if)
wenzelm@53406
  2354
  {
wenzelm@53406
  2355
    fix x k
wenzelm@49522
  2356
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  2357
      by (simp add: field_simps)
huffman@44133
  2358
    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
  2359
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  2360
      apply (rule span_add_eq)
huffman@44133
  2361
      apply (rule span_mul)
nipkow@64267
  2362
      apply (rule span_sum)
huffman@44133
  2363
      apply (rule span_mul)
wenzelm@49522
  2364
      apply (rule span_superset)
wenzelm@49522
  2365
      apply assumption
wenzelm@53406
  2366
      done
wenzelm@53406
  2367
  }
huffman@44133
  2368
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  2369
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  2370
  {
wenzelm@53406
  2371
    fix y
wenzelm@53406
  2372
    assume yC: "y \<in> C"
wenzelm@53406
  2373
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  2374
      by blast
wenzelm@53406
  2375
    have fth: "finite (C - {y})"
wenzelm@53406
  2376
      using C by simp
huffman@44528
  2377
    have "orthogonal ?a y"
huffman@44528
  2378
      unfolding orthogonal_def
nipkow@64267
  2379
      unfolding inner_diff inner_sum_left right_minus_eq
nipkow@64267
  2380
      unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
huffman@44528
  2381
      apply (clarsimp simp add: inner_commute[of y a])
nipkow@64267
  2382
      apply (rule sum.neutral)
huffman@44528
  2383
      apply clarsimp
huffman@44528
  2384
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@60420
  2385
      using \<open>y \<in> C\<close> by auto
wenzelm@53406
  2386
  }
wenzelm@60420
  2387
  with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
huffman@44528
  2388
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  2389
  from fC cC SC CPO have "?P (insert a B) ?C"
wenzelm@53406
  2390
    by blast
huffman@44133
  2391
  then show ?case by blast
huffman@44133
  2392
qed
huffman@44133
  2393