src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author wenzelm
Fri Aug 30 18:22:17 2013 +0200 (2013-08-30)
changeset 53333 e9dba6602a84
parent 53302 98fdf6c34142
child 53339 0dc28fd72c7d
permissions -rw-r--r--
tuned proofs;
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(*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
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    Author:     Robert Himmelmann, TU Muenchen
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    Author:     Bogdan Grechuk, University of Edinburgh
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*)
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header {* Convex sets, functions and related things. *}
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theory Convex_Euclidean_Space
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imports
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  Topology_Euclidean_Space
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  "~~/src/HOL/Library/Convex"
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  "~~/src/HOL/Library/Set_Algebras"
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begin
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(* ------------------------------------------------------------------------- *)
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(* To be moved elsewhere                                                     *)
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(* ------------------------------------------------------------------------- *)
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lemma linear_scaleR: "linear (\<lambda>x. scaleR c x)"
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  by (simp add: linear_def scaleR_add_right)
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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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lemma mem_convex_2:
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  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v=1"
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  shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
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  using assms convex_def[of S] by auto
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lemma mem_convex_alt:
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  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
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  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
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  apply (subst mem_convex_2)
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  using assms
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  apply (auto simp add: algebra_simps zero_le_divide_iff)
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  using add_divide_distrib[of u v "u+v"]
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  apply auto
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  done
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lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
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  by (blast dest: inj_onD)
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lemma independent_injective_on_span_image:
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  assumes iS: "independent S"
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    and lf: "linear f"
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    and fi: "inj_on f (span S)"
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  shows "independent (f ` S)"
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proof -
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  {
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    fix a
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    assume a: "a : S" "f a : span (f ` S - {f a})"
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    have eq: "f ` S - {f a} = f ` (S - {a})"
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      using fi a span_inc by (auto simp add: inj_on_def)
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    from a have "f a : f ` span (S -{a})"
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      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
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    moreover have "span (S -{a}) <= span S"
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      using span_mono[of "S-{a}" S] by auto
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    ultimately have "a : span (S -{a})"
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      using fi a span_inc by (auto simp add: inj_on_def)
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    with a(1) iS have False
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      by (simp add: dependent_def)
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  }
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  then show ?thesis
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    unfolding dependent_def by blast
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qed
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lemma dim_image_eq:
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  fixes f :: "'n::euclidean_space => 'm::euclidean_space"
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  assumes lf: "linear f"
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    and fi: "inj_on f (span S)"
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  shows "dim (f ` S) = dim (S::('n::euclidean_space) set)"
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proof -
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  obtain B where B_def: "B \<subseteq> S & independent B & S \<subseteq> span B & card B = dim S"
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    using basis_exists[of S] by auto
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  then have "span S = span B"
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    using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
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  then have "independent (f ` B)"
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    using independent_injective_on_span_image[of B f] B_def assms by auto
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  moreover have "card (f ` B) = card B"
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    using assms card_image[of f B] subset_inj_on[of f "span S" B] B_def span_inc by auto
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  moreover have "(f ` B) \<subseteq> (f ` S)"
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    using B_def by auto
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  ultimately have "dim (f ` S) \<ge> dim S"
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    using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
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  then show ?thesis
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    using dim_image_le[of f S] assms by auto
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qed
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lemma linear_injective_on_subspace_0:
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  assumes lf: "linear f"
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    and "subspace S"
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  shows "inj_on f S \<longleftrightarrow> (\<forall>x \<in> S. f x = 0 \<longrightarrow> x = 0)"
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proof -
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  have "inj_on f S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. 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 \<in> S. \<forall>y \<in> S. 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 \<in> S. \<forall>y \<in> S. f (x - y) = 0 \<longrightarrow> x - y = 0)"
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    by (simp add: linear_sub[OF lf])
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  also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. f x = 0 \<longrightarrow> x = 0)"
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    using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
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  finally show ?thesis .
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qed
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lemma subspace_Inter: "\<forall>s \<in> f. subspace s \<Longrightarrow> subspace (Inter f)"
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  unfolding subspace_def by auto
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lemma span_eq[simp]: "span s = s \<longleftrightarrow> subspace s"
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  unfolding span_def by (rule hull_eq) (rule subspace_Inter)
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lemma substdbasis_expansion_unique:
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  assumes d: "d \<subseteq> Basis"
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  shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
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    (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
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proof -
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  have *: "\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)"
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    by auto
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  have **: "finite d"
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    by (auto intro: finite_subset[OF assms])
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  have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
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    using d
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    by (auto intro!: setsum_cong simp: inner_Basis inner_setsum_left)
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  show ?thesis
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    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum_delta[OF **] ***)
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qed
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lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
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  by (rule independent_mono[OF independent_Basis])
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lemma dim_cball:
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  assumes "e > 0"
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  shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
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proof -
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  {
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    fix x :: "'n::euclidean_space"
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    def y \<equiv> "(e / norm x) *\<^sub>R x"
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    then have "y : cball 0 e"
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      using cball_def dist_norm[of 0 y] assms by auto
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    moreover have *: "x = (norm x/e) *\<^sub>R y"
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      using y_def assms by simp
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    moreover from * have "x = (norm x/e) *\<^sub>R y"
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      by auto
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    ultimately have "x : span (cball 0 e)"
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      using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
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  }
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  then have "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)"
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    by auto
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  then show ?thesis
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    using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
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qed
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lemma indep_card_eq_dim_span:
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  fixes B :: "('n::euclidean_space) set"
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  assumes "independent B"
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  shows "finite B & card B = dim (span B)"
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  using assms basis_card_eq_dim[of B "span B"] span_inc by auto
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lemma setsum_not_0: "setsum f A \<noteq> 0 \<Longrightarrow> \<exists> a\<in>A. f a \<noteq> 0"
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  by (rule ccontr) auto
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lemma translate_inj_on:
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  fixes A :: "('a::ab_group_add) set"
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  shows "inj_on (\<lambda>x. a+x) A"
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  unfolding inj_on_def by auto
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lemma translation_assoc:
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  fixes a b :: "'a::ab_group_add"
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  shows "(\<lambda>x. b+x) ` ((\<lambda>x. a+x) ` S) = (\<lambda>x. (a+b)+x) ` S"
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  by auto
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lemma translation_invert:
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  fixes a :: "'a::ab_group_add"
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  assumes "(\<lambda>x. a+x) ` A = (\<lambda>x. a+x) ` B"
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  shows "A = B"
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proof -
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  have "(\<lambda>x. -a+x) ` ((\<lambda>x. a+x) ` A) = (\<lambda>x. -a+x) ` ((\<lambda>x. a+x) ` B)"
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    using assms by auto
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  then show ?thesis
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    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
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qed
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lemma translation_galois:
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  fixes a :: "'a::ab_group_add"
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  shows "T = ((\<lambda>x. a+x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (-a)+x) ` T)"
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  using translation_assoc[of "-a" a S]
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  apply auto
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  using translation_assoc[of a "-a" T]
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  apply auto
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  done
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lemma translation_inverse_subset:
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  assumes "((%x. -a+x) ` V) \<le> (S :: 'n::ab_group_add set)"
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  shows "V \<le> ((%x. a+x) ` S)"
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proof -
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  {
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    fix x
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    assume "x \<in> V"
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    then have "x-a \<in> S" using assms by auto
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    then have "x \<in> {a + v |v. v \<in> S}"
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      apply auto
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      apply (rule exI[of _ "x-a"])
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      apply simp
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      done
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    then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
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  }
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  then show ?thesis by auto
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qed
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lemma basis_to_basis_subspace_isomorphism:
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  assumes s: "subspace (S:: ('n::euclidean_space) set)"
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    and t: "subspace (T :: ('m::euclidean_space) set)"
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    and d: "dim S = dim T"
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    and B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
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    and C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T"
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  shows "\<exists>f. linear f \<and> f ` B = C \<and> f ` S = T \<and> inj_on f S"
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proof -
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(* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
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*)
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  from B independent_bound have fB: "finite B"
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    by blast
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  from C independent_bound have fC: "finite C"
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    by blast
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  from B(4) C(4) card_le_inj[of B C] d obtain f where
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    f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
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  from linear_independent_extend[OF B(2)] obtain g where
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    g: "linear g" "\<forall>x \<in> B. g x = f x" by blast
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  from inj_on_iff_eq_card[OF fB, of f] f(2)
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  have "card (f ` B) = card B" by simp
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  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
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    by simp
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  have "g ` B = f ` B" using g(2)
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    by (auto simp add: image_iff)
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  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
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  finally have gBC: "g ` B = C" .
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  have gi: "inj_on g B" using f(2) g(2)
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    by (auto simp add: inj_on_def)
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  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
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  {
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    fix x y
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    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
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    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
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      by blast+
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    from gxy have th0: "g (x - y) = 0"
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      by (simp add: linear_sub[OF g(1)])
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    have th1: "x - y \<in> span B" using x' y'
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      by (metis span_sub)
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    have "x = y" using g0[OF th1 th0] by simp
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  }
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  then have giS: "inj_on g S" unfolding inj_on_def by blast
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  from span_subspace[OF B(1,3) s]
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  have "g ` S = span (g ` B)"
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    by (simp add: span_linear_image[OF g(1)])
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  also have "\<dots> = span C"
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    unfolding gBC ..
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  also have "\<dots> = T"
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    using span_subspace[OF C(1,3) t] .
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  finally have gS: "g ` S = T" .
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  from g(1) gS giS gBC show ?thesis
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    by blast
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qed
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lemma closure_bounded_linear_image:
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  assumes f: "bounded_linear f"
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  shows "f ` closure S \<subseteq> closure (f ` S)"
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  using linear_continuous_on [OF f] closed_closure closure_subset
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  by (rule image_closure_subset)
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lemma closure_linear_image:
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  fixes f :: "('m::euclidean_space) \<Rightarrow> ('n::real_normed_vector)"
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  assumes "linear f"
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  shows "f ` (closure S) \<le> closure (f ` S)"
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  using assms unfolding linear_conv_bounded_linear
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  by (rule closure_bounded_linear_image)
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lemma closure_injective_linear_image:
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  fixes f :: "('n::euclidean_space) \<Rightarrow> ('n::euclidean_space)"
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  assumes "linear f" "inj f"
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  shows "f ` (closure S) = closure (f ` S)"
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proof -
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  obtain f' where f'_def: "linear f' \<and> f o f' = id \<and> f' o f = id"
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    using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
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  then have "f' ` closure (f ` S) \<le> closure (S)"
wenzelm@49529
   289
    using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto
wenzelm@53333
   290
  then have "f ` f' ` closure (f ` S) \<le> f ` closure (S)" by auto
wenzelm@53333
   291
  then have "closure (f ` S) \<le> f ` closure (S)"
wenzelm@49529
   292
    using image_compose[of f f' "closure (f ` S)"] f'_def by auto
wenzelm@49529
   293
  then show ?thesis using closure_linear_image[of f S] assms by auto
hoelzl@40377
   294
qed
hoelzl@40377
   295
wenzelm@53333
   296
lemma closure_direct_sum: "closure (S \<times> T) = closure S \<times> closure T"
huffman@44365
   297
  by (rule closure_Times)
hoelzl@40377
   298
huffman@44524
   299
lemma closure_scaleR:
huffman@44524
   300
  fixes S :: "('a::real_normed_vector) set"
huffman@44524
   301
  shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
huffman@44524
   302
proof
huffman@44524
   303
  show "(op *\<^sub>R c) ` (closure S) \<subseteq> closure ((op *\<^sub>R c) ` S)"
wenzelm@53333
   304
    using bounded_linear_scaleR_right
wenzelm@53333
   305
    by (rule closure_bounded_linear_image)
huffman@44524
   306
  show "closure ((op *\<^sub>R c) ` S) \<subseteq> (op *\<^sub>R c) ` (closure S)"
wenzelm@49529
   307
    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
wenzelm@49529
   308
qed
wenzelm@49529
   309
wenzelm@49529
   310
lemma fst_linear: "linear fst"
wenzelm@49529
   311
  unfolding linear_def by (simp add: algebra_simps)
wenzelm@49529
   312
wenzelm@49529
   313
lemma snd_linear: "linear snd"
wenzelm@49529
   314
  unfolding linear_def by (simp add: algebra_simps)
wenzelm@49529
   315
wenzelm@49529
   316
lemma fst_snd_linear: "linear (%(x,y). x + y)"
wenzelm@49529
   317
  unfolding linear_def by (simp add: algebra_simps)
hoelzl@40377
   318
hoelzl@37489
   319
lemma scaleR_2:
hoelzl@37489
   320
  fixes x :: "'a::real_vector"
hoelzl@37489
   321
  shows "scaleR 2 x = x + x"
wenzelm@49529
   322
  unfolding one_add_one [symmetric] scaleR_left_distrib by simp
wenzelm@49529
   323
wenzelm@49529
   324
lemma vector_choose_size:
wenzelm@53333
   325
  "0 \<le> c \<Longrightarrow> \<exists>x::'a::euclidean_space. norm x = c"
wenzelm@53333
   326
  apply (rule exI [where x="c *\<^sub>R (SOME i. i \<in> Basis)"])
hoelzl@50526
   327
  apply (auto simp: SOME_Basis)
wenzelm@49529
   328
  done
wenzelm@49529
   329
wenzelm@49529
   330
lemma setsum_delta_notmem:
wenzelm@49529
   331
  assumes "x \<notin> s"
himmelma@33175
   332
  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
wenzelm@49529
   333
    and "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
wenzelm@49529
   334
    and "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
wenzelm@49529
   335
    and "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
wenzelm@50804
   336
  apply (rule_tac [!] setsum_cong2)
wenzelm@53333
   337
  using assms
wenzelm@53333
   338
  apply auto
wenzelm@49529
   339
  done
himmelma@33175
   340
himmelma@33175
   341
lemma setsum_delta'':
wenzelm@49529
   342
  fixes s::"'a::real_vector set"
wenzelm@49529
   343
  assumes "finite s"
himmelma@33175
   344
  shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
wenzelm@49529
   345
proof -
wenzelm@49529
   346
  have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
wenzelm@49529
   347
    by auto
wenzelm@49529
   348
  show ?thesis
wenzelm@49529
   349
    unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
himmelma@33175
   350
qed
himmelma@33175
   351
wenzelm@53333
   352
lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
wenzelm@53333
   353
  by auto
himmelma@33175
   354
wenzelm@49529
   355
lemma image_smult_interval:
wenzelm@49529
   356
  "(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
wenzelm@49529
   357
    (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
himmelma@33175
   358
  using image_affinity_interval[of m 0 a b] by auto
himmelma@33175
   359
himmelma@33175
   360
lemma dist_triangle_eq:
huffman@44361
   361
  fixes x y z :: "'a::real_inner"
wenzelm@53333
   362
  shows "dist x z = dist x y + dist y z \<longleftrightarrow>
wenzelm@53333
   363
    norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
wenzelm@49529
   364
proof -
wenzelm@49529
   365
  have *: "x - y + (y - z) = x - z" by auto
hoelzl@37489
   366
  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
wenzelm@49529
   367
    by (auto simp add:norm_minus_commute)
wenzelm@49529
   368
qed
himmelma@33175
   369
hoelzl@37489
   370
lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
himmelma@33175
   371
wenzelm@49529
   372
lemma Min_grI:
wenzelm@49529
   373
  assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
wenzelm@49529
   374
  shows "x < Min A"
himmelma@33175
   375
  unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
himmelma@33175
   376
hoelzl@37489
   377
lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
hoelzl@37489
   378
  unfolding norm_eq_sqrt_inner by simp
himmelma@33175
   379
hoelzl@37489
   380
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
hoelzl@37489
   381
  unfolding norm_eq_sqrt_inner by simp
hoelzl@37489
   382
hoelzl@37489
   383
huffman@44467
   384
subsection {* Affine set and affine hull *}
himmelma@33175
   385
wenzelm@49529
   386
definition affine :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@49529
   387
  where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
himmelma@33175
   388
himmelma@33175
   389
lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
wenzelm@49529
   390
  unfolding affine_def by (metis eq_diff_eq')
himmelma@33175
   391
himmelma@33175
   392
lemma affine_empty[intro]: "affine {}"
himmelma@33175
   393
  unfolding affine_def by auto
himmelma@33175
   394
himmelma@33175
   395
lemma affine_sing[intro]: "affine {x}"
himmelma@33175
   396
  unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
himmelma@33175
   397
himmelma@33175
   398
lemma affine_UNIV[intro]: "affine UNIV"
himmelma@33175
   399
  unfolding affine_def by auto
himmelma@33175
   400
himmelma@33175
   401
lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
wenzelm@49531
   402
  unfolding affine_def by auto
himmelma@33175
   403
himmelma@33175
   404
lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
himmelma@33175
   405
  unfolding affine_def by auto
himmelma@33175
   406
himmelma@33175
   407
lemma affine_affine_hull: "affine(affine hull s)"
wenzelm@49529
   408
  unfolding hull_def
wenzelm@49529
   409
  using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
himmelma@33175
   410
himmelma@33175
   411
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
wenzelm@49529
   412
  by (metis affine_affine_hull hull_same)
wenzelm@49529
   413
himmelma@33175
   414
huffman@44467
   415
subsubsection {* Some explicit formulations (from Lars Schewe) *}
himmelma@33175
   416
wenzelm@49529
   417
lemma affine:
wenzelm@49529
   418
  fixes V::"'a::real_vector set"
wenzelm@49529
   419
  shows "affine V \<longleftrightarrow>
wenzelm@49529
   420
    (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
wenzelm@49529
   421
  unfolding affine_def
wenzelm@49529
   422
  apply rule
wenzelm@49529
   423
  apply(rule, rule, rule)
wenzelm@49531
   424
  apply(erule conjE)+
wenzelm@49529
   425
  defer
wenzelm@49529
   426
  apply (rule, rule, rule, rule, rule)
wenzelm@49529
   427
proof -
wenzelm@49529
   428
  fix x y u v
wenzelm@49529
   429
  assume as: "x \<in> V" "y \<in> V" "u + v = (1::real)"
himmelma@33175
   430
    "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
wenzelm@49529
   431
  then show "u *\<^sub>R x + v *\<^sub>R y \<in> V"
wenzelm@49529
   432
    apply (cases "x = y")
wenzelm@49529
   433
    using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]]
wenzelm@49529
   434
      and as(1-3)
wenzelm@53333
   435
    apply (auto simp add: scaleR_left_distrib[symmetric])
wenzelm@53333
   436
    done
himmelma@33175
   437
next
wenzelm@49529
   438
  fix s u
wenzelm@49529
   439
  assume as: "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
himmelma@33175
   440
    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
himmelma@33175
   441
  def n \<equiv> "card s"
himmelma@33175
   442
  have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
wenzelm@49529
   443
  then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
wenzelm@49529
   444
  proof (auto simp only: disjE)
wenzelm@49529
   445
    assume "card s = 2"
wenzelm@53333
   446
    then have "card s = Suc (Suc 0)"
wenzelm@53333
   447
      by auto
wenzelm@53333
   448
    then obtain a b where "s = {a, b}"
wenzelm@53333
   449
      unfolding card_Suc_eq by auto
wenzelm@49529
   450
    then show ?thesis
wenzelm@49529
   451
      using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
wenzelm@49529
   452
      by (auto simp add: setsum_clauses(2))
wenzelm@49529
   453
  next
wenzelm@49529
   454
    assume "card s > 2"
wenzelm@49529
   455
    then show ?thesis using as and n_def
wenzelm@49529
   456
    proof (induct n arbitrary: u s)
wenzelm@49529
   457
      case 0
wenzelm@49529
   458
      then show ?case by auto
wenzelm@49529
   459
    next
wenzelm@49529
   460
      case (Suc n)
wenzelm@49529
   461
      fix s :: "'a set" and u :: "'a \<Rightarrow> real"
wenzelm@49529
   462
      assume IA:
wenzelm@49529
   463
        "\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
wenzelm@49529
   464
          s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
wenzelm@49529
   465
        and as:
wenzelm@49529
   466
          "Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
himmelma@33175
   467
           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
wenzelm@49529
   468
      have "\<exists>x\<in>s. u x \<noteq> 1"
wenzelm@49529
   469
      proof (rule ccontr)
wenzelm@49529
   470
        assume "\<not> ?thesis"
wenzelm@53333
   471
        then have "setsum u s = real_of_nat (card s)"
wenzelm@53333
   472
          unfolding card_eq_setsum by auto
wenzelm@49529
   473
        then show False
wenzelm@49529
   474
          using as(7) and `card s > 2`
wenzelm@49529
   475
          by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
huffman@45498
   476
      qed
himmelma@33175
   477
      then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
himmelma@33175
   478
wenzelm@49529
   479
      have c: "card (s - {x}) = card s - 1"
wenzelm@53333
   480
        apply (rule card_Diff_singleton)
wenzelm@53333
   481
        using `x\<in>s` as(4)
wenzelm@53333
   482
        apply auto
wenzelm@53333
   483
        done
wenzelm@49529
   484
      have *: "s = insert x (s - {x})" "finite (s - {x})"
wenzelm@49529
   485
        using `x\<in>s` and as(4) by auto
wenzelm@49529
   486
      have **: "setsum u (s - {x}) = 1 - u x"
wenzelm@49530
   487
        using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
wenzelm@49529
   488
      have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"
wenzelm@49529
   489
        unfolding ** using `u x \<noteq> 1` by auto
wenzelm@49529
   490
      have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V"
wenzelm@49529
   491
      proof (cases "card (s - {x}) > 2")
wenzelm@49529
   492
        case True
wenzelm@49529
   493
        then have "s - {x} \<noteq> {}" "card (s - {x}) = n"
wenzelm@49529
   494
          unfolding c and as(1)[symmetric]
wenzelm@49531
   495
        proof (rule_tac ccontr)
wenzelm@49529
   496
          assume "\<not> s - {x} \<noteq> {}"
wenzelm@49531
   497
          then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
wenzelm@49529
   498
          then show False using True by auto
wenzelm@49529
   499
        qed auto
wenzelm@49529
   500
        then show ?thesis
wenzelm@49529
   501
          apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
wenzelm@53333
   502
          unfolding setsum_right_distrib[symmetric]
wenzelm@53333
   503
          using as and *** and True
wenzelm@49529
   504
          apply auto
wenzelm@49529
   505
          done
wenzelm@49529
   506
      next
wenzelm@49529
   507
        case False
wenzelm@53333
   508
        then have "card (s - {x}) = Suc (Suc 0)"
wenzelm@53333
   509
          using as(2) and c by auto
wenzelm@53333
   510
        then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b"
wenzelm@53333
   511
          unfolding card_Suc_eq by auto
wenzelm@53333
   512
        then show ?thesis
wenzelm@53333
   513
          using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
wenzelm@49529
   514
          using *** *(2) and `s \<subseteq> V`
wenzelm@53333
   515
          unfolding setsum_right_distrib
wenzelm@53333
   516
          by (auto simp add: setsum_clauses(2))
wenzelm@49529
   517
      qed
wenzelm@49529
   518
      then have "u x + (1 - u x) = 1 \<Longrightarrow>
wenzelm@49529
   519
          u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa) /\<^sub>R (1 - u x)) \<in> V"
wenzelm@49529
   520
        apply -
wenzelm@49529
   521
        apply (rule as(3)[rule_format])
hoelzl@51524
   522
        unfolding  Real_Vector_Spaces.scaleR_right.setsum
wenzelm@53333
   523
        using x(1) as(6)
wenzelm@53333
   524
        apply auto
wenzelm@49529
   525
        done
wenzelm@49529
   526
      then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
wenzelm@49530
   527
        unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
wenzelm@49529
   528
        apply (subst *)
wenzelm@49529
   529
        unfolding setsum_clauses(2)[OF *(2)]
wenzelm@53333
   530
        using `u x \<noteq> 1`
wenzelm@53333
   531
        apply auto
wenzelm@49529
   532
        done
wenzelm@49529
   533
    qed
wenzelm@49529
   534
  next
wenzelm@49529
   535
    assume "card s = 1"
wenzelm@53333
   536
    then obtain a where "s={a}"
wenzelm@53333
   537
      by (auto simp add: card_Suc_eq)
wenzelm@53333
   538
    then show ?thesis
wenzelm@53333
   539
      using as(4,5) by simp
wenzelm@49529
   540
  qed (insert `s\<noteq>{}` `finite s`, auto)
himmelma@33175
   541
qed
himmelma@33175
   542
himmelma@33175
   543
lemma affine_hull_explicit:
wenzelm@53333
   544
  "affine hull p =
wenzelm@53333
   545
    {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
wenzelm@49529
   546
  apply (rule hull_unique)
wenzelm@49529
   547
  apply (subst subset_eq)
wenzelm@49529
   548
  prefer 3
wenzelm@49529
   549
  apply rule
wenzelm@49529
   550
  unfolding mem_Collect_eq
wenzelm@49529
   551
  apply (erule exE)+
wenzelm@49529
   552
  apply (erule conjE)+
wenzelm@49529
   553
  prefer 2
wenzelm@49529
   554
  apply rule
wenzelm@49529
   555
proof -
wenzelm@49529
   556
  fix x
wenzelm@49529
   557
  assume "x\<in>p"
wenzelm@49529
   558
  then show "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@53333
   559
    apply (rule_tac x="{x}" in exI)
wenzelm@53333
   560
    apply (rule_tac x="\<lambda>x. 1" in exI)
wenzelm@49529
   561
    apply auto
wenzelm@49529
   562
    done
himmelma@33175
   563
next
wenzelm@49529
   564
  fix t x s u
wenzelm@53333
   565
  assume as: "p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}"
wenzelm@53333
   566
    "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   567
  then show "x \<in> t"
wenzelm@53333
   568
    using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]]
wenzelm@53333
   569
    by auto
himmelma@33175
   570
next
wenzelm@49529
   571
  show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}"
wenzelm@49529
   572
    unfolding affine_def
wenzelm@49529
   573
    apply (rule, rule, rule, rule, rule)
wenzelm@49529
   574
    unfolding mem_Collect_eq
wenzelm@49529
   575
  proof -
wenzelm@49529
   576
    fix u v :: real
wenzelm@49529
   577
    assume uv: "u + v = 1"
wenzelm@49529
   578
    fix x
wenzelm@49529
   579
    assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   580
    then obtain sx ux where
wenzelm@53333
   581
      x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x"
wenzelm@53333
   582
      by auto
wenzelm@53333
   583
    fix y
wenzelm@53333
   584
    assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
wenzelm@49529
   585
    then obtain sy uy where
wenzelm@49529
   586
      y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
wenzelm@53333
   587
    have xy: "finite (sx \<union> sy)"
wenzelm@53333
   588
      using x(1) y(1) by auto
wenzelm@53333
   589
    have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
wenzelm@53333
   590
      by auto
wenzelm@49529
   591
    show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and>
wenzelm@49529
   592
        setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
wenzelm@49529
   593
      apply (rule_tac x="sx \<union> sy" in exI)
wenzelm@49529
   594
      apply (rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>sy then v * uy a else 0)" in exI)
wenzelm@53333
   595
      unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left
wenzelm@53333
   596
        ** setsum_restrict_set[OF xy, symmetric]
wenzelm@53333
   597
      unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
wenzelm@53333
   598
        and setsum_right_distrib[symmetric]
wenzelm@49529
   599
      unfolding x y
wenzelm@53333
   600
      using x(1-3) y(1-3) uv
wenzelm@53333
   601
      apply simp
wenzelm@49529
   602
      done
wenzelm@49529
   603
  qed
wenzelm@49529
   604
qed
himmelma@33175
   605
himmelma@33175
   606
lemma affine_hull_finite:
himmelma@33175
   607
  assumes "finite s"
himmelma@33175
   608
  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
wenzelm@53333
   609
  unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq
wenzelm@53333
   610
  apply (rule, rule)
wenzelm@53333
   611
  apply (erule exE)+
wenzelm@53333
   612
  apply (erule conjE)+
wenzelm@49529
   613
  defer
wenzelm@49529
   614
  apply (erule exE)
wenzelm@49529
   615
  apply (erule conjE)
wenzelm@49529
   616
proof -
wenzelm@49529
   617
  fix x u
wenzelm@49529
   618
  assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   619
  then show "\<exists>sa u. finite sa \<and>
wenzelm@49529
   620
      \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
wenzelm@49529
   621
    apply (rule_tac x=s in exI, rule_tac x=u in exI)
wenzelm@53333
   622
    using assms
wenzelm@53333
   623
    apply auto
wenzelm@49529
   624
    done
himmelma@33175
   625
next
wenzelm@49529
   626
  fix x t u
wenzelm@49529
   627
  assume "t \<subseteq> s"
wenzelm@53333
   628
  then have *: "s \<inter> t = t"
wenzelm@53333
   629
    by auto
himmelma@33175
   630
  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
wenzelm@49529
   631
  then show "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   632
    apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
wenzelm@49530
   633
    unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, symmetric] and *
wenzelm@49529
   634
    apply auto
wenzelm@49529
   635
    done
wenzelm@49529
   636
qed
wenzelm@49529
   637
himmelma@33175
   638
huffman@44467
   639
subsubsection {* Stepping theorems and hence small special cases *}
himmelma@33175
   640
himmelma@33175
   641
lemma affine_hull_empty[simp]: "affine hull {} = {}"
wenzelm@49529
   642
  by (rule hull_unique) auto
himmelma@33175
   643
himmelma@33175
   644
lemma affine_hull_finite_step:
himmelma@33175
   645
  fixes y :: "'a::real_vector"
wenzelm@49529
   646
  shows
wenzelm@49529
   647
    "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
wenzelm@49529
   648
    "finite s \<Longrightarrow>
wenzelm@49529
   649
      (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
wenzelm@49529
   650
      (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
wenzelm@49529
   651
proof -
himmelma@33175
   652
  show ?th1 by simp
wenzelm@49529
   653
  assume ?as
wenzelm@53302
   654
  {
wenzelm@53302
   655
    assume ?lhs
wenzelm@53302
   656
    then obtain u where u: "setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
wenzelm@53302
   657
      by auto
wenzelm@49529
   658
    have ?rhs
wenzelm@49529
   659
    proof (cases "a \<in> s")
wenzelm@49529
   660
      case True
wenzelm@49529
   661
      then have *: "insert a s = s" by auto
wenzelm@53302
   662
      show ?thesis
wenzelm@53302
   663
        using u[unfolded *]
wenzelm@53302
   664
        apply(rule_tac x=0 in exI)
wenzelm@53302
   665
        apply auto
wenzelm@53302
   666
        done
himmelma@33175
   667
    next
wenzelm@49529
   668
      case False
wenzelm@49529
   669
      then show ?thesis
wenzelm@49529
   670
        apply (rule_tac x="u a" in exI)
wenzelm@53302
   671
        using u and `?as`
wenzelm@53302
   672
        apply auto
wenzelm@49529
   673
        done
wenzelm@53302
   674
    qed
wenzelm@53302
   675
  }
wenzelm@49529
   676
  moreover
wenzelm@53302
   677
  {
wenzelm@53302
   678
    assume ?rhs
wenzelm@53302
   679
    then obtain v u where vu: "setsum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
wenzelm@53302
   680
      by auto
wenzelm@53302
   681
    have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
wenzelm@53302
   682
      by auto
wenzelm@49529
   683
    have ?lhs
wenzelm@49529
   684
    proof (cases "a \<in> s")
wenzelm@49529
   685
      case True
wenzelm@49529
   686
      then show ?thesis
wenzelm@49529
   687
        apply (rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
wenzelm@53333
   688
        unfolding setsum_clauses(2)[OF `?as`]
wenzelm@53333
   689
        apply simp
wenzelm@49531
   690
        unfolding scaleR_left_distrib and setsum_addf
himmelma@33175
   691
        unfolding vu and * and scaleR_zero_left
wenzelm@49529
   692
        apply (auto simp add: setsum_delta[OF `?as`])
wenzelm@49529
   693
        done
himmelma@33175
   694
    next
wenzelm@49531
   695
      case False
wenzelm@49529
   696
      then have **:
wenzelm@49529
   697
        "\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
wenzelm@49529
   698
        "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
himmelma@33175
   699
      from False show ?thesis
wenzelm@49529
   700
        apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
himmelma@33175
   701
        unfolding setsum_clauses(2)[OF `?as`] and * using vu
himmelma@33175
   702
        using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
wenzelm@49529
   703
        using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)]
wenzelm@49529
   704
        apply auto
wenzelm@49529
   705
        done
wenzelm@49529
   706
    qed
wenzelm@49529
   707
  }
himmelma@33175
   708
  ultimately show "?lhs = ?rhs" by blast
himmelma@33175
   709
qed
himmelma@33175
   710
himmelma@33175
   711
lemma affine_hull_2:
himmelma@33175
   712
  fixes a b :: "'a::real_vector"
wenzelm@53302
   713
  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
wenzelm@53302
   714
  (is "?lhs = ?rhs")
wenzelm@49529
   715
proof -
wenzelm@49529
   716
  have *:
wenzelm@49531
   717
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
wenzelm@49529
   718
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
himmelma@33175
   719
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
himmelma@33175
   720
    using affine_hull_finite[of "{a,b}"] by auto
himmelma@33175
   721
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
wenzelm@49529
   722
    by (simp add: affine_hull_finite_step(2)[of "{b}" a])
himmelma@33175
   723
  also have "\<dots> = ?rhs" unfolding * by auto
himmelma@33175
   724
  finally show ?thesis by auto
himmelma@33175
   725
qed
himmelma@33175
   726
himmelma@33175
   727
lemma affine_hull_3:
himmelma@33175
   728
  fixes a b c :: "'a::real_vector"
wenzelm@53302
   729
  shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
wenzelm@49529
   730
proof -
wenzelm@49529
   731
  have *:
wenzelm@49531
   732
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
wenzelm@49529
   733
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
wenzelm@49529
   734
  show ?thesis
wenzelm@49529
   735
    apply (simp add: affine_hull_finite affine_hull_finite_step)
wenzelm@49529
   736
    unfolding *
wenzelm@49529
   737
    apply auto
wenzelm@53302
   738
    apply (rule_tac x=v in exI)
wenzelm@53302
   739
    apply (rule_tac x=va in exI)
wenzelm@53302
   740
    apply auto
wenzelm@53302
   741
    apply (rule_tac x=u in exI)
wenzelm@53302
   742
    apply force
wenzelm@49529
   743
    done
himmelma@33175
   744
qed
himmelma@33175
   745
hoelzl@40377
   746
lemma mem_affine:
wenzelm@53333
   747
  assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
wenzelm@53333
   748
  shows "(u *\<^sub>R x + v *\<^sub>R y) \<in> S"
hoelzl@40377
   749
  using assms affine_def[of S] by auto
hoelzl@40377
   750
hoelzl@40377
   751
lemma mem_affine_3:
wenzelm@53333
   752
  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
wenzelm@53333
   753
  shows "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) \<in> S"
wenzelm@49529
   754
proof -
wenzelm@53333
   755
  have "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) \<in> affine hull {x, y, z}"
wenzelm@49529
   756
    using affine_hull_3[of x y z] assms by auto
wenzelm@49529
   757
  moreover
wenzelm@49529
   758
  have "affine hull {x, y, z} <= affine hull S"
wenzelm@49529
   759
    using hull_mono[of "{x, y, z}" "S"] assms by auto
wenzelm@49529
   760
  moreover
wenzelm@49529
   761
  have "affine hull S = S" using assms affine_hull_eq[of S] by auto
wenzelm@49531
   762
  ultimately show ?thesis by auto
hoelzl@40377
   763
qed
hoelzl@40377
   764
hoelzl@40377
   765
lemma mem_affine_3_minus:
wenzelm@53333
   766
  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
wenzelm@53333
   767
  shows "x + v *\<^sub>R (y-z) \<in> S"
wenzelm@53333
   768
  using mem_affine_3[of S x y z 1 v "-v"] assms
wenzelm@53333
   769
  by (simp add: algebra_simps)
hoelzl@40377
   770
hoelzl@40377
   771
huffman@44467
   772
subsubsection {* Some relations between affine hull and subspaces *}
himmelma@33175
   773
himmelma@33175
   774
lemma affine_hull_insert_subset_span:
wenzelm@49529
   775
  "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
wenzelm@49529
   776
  unfolding subset_eq Ball_def
wenzelm@49529
   777
  unfolding affine_hull_explicit span_explicit mem_Collect_eq
wenzelm@50804
   778
  apply (rule, rule)
wenzelm@50804
   779
  apply (erule exE)+
wenzelm@50804
   780
  apply (erule conjE)+
wenzelm@49529
   781
proof -
wenzelm@49529
   782
  fix x t u
wenzelm@49529
   783
  assume as: "finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
wenzelm@53333
   784
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}"
wenzelm@53333
   785
    using as(3) by auto
wenzelm@49529
   786
  then show "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
wenzelm@49529
   787
    apply (rule_tac x="x - a" in exI)
himmelma@33175
   788
    apply (rule conjI, simp)
wenzelm@49529
   789
    apply (rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
wenzelm@49529
   790
    apply (rule_tac x="\<lambda>x. u (x + a)" in exI)
himmelma@33175
   791
    apply (rule conjI) using as(1) apply simp
himmelma@33175
   792
    apply (erule conjI)
himmelma@33175
   793
    using as(1)
wenzelm@49529
   794
    apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib
wenzelm@49530
   795
      setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)
wenzelm@49529
   796
    unfolding as
wenzelm@49529
   797
    apply simp
wenzelm@49529
   798
    done
wenzelm@49529
   799
qed
himmelma@33175
   800
himmelma@33175
   801
lemma affine_hull_insert_span:
himmelma@33175
   802
  assumes "a \<notin> s"
wenzelm@49529
   803
  shows "affine hull (insert a s) = {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
wenzelm@49529
   804
  apply (rule, rule affine_hull_insert_subset_span)
wenzelm@49529
   805
  unfolding subset_eq Ball_def
wenzelm@49529
   806
  unfolding affine_hull_explicit and mem_Collect_eq
wenzelm@49529
   807
proof (rule, rule, erule exE, erule conjE)
wenzelm@49531
   808
  fix y v
wenzelm@49529
   809
  assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
wenzelm@49529
   810
  then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y"
wenzelm@49529
   811
    unfolding span_explicit by auto
himmelma@33175
   812
  def f \<equiv> "(\<lambda>x. x + a) ` t"
wenzelm@53333
   813
  have f: "finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a"
wenzelm@49529
   814
    unfolding f_def using obt by (auto simp add: setsum_reindex[unfolded inj_on_def])
wenzelm@53333
   815
  have *: "f \<inter> {a} = {}" "f \<inter> - {a} = f"
wenzelm@53333
   816
    using f(2) assms by auto
himmelma@33175
   817
  show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
wenzelm@49529
   818
    apply (rule_tac x = "insert a f" in exI)
wenzelm@49529
   819
    apply (rule_tac x = "\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
himmelma@33175
   820
    using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
hoelzl@35577
   821
    unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
wenzelm@49529
   822
    apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)
wenzelm@49529
   823
    done
wenzelm@49529
   824
qed
himmelma@33175
   825
himmelma@33175
   826
lemma affine_hull_span:
himmelma@33175
   827
  assumes "a \<in> s"
himmelma@33175
   828
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
himmelma@33175
   829
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
himmelma@33175
   830
wenzelm@49529
   831
huffman@44467
   832
subsubsection {* Parallel affine sets *}
hoelzl@40377
   833
hoelzl@40377
   834
definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
wenzelm@53333
   835
  where "affine_parallel S T = (? a. T = ((\<lambda>x. a + x) ` S))"
hoelzl@40377
   836
hoelzl@40377
   837
lemma affine_parallel_expl_aux:
wenzelm@49529
   838
  fixes S T :: "'a::real_vector set"
wenzelm@53333
   839
  assumes "\<forall>x. (x : S \<longleftrightarrow> (a+x) \<in> T)"
wenzelm@53333
   840
  shows "T = ((\<lambda>x. a + x) ` S)"
wenzelm@49529
   841
proof -
wenzelm@53302
   842
  {
wenzelm@53302
   843
    fix x
wenzelm@49529
   844
    assume "x : T"
wenzelm@53333
   845
    then have "(-a)+x \<in> S" using assms by auto
wenzelm@53333
   846
    then have "x : ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   847
      using imageI[of "-a+x" S "(\<lambda>x. a+x)"] by auto
wenzelm@53302
   848
  }
wenzelm@53333
   849
  moreover have "T >= ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   850
    using assms by auto
wenzelm@49529
   851
  ultimately show ?thesis by auto
wenzelm@49529
   852
qed
wenzelm@49529
   853
wenzelm@53333
   854
lemma affine_parallel_expl: "affine_parallel S T = (\<exists>a. \<forall>x. (x \<in> S \<longleftrightarrow> (a+x) \<in> T))"
wenzelm@49529
   855
  unfolding affine_parallel_def
wenzelm@49529
   856
  using affine_parallel_expl_aux[of S _ T] by auto
wenzelm@49529
   857
wenzelm@49529
   858
lemma affine_parallel_reflex: "affine_parallel S S"
wenzelm@53302
   859
  unfolding affine_parallel_def
wenzelm@53302
   860
  apply (rule exI[of _ "0"])
wenzelm@53302
   861
  apply auto
wenzelm@53302
   862
  done
hoelzl@40377
   863
hoelzl@40377
   864
lemma affine_parallel_commut:
wenzelm@49529
   865
  assumes "affine_parallel A B"
wenzelm@49529
   866
  shows "affine_parallel B A"
wenzelm@49529
   867
proof -
wenzelm@53333
   868
  from assms obtain a where "B = (\<lambda>x. a + x) ` A"
wenzelm@49529
   869
    unfolding affine_parallel_def by auto
wenzelm@49529
   870
  then show ?thesis
wenzelm@53333
   871
    using translation_galois [of B a A]
wenzelm@53333
   872
    unfolding affine_parallel_def by auto
hoelzl@40377
   873
qed
hoelzl@40377
   874
hoelzl@40377
   875
lemma affine_parallel_assoc:
wenzelm@49529
   876
  assumes "affine_parallel A B" "affine_parallel B C"
wenzelm@49531
   877
  shows "affine_parallel A C"
wenzelm@49529
   878
proof -
wenzelm@53333
   879
  from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
wenzelm@49531
   880
    unfolding affine_parallel_def by auto
wenzelm@49531
   881
  moreover
wenzelm@53333
   882
  from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
wenzelm@49529
   883
    unfolding affine_parallel_def by auto
wenzelm@49529
   884
  ultimately show ?thesis
wenzelm@49529
   885
    using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
hoelzl@40377
   886
qed
hoelzl@40377
   887
hoelzl@40377
   888
lemma affine_translation_aux:
hoelzl@40377
   889
  fixes a :: "'a::real_vector"
wenzelm@53333
   890
  assumes "affine ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   891
  shows "affine S"
wenzelm@53302
   892
proof -
wenzelm@53302
   893
  {
wenzelm@53302
   894
    fix x y u v
wenzelm@53333
   895
    assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
wenzelm@53333
   896
    then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   897
      by auto
wenzelm@53333
   898
    then have h1: "u *\<^sub>R  (a+x) + v *\<^sub>R (a+y) \<in> ((\<lambda>x. a + x) ` S)"
wenzelm@49529
   899
      using xy assms unfolding affine_def by auto
wenzelm@49529
   900
    have "u *\<^sub>R (a+x) + v *\<^sub>R (a+y) = (u+v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
wenzelm@49529
   901
      by (simp add: algebra_simps)
wenzelm@53333
   902
    also have "...= a + (u *\<^sub>R x + v *\<^sub>R y)"
wenzelm@53333
   903
      using `u+v=1` by auto
wenzelm@53333
   904
    ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) : ((%x. a + x) ` S)"
wenzelm@53333
   905
      using h1 by auto
wenzelm@49529
   906
    then have "u *\<^sub>R x + v *\<^sub>R y : S" by auto
wenzelm@49529
   907
  }
wenzelm@49529
   908
  then show ?thesis unfolding affine_def by auto
hoelzl@40377
   909
qed
hoelzl@40377
   910
hoelzl@40377
   911
lemma affine_translation:
hoelzl@40377
   912
  fixes a :: "'a::real_vector"
wenzelm@53333
   913
  shows "affine S \<longleftrightarrow> affine ((%x. a + x) ` S)"
wenzelm@49529
   914
proof -
wenzelm@53333
   915
  have "affine S \<Longrightarrow> affine ((%x. a + x) ` S)"
wenzelm@49529
   916
    using affine_translation_aux[of "-a" "((%x. a + x) ` S)"]
wenzelm@49529
   917
    using translation_assoc[of "-a" a S] by auto
wenzelm@49529
   918
  then show ?thesis using affine_translation_aux by auto
hoelzl@40377
   919
qed
hoelzl@40377
   920
hoelzl@40377
   921
lemma parallel_is_affine:
wenzelm@49529
   922
  fixes S T :: "'a::real_vector set"
wenzelm@49529
   923
  assumes "affine S" "affine_parallel S T"
wenzelm@49529
   924
  shows "affine T"
wenzelm@49529
   925
proof -
wenzelm@49529
   926
  from assms obtain a where "T=((%x. a + x) ` S)"
wenzelm@49531
   927
    unfolding affine_parallel_def by auto
wenzelm@49529
   928
  then show ?thesis using affine_translation assms by auto
hoelzl@40377
   929
qed
hoelzl@40377
   930
huffman@44361
   931
lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
hoelzl@40377
   932
  unfolding subspace_def affine_def by auto
hoelzl@40377
   933
wenzelm@49529
   934
huffman@44467
   935
subsubsection {* Subspace parallel to an affine set *}
hoelzl@40377
   936
wenzelm@53333
   937
lemma subspace_affine: "subspace S \<longleftrightarrow> (affine S \<and> 0 : S)"
wenzelm@49529
   938
proof -
wenzelm@53333
   939
  have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
wenzelm@49529
   940
    using subspace_imp_affine[of S] subspace_0 by auto
wenzelm@53302
   941
  {
wenzelm@53333
   942
    assume assm: "affine S \<and> 0 \<in> S"
wenzelm@53302
   943
    {
wenzelm@53302
   944
      fix c :: real
wenzelm@53333
   945
      fix x assume x_def: "x \<in> S"
wenzelm@49529
   946
      have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
wenzelm@49529
   947
      moreover
wenzelm@53302
   948
      have "(1-c) *\<^sub>R 0 + c *\<^sub>R x : S"
wenzelm@53302
   949
        using affine_alt[of S] assm x_def by auto
wenzelm@53333
   950
      ultimately have "c *\<^sub>R x \<in> S" by auto
wenzelm@49529
   951
    }
wenzelm@53333
   952
    then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
wenzelm@49529
   953
wenzelm@53302
   954
    {
wenzelm@53302
   955
      fix x y
wenzelm@53302
   956
      assume xy_def: "x \<in> S" "y \<in> S"
wenzelm@49529
   957
      def u == "(1 :: real)/2"
wenzelm@53302
   958
      have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
wenzelm@53302
   959
        by auto
wenzelm@49529
   960
      moreover
wenzelm@53302
   961
      have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
wenzelm@53302
   962
        by (simp add: algebra_simps)
wenzelm@49529
   963
      moreover
wenzelm@53333
   964
      have "(1-u) *\<^sub>R x + u *\<^sub>R y \<in> S"
wenzelm@53302
   965
        using affine_alt[of S] assm xy_def by auto
wenzelm@49529
   966
      ultimately
wenzelm@53333
   967
      have "(1/2) *\<^sub>R (x+y) \<in> S"
wenzelm@53302
   968
        using u_def by auto
wenzelm@49529
   969
      moreover
wenzelm@53302
   970
      have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
wenzelm@53302
   971
        by auto
wenzelm@49529
   972
      ultimately
wenzelm@53333
   973
      have "(x+y) \<in> S"
wenzelm@53302
   974
        using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
wenzelm@49529
   975
    }
wenzelm@53302
   976
    then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
wenzelm@53302
   977
      by auto
wenzelm@53302
   978
    then have "subspace S"
wenzelm@53302
   979
      using h1 assm unfolding subspace_def by auto
wenzelm@49529
   980
  }
wenzelm@49529
   981
  then show ?thesis using h0 by metis
hoelzl@40377
   982
qed
hoelzl@40377
   983
hoelzl@40377
   984
lemma affine_diffs_subspace:
wenzelm@53333
   985
  assumes "affine S" "a \<in> S"
wenzelm@53302
   986
  shows "subspace ((\<lambda>x. (-a)+x) ` S)"
wenzelm@49529
   987
proof -
wenzelm@53302
   988
  have "affine ((\<lambda>x. (-a)+x) ` S)"
wenzelm@49531
   989
    using  affine_translation assms by auto
wenzelm@53302
   990
  moreover have "0 : ((\<lambda>x. (-a)+x) ` S)"
wenzelm@53333
   991
    using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
wenzelm@49531
   992
  ultimately show ?thesis using subspace_affine by auto
hoelzl@40377
   993
qed
hoelzl@40377
   994
hoelzl@40377
   995
lemma parallel_subspace_explicit:
wenzelm@49529
   996
  assumes "affine S" "a : S"
wenzelm@53302
   997
  assumes "L \<equiv> {y. \<exists>x \<in> S. (-a)+x=y}"
wenzelm@49531
   998
  shows "subspace L & affine_parallel S L"
wenzelm@49529
   999
proof -
wenzelm@49529
  1000
  have par: "affine_parallel S L"
wenzelm@49529
  1001
    unfolding affine_parallel_def using assms by auto
wenzelm@49531
  1002
  then have "affine L" using assms parallel_is_affine by auto
wenzelm@53302
  1003
  moreover have "0 \<in> L"
wenzelm@53302
  1004
    using assms
wenzelm@53302
  1005
    apply auto
wenzelm@53302
  1006
    using exI[of "(%x. x:S & -a+x=0)" a]
wenzelm@53302
  1007
    apply auto
wenzelm@49529
  1008
    done
wenzelm@53302
  1009
  ultimately show ?thesis
wenzelm@53302
  1010
    using subspace_affine par by auto
hoelzl@40377
  1011
qed
hoelzl@40377
  1012
hoelzl@40377
  1013
lemma parallel_subspace_aux:
wenzelm@53302
  1014
  assumes "subspace A"
wenzelm@53302
  1015
    and "subspace B"
wenzelm@53302
  1016
    and "affine_parallel A B"
wenzelm@53302
  1017
  shows "A \<supseteq> B"
wenzelm@49529
  1018
proof -
wenzelm@53302
  1019
  from assms obtain a where a_def: "\<forall>x. (x \<in> A \<longleftrightarrow> (a+x) \<in> B)"
wenzelm@49529
  1020
    using affine_parallel_expl[of A B] by auto
wenzelm@53302
  1021
  then have "-a \<in> A"
wenzelm@53302
  1022
    using assms subspace_0[of B] by auto
wenzelm@53302
  1023
  then have "a \<in> A"
wenzelm@53302
  1024
    using assms subspace_neg[of A "-a"] by auto
wenzelm@53302
  1025
  then show ?thesis
wenzelm@53302
  1026
    using assms a_def unfolding subspace_def by auto
hoelzl@40377
  1027
qed
hoelzl@40377
  1028
hoelzl@40377
  1029
lemma parallel_subspace:
wenzelm@53302
  1030
  assumes "subspace A"
wenzelm@53302
  1031
    and "subspace B"
wenzelm@53302
  1032
    and "affine_parallel A B"
wenzelm@49529
  1033
  shows "A = B"
wenzelm@49529
  1034
proof
wenzelm@53302
  1035
  show "A \<supseteq> B"
wenzelm@49529
  1036
    using assms parallel_subspace_aux by auto
wenzelm@53302
  1037
  show "A \<subseteq> B"
wenzelm@49529
  1038
    using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
hoelzl@40377
  1039
qed
hoelzl@40377
  1040
hoelzl@40377
  1041
lemma affine_parallel_subspace:
wenzelm@53302
  1042
  assumes "affine S" "S \<noteq> {}"
wenzelm@53302
  1043
  shows "\<exists>!L. subspace L & affine_parallel S L"
wenzelm@49529
  1044
proof -
wenzelm@53302
  1045
  have ex: "\<exists>L. subspace L & affine_parallel S L"
wenzelm@49531
  1046
    using assms parallel_subspace_explicit by auto
wenzelm@53302
  1047
  {
wenzelm@53302
  1048
    fix L1 L2
wenzelm@49529
  1049
    assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"
wenzelm@49529
  1050
    then have "affine_parallel L1 L2"
wenzelm@49529
  1051
      using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
wenzelm@49529
  1052
    then have "L1 = L2"
wenzelm@49529
  1053
      using ass parallel_subspace by auto
wenzelm@49529
  1054
  }
wenzelm@49529
  1055
  then show ?thesis using ex by auto
wenzelm@49529
  1056
qed
wenzelm@49529
  1057
hoelzl@40377
  1058
huffman@44467
  1059
subsection {* Cones *}
himmelma@33175
  1060
wenzelm@49529
  1061
definition cone :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@49529
  1062
  where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
himmelma@33175
  1063
himmelma@33175
  1064
lemma cone_empty[intro, simp]: "cone {}"
himmelma@33175
  1065
  unfolding cone_def by auto
himmelma@33175
  1066
himmelma@33175
  1067
lemma cone_univ[intro, simp]: "cone UNIV"
himmelma@33175
  1068
  unfolding cone_def by auto
himmelma@33175
  1069
wenzelm@53302
  1070
lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone(\<Inter> f)"
himmelma@33175
  1071
  unfolding cone_def by auto
himmelma@33175
  1072
wenzelm@49529
  1073
huffman@44467
  1074
subsubsection {* Conic hull *}
himmelma@33175
  1075
himmelma@33175
  1076
lemma cone_cone_hull: "cone (cone hull s)"
huffman@44170
  1077
  unfolding hull_def by auto
himmelma@33175
  1078
wenzelm@53302
  1079
lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
wenzelm@49529
  1080
  apply (rule hull_eq)
wenzelm@53302
  1081
  using cone_Inter
wenzelm@53302
  1082
  unfolding subset_eq
wenzelm@53302
  1083
  apply auto
wenzelm@49529
  1084
  done
himmelma@33175
  1085
hoelzl@40377
  1086
lemma mem_cone:
wenzelm@53302
  1087
  assumes "cone S" "x \<in> S" "c \<ge> 0"
hoelzl@40377
  1088
  shows "c *\<^sub>R x : S"
hoelzl@40377
  1089
  using assms cone_def[of S] by auto
hoelzl@40377
  1090
hoelzl@40377
  1091
lemma cone_contains_0:
wenzelm@49529
  1092
  assumes "cone S"
wenzelm@53302
  1093
  shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
wenzelm@49529
  1094
proof -
wenzelm@53302
  1095
  {
wenzelm@53302
  1096
    assume "S \<noteq> {}"
wenzelm@53302
  1097
    then obtain a where "a \<in> S" by auto
wenzelm@53302
  1098
    then have "0 \<in> S"
wenzelm@53302
  1099
      using assms mem_cone[of S a 0] by auto
wenzelm@49529
  1100
  }
wenzelm@49529
  1101
  then show ?thesis by auto
hoelzl@40377
  1102
qed
hoelzl@40377
  1103
huffman@44361
  1104
lemma cone_0: "cone {0}"
wenzelm@49529
  1105
  unfolding cone_def by auto
hoelzl@40377
  1106
wenzelm@53302
  1107
lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (Union f)"
hoelzl@40377
  1108
  unfolding cone_def by blast
hoelzl@40377
  1109
hoelzl@40377
  1110
lemma cone_iff:
wenzelm@49529
  1111
  assumes "S ~= {}"
wenzelm@53302
  1112
  shows "cone S \<longleftrightarrow> 0 \<in> S & (\<forall>c. c>0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
wenzelm@49529
  1113
proof -
wenzelm@53302
  1114
  {
wenzelm@53302
  1115
    assume "cone S"
wenzelm@53302
  1116
    {
wenzelm@53302
  1117
      fix c
wenzelm@49529
  1118
      assume "(c :: real) > 0"
wenzelm@53302
  1119
      {
wenzelm@53302
  1120
        fix x
wenzelm@49529
  1121
        assume "x : S"
wenzelm@49529
  1122
        then have "x : (op *\<^sub>R c) ` S"
wenzelm@49529
  1123
          unfolding image_def
wenzelm@49529
  1124
          using `cone S` `c>0` mem_cone[of S x "1/c"]
wenzelm@53302
  1125
            exI[of "(%t. t:S & x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
wenzelm@53302
  1126
          apply auto
wenzelm@49529
  1127
          done
wenzelm@49529
  1128
      }
wenzelm@49529
  1129
      moreover
wenzelm@53302
  1130
      {
wenzelm@53302
  1131
        fix x
wenzelm@53302
  1132
        assume "x : (op *\<^sub>R c) ` S"
wenzelm@49529
  1133
        (*from this obtain t where "t:S & x = c *\<^sub>R t" by auto*)
wenzelm@49529
  1134
        then have "x:S"
wenzelm@49529
  1135
          using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto
wenzelm@49529
  1136
      }
wenzelm@53302
  1137
      ultimately have "(op *\<^sub>R c) ` S = S" by auto
hoelzl@40377
  1138
    }
wenzelm@53302
  1139
    then have "0 \<in> S & (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
wenzelm@49529
  1140
      using `cone S` cone_contains_0[of S] assms by auto
wenzelm@49529
  1141
  }
wenzelm@49529
  1142
  moreover
wenzelm@53302
  1143
  {
wenzelm@53302
  1144
    assume a: "0 \<in> S & (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
wenzelm@53302
  1145
    {
wenzelm@53302
  1146
      fix x
wenzelm@53302
  1147
      assume "x \<in> S"
wenzelm@49529
  1148
      fix c1
wenzelm@53302
  1149
      assume "(c1 :: real) \<ge> 0"
wenzelm@53302
  1150
      then have "c1 = 0 | c1 > 0" by auto
wenzelm@53302
  1151
      then have "c1 *\<^sub>R x : S" using a `x \<in> S` by auto
wenzelm@49529
  1152
    }
wenzelm@49529
  1153
    then have "cone S" unfolding cone_def by auto
hoelzl@40377
  1154
  }
wenzelm@49529
  1155
  ultimately show ?thesis by blast
wenzelm@49529
  1156
qed
wenzelm@49529
  1157
wenzelm@49529
  1158
lemma cone_hull_empty: "cone hull {} = {}"
wenzelm@49529
  1159
  by (metis cone_empty cone_hull_eq)
wenzelm@49529
  1160
wenzelm@53302
  1161
lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
wenzelm@49529
  1162
  by (metis bot_least cone_hull_empty hull_subset xtrans(5))
wenzelm@49529
  1163
wenzelm@53302
  1164
lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
wenzelm@49529
  1165
  using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
wenzelm@49529
  1166
  by auto
hoelzl@40377
  1167
hoelzl@40377
  1168
lemma mem_cone_hull:
wenzelm@53302
  1169
  assumes "x : S" "c \<ge> 0"
wenzelm@53302
  1170
  shows "c *\<^sub>R x \<in> cone hull S"
wenzelm@49529
  1171
  by (metis assms cone_cone_hull hull_inc mem_cone)
wenzelm@49529
  1172
wenzelm@53302
  1173
lemma cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 & x \<in> S}" (is "?lhs = ?rhs")
wenzelm@49529
  1174
proof -
wenzelm@53302
  1175
  {
wenzelm@53302
  1176
    fix x
wenzelm@53302
  1177
    assume "x \<in> ?rhs"
wenzelm@53302
  1178
    then obtain cx xx where x_def: "x = cx *\<^sub>R xx & (cx :: real) \<ge> 0 & xx \<in> S"
wenzelm@49529
  1179
      by auto
wenzelm@49529
  1180
    fix c
wenzelm@53302
  1181
    assume c_def: "(c :: real) \<ge> 0"
wenzelm@49529
  1182
    then have "c *\<^sub>R x = (c*cx) *\<^sub>R xx"
wenzelm@49529
  1183
      using x_def by (simp add: algebra_simps)
wenzelm@49529
  1184
    moreover
wenzelm@53302
  1185
    have "c * cx \<ge> 0"
wenzelm@49529
  1186
      using c_def x_def using mult_nonneg_nonneg by auto
wenzelm@49529
  1187
    ultimately
wenzelm@53302
  1188
    have "c *\<^sub>R x \<in> ?rhs" using x_def by auto
wenzelm@53302
  1189
  }
wenzelm@53302
  1190
  then have "cone ?rhs" unfolding cone_def by auto
wenzelm@49529
  1191
  then have "?rhs : Collect cone" unfolding mem_Collect_eq by auto
wenzelm@53302
  1192
  {
wenzelm@53302
  1193
    fix x
wenzelm@53302
  1194
    assume "x \<in> S"
wenzelm@53302
  1195
    then have "1 *\<^sub>R x \<in> ?rhs"
wenzelm@49531
  1196
      apply auto
wenzelm@49529
  1197
      apply (rule_tac x="1" in exI)
wenzelm@49529
  1198
      apply auto
wenzelm@49529
  1199
      done
wenzelm@53302
  1200
    then have "x \<in> ?rhs" by auto
wenzelm@53302
  1201
  } then have "S \<subseteq> ?rhs" by auto
wenzelm@53302
  1202
  then have "?lhs \<subseteq> ?rhs"
wenzelm@53302
  1203
    using `?rhs \<in> Collect cone` hull_minimal[of S "?rhs" "cone"] by auto
wenzelm@49529
  1204
  moreover
wenzelm@53302
  1205
  {
wenzelm@53302
  1206
    fix x
wenzelm@53302
  1207
    assume "x \<in> ?rhs"
wenzelm@53302
  1208
    then obtain cx xx where x_def: "x = cx *\<^sub>R xx & (cx :: real) \<ge> 0 & xx \<in> S" by auto
wenzelm@53302
  1209
    then have "xx \<in> cone hull S" using hull_subset[of S] by auto
wenzelm@53302
  1210
    then have "x \<in> ?lhs"
wenzelm@49529
  1211
      using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
wenzelm@49529
  1212
  }
wenzelm@49529
  1213
  ultimately show ?thesis by auto
hoelzl@40377
  1214
qed
hoelzl@40377
  1215
hoelzl@40377
  1216
lemma cone_closure:
huffman@44524
  1217
  fixes S :: "('a::real_normed_vector) set"
wenzelm@49529
  1218
  assumes "cone S"
wenzelm@49529
  1219
  shows "cone (closure S)"
wenzelm@49529
  1220
proof (cases "S = {}")
wenzelm@49529
  1221
  case True
wenzelm@49529
  1222
  then show ?thesis by auto
wenzelm@49529
  1223
next
wenzelm@49529
  1224
  case False
wenzelm@53302
  1225
  then have "0 \<in> S & (!c. c>0 --> op *\<^sub>R c ` S = S)"
wenzelm@49529
  1226
    using cone_iff[of S] assms by auto
wenzelm@53302
  1227
  then have "0 \<in> closure S & (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` closure S = closure S)"
wenzelm@49529
  1228
    using closure_subset by (auto simp add: closure_scaleR)
wenzelm@49529
  1229
  then show ?thesis using cone_iff[of "closure S"] by auto
wenzelm@49529
  1230
qed
wenzelm@49529
  1231
hoelzl@40377
  1232
huffman@44467
  1233
subsection {* Affine dependence and consequential theorems (from Lars Schewe) *}
himmelma@33175
  1234
wenzelm@49529
  1235
definition affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@49529
  1236
  where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
himmelma@33175
  1237
himmelma@33175
  1238
lemma affine_dependent_explicit:
himmelma@33175
  1239
  "affine_dependent p \<longleftrightarrow>
himmelma@33175
  1240
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
himmelma@33175
  1241
    (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
wenzelm@49529
  1242
  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq
wenzelm@49529
  1243
  apply rule
wenzelm@49529
  1244
  apply (erule bexE, erule exE, erule exE)
wenzelm@49529
  1245
  apply (erule conjE)+
wenzelm@49529
  1246
  defer
wenzelm@49529
  1247
  apply (erule exE, erule exE)
wenzelm@49529
  1248
  apply (erule conjE)+
wenzelm@49529
  1249
  apply (erule bexE)
wenzelm@49529
  1250
proof -
wenzelm@49529
  1251
  fix x s u
wenzelm@49529
  1252
  assume as: "x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@53302
  1253
  have "x \<notin> s" using as(1,4) by auto
himmelma@33175
  1254
  show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
wenzelm@49529
  1255
    apply (rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
wenzelm@49529
  1256
    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as
wenzelm@49529
  1257
    using as apply auto
wenzelm@49529
  1258
    done
himmelma@33175
  1259
next
wenzelm@49529
  1260
  fix s u v
wenzelm@53302
  1261
  assume as: "finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
himmelma@33175
  1262
  have "s \<noteq> {v}" using as(3,6) by auto
wenzelm@49529
  1263
  then show "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@53302
  1264
    apply (rule_tac x=v in bexI)
wenzelm@53302
  1265
    apply (rule_tac x="s - {v}" in exI)
wenzelm@53302
  1266
    apply (rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
wenzelm@49530
  1267
    unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
wenzelm@49530
  1268
    unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]
wenzelm@53302
  1269
    using as
wenzelm@53302
  1270
    apply auto
wenzelm@49529
  1271
    done
himmelma@33175
  1272
qed
himmelma@33175
  1273
himmelma@33175
  1274
lemma affine_dependent_explicit_finite:
wenzelm@49529
  1275
  fixes s :: "'a::real_vector set"
wenzelm@49529
  1276
  assumes "finite s"
wenzelm@53302
  1277
  shows "affine_dependent s \<longleftrightarrow>
wenzelm@53302
  1278
    (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
himmelma@33175
  1279
  (is "?lhs = ?rhs")
himmelma@33175
  1280
proof
wenzelm@49529
  1281
  have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::'a))"
wenzelm@49529
  1282
    by auto
himmelma@33175
  1283
  assume ?lhs
wenzelm@49529
  1284
  then obtain t u v where
wenzelm@49529
  1285
      "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
himmelma@33175
  1286
    unfolding affine_dependent_explicit by auto
wenzelm@49529
  1287
  then show ?rhs
wenzelm@49529
  1288
    apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
wenzelm@49530
  1289
    apply auto unfolding * and setsum_restrict_set[OF assms, symmetric]
wenzelm@49529
  1290
    unfolding Int_absorb1[OF `t\<subseteq>s`]
wenzelm@49529
  1291
    apply auto
wenzelm@49529
  1292
    done
himmelma@33175
  1293
next
himmelma@33175
  1294
  assume ?rhs
himmelma@33175
  1295
  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
wenzelm@49529
  1296
  then show ?lhs unfolding affine_dependent_explicit
wenzelm@49529
  1297
    using assms by auto
wenzelm@49529
  1298
qed
wenzelm@49529
  1299
himmelma@33175
  1300
huffman@44465
  1301
subsection {* Connectedness of convex sets *}
huffman@44465
  1302
hoelzl@51480
  1303
lemma connectedD:
hoelzl@51480
  1304
  "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
hoelzl@51480
  1305
  by (metis connected_def)
himmelma@33175
  1306
himmelma@33175
  1307
lemma convex_connected:
himmelma@33175
  1308
  fixes s :: "'a::real_normed_vector set"
wenzelm@53302
  1309
  assumes "convex s"
wenzelm@53302
  1310
  shows "connected s"
hoelzl@51480
  1311
proof (rule connectedI)
hoelzl@51480
  1312
  fix A B
hoelzl@51480
  1313
  assume "open A" "open B" "A \<inter> B \<inter> s = {}" "s \<subseteq> A \<union> B"
hoelzl@51480
  1314
  moreover
hoelzl@51480
  1315
  assume "A \<inter> s \<noteq> {}" "B \<inter> s \<noteq> {}"
hoelzl@51480
  1316
  then obtain a b where a: "a \<in> A" "a \<in> s" and b: "b \<in> B" "b \<in> s" by auto
hoelzl@51480
  1317
  def f \<equiv> "\<lambda>u. u *\<^sub>R a + (1 - u) *\<^sub>R b"
hoelzl@51480
  1318
  then have "continuous_on {0 .. 1} f"
hoelzl@51480
  1319
    by (auto intro!: continuous_on_intros)
hoelzl@51480
  1320
  then have "connected (f ` {0 .. 1})"
hoelzl@51480
  1321
    by (auto intro!: connected_continuous_image)
hoelzl@51480
  1322
  note connectedD[OF this, of A B]
hoelzl@51480
  1323
  moreover have "a \<in> A \<inter> f ` {0 .. 1}"
hoelzl@51480
  1324
    using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
hoelzl@51480
  1325
  moreover have "b \<in> B \<inter> f ` {0 .. 1}"
hoelzl@51480
  1326
    using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
hoelzl@51480
  1327
  moreover have "f ` {0 .. 1} \<subseteq> s"
hoelzl@51480
  1328
    using `convex s` a b unfolding convex_def f_def by auto
hoelzl@51480
  1329
  ultimately show False by auto
himmelma@33175
  1330
qed
himmelma@33175
  1331
huffman@44467
  1332
text {* One rather trivial consequence. *}
himmelma@33175
  1333
hoelzl@34964
  1334
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
himmelma@33175
  1335
  by(simp add: convex_connected convex_UNIV)
himmelma@33175
  1336
huffman@44467
  1337
text {* Balls, being convex, are connected. *}
himmelma@33175
  1338
wenzelm@53302
  1339
lemma convex_box:
wenzelm@53302
  1340
  fixes a::"'a::euclidean_space"
hoelzl@50526
  1341
  assumes "\<And>i. i\<in>Basis \<Longrightarrow> convex {x. P i x}"
hoelzl@50526
  1342
  shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
hoelzl@50526
  1343
  using assms unfolding convex_def
hoelzl@50526
  1344
  by (auto simp: inner_add_left)
hoelzl@50526
  1345
hoelzl@50526
  1346
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
hoelzl@36623
  1347
  by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
himmelma@33175
  1348
himmelma@33175
  1349
lemma convex_local_global_minimum:
himmelma@33175
  1350
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1351
  assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
himmelma@33175
  1352
  shows "\<forall>y\<in>s. f x \<le> f y"
wenzelm@53302
  1353
proof (rule ccontr)
wenzelm@53302
  1354
  have "x \<in> s" using assms(1,3) by auto
wenzelm@53302
  1355
  assume "\<not> ?thesis"
wenzelm@53302
  1356
  then obtain y where "y\<in>s" and y: "f x > f y" by auto
wenzelm@53302
  1357
  hence xy: "0 < dist x y" by (auto simp add: dist_nz[symmetric])
himmelma@33175
  1358
himmelma@33175
  1359
  then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
wenzelm@50804
  1360
    using real_lbound_gt_zero[of 1 "e / dist x y"]
wenzelm@50804
  1361
    using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
wenzelm@53302
  1362
  then have "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y"
wenzelm@53302
  1363
    using `x\<in>s` `y\<in>s`
wenzelm@53302
  1364
    using assms(2)[unfolded convex_on_def,
wenzelm@53302
  1365
      THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
wenzelm@50804
  1366
    by auto
himmelma@33175
  1367
  moreover
wenzelm@50804
  1368
  have *: "x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)"
wenzelm@50804
  1369
    by (simp add: algebra_simps)
wenzelm@50804
  1370
  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e"
wenzelm@53302
  1371
    unfolding mem_ball dist_norm
wenzelm@53302
  1372
    unfolding * and norm_scaleR and abs_of_pos[OF `0<u`]
wenzelm@50804
  1373
    unfolding dist_norm[symmetric]
wenzelm@53302
  1374
    using u
wenzelm@53302
  1375
    unfolding pos_less_divide_eq[OF xy]
wenzelm@53302
  1376
    by auto
wenzelm@53302
  1377
  then have "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)"
wenzelm@53302
  1378
    using assms(4) by auto
wenzelm@50804
  1379
  ultimately show False
wenzelm@53302
  1380
    using mult_strict_left_mono[OF y `u>0`]
wenzelm@53302
  1381
    unfolding left_diff_distrib
wenzelm@53302
  1382
    by auto
himmelma@33175
  1383
qed
himmelma@33175
  1384
himmelma@33175
  1385
lemma convex_ball:
himmelma@33175
  1386
  fixes x :: "'a::real_normed_vector"
wenzelm@49531
  1387
  shows "convex (ball x e)"
wenzelm@50804
  1388
proof (auto simp add: convex_def)
wenzelm@50804
  1389
  fix y z
wenzelm@50804
  1390
  assume yz: "dist x y < e" "dist x z < e"
wenzelm@50804
  1391
  fix u v :: real
wenzelm@50804
  1392
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@50804
  1393
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
wenzelm@50804
  1394
    using uv yz
wenzelm@53302
  1395
    using convex_distance[of "ball x e" x, unfolded convex_on_def,
wenzelm@53302
  1396
      THEN bspec[where x=y], THEN bspec[where x=z]]
wenzelm@50804
  1397
    by auto
wenzelm@50804
  1398
  then show "dist x (u *\<^sub>R y + v *\<^sub>R z) < e"
wenzelm@50804
  1399
    using convex_bound_lt[OF yz uv] by auto
himmelma@33175
  1400
qed
himmelma@33175
  1401
himmelma@33175
  1402
lemma convex_cball:
himmelma@33175
  1403
  fixes x :: "'a::real_normed_vector"
himmelma@33175
  1404
  shows "convex(cball x e)"
wenzelm@50804
  1405
proof (auto simp add: convex_def Ball_def)
wenzelm@50804
  1406
  fix y z
wenzelm@50804
  1407
  assume yz: "dist x y \<le> e" "dist x z \<le> e"
wenzelm@50804
  1408
  fix u v :: real
wenzelm@50804
  1409
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@50804
  1410
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
wenzelm@50804
  1411
    using uv yz
wenzelm@53302
  1412
    using convex_distance[of "cball x e" x, unfolded convex_on_def,
wenzelm@53302
  1413
      THEN bspec[where x=y], THEN bspec[where x=z]]
wenzelm@50804
  1414
    by auto
wenzelm@50804
  1415
  then show "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e"
wenzelm@50804
  1416
    using convex_bound_le[OF yz uv] by auto
himmelma@33175
  1417
qed
himmelma@33175
  1418
himmelma@33175
  1419
lemma connected_ball:
himmelma@33175
  1420
  fixes x :: "'a::real_normed_vector"
himmelma@33175
  1421
  shows "connected (ball x e)"
himmelma@33175
  1422
  using convex_connected convex_ball by auto
himmelma@33175
  1423
himmelma@33175
  1424
lemma connected_cball:
himmelma@33175
  1425
  fixes x :: "'a::real_normed_vector"
wenzelm@53302
  1426
  shows "connected (cball x e)"
himmelma@33175
  1427
  using convex_connected convex_cball by auto
himmelma@33175
  1428
wenzelm@50804
  1429
huffman@44467
  1430
subsection {* Convex hull *}
himmelma@33175
  1431
wenzelm@53302
  1432
lemma convex_convex_hull: "convex (convex hull s)"
wenzelm@53302
  1433
  unfolding hull_def
wenzelm@53302
  1434
  using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
huffman@44170
  1435
  by auto
himmelma@33175
  1436
haftmann@34064
  1437
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
wenzelm@50804
  1438
  by (metis convex_convex_hull hull_same)
himmelma@33175
  1439
himmelma@33175
  1440
lemma bounded_convex_hull:
himmelma@33175
  1441
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1442
  assumes "bounded s" shows "bounded(convex hull s)"
wenzelm@50804
  1443
proof -
wenzelm@50804
  1444
  from assms obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
wenzelm@50804
  1445
    unfolding bounded_iff by auto
wenzelm@50804
  1446
  show ?thesis
wenzelm@50804
  1447
    apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
huffman@44170
  1448
    unfolding subset_hull[of convex, OF convex_cball]
wenzelm@53302
  1449
    unfolding subset_eq mem_cball dist_norm using B
wenzelm@53302
  1450
    apply auto
wenzelm@50804
  1451
    done
wenzelm@50804
  1452
qed
himmelma@33175
  1453
himmelma@33175
  1454
lemma finite_imp_bounded_convex_hull:
himmelma@33175
  1455
  fixes s :: "'a::real_normed_vector set"
wenzelm@53302
  1456
  shows "finite s \<Longrightarrow> bounded (convex hull s)"
wenzelm@53302
  1457
  using bounded_convex_hull finite_imp_bounded
wenzelm@53302
  1458
  by auto
himmelma@33175
  1459
wenzelm@50804
  1460
huffman@44467
  1461
subsubsection {* Convex hull is "preserved" by a linear function *}
hoelzl@40377
  1462
hoelzl@40377
  1463
lemma convex_hull_linear_image:
hoelzl@40377
  1464
  assumes "bounded_linear f"
hoelzl@40377
  1465
  shows "f ` (convex hull s) = convex hull (f ` s)"
wenzelm@50804
  1466
  apply rule
wenzelm@50804
  1467
  unfolding subset_eq ball_simps
wenzelm@50804
  1468
  apply (rule_tac[!] hull_induct, rule hull_inc)
wenzelm@50804
  1469
  prefer 3
wenzelm@50804
  1470
  apply (erule imageE)
wenzelm@50804
  1471
  apply (rule_tac x=xa in image_eqI)
wenzelm@50804
  1472
  apply assumption
wenzelm@50804
  1473
  apply (rule hull_subset[unfolded subset_eq, rule_format])
wenzelm@50804
  1474
  apply assumption
wenzelm@50804
  1475
proof -
hoelzl@40377
  1476
  interpret f: bounded_linear f by fact
wenzelm@49531
  1477
  show "convex {x. f x \<in> convex hull f ` s}"
wenzelm@50804
  1478
    unfolding convex_def
wenzelm@50804
  1479
    by (auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format])
wenzelm@50804
  1480
  show "convex {x. x \<in> f ` (convex hull s)}"
wenzelm@50804
  1481
    using  convex_convex_hull[unfolded convex_def, of s]
hoelzl@40377
  1482
    unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
hoelzl@40377
  1483
qed auto
hoelzl@40377
  1484
hoelzl@40377
  1485
lemma in_convex_hull_linear_image:
hoelzl@40377
  1486
  assumes "bounded_linear f" "x \<in> convex hull s"
hoelzl@40377
  1487
  shows "(f x) \<in> convex hull (f ` s)"
wenzelm@50804
  1488
  using convex_hull_linear_image[OF assms(1)] assms(2) by auto
wenzelm@50804
  1489
hoelzl@40377
  1490
huffman@44467
  1491
subsubsection {* Stepping theorems for convex hulls of finite sets *}
himmelma@33175
  1492
himmelma@33175
  1493
lemma convex_hull_empty[simp]: "convex hull {} = {}"
wenzelm@50804
  1494
  by (rule hull_unique) auto
himmelma@33175
  1495
himmelma@33175
  1496
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
wenzelm@50804
  1497
  by (rule hull_unique) auto
himmelma@33175
  1498
himmelma@33175
  1499
lemma convex_hull_insert:
himmelma@33175
  1500
  fixes s :: "'a::real_vector set"
himmelma@33175
  1501
  assumes "s \<noteq> {}"
wenzelm@50804
  1502
  shows "convex hull (insert a s) =
wenzelm@50804
  1503
    {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
wenzelm@50804
  1504
  (is "?xyz = ?hull")
wenzelm@50804
  1505
  apply (rule, rule hull_minimal, rule)
wenzelm@50804
  1506
  unfolding insert_iff
wenzelm@50804
  1507
  prefer 3
wenzelm@50804
  1508
  apply rule
wenzelm@50804
  1509
proof -
wenzelm@50804
  1510
  fix x
wenzelm@50804
  1511
  assume x: "x = a \<or> x \<in> s"
wenzelm@50804
  1512
  then show "x \<in> ?hull"
wenzelm@50804
  1513
    apply rule
wenzelm@50804
  1514
    unfolding mem_Collect_eq
wenzelm@50804
  1515
    apply (rule_tac x=1 in exI)
wenzelm@50804
  1516
    defer
wenzelm@50804
  1517
    apply (rule_tac x=0 in exI)
wenzelm@50804
  1518
    using assms hull_subset[of s convex]
wenzelm@50804
  1519
    apply auto
wenzelm@50804
  1520
    done
himmelma@33175
  1521
next
wenzelm@50804
  1522
  fix x
wenzelm@50804
  1523
  assume "x \<in> ?hull"
wenzelm@50804
  1524
  then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b"
wenzelm@50804
  1525
    by auto
wenzelm@50804
  1526
  have "a \<in> convex hull insert a s" "b\<in>convex hull insert a s"
wenzelm@50804
  1527
    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4)
wenzelm@50804
  1528
    by auto
wenzelm@50804
  1529
  then show "x \<in> convex hull insert a s"
wenzelm@50804
  1530
    unfolding obt(5)
wenzelm@50804
  1531
    using convex_convex_hull[of "insert a s", unfolded convex_def]
wenzelm@50804
  1532
    apply (erule_tac x = a in ballE)
wenzelm@50804
  1533
    apply (erule_tac x = b in ballE)
wenzelm@50804
  1534
    apply (erule_tac x = u in allE)
wenzelm@53302
  1535
    using obt
wenzelm@53302
  1536
    apply auto
wenzelm@50804
  1537
    done
himmelma@33175
  1538
next
wenzelm@50804
  1539
  show "convex ?hull"
wenzelm@50804
  1540
    unfolding convex_def
wenzelm@50804
  1541
    apply (rule, rule, rule, rule, rule, rule, rule)
wenzelm@50804
  1542
  proof -
wenzelm@50804
  1543
    fix x y u v
wenzelm@50804
  1544
    assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
wenzelm@50804
  1545
    from as(4) obtain u1 v1 b1
wenzelm@50804
  1546
      where obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
wenzelm@50804
  1547
    from as(5) obtain u2 v2 b2
wenzelm@50804
  1548
      where obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
wenzelm@50804
  1549
    have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
wenzelm@50804
  1550
      by (auto simp add: algebra_simps)
wenzelm@50804
  1551
    have **: "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y =
wenzelm@50804
  1552
      (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
wenzelm@50804
  1553
    proof (cases "u * v1 + v * v2 = 0")
wenzelm@50804
  1554
      case True
wenzelm@50804
  1555
      have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
wenzelm@50804
  1556
        by (auto simp add: algebra_simps)
wenzelm@50804
  1557
      from True have ***: "u * v1 = 0" "v * v2 = 0"
wenzelm@53302
  1558
        using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`]
wenzelm@53302
  1559
        by arith+
wenzelm@50804
  1560
      then have "u * u1 + v * u2 = 1"
wenzelm@50804
  1561
        using as(3) obt1(3) obt2(3) by auto
wenzelm@50804
  1562
      then show ?thesis
wenzelm@50804
  1563
        unfolding obt1(5) obt2(5) *
wenzelm@50804
  1564
        using assms hull_subset[of s convex]
wenzelm@50804
  1565
        by (auto simp add: *** scaleR_right_distrib)
himmelma@33175
  1566
    next
wenzelm@50804
  1567
      case False
wenzelm@50804
  1568
      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
wenzelm@50804
  1569
        using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
wenzelm@50804
  1570
      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
wenzelm@50804
  1571
        using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
wenzelm@50804
  1572
      also have "\<dots> = u * v1 + v * v2"
wenzelm@50804
  1573
        by simp
wenzelm@50804
  1574
      finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
wenzelm@50804
  1575
      have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
wenzelm@50804
  1576
        apply (rule add_nonneg_nonneg)
wenzelm@50804
  1577
        prefer 4
wenzelm@50804
  1578
        apply (rule add_nonneg_nonneg)
wenzelm@50804
  1579
        apply (rule_tac [!] mult_nonneg_nonneg)
wenzelm@53302
  1580
        using as(1,2) obt1(1,2) obt2(1,2)
wenzelm@53302
  1581
        apply auto
wenzelm@50804
  1582
        done
wenzelm@50804
  1583
      then show ?thesis
wenzelm@50804
  1584
        unfolding obt1(5) obt2(5)
wenzelm@50804
  1585
        unfolding * and **
wenzelm@50804
  1586
        using False
wenzelm@50804
  1587
        apply (rule_tac x = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI)
wenzelm@50804
  1588
        defer
wenzelm@50804
  1589
        apply (rule convex_convex_hull[of s, unfolded convex_def, rule_format])
wenzelm@50804
  1590
        using obt1(4) obt2(4)
wenzelm@49530
  1591
        unfolding add_divide_distrib[symmetric] and zero_le_divide_iff
wenzelm@50804
  1592
        apply (auto simp add: scaleR_left_distrib scaleR_right_distrib)
wenzelm@50804
  1593
        done
wenzelm@50804
  1594
    qed
wenzelm@50804
  1595
    have u1: "u1 \<le> 1"
wenzelm@50804
  1596
      unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
wenzelm@50804
  1597
    have u2: "u2 \<le> 1"
wenzelm@50804
  1598
      unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
wenzelm@50804
  1599
    have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v"
wenzelm@50804
  1600
      apply (rule add_mono)
wenzelm@50804
  1601
      apply (rule_tac [!] mult_right_mono)
wenzelm@50804
  1602
      using as(1,2) obt1(1,2) obt2(1,2)
wenzelm@50804
  1603
      apply auto
wenzelm@50804
  1604
      done
wenzelm@50804
  1605
    also have "\<dots> \<le> 1"
wenzelm@50804
  1606
      unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
wenzelm@50804
  1607
    finally show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
wenzelm@50804
  1608
      unfolding mem_Collect_eq
wenzelm@50804
  1609
      apply (rule_tac x="u * u1 + v * u2" in exI)
wenzelm@50804
  1610
      apply (rule conjI)
wenzelm@50804
  1611
      defer
wenzelm@50804
  1612
      apply (rule_tac x="1 - u * u1 - v * u2" in exI)
wenzelm@50804
  1613
      unfolding Bex_def
wenzelm@50804
  1614
      using as(1,2) obt1(1,2) obt2(1,2) **
wenzelm@50804
  1615
      apply (auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
wenzelm@50804
  1616
      done
himmelma@33175
  1617
  qed
himmelma@33175
  1618
qed
himmelma@33175
  1619
himmelma@33175
  1620
huffman@44467
  1621
subsubsection {* Explicit expression for convex hull *}
himmelma@33175
  1622
himmelma@33175
  1623
lemma convex_hull_indexed:
himmelma@33175
  1624
  fixes s :: "'a::real_vector set"
wenzelm@50804
  1625
  shows "convex hull s =
wenzelm@50804
  1626
    {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
wenzelm@50804
  1627
        (setsum u {1..k} = 1) \<and>
wenzelm@50804
  1628
        (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
wenzelm@50804
  1629
  apply (rule hull_unique)
wenzelm@50804
  1630
  apply rule
wenzelm@50804
  1631
  defer
wenzelm@50804
  1632
  apply (subst convex_def)
wenzelm@50804
  1633
  apply (rule, rule, rule, rule, rule, rule, rule)
wenzelm@50804
  1634
proof -
wenzelm@50804
  1635
  fix x
wenzelm@50804
  1636
  assume "x\<in>s"
wenzelm@50804
  1637
  then show "x \<in> ?hull"
wenzelm@50804
  1638
    unfolding mem_Collect_eq
wenzelm@50804
  1639
    apply (rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI)
wenzelm@50804
  1640
    apply auto
wenzelm@50804
  1641
    done
himmelma@33175
  1642
next
wenzelm@50804
  1643
  fix t
wenzelm@50804
  1644
  assume as: "s \<subseteq> t" "convex t"
wenzelm@50804
  1645
  show "?hull \<subseteq> t"
wenzelm@50804
  1646
    apply rule
wenzelm@50804
  1647
    unfolding mem_Collect_eq
wenzelm@53302
  1648
    apply (elim exE conjE)
wenzelm@50804
  1649
  proof -
wenzelm@50804
  1650
    fix x k u y
wenzelm@50804
  1651
    assume assm:
wenzelm@50804
  1652
      "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s"
wenzelm@50804
  1653
      "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
wenzelm@50804
  1654
    show "x\<in>t"
wenzelm@50804
  1655
      unfolding assm(3) [symmetric]
wenzelm@50804
  1656
      apply (rule as(2)[unfolded convex, rule_format])
wenzelm@50804
  1657
      using assm(1,2) as(1) apply auto
wenzelm@50804
  1658
      done
wenzelm@50804
  1659
  qed
himmelma@33175
  1660
next
wenzelm@50804
  1661
  fix x y u v
wenzelm@53302
  1662
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)" and xy: "x \<in> ?hull" "y \<in> ?hull"
wenzelm@50804
  1663
  from xy obtain k1 u1 x1 where
wenzelm@50804
  1664
      x: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
wenzelm@50804
  1665
    by auto
wenzelm@50804
  1666
  from xy obtain k2 u2 x2 where
wenzelm@50804
  1667
      y: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
wenzelm@50804
  1668
    by auto
wenzelm@50804
  1669
  have *: "\<And>P (x1::'a) x2 s1 s2 i.
wenzelm@50804
  1670
    (if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
himmelma@33175
  1671
    "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
wenzelm@50804
  1672
    prefer 3
wenzelm@50804
  1673
    apply (rule, rule)
wenzelm@50804
  1674
    unfolding image_iff
wenzelm@50804
  1675
    apply (rule_tac x = "x - k1" in bexI)
wenzelm@50804
  1676
    apply (auto simp add: not_le)
wenzelm@50804
  1677
    done
wenzelm@50804
  1678
  have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
wenzelm@50804
  1679
    unfolding inj_on_def by auto
wenzelm@50804
  1680
  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
wenzelm@50804
  1681
    apply rule
wenzelm@50804
  1682
    apply (rule_tac x="k1 + k2" in exI)
wenzelm@50804
  1683
    apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
wenzelm@50804
  1684
    apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI)
wenzelm@50804
  1685
    apply (rule, rule)
wenzelm@50804
  1686
    defer
wenzelm@50804
  1687
    apply rule
wenzelm@50804
  1688
    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
wenzelm@50804
  1689
      setsum_reindex[OF inj] and o_def Collect_mem_eq
wenzelm@50804
  1690
    unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]
wenzelm@50804
  1691
  proof -
wenzelm@50804
  1692
    fix i
wenzelm@50804
  1693
    assume i: "i \<in> {1..k1+k2}"
wenzelm@50804
  1694
    show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and>
wenzelm@50804
  1695
      (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
wenzelm@50804
  1696
    proof (cases "i\<in>{1..k1}")
wenzelm@50804
  1697
      case True
wenzelm@50804
  1698
      then show ?thesis
wenzelm@50804
  1699
        using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
wenzelm@50804
  1700
    next
wenzelm@50804
  1701
      case False
wenzelm@50804
  1702
      def j \<equiv> "i - k1"
wenzelm@50804
  1703
      from i False have "j \<in> {1..k2}" unfolding j_def by auto
wenzelm@50804
  1704
      then show ?thesis
wenzelm@50804
  1705
        unfolding j_def[symmetric]
wenzelm@50804
  1706
        using False
wenzelm@50804
  1707
        using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]]
wenzelm@50804
  1708
        apply auto
wenzelm@50804
  1709
        done
wenzelm@50804
  1710
    qed
wenzelm@50804
  1711
  qed (auto simp add: not_le x(2,3) y(2,3) uv(3))
himmelma@33175
  1712
qed
himmelma@33175
  1713
himmelma@33175
  1714
lemma convex_hull_finite:
himmelma@33175
  1715
  fixes s :: "'a::real_vector set"
himmelma@33175
  1716
  assumes "finite s"
himmelma@33175
  1717
  shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
wenzelm@50804
  1718
    setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
wenzelm@50804
  1719
proof (rule hull_unique, auto simp add: convex_def[of ?set])
wenzelm@50804
  1720
  fix x
wenzelm@50804
  1721
  assume "x \<in> s"
wenzelm@50804
  1722
  then show "\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x"
wenzelm@50804
  1723
    apply (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI)
wenzelm@50804
  1724
    apply auto
wenzelm@50804
  1725
    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms]
wenzelm@50804
  1726
    apply auto
wenzelm@50804
  1727
    done
himmelma@33175
  1728
next
wenzelm@50804
  1729
  fix u v :: real
wenzelm@50804
  1730
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@50804
  1731
  fix ux assume ux: "\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
wenzelm@50804
  1732
  fix uy assume uy: "\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
wenzelm@50804
  1733
  { fix x
wenzelm@50804
  1734
    assume "x\<in>s"
wenzelm@50804
  1735
    then have "0 \<le> u * ux x + v * uy x"
wenzelm@50804
  1736
      using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
wenzelm@50804
  1737
      apply auto
wenzelm@50804
  1738
      apply (metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))
wenzelm@50804
  1739
      done
wenzelm@50804
  1740
  }
wenzelm@50804
  1741
  moreover
wenzelm@50804
  1742
  have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
wenzelm@53302
  1743
    unfolding setsum_addf and setsum_right_distrib[symmetric] and ux(2) uy(2)
wenzelm@53302
  1744
    using uv(3) by auto
wenzelm@50804
  1745
  moreover
wenzelm@50804
  1746
  have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
wenzelm@50804
  1747
    unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
wenzelm@50804
  1748
    by auto
wenzelm@50804
  1749
  ultimately
wenzelm@50804
  1750
  show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and>
wenzelm@50804
  1751
      (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
wenzelm@50804
  1752
    apply (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI)
wenzelm@50804
  1753
    apply auto
wenzelm@50804
  1754
    done
himmelma@33175
  1755
next
wenzelm@50804
  1756
  fix t
wenzelm@50804
  1757
  assume t: "s \<subseteq> t" "convex t"
wenzelm@50804
  1758
  fix u
wenzelm@50804
  1759
  assume u: "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
wenzelm@50804
  1760
  then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t"
wenzelm@50804
  1761
    using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
himmelma@33175
  1762
    using assms and t(1) by auto
himmelma@33175
  1763
qed
himmelma@33175
  1764
wenzelm@50804
  1765
huffman@44467
  1766
subsubsection {* Another formulation from Lars Schewe *}
himmelma@33175
  1767
himmelma@33175
  1768
lemma setsum_constant_scaleR:
himmelma@33175
  1769
  fixes y :: "'a::real_vector"
himmelma@33175
  1770
  shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
wenzelm@50804
  1771
  apply (cases "finite A")
wenzelm@50804
  1772
  apply (induct set: finite)
wenzelm@50804
  1773
  apply (simp_all add: algebra_simps)
wenzelm@50804
  1774
  done
himmelma@33175
  1775
himmelma@33175
  1776
lemma convex_hull_explicit:
himmelma@33175
  1777
  fixes p :: "'a::real_vector set"
himmelma@33175
  1778
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
wenzelm@50804
  1779
    (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
wenzelm@50804
  1780
proof -
wenzelm@53302
  1781
  {
wenzelm@53302
  1782
    fix x
wenzelm@53302
  1783
    assume "x\<in>?lhs"
wenzelm@50804
  1784
    then obtain k u y where
wenzelm@50804
  1785
        obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
himmelma@33175
  1786
      unfolding convex_hull_indexed by auto
himmelma@33175
  1787
wenzelm@50804
  1788
    have fin: "finite {1..k}" by auto
wenzelm@50804
  1789
    have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
wenzelm@53302
  1790
    {
wenzelm@53302
  1791
      fix j
wenzelm@50804
  1792
      assume "j\<in>{1..k}"
wenzelm@50804
  1793
      then have "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
wenzelm@50804
  1794
        using obt(1)[THEN bspec[where x=j]] and obt(2)
wenzelm@50804
  1795
        apply simp
wenzelm@50804
  1796
        apply (rule setsum_nonneg)
wenzelm@50804
  1797
        using obt(1)
wenzelm@50804
  1798
        apply auto
wenzelm@50804
  1799
        done
wenzelm@50804
  1800
    }
himmelma@33175
  1801
    moreover
wenzelm@49531
  1802
    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
wenzelm@49530
  1803
      unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
himmelma@33175
  1804
    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
wenzelm@49530
  1805
      using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
himmelma@33175
  1806
      unfolding scaleR_left.setsum using obt(3) by auto
wenzelm@50804
  1807
    ultimately
wenzelm@50804
  1808
    have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@50804
  1809
      apply (rule_tac x="y ` {1..k}" in exI)
wenzelm@50804
  1810
      apply (rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI)
wenzelm@50804
  1811
      apply auto
wenzelm@50804
  1812
      done
wenzelm@50804
  1813
    then have "x\<in>?rhs" by auto
wenzelm@50804
  1814
  }
himmelma@33175
  1815
  moreover
wenzelm@53302
  1816
  {
wenzelm@53302
  1817
    fix y
wenzelm@53302
  1818
    assume "y\<in>?rhs"
wenzelm@50804
  1819
    then obtain s u where
wenzelm@50804
  1820
      obt: "finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
wenzelm@50804
  1821
wenzelm@50804
  1822
    obtain f where f: "inj_on f {1..card s}" "f ` {1..card s} = s"
wenzelm@50804
  1823
      using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
wenzelm@50804
  1824
wenzelm@53302
  1825
    {
wenzelm@53302
  1826
      fix i :: nat
wenzelm@50804
  1827
      assume "i\<in>{1..card s}"
wenzelm@50804
  1828
      then have "f i \<in> s"
wenzelm@50804
  1829
        apply (subst f(2)[symmetric])
wenzelm@50804
  1830
        apply auto
wenzelm@50804
  1831
        done
wenzelm@50804
  1832
      then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
wenzelm@50804
  1833
    }
himmelma@33175
  1834
    moreover have *:"finite {1..card s}" by auto
wenzelm@53302
  1835
    {
wenzelm@53302
  1836
      fix y
wenzelm@50804
  1837
      assume "y\<in>s"
wenzelm@53302
  1838
      then obtain i where "i\<in>{1..card s}" "f i = y"
wenzelm@53302
  1839
        using f using image_iff[of y f "{1..card s}"]
wenzelm@50804
  1840
        by auto
wenzelm@50804
  1841
      then have "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}"
wenzelm@50804
  1842
        apply auto
wenzelm@50804
  1843
        using f(1)[unfolded inj_on_def]
wenzelm@50804
  1844
        apply(erule_tac x=x in ballE)
wenzelm@50804
  1845
        apply auto
wenzelm@50804
  1846
        done
wenzelm@50804
  1847
      then have "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
wenzelm@50804
  1848
      then have "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
wenzelm@50804
  1849
          "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
wenzelm@50804
  1850
        by (auto simp add: setsum_constant_scaleR)
wenzelm@50804
  1851
    }
wenzelm@50804
  1852
    then have "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
wenzelm@49531
  1853
      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
himmelma@33175
  1854
      unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
wenzelm@50804
  1855
      using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u]
wenzelm@53302
  1856
      unfolding obt(4,5)
wenzelm@53302
  1857
      by auto
wenzelm@50804
  1858
    ultimately
wenzelm@50804
  1859
    have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and>
wenzelm@50804
  1860
        (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
wenzelm@50804
  1861
      apply (rule_tac x="card s" in exI)
wenzelm@50804
  1862
      apply (rule_tac x="u \<circ> f" in exI)
wenzelm@50804
  1863
      apply (rule_tac x=f in exI)
wenzelm@50804
  1864
      apply fastforce
wenzelm@50804
  1865
      done
wenzelm@53302
  1866
    then have "y \<in> ?lhs"
wenzelm@53302
  1867
      unfolding convex_hull_indexed by auto
wenzelm@50804
  1868
  }
wenzelm@53302
  1869
  ultimately show ?thesis
wenzelm@53302
  1870
    unfolding set_eq_iff by blast
himmelma@33175
  1871
qed
himmelma@33175
  1872
wenzelm@50804
  1873
huffman@44467
  1874
subsubsection {* A stepping theorem for that expansion *}
himmelma@33175
  1875
himmelma@33175
  1876
lemma convex_hull_finite_step:
wenzelm@50804
  1877
  fixes s :: "'a::real_vector set"
wenzelm@50804
  1878
  assumes "finite s"
wenzelm@53302
  1879
  shows
wenzelm@53302
  1880
    "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
wenzelm@53302
  1881
      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)"
wenzelm@53302
  1882
  (is "?lhs = ?rhs")
wenzelm@50804
  1883
proof (rule, case_tac[!] "a\<in>s")
wenzelm@53302
  1884
  assume "a \<in> s"
wenzelm@50804
  1885
  then have *:" insert a s = s" by auto
wenzelm@50804
  1886
  assume ?lhs
wenzelm@50804
  1887
  then show ?rhs
wenzelm@50804
  1888
    unfolding *
wenzelm@50804
  1889
    apply (rule_tac x=0 in exI)
wenzelm@50804
  1890
    apply auto
wenzelm@50804
  1891
    done
himmelma@33175
  1892
next
wenzelm@50804
  1893
  assume ?lhs
wenzelm@53302
  1894
  then obtain u where
wenzelm@53302
  1895
      u: "\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
wenzelm@50804
  1896
    by auto
wenzelm@50804
  1897
  assume "a \<notin> s"
wenzelm@50804
  1898
  then show ?rhs
wenzelm@50804
  1899
    apply (rule_tac x="u a" in exI)
wenzelm@50804
  1900
    using u(1)[THEN bspec[where x=a]]
wenzelm@50804
  1901
    apply simp
wenzelm@50804
  1902
    apply (rule_tac x=u in exI)
wenzelm@50804
  1903
    using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s`
wenzelm@50804
  1904
    apply auto
wenzelm@50804
  1905
    done
himmelma@33175
  1906
next
wenzelm@50804
  1907
  assume "a \<in> s"
wenzelm@50804
  1908
  then have *: "insert a s = s" by auto
wenzelm@50804
  1909
  have fin: "finite (insert a s)" using assms by auto
wenzelm@50804
  1910
  assume ?rhs
wenzelm@50804
  1911
  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
wenzelm@50804
  1912
    by auto
wenzelm@50804
  1913
  show ?lhs
wenzelm@50804
  1914
    apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
wenzelm@50804
  1915
    unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
wenzelm@50804
  1916
    unfolding setsum_clauses(2)[OF assms]
wenzelm@50804
  1917
    using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s`
wenzelm@50804
  1918
    apply auto
wenzelm@50804
  1919
    done
himmelma@33175
  1920
next
wenzelm@50804
  1921
  assume ?rhs
wenzelm@50804
  1922
  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
wenzelm@50804
  1923
    by auto
wenzelm@50804
  1924
  moreover
wenzelm@50804
  1925
  assume "a \<notin> s"
wenzelm@50804
  1926
  moreover
wenzelm@53302
  1927
  have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s"
wenzelm@53302
  1928
    and "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
wenzelm@50804
  1929
    apply (rule_tac setsum_cong2)
wenzelm@50804
  1930
    defer
wenzelm@50804
  1931
    apply (rule_tac setsum_cong2)
wenzelm@50804
  1932
    using `a \<notin> s`
wenzelm@50804
  1933
    apply auto
wenzelm@50804
  1934
    done
wenzelm@50804
  1935
  ultimately show ?lhs
wenzelm@50804
  1936
    apply (rule_tac x="\<lambda>x. if a = x then v else u x" in exI)
wenzelm@50804
  1937
    unfolding setsum_clauses(2)[OF assms]
wenzelm@50804
  1938
    apply auto
wenzelm@50804
  1939
    done
wenzelm@50804
  1940
qed
wenzelm@50804
  1941
himmelma@33175
  1942
huffman@44467
  1943
subsubsection {* Hence some special cases *}
himmelma@33175
  1944
himmelma@33175
  1945
lemma convex_hull_2:
himmelma@33175
  1946
  "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
wenzelm@53302
  1947
proof -
wenzelm@53302
  1948
  have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
wenzelm@53302
  1949
    by auto
wenzelm@53302
  1950
  have **: "finite {b}" by auto
wenzelm@53302
  1951
  show ?thesis
wenzelm@53302
  1952
    apply (simp add: convex_hull_finite)
wenzelm@53302
  1953
    unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
wenzelm@53302
  1954
    apply auto
wenzelm@53302
  1955
    apply (rule_tac x=v in exI)
wenzelm@53302
  1956
    apply (rule_tac x="1 - v" in exI)
wenzelm@53302
  1957
    apply simp
wenzelm@53302
  1958
    apply (rule_tac x=u in exI)
wenzelm@53302
  1959
    apply simp
wenzelm@53302
  1960
    apply (rule_tac x="\<lambda>x. v" in exI)
wenzelm@53302
  1961
    apply simp
wenzelm@53302
  1962
    done
wenzelm@53302
  1963
qed
himmelma@33175
  1964
himmelma@33175
  1965
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
huffman@44170
  1966
  unfolding convex_hull_2
wenzelm@53302
  1967
proof (rule Collect_cong)
wenzelm@53302
  1968
  have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
wenzelm@53302
  1969
    by auto
wenzelm@53302
  1970
  fix x
wenzelm@53302
  1971
  show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
wenzelm@53302
  1972
    (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
wenzelm@53302
  1973
    unfolding *
wenzelm@53302
  1974
    apply auto
wenzelm@53302
  1975
    apply (rule_tac[!] x=u in exI)
wenzelm@53302
  1976
    apply (auto simp add: algebra_simps)
wenzelm@53302
  1977
    done
wenzelm@53302
  1978
qed
himmelma@33175
  1979
himmelma@33175
  1980
lemma convex_hull_3:
himmelma@33175
  1981
  "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
wenzelm@53302
  1982
proof -
wenzelm@53302
  1983
  have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
wenzelm@53302
  1984
    by auto
wenzelm@53302
  1985
  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
huffman@44361
  1986
    by (auto simp add: field_simps)
wenzelm@53302
  1987
  show ?thesis
wenzelm@53302
  1988
    unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
wenzelm@53302
  1989
    unfolding convex_hull_finite_step[OF fin(3)]
wenzelm@53302
  1990
    apply (rule Collect_cong)
wenzelm@53302
  1991
    apply simp
wenzelm@53302
  1992
    apply auto
wenzelm@53302
  1993
    apply (rule_tac x=va in exI)
wenzelm@53302
  1994
    apply (rule_tac x="u c" in exI)
wenzelm@53302
  1995
    apply simp
wenzelm@53302
  1996
    apply (rule_tac x="1 - v - w" in exI)
wenzelm@53302
  1997
    apply simp
wenzelm@53302
  1998
    apply (rule_tac x=v in exI)
wenzelm@53302
  1999
    apply simp
wenzelm@53302
  2000
    apply (rule_tac x="\<lambda>x. w" in exI)
wenzelm@53302
  2001
    apply simp
wenzelm@53302
  2002
    done
wenzelm@53302
  2003
qed
himmelma@33175
  2004
himmelma@33175
  2005
lemma convex_hull_3_alt:
himmelma@33175
  2006
  "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
wenzelm@53302
  2007
proof -
wenzelm@53302
  2008
  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
wenzelm@53302
  2009
    by auto
wenzelm@53302
  2010
  show ?thesis
wenzelm@53302
  2011
    unfolding convex_hull_3
wenzelm@53302
  2012
    apply (auto simp add: *)
wenzelm@53302
  2013
    apply (rule_tac x=v in exI)
wenzelm@53302
  2014
    apply (rule_tac x=w in exI)
wenzelm@53302
  2015
    apply (simp add: algebra_simps)
wenzelm@53302
  2016
    apply (rule_tac x=u in exI)
wenzelm@53302
  2017
    apply (rule_tac x=v in exI)
wenzelm@53302
  2018
    apply (simp add: algebra_simps)
wenzelm@53302
  2019
    done
wenzelm@53302
  2020
qed
wenzelm@53302
  2021
himmelma@33175
  2022
huffman@44467
  2023
subsection {* Relations among closure notions and corresponding hulls *}
himmelma@33175
  2024
himmelma@33175
  2025
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
himmelma@33175
  2026
  unfolding affine_def convex_def by auto
himmelma@33175
  2027
huffman@44361
  2028
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
himmelma@33175
  2029
  using subspace_imp_affine affine_imp_convex by auto
himmelma@33175
  2030
huffman@44361
  2031
lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
wenzelm@53302
  2032
  by (metis hull_minimal span_inc subspace_imp_affine subspace_span)
himmelma@33175
  2033
huffman@44361
  2034
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
wenzelm@53302
  2035
  by (metis hull_minimal span_inc subspace_imp_convex subspace_span)
himmelma@33175
  2036
himmelma@33175
  2037
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
wenzelm@53302
  2038
  by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
wenzelm@53302
  2039
wenzelm@53302
  2040
wenzelm@53302
  2041
lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
wenzelm@49531
  2042
  unfolding affine_dependent_def dependent_def
himmelma@33175
  2043
  using affine_hull_subset_span by auto
himmelma@33175
  2044
himmelma@33175
  2045
lemma dependent_imp_affine_dependent:
wenzelm@53302
  2046
  assumes "dependent {x - a| x . x \<in> s}"
wenzelm@53302
  2047
    and "a \<notin> s"
himmelma@33175
  2048
  shows "affine_dependent (insert a s)"
wenzelm@53302
  2049
proof -
wenzelm@49531
  2050
  from assms(1)[unfolded dependent_explicit] obtain S u v
himmelma@33175
  2051
    where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
himmelma@33175
  2052
  def t \<equiv> "(\<lambda>x. x + a) ` S"
himmelma@33175
  2053
wenzelm@53302
  2054
  have inj:"inj_on (\<lambda>x. x + a) S"
wenzelm@53302
  2055
    unfolding inj_on_def by auto
wenzelm@53302
  2056
  have "0 \<notin> S"
wenzelm@53302
  2057
    using obt(2) assms(2) unfolding subset_eq by auto
wenzelm@53302
  2058
  have fin: "finite t" and  "t \<subseteq> s"
wenzelm@53302
  2059
    unfolding t_def using obt(1,2) by auto
wenzelm@53302
  2060
  then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
wenzelm@53302
  2061
    by auto
wenzelm@53302
  2062
  moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
wenzelm@53302
  2063
    apply (rule setsum_cong2)
wenzelm@53302
  2064
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2065
    apply auto
wenzelm@53302
  2066
    done
himmelma@33175
  2067
  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
wenzelm@53302
  2068
    unfolding setsum_clauses(2)[OF fin]
wenzelm@53302
  2069
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2070
    apply auto
wenzelm@53302
  2071
    unfolding *
wenzelm@53302
  2072
    apply auto
wenzelm@53302
  2073
    done
himmelma@33175
  2074
  moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
wenzelm@53302
  2075
    apply (rule_tac x="v + a" in bexI)
wenzelm@53302
  2076
    using obt(3,4) and `0\<notin>S`
wenzelm@53302
  2077
    unfolding t_def
wenzelm@53302
  2078
    apply auto
wenzelm@53302
  2079
    done
wenzelm@53302
  2080
  moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
wenzelm@53302
  2081
    apply (rule setsum_cong2)
wenzelm@53302
  2082
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2083
    apply auto
wenzelm@53302
  2084
    done
wenzelm@49531
  2085
  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
wenzelm@53302
  2086
    unfolding scaleR_left.setsum
wenzelm@53302
  2087
    unfolding t_def and setsum_reindex[OF inj] and o_def
wenzelm@53302
  2088
    using obt(5)
wenzelm@53302
  2089
    by (auto simp add: setsum_addf scaleR_right_distrib)
wenzelm@53302
  2090
  then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
wenzelm@53302
  2091
    unfolding setsum_clauses(2)[OF fin]
wenzelm@53302
  2092
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2093
    by (auto simp add: *)
wenzelm@53302
  2094
  ultimately show ?thesis
wenzelm@53302
  2095
    unfolding affine_dependent_explicit
wenzelm@53302
  2096
    apply (rule_tac x="insert a t" in exI)
wenzelm@53302
  2097
    apply auto
wenzelm@53302
  2098
    done
himmelma@33175
  2099
qed
himmelma@33175
  2100
himmelma@33175
  2101
lemma convex_cone:
wenzelm@53302
  2102
  "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
wenzelm@53302
  2103
  (is "?lhs = ?rhs")
wenzelm@53302
  2104
proof -
wenzelm@53302
  2105
  {
wenzelm@53302
  2106
    fix x y
wenzelm@53302
  2107
    assume "x\<in>s" "y\<in>s" and ?lhs
wenzelm@53302
  2108
    then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
wenzelm@53302
  2109
      unfolding cone_def by auto
wenzelm@53302
  2110
    then have "x + y \<in> s"
wenzelm@53302
  2111
      using `?lhs`[unfolded convex_def, THEN conjunct1]
wenzelm@53302
  2112
      apply (erule_tac x="2*\<^sub>R x" in ballE)
wenzelm@53302
  2113
      apply (erule_tac x="2*\<^sub>R y" in ballE)
wenzelm@53302
  2114
      apply (erule_tac x="1/2" in allE)
wenzelm@53302
  2115
      apply simp
wenzelm@53302
  2116
      apply (erule_tac x="1/2" in allE)
wenzelm@53302
  2117
      apply auto
wenzelm@53302
  2118
      done
wenzelm@53302
  2119
  }
wenzelm@53302
  2120
  then show ?thesis
wenzelm@53302
  2121
    unfolding convex_def cone_def by blast
wenzelm@53302
  2122
qed
wenzelm@53302
  2123
wenzelm@53302
  2124
lemma affine_dependent_biggerset:
wenzelm@53302
  2125
  fixes s::"('a::euclidean_space) set"
hoelzl@37489
  2126
  assumes "finite s" "card s \<ge> DIM('a) + 2"
himmelma@33175
  2127
  shows "affine_dependent s"
wenzelm@53302
  2128
proof -
wenzelm@53302
  2129
  have "s \<noteq> {}" using assms by auto
wenzelm@53302
  2130
  then obtain a where "a\<in>s" by auto
wenzelm@53302
  2131
  have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
wenzelm@53302
  2132
    by auto
wenzelm@53302
  2133
  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
wenzelm@53302
  2134
    unfolding *
wenzelm@53302
  2135
    apply (rule card_image)
wenzelm@53302
  2136
    unfolding inj_on_def
wenzelm@53302
  2137
    apply auto
wenzelm@53302
  2138
    done
hoelzl@37489
  2139
  also have "\<dots> > DIM('a)" using assms(2)
himmelma@33175
  2140
    unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
wenzelm@53302
  2141
  finally show ?thesis
wenzelm@53302
  2142
    apply (subst insert_Diff[OF `a\<in>s`, symmetric])
wenzelm@53302
  2143
    apply (rule dependent_imp_affine_dependent)
wenzelm@53302
  2144
    apply (rule dependent_biggerset)
wenzelm@53302
  2145
    apply auto
wenzelm@53302
  2146
    done
wenzelm@53302
  2147
qed
himmelma@33175
  2148
himmelma@33175
  2149
lemma affine_dependent_biggerset_general:
hoelzl@37489
  2150
  assumes "finite (s::('a::euclidean_space) set)" "card s \<ge> dim s + 2"
himmelma@33175
  2151
  shows "affine_dependent s"
wenzelm@53302
  2152
proof -
himmelma@33175
  2153
  from assms(2) have "s \<noteq> {}" by auto
himmelma@33175
  2154
  then obtain a where "a\<in>s" by auto
wenzelm@53302
  2155
  have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
wenzelm@53302
  2156
    by auto
wenzelm@53302
  2157
  have **: "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
wenzelm@53302
  2158
    unfolding *
wenzelm@53302
  2159
    apply (rule card_image)
wenzelm@53302
  2160
    unfolding inj_on_def
wenzelm@53302
  2161
    apply auto
wenzelm@53302
  2162
    done
himmelma@33175
  2163
  have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
wenzelm@53302
  2164
    apply (rule subset_le_dim)
wenzelm@53302
  2165
    unfolding subset_eq
wenzelm@53302
  2166
    using `a\<in>s`
wenzelm@53302
  2167
    apply (auto simp add:span_superset span_sub)
wenzelm@53302
  2168
    done
himmelma@33175
  2169
  also have "\<dots> < dim s + 1" by auto
wenzelm@53302
  2170
  also have "\<dots> \<le> card (s - {a})"
wenzelm@53302
  2171
    using assms
wenzelm@53302
  2172
    using card_Diff_singleton[OF assms(1) `a\<in>s`]
wenzelm@53302
  2173
    by auto
wenzelm@53302
  2174
  finally show ?thesis
wenzelm@53302
  2175
    apply (subst insert_Diff[OF `a\<in>s`, symmetric])
wenzelm@53302
  2176
    apply (rule dependent_imp_affine_dependent)
wenzelm@53302
  2177
    apply (rule dependent_biggerset_general)
wenzelm@53302
  2178
    unfolding **
wenzelm@53302
  2179
    apply auto
wenzelm@53302
  2180
    done
wenzelm@53302
  2181
qed
wenzelm@53302
  2182
himmelma@33175
  2183
himmelma@33175
  2184
subsection {* Caratheodory's theorem. *}
himmelma@33175
  2185
wenzelm@53302
  2186
lemma convex_hull_caratheodory:
wenzelm@53302
  2187
  fixes p :: "('a::euclidean_space) set"
wenzelm@53302
  2188
  shows "convex hull p =
wenzelm@53302
  2189
    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
wenzelm@53302
  2190
      (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
nipkow@39302
  2191
  unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
wenzelm@53302
  2192
proof (rule, rule)
wenzelm@53302
  2193
  fix y
wenzelm@53302
  2194
  let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
wenzelm@53302
  2195
    setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
  2196
  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
  2197
  then obtain N where "?P N" by auto
wenzelm@53302
  2198
  then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
wenzelm@53302
  2199
    apply (rule_tac ex_least_nat_le)
wenzelm@53302
  2200
    apply auto
wenzelm@53302
  2201
    done
wenzelm@53302
  2202
  then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
wenzelm@53302
  2203
    by blast
wenzelm@53302
  2204
  then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
wenzelm@53302
  2205
    "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
wenzelm@53302
  2206
wenzelm@53302
  2207
  have "card s \<le> DIM('a) + 1"
wenzelm@53302
  2208
  proof (rule ccontr, simp only: not_le)
hoelzl@37489
  2209
    assume "DIM('a) + 1 < card s"
wenzelm@53302
  2210
    then have "affine_dependent s"
wenzelm@53302
  2211
      using affine_dependent_biggerset[OF obt(1)] by auto
wenzelm@53302
  2212
    then obtain w v where wv: "setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
himmelma@33175
  2213
      using affine_dependent_explicit_finite[OF obt(1)] by auto
wenzelm@53302
  2214
    def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
wenzelm@53302
  2215
    def t \<equiv> "Min i"
wenzelm@53302
  2216
    have "\<exists>x\<in>s. w x < 0"
wenzelm@53302
  2217
    proof (rule ccontr, simp add: not_less)
himmelma@33175
  2218
      assume as:"\<forall>x\<in>s. 0 \<le> w x"
wenzelm@53302
  2219
      then have "setsum w (s - {v}) \<ge> 0"
wenzelm@53302
  2220
        apply (rule_tac setsum_nonneg)
wenzelm@53302
  2221
        apply auto
wenzelm@53302
  2222
        done
wenzelm@53302
  2223
      then have "setsum w s > 0"
wenzelm@53302
  2224
        unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
wenzelm@53302
  2225
        using as[THEN bspec[where x=v]] and `v\<in>s`
wenzelm@53302
  2226
        using `w v \<noteq> 0`
wenzelm@53302
  2227
        by auto
wenzelm@53302
  2228
      then show False using wv(1) by auto
wenzelm@53302
  2229
    qed
wenzelm@53302
  2230
    then have "i \<noteq> {}" unfolding i_def by auto
wenzelm@53302
  2231
wenzelm@53302
  2232
    then have "t \<ge> 0"
wenzelm@53302
  2233
      using Min_ge_iff[of i 0 ] and obt(1)
wenzelm@53302
  2234
      unfolding t_def i_def
wenzelm@53302
  2235
      using obt(4)[unfolded le_less]
wenzelm@53302
  2236
      apply auto
wenzelm@53302
  2237
      unfolding divide_le_0_iff
wenzelm@53302
  2238
      apply auto
wenzelm@53302
  2239
      done
wenzelm@53302
  2240
    have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
wenzelm@53302
  2241
    proof
wenzelm@53302
  2242
      fix v
wenzelm@53302
  2243
      assume "v \<in> s"
wenzelm@53302
  2244
      then have v: "0 \<le> u v"
wenzelm@53302
  2245
        using obt(4)[THEN bspec[where x=v]] by auto
wenzelm@53302
  2246
      show "0 \<le> u v + t * w v"
wenzelm@53302
  2247
      proof (cases "w v < 0")
wenzelm@53302
  2248
        case False
wenzelm@53302
  2249
        then show ?thesis
wenzelm@53302
  2250
          apply (rule_tac add_nonneg_nonneg)
wenzelm@53302
  2251
          using v
wenzelm@53302
  2252
          apply simp
wenzelm@53302
  2253
          apply (rule mult_nonneg_nonneg)
wenzelm@53302
  2254
          using `t\<ge>0`
wenzelm@53302
  2255
          apply auto
wenzelm@53302
  2256
          done
wenzelm@53302
  2257
      next
wenzelm@53302
  2258
        case True
wenzelm@53302
  2259
        then have "t \<le> u v / (- w v)"
wenzelm@53302
  2260
          using `v\<in>s`
wenzelm@53302
  2261
          unfolding t_def i_def
wenzelm@53302
  2262
          apply (rule_tac Min_le)
wenzelm@53302
  2263
          using obt(1)
wenzelm@53302
  2264
          apply auto
wenzelm@53302
  2265
          done
wenzelm@53302
  2266
        then show ?thesis
wenzelm@53302
  2267
          unfolding real_0_le_add_iff
wenzelm@53302
  2268
          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
wenzelm@53302
  2269
          by auto
wenzelm@53302
  2270
      qed
wenzelm@53302
  2271
    qed
wenzelm@53302
  2272
wenzelm@53302
  2273
    obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
himmelma@33175
  2274
      using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
wenzelm@53302
  2275
    then have a: "a \<in> s" "u a + t * w a = 0" by auto
wenzelm@53302
  2276
    have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
wenzelm@49531
  2277
      unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto
himmelma@33175
  2278
    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
wenzelm@49530
  2279
      unfolding setsum_addf wv(1) setsum_right_distrib[symmetric] obt(5) by auto
wenzelm@49531
  2280
    moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
wenzelm@49530
  2281
      unfolding setsum_addf obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
hoelzl@37489
  2282
      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
wenzelm@53302
  2283
    ultimately have "?P (n - 1)"
wenzelm@53302
  2284
      apply (rule_tac x="(s - {a})" in exI)
wenzelm@53302
  2285
      apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
wenzelm@53302
  2286
      using obt(1-3) and t and a
wenzelm@53302
  2287
      apply (auto simp add: * scaleR_left_distrib)
wenzelm@53302
  2288
      done
wenzelm@53302
  2289
    then show False
wenzelm@53302
  2290
      using smallest[THEN spec[where x="n - 1"]] by auto
wenzelm@53302
  2291
  qed
wenzelm@53302
  2292
  then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
wenzelm@53302
  2293
    (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" using obt by auto
himmelma@33175
  2294
qed auto
himmelma@33175
  2295
himmelma@33175
  2296
lemma caratheodory:
wenzelm@53302
  2297
  "convex hull p =
wenzelm@53302
  2298
    {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
hoelzl@37489
  2299
      card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
wenzelm@53302
  2300
  unfolding set_eq_iff
wenzelm@53302
  2301
  apply rule
wenzelm@53302
  2302
  apply rule
wenzelm@53302
  2303
  unfolding mem_Collect_eq
wenzelm@53302
  2304
proof -
wenzelm@53302
  2305
  fix x
wenzelm@53302
  2306
  assume "x \<in> convex hull p"
hoelzl@37489
  2307
  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
wenzelm@53302
  2308
    "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@53302
  2309
    unfolding convex_hull_caratheodory by auto
wenzelm@53302
  2310
  then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
wenzelm@53302
  2311
    apply (rule_tac x=s in exI)
wenzelm@53302
  2312
    using hull_subset[of s convex]
wenzelm@53302
  2313
    using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]]
wenzelm@53302
  2314
    apply auto
wenzelm@53302
  2315
    done
himmelma@33175
  2316
next
wenzelm@53302
  2317
  fix x
wenzelm@53302
  2318
  assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
wenzelm@53302
  2319
  then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
wenzelm@53302
  2320
    by auto
wenzelm@53302
  2321
  then show "x \<in> convex hull p"
wenzelm@53302
  2322
    using hull_mono[OF `s\<subseteq>p`] by auto
himmelma@33175
  2323
qed
himmelma@33175
  2324
hoelzl@40377
  2325
hoelzl@40377
  2326
subsection {* Some Properties of Affine Dependent Sets *}
hoelzl@40377
  2327
hoelzl@40377
  2328
lemma affine_independent_empty: "~(affine_dependent {})"
hoelzl@40377
  2329
  by (simp add: affine_dependent_def)
hoelzl@40377
  2330
wenzelm@53302
  2331
lemma affine_independent_sing: "\<not> affine_dependent {a}"
wenzelm@53302
  2332
  by (simp add: affine_dependent_def)
wenzelm@53302
  2333
wenzelm@53302
  2334
lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2335
proof -
wenzelm@53302
  2336
  have "affine ((\<lambda>x. a + x) ` (affine hull S))"
wenzelm@53302
  2337
    using affine_translation affine_affine_hull by auto
wenzelm@53302
  2338
  moreover have "(\<lambda>x. a + x) `  S <= (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2339
    using hull_subset[of S] by auto
wenzelm@53302
  2340
  ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) <= (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2341
    by (metis hull_minimal)
wenzelm@53302
  2342
  have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
wenzelm@53302
  2343
    using affine_translation affine_affine_hull by auto
wenzelm@53302
  2344
  moreover have "(\<lambda>x. -a + x) ` (%x. a + x) `  S <= (\<lambda>x. -a + x) ` (affine hull ((%x. a + x) `  S))"
wenzelm@53302
  2345
    using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
wenzelm@53302
  2346
  moreover have "S = (\<lambda>x. -a + x) ` (%x. a + x) `  S"
wenzelm@53302
  2347
    using translation_assoc[of "-a" a] by auto
wenzelm@53302
  2348
  ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
wenzelm@53302
  2349
    by (metis hull_minimal)
wenzelm@53302
  2350
  then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2351
    by auto
wenzelm@53302
  2352
  from this show ?thesis using h1 by auto
hoelzl@40377
  2353
qed
hoelzl@40377
  2354
hoelzl@40377
  2355
lemma affine_dependent_translation:
hoelzl@40377
  2356
  assumes "affine_dependent S"
hoelzl@40377
  2357
  shows "affine_dependent ((%x. a + x) ` S)"
wenzelm@53302
  2358
proof -
wenzelm@53302
  2359
  obtain x where x_def: "x : S & x : affine hull (S - {x})"
wenzelm@53302
  2360
    using assms affine_dependent_def by auto
wenzelm@53302
  2361
  have "op + a ` (S - {x}) = op + a ` S - {a + x}"
wenzelm@53302
  2362
    by auto
wenzelm@53302
  2363
  then have "a+x \<in> affine hull ((%x. a + x) ` S - {a+x})"
wenzelm@53302
  2364
    using affine_hull_translation[of a "S-{x}"] x_def by auto
wenzelm@53302
  2365
  moreover have "a+x : (\<lambda>x. a + x) ` S"
wenzelm@53302
  2366
    using x_def by auto
wenzelm@53302
  2367
  ultimately show ?thesis
wenzelm@53302
  2368
    unfolding affine_dependent_def by auto
hoelzl@40377
  2369
qed
hoelzl@40377
  2370
hoelzl@40377
  2371
lemma affine_dependent_translation_eq:
hoelzl@40377
  2372
  "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"
wenzelm@53302
  2373
proof -
wenzelm@53302
  2374
  {
wenzelm@53302
  2375
    assume "affine_dependent ((%x. a + x) ` S)"
wenzelm@53302
  2376
    then have "affine_dependent S"
wenzelm@53302
  2377
      using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
wenzelm@53302
  2378
      by auto
wenzelm@53302
  2379
  }
wenzelm@53302
  2380
  then show ?thesis
wenzelm@53302
  2381
    using affine_dependent_translation by auto
hoelzl@40377
  2382
qed
hoelzl@40377
  2383
hoelzl@40377
  2384
lemma affine_hull_0_dependent:
hoelzl@40377
  2385
  assumes "0 : affine hull S"
hoelzl@40377
  2386
  shows "dependent S"
wenzelm@53302
  2387
proof -
wenzelm@53302
  2388
  obtain s u where s_u_def: "finite s & s ~= {} & s <= S & setsum u s = 1 & (SUM v:s. u v *\<^sub>R v) = 0"
wenzelm@53302
  2389
    using assms affine_hull_explicit[of S] by auto
wenzelm@53302
  2390
  then have "EX v:s. u v \<noteq> 0"
wenzelm@53302
  2391
    using setsum_not_0[of "u" "s"] by auto
wenzelm@53302
  2392
  then have "finite s & s <= S & (EX v:s. u v ~= 0 & (SUM v:s. u v *\<^sub>R v) = 0)"
wenzelm@53302
  2393
    using s_u_def by auto
wenzelm@53302
  2394
  then show ?thesis
wenzelm@53302
  2395
    unfolding dependent_explicit[of S] by auto
hoelzl@40377
  2396
qed
hoelzl@40377
  2397
hoelzl@40377
  2398
lemma affine_dependent_imp_dependent2:
hoelzl@40377
  2399
  assumes "affine_dependent (insert 0 S)"
hoelzl@40377
  2400
  shows "dependent S"
wenzelm@53302
  2401
proof -
wenzelm@53302
  2402
  obtain x where x_def: "x:insert 0 S & x : affine hull (insert 0 S - {x})"
wenzelm@53302
  2403
    using affine_dependent_def[of "(insert 0 S)"] assms by blast
wenzelm@53302
  2404
  then have "x \<in> span (insert 0 S - {x})"
wenzelm@53302
  2405
    using affine_hull_subset_span by auto
wenzelm@53302
  2406
  moreover have "span (insert 0 S - {x}) = span (S - {x})"
wenzelm@53302
  2407
    using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
wenzelm@53302
  2408
  ultimately have "x \<in> span (S - {x})" by auto
wenzelm@53302
  2409
  then have "x \<noteq> 0 \<Longrightarrow> dependent S"
wenzelm@53302
  2410
    using x_def dependent_def by auto
wenzelm@53302
  2411
  moreover
wenzelm@53302
  2412
  {
wenzelm@53302
  2413
    assume "x = 0"
wenzelm@53302
  2414
    then have "0 \<in> affine hull S"
wenzelm@53302
  2415
      using x_def hull_mono[of "S - {0}" S] by auto
wenzelm@53302
  2416
    then have "dependent S"
wenzelm@53302
  2417
      using affine_hull_0_dependent by auto
wenzelm@53302
  2418
  }
wenzelm@53302
  2419
  ultimately show ?thesis by auto
hoelzl@40377
  2420
qed
hoelzl@40377
  2421
hoelzl@40377
  2422
lemma affine_dependent_iff_dependent:
wenzelm@53302
  2423
  assumes "a \<notin> S"
wenzelm@53302
  2424
  shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
wenzelm@53302
  2425
proof -
wenzelm@53302
  2426
  have "(op + (- a) ` S) = {x - a| x . x : S}" by auto
wenzelm@53302
  2427
  then show ?thesis
wenzelm@53302
  2428
    using affine_dependent_translation_eq[of "(insert a S)" "-a"]
wenzelm@49531
  2429
      affine_dependent_imp_dependent2 assms
wenzelm@53302
  2430
      dependent_imp_affine_dependent[of a S]
wenzelm@53302
  2431
    by auto
hoelzl@40377
  2432
qed
hoelzl@40377
  2433
hoelzl@40377
  2434
lemma affine_dependent_iff_dependent2:
hoelzl@40377
  2435
  assumes "a : S"
hoelzl@40377
  2436
  shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"
wenzelm@53302
  2437
proof -
wenzelm@53302
  2438
  have "insert a (S - {a})=S"
wenzelm@53302
  2439
    using assms by auto
wenzelm@53302
  2440
  then show ?thesis
wenzelm@53302
  2441
    using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
hoelzl@40377
  2442
qed
hoelzl@40377
  2443
hoelzl@40377
  2444
lemma affine_hull_insert_span_gen:
wenzelm@53302
  2445
  "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)"
wenzelm@53302
  2446
proof -
wenzelm@53302
  2447
  have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)"
wenzelm@53302
  2448
    by auto
wenzelm@53302
  2449
  {
wenzelm@53302
  2450
    assume "a \<notin> s"
wenzelm@53302
  2451
    then have ?thesis
wenzelm@53302
  2452
      using affine_hull_insert_span[of a s] h1 by auto
wenzelm@53302
  2453
  }
wenzelm@53302
  2454
  moreover
wenzelm@53302
  2455
  {
wenzelm@53302
  2456
    assume a1: "a \<in> s"
wenzelm@53302
  2457
    have "\<exists>x. x \<in> s & -a+x=0"
wenzelm@53302
  2458
      apply (rule exI[of _ a])
wenzelm@53302
  2459
      using a1
wenzelm@53302
  2460
      apply auto
wenzelm@53302
  2461
      done
wenzelm@53302
  2462
    then have "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s"
wenzelm@53302
  2463
      by auto
wenzelm@53302
  2464
    then have "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)"
wenzelm@53302
  2465
      using span_insert_0[of "op + (- a) ` (s - {a})"] by auto
wenzelm@53302
  2466
    moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))"
wenzelm@53302
  2467
      by auto
wenzelm@53302
  2468
    moreover have "insert a (s - {a})=(insert a s)"