src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author wenzelm
Sun Nov 17 21:46:12 2013 +0100 (2013-11-17)
changeset 54465 2f7867850cc3
parent 54263 c4159fe6fa46
child 54775 2d3df8633dad
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_iff scaleR_add_right)
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lemma linear_scaleR_left: "linear (\<lambda>r. scaleR r x)"
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  by (simp add: linear_iff scaleR_add_left)
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lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
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  by (simp add: inj_on_def)
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lemma linear_add_cmul:
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  assumes "linear f"
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  shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
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  using linear_add[of f] linear_cmul[of f] assms by simp
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lemma mem_convex_alt:
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  assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
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  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
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  apply (rule convexD)
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  using assms
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  apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
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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 \<in> S" "f a \<in> 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 \<in> 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}) \<subseteq> span S"
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      using span_mono[of "S - {a}" S] by auto
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    ultimately have "a \<in> 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 \<Rightarrow> '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: "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 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 span_inc by auto
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  moreover have "(f ` B) \<subseteq> (f ` S)"
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    using B 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 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 \<in> 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 \<in> 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 \<and> 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 "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
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  shows "V \<le> ((\<lambda>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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  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': "linear f' \<and> f \<circ> f' = id \<and> f' \<circ> 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)"
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    using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto
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  then have "f ` f' ` closure (f ` S) \<le> f ` closure S" by auto
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  then have "closure (f ` S) \<le> f ` closure S"
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    using image_compose[of f f' "closure (f ` S)"] f' by auto
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   287
  then show ?thesis using closure_linear_image[of f S] assms by auto
hoelzl@40377
   288
qed
hoelzl@40377
   289
huffman@44524
   290
lemma closure_scaleR:
wenzelm@53339
   291
  fixes S :: "'a::real_normed_vector set"
huffman@44524
   292
  shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
huffman@44524
   293
proof
huffman@44524
   294
  show "(op *\<^sub>R c) ` (closure S) \<subseteq> closure ((op *\<^sub>R c) ` S)"
wenzelm@53333
   295
    using bounded_linear_scaleR_right
wenzelm@53333
   296
    by (rule closure_bounded_linear_image)
huffman@44524
   297
  show "closure ((op *\<^sub>R c) ` S) \<subseteq> (op *\<^sub>R c) ` (closure S)"
wenzelm@49529
   298
    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
wenzelm@49529
   299
qed
wenzelm@49529
   300
wenzelm@49529
   301
lemma fst_linear: "linear fst"
huffman@53600
   302
  unfolding linear_iff by (simp add: algebra_simps)
wenzelm@49529
   303
wenzelm@49529
   304
lemma snd_linear: "linear snd"
huffman@53600
   305
  unfolding linear_iff by (simp add: algebra_simps)
wenzelm@49529
   306
wenzelm@54465
   307
lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
huffman@53600
   308
  unfolding linear_iff by (simp add: algebra_simps)
hoelzl@40377
   309
hoelzl@37489
   310
lemma scaleR_2:
hoelzl@37489
   311
  fixes x :: "'a::real_vector"
hoelzl@37489
   312
  shows "scaleR 2 x = x + x"
wenzelm@49529
   313
  unfolding one_add_one [symmetric] scaleR_left_distrib by simp
wenzelm@49529
   314
wenzelm@49529
   315
lemma vector_choose_size:
wenzelm@53333
   316
  "0 \<le> c \<Longrightarrow> \<exists>x::'a::euclidean_space. norm x = c"
wenzelm@53333
   317
  apply (rule exI [where x="c *\<^sub>R (SOME i. i \<in> Basis)"])
hoelzl@50526
   318
  apply (auto simp: SOME_Basis)
wenzelm@49529
   319
  done
wenzelm@49529
   320
wenzelm@49529
   321
lemma setsum_delta_notmem:
wenzelm@49529
   322
  assumes "x \<notin> s"
himmelma@33175
   323
  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
wenzelm@49529
   324
    and "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
wenzelm@49529
   325
    and "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
wenzelm@49529
   326
    and "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
wenzelm@50804
   327
  apply (rule_tac [!] setsum_cong2)
wenzelm@53333
   328
  using assms
wenzelm@53333
   329
  apply auto
wenzelm@49529
   330
  done
himmelma@33175
   331
himmelma@33175
   332
lemma setsum_delta'':
wenzelm@49529
   333
  fixes s::"'a::real_vector set"
wenzelm@49529
   334
  assumes "finite s"
himmelma@33175
   335
  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
   336
proof -
wenzelm@49529
   337
  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
   338
    by auto
wenzelm@49529
   339
  show ?thesis
wenzelm@49529
   340
    unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
himmelma@33175
   341
qed
himmelma@33175
   342
wenzelm@53333
   343
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
   344
  by auto
himmelma@33175
   345
wenzelm@49529
   346
lemma image_smult_interval:
wenzelm@49529
   347
  "(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
wenzelm@49529
   348
    (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
   349
  using image_affinity_interval[of m 0 a b] by auto
himmelma@33175
   350
himmelma@33175
   351
lemma dist_triangle_eq:
huffman@44361
   352
  fixes x y z :: "'a::real_inner"
wenzelm@53333
   353
  shows "dist x z = dist x y + dist y z \<longleftrightarrow>
wenzelm@53333
   354
    norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
wenzelm@49529
   355
proof -
wenzelm@49529
   356
  have *: "x - y + (y - z) = x - z" by auto
hoelzl@37489
   357
  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
wenzelm@49529
   358
    by (auto simp add:norm_minus_commute)
wenzelm@49529
   359
qed
himmelma@33175
   360
wenzelm@53406
   361
lemma norm_minus_eqI: "x = - y \<Longrightarrow> norm x = norm y" by auto
himmelma@33175
   362
wenzelm@49529
   363
lemma Min_grI:
wenzelm@49529
   364
  assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
wenzelm@49529
   365
  shows "x < Min A"
himmelma@33175
   366
  unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
himmelma@33175
   367
hoelzl@37489
   368
lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
hoelzl@37489
   369
  unfolding norm_eq_sqrt_inner by simp
himmelma@33175
   370
hoelzl@37489
   371
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
hoelzl@37489
   372
  unfolding norm_eq_sqrt_inner by simp
hoelzl@37489
   373
hoelzl@37489
   374
huffman@44467
   375
subsection {* Affine set and affine hull *}
himmelma@33175
   376
wenzelm@49529
   377
definition affine :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@49529
   378
  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
   379
himmelma@33175
   380
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
   381
  unfolding affine_def by (metis eq_diff_eq')
himmelma@33175
   382
himmelma@33175
   383
lemma affine_empty[intro]: "affine {}"
himmelma@33175
   384
  unfolding affine_def by auto
himmelma@33175
   385
himmelma@33175
   386
lemma affine_sing[intro]: "affine {x}"
himmelma@33175
   387
  unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
himmelma@33175
   388
himmelma@33175
   389
lemma affine_UNIV[intro]: "affine UNIV"
himmelma@33175
   390
  unfolding affine_def by auto
himmelma@33175
   391
himmelma@33175
   392
lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
wenzelm@49531
   393
  unfolding affine_def by auto
himmelma@33175
   394
himmelma@33175
   395
lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
himmelma@33175
   396
  unfolding affine_def by auto
himmelma@33175
   397
himmelma@33175
   398
lemma affine_affine_hull: "affine(affine hull s)"
wenzelm@49529
   399
  unfolding hull_def
wenzelm@49529
   400
  using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
himmelma@33175
   401
himmelma@33175
   402
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
wenzelm@49529
   403
  by (metis affine_affine_hull hull_same)
wenzelm@49529
   404
himmelma@33175
   405
huffman@44467
   406
subsubsection {* Some explicit formulations (from Lars Schewe) *}
himmelma@33175
   407
wenzelm@49529
   408
lemma affine:
wenzelm@49529
   409
  fixes V::"'a::real_vector set"
wenzelm@49529
   410
  shows "affine V \<longleftrightarrow>
wenzelm@49529
   411
    (\<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
   412
  unfolding affine_def
wenzelm@49529
   413
  apply rule
wenzelm@49529
   414
  apply(rule, rule, rule)
wenzelm@49531
   415
  apply(erule conjE)+
wenzelm@49529
   416
  defer
wenzelm@49529
   417
  apply (rule, rule, rule, rule, rule)
wenzelm@49529
   418
proof -
wenzelm@49529
   419
  fix x y u v
wenzelm@49529
   420
  assume as: "x \<in> V" "y \<in> V" "u + v = (1::real)"
himmelma@33175
   421
    "\<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
   422
  then show "u *\<^sub>R x + v *\<^sub>R y \<in> V"
wenzelm@49529
   423
    apply (cases "x = y")
wenzelm@49529
   424
    using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]]
wenzelm@49529
   425
      and as(1-3)
wenzelm@53333
   426
    apply (auto simp add: scaleR_left_distrib[symmetric])
wenzelm@53333
   427
    done
himmelma@33175
   428
next
wenzelm@49529
   429
  fix s u
wenzelm@49529
   430
  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
   431
    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
himmelma@33175
   432
  def n \<equiv> "card s"
himmelma@33175
   433
  have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
wenzelm@49529
   434
  then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
wenzelm@49529
   435
  proof (auto simp only: disjE)
wenzelm@49529
   436
    assume "card s = 2"
wenzelm@53333
   437
    then have "card s = Suc (Suc 0)"
wenzelm@53333
   438
      by auto
wenzelm@53333
   439
    then obtain a b where "s = {a, b}"
wenzelm@53333
   440
      unfolding card_Suc_eq by auto
wenzelm@49529
   441
    then show ?thesis
wenzelm@49529
   442
      using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
wenzelm@49529
   443
      by (auto simp add: setsum_clauses(2))
wenzelm@49529
   444
  next
wenzelm@49529
   445
    assume "card s > 2"
wenzelm@49529
   446
    then show ?thesis using as and n_def
wenzelm@49529
   447
    proof (induct n arbitrary: u s)
wenzelm@49529
   448
      case 0
wenzelm@49529
   449
      then show ?case by auto
wenzelm@49529
   450
    next
wenzelm@49529
   451
      case (Suc n)
wenzelm@49529
   452
      fix s :: "'a set" and u :: "'a \<Rightarrow> real"
wenzelm@49529
   453
      assume IA:
wenzelm@49529
   454
        "\<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
   455
          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
   456
        and as:
wenzelm@49529
   457
          "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
   458
           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
wenzelm@49529
   459
      have "\<exists>x\<in>s. u x \<noteq> 1"
wenzelm@49529
   460
      proof (rule ccontr)
wenzelm@49529
   461
        assume "\<not> ?thesis"
wenzelm@53333
   462
        then have "setsum u s = real_of_nat (card s)"
wenzelm@53333
   463
          unfolding card_eq_setsum by auto
wenzelm@49529
   464
        then show False
wenzelm@49529
   465
          using as(7) and `card s > 2`
wenzelm@49529
   466
          by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
huffman@45498
   467
      qed
wenzelm@53339
   468
      then obtain x where x:"x \<in> s" "u x \<noteq> 1" by auto
himmelma@33175
   469
wenzelm@49529
   470
      have c: "card (s - {x}) = card s - 1"
wenzelm@53333
   471
        apply (rule card_Diff_singleton)
wenzelm@53333
   472
        using `x\<in>s` as(4)
wenzelm@53333
   473
        apply auto
wenzelm@53333
   474
        done
wenzelm@49529
   475
      have *: "s = insert x (s - {x})" "finite (s - {x})"
wenzelm@49529
   476
        using `x\<in>s` and as(4) by auto
wenzelm@49529
   477
      have **: "setsum u (s - {x}) = 1 - u x"
wenzelm@49530
   478
        using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
wenzelm@49529
   479
      have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"
wenzelm@49529
   480
        unfolding ** using `u x \<noteq> 1` by auto
wenzelm@49529
   481
      have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V"
wenzelm@49529
   482
      proof (cases "card (s - {x}) > 2")
wenzelm@49529
   483
        case True
wenzelm@49529
   484
        then have "s - {x} \<noteq> {}" "card (s - {x}) = n"
wenzelm@49529
   485
          unfolding c and as(1)[symmetric]
wenzelm@49531
   486
        proof (rule_tac ccontr)
wenzelm@49529
   487
          assume "\<not> s - {x} \<noteq> {}"
wenzelm@49531
   488
          then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
wenzelm@49529
   489
          then show False using True by auto
wenzelm@49529
   490
        qed auto
wenzelm@49529
   491
        then show ?thesis
wenzelm@49529
   492
          apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
wenzelm@53333
   493
          unfolding setsum_right_distrib[symmetric]
wenzelm@53333
   494
          using as and *** and True
wenzelm@49529
   495
          apply auto
wenzelm@49529
   496
          done
wenzelm@49529
   497
      next
wenzelm@49529
   498
        case False
wenzelm@53333
   499
        then have "card (s - {x}) = Suc (Suc 0)"
wenzelm@53333
   500
          using as(2) and c by auto
wenzelm@53333
   501
        then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b"
wenzelm@53333
   502
          unfolding card_Suc_eq by auto
wenzelm@53333
   503
        then show ?thesis
wenzelm@53333
   504
          using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
wenzelm@49529
   505
          using *** *(2) and `s \<subseteq> V`
wenzelm@53333
   506
          unfolding setsum_right_distrib
wenzelm@53333
   507
          by (auto simp add: setsum_clauses(2))
wenzelm@49529
   508
      qed
wenzelm@49529
   509
      then have "u x + (1 - u x) = 1 \<Longrightarrow>
wenzelm@49529
   510
          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
   511
        apply -
wenzelm@49529
   512
        apply (rule as(3)[rule_format])
hoelzl@51524
   513
        unfolding  Real_Vector_Spaces.scaleR_right.setsum
wenzelm@53333
   514
        using x(1) as(6)
wenzelm@53333
   515
        apply auto
wenzelm@49529
   516
        done
wenzelm@49529
   517
      then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
wenzelm@49530
   518
        unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
wenzelm@49529
   519
        apply (subst *)
wenzelm@49529
   520
        unfolding setsum_clauses(2)[OF *(2)]
wenzelm@53333
   521
        using `u x \<noteq> 1`
wenzelm@53333
   522
        apply auto
wenzelm@49529
   523
        done
wenzelm@49529
   524
    qed
wenzelm@49529
   525
  next
wenzelm@49529
   526
    assume "card s = 1"
wenzelm@53333
   527
    then obtain a where "s={a}"
wenzelm@53333
   528
      by (auto simp add: card_Suc_eq)
wenzelm@53333
   529
    then show ?thesis
wenzelm@53333
   530
      using as(4,5) by simp
wenzelm@49529
   531
  qed (insert `s\<noteq>{}` `finite s`, auto)
himmelma@33175
   532
qed
himmelma@33175
   533
himmelma@33175
   534
lemma affine_hull_explicit:
wenzelm@53333
   535
  "affine hull p =
wenzelm@53333
   536
    {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
   537
  apply (rule hull_unique)
wenzelm@49529
   538
  apply (subst subset_eq)
wenzelm@49529
   539
  prefer 3
wenzelm@49529
   540
  apply rule
wenzelm@49529
   541
  unfolding mem_Collect_eq
wenzelm@49529
   542
  apply (erule exE)+
wenzelm@49529
   543
  apply (erule conjE)+
wenzelm@49529
   544
  prefer 2
wenzelm@49529
   545
  apply rule
wenzelm@49529
   546
proof -
wenzelm@49529
   547
  fix x
wenzelm@49529
   548
  assume "x\<in>p"
wenzelm@49529
   549
  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
   550
    apply (rule_tac x="{x}" in exI)
wenzelm@53333
   551
    apply (rule_tac x="\<lambda>x. 1" in exI)
wenzelm@49529
   552
    apply auto
wenzelm@49529
   553
    done
himmelma@33175
   554
next
wenzelm@49529
   555
  fix t x s u
wenzelm@53333
   556
  assume as: "p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}"
wenzelm@53333
   557
    "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   558
  then show "x \<in> t"
wenzelm@53333
   559
    using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]]
wenzelm@53333
   560
    by auto
himmelma@33175
   561
next
wenzelm@49529
   562
  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
   563
    unfolding affine_def
wenzelm@49529
   564
    apply (rule, rule, rule, rule, rule)
wenzelm@49529
   565
    unfolding mem_Collect_eq
wenzelm@49529
   566
  proof -
wenzelm@49529
   567
    fix u v :: real
wenzelm@49529
   568
    assume uv: "u + v = 1"
wenzelm@49529
   569
    fix x
wenzelm@49529
   570
    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
   571
    then obtain sx ux where
wenzelm@53333
   572
      x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x"
wenzelm@53333
   573
      by auto
wenzelm@53333
   574
    fix y
wenzelm@53333
   575
    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
   576
    then obtain sy uy where
wenzelm@49529
   577
      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
   578
    have xy: "finite (sx \<union> sy)"
wenzelm@53333
   579
      using x(1) y(1) by auto
wenzelm@53333
   580
    have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
wenzelm@53333
   581
      by auto
wenzelm@49529
   582
    show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and>
wenzelm@49529
   583
        setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
wenzelm@49529
   584
      apply (rule_tac x="sx \<union> sy" in exI)
wenzelm@49529
   585
      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
   586
      unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left
wenzelm@53333
   587
        ** setsum_restrict_set[OF xy, symmetric]
wenzelm@53333
   588
      unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
wenzelm@53333
   589
        and setsum_right_distrib[symmetric]
wenzelm@49529
   590
      unfolding x y
wenzelm@53333
   591
      using x(1-3) y(1-3) uv
wenzelm@53333
   592
      apply simp
wenzelm@49529
   593
      done
wenzelm@49529
   594
  qed
wenzelm@49529
   595
qed
himmelma@33175
   596
himmelma@33175
   597
lemma affine_hull_finite:
himmelma@33175
   598
  assumes "finite s"
himmelma@33175
   599
  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
wenzelm@53333
   600
  unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq
wenzelm@53333
   601
  apply (rule, rule)
wenzelm@53333
   602
  apply (erule exE)+
wenzelm@53333
   603
  apply (erule conjE)+
wenzelm@49529
   604
  defer
wenzelm@49529
   605
  apply (erule exE)
wenzelm@49529
   606
  apply (erule conjE)
wenzelm@49529
   607
proof -
wenzelm@49529
   608
  fix x u
wenzelm@49529
   609
  assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   610
  then show "\<exists>sa u. finite sa \<and>
wenzelm@49529
   611
      \<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
   612
    apply (rule_tac x=s in exI, rule_tac x=u in exI)
wenzelm@53333
   613
    using assms
wenzelm@53333
   614
    apply auto
wenzelm@49529
   615
    done
himmelma@33175
   616
next
wenzelm@49529
   617
  fix x t u
wenzelm@49529
   618
  assume "t \<subseteq> s"
wenzelm@53333
   619
  then have *: "s \<inter> t = t"
wenzelm@53333
   620
    by auto
himmelma@33175
   621
  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
   622
  then show "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@49529
   623
    apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
wenzelm@49530
   624
    unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, symmetric] and *
wenzelm@49529
   625
    apply auto
wenzelm@49529
   626
    done
wenzelm@49529
   627
qed
wenzelm@49529
   628
himmelma@33175
   629
huffman@44467
   630
subsubsection {* Stepping theorems and hence small special cases *}
himmelma@33175
   631
himmelma@33175
   632
lemma affine_hull_empty[simp]: "affine hull {} = {}"
wenzelm@49529
   633
  by (rule hull_unique) auto
himmelma@33175
   634
himmelma@33175
   635
lemma affine_hull_finite_step:
himmelma@33175
   636
  fixes y :: "'a::real_vector"
wenzelm@49529
   637
  shows
wenzelm@49529
   638
    "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
wenzelm@53347
   639
    and
wenzelm@49529
   640
    "finite s \<Longrightarrow>
wenzelm@49529
   641
      (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
wenzelm@53347
   642
      (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
wenzelm@49529
   643
proof -
himmelma@33175
   644
  show ?th1 by simp
wenzelm@53347
   645
  assume fin: "finite s"
wenzelm@53347
   646
  show "?lhs = ?rhs"
wenzelm@53347
   647
  proof
wenzelm@53302
   648
    assume ?lhs
wenzelm@53302
   649
    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
   650
      by auto
wenzelm@53347
   651
    show ?rhs
wenzelm@49529
   652
    proof (cases "a \<in> s")
wenzelm@49529
   653
      case True
wenzelm@49529
   654
      then have *: "insert a s = s" by auto
wenzelm@53302
   655
      show ?thesis
wenzelm@53302
   656
        using u[unfolded *]
wenzelm@53302
   657
        apply(rule_tac x=0 in exI)
wenzelm@53302
   658
        apply auto
wenzelm@53302
   659
        done
himmelma@33175
   660
    next
wenzelm@49529
   661
      case False
wenzelm@49529
   662
      then show ?thesis
wenzelm@49529
   663
        apply (rule_tac x="u a" in exI)
wenzelm@53347
   664
        using u and fin
wenzelm@53302
   665
        apply auto
wenzelm@49529
   666
        done
wenzelm@53302
   667
    qed
wenzelm@53347
   668
  next
wenzelm@53302
   669
    assume ?rhs
wenzelm@53302
   670
    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
   671
      by auto
wenzelm@53302
   672
    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
   673
      by auto
wenzelm@53347
   674
    show ?lhs
wenzelm@49529
   675
    proof (cases "a \<in> s")
wenzelm@49529
   676
      case True
wenzelm@49529
   677
      then show ?thesis
wenzelm@49529
   678
        apply (rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
wenzelm@53347
   679
        unfolding setsum_clauses(2)[OF fin]
wenzelm@53333
   680
        apply simp
wenzelm@49531
   681
        unfolding scaleR_left_distrib and setsum_addf
himmelma@33175
   682
        unfolding vu and * and scaleR_zero_left
wenzelm@53347
   683
        apply (auto simp add: setsum_delta[OF fin])
wenzelm@49529
   684
        done
himmelma@33175
   685
    next
wenzelm@49531
   686
      case False
wenzelm@49529
   687
      then have **:
wenzelm@49529
   688
        "\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
wenzelm@49529
   689
        "\<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
   690
      from False show ?thesis
wenzelm@49529
   691
        apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
wenzelm@53347
   692
        unfolding setsum_clauses(2)[OF fin] and * using vu
himmelma@33175
   693
        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
   694
        using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)]
wenzelm@49529
   695
        apply auto
wenzelm@49529
   696
        done
wenzelm@49529
   697
    qed
wenzelm@53347
   698
  qed
himmelma@33175
   699
qed
himmelma@33175
   700
himmelma@33175
   701
lemma affine_hull_2:
himmelma@33175
   702
  fixes a b :: "'a::real_vector"
wenzelm@53302
   703
  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
wenzelm@53302
   704
  (is "?lhs = ?rhs")
wenzelm@49529
   705
proof -
wenzelm@49529
   706
  have *:
wenzelm@49531
   707
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
wenzelm@49529
   708
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
himmelma@33175
   709
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
himmelma@33175
   710
    using affine_hull_finite[of "{a,b}"] by auto
himmelma@33175
   711
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
wenzelm@49529
   712
    by (simp add: affine_hull_finite_step(2)[of "{b}" a])
himmelma@33175
   713
  also have "\<dots> = ?rhs" unfolding * by auto
himmelma@33175
   714
  finally show ?thesis by auto
himmelma@33175
   715
qed
himmelma@33175
   716
himmelma@33175
   717
lemma affine_hull_3:
himmelma@33175
   718
  fixes a b c :: "'a::real_vector"
wenzelm@53302
   719
  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
   720
proof -
wenzelm@49529
   721
  have *:
wenzelm@49531
   722
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
wenzelm@49529
   723
    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
wenzelm@49529
   724
  show ?thesis
wenzelm@49529
   725
    apply (simp add: affine_hull_finite affine_hull_finite_step)
wenzelm@49529
   726
    unfolding *
wenzelm@49529
   727
    apply auto
wenzelm@53302
   728
    apply (rule_tac x=v in exI)
wenzelm@53302
   729
    apply (rule_tac x=va in exI)
wenzelm@53302
   730
    apply auto
wenzelm@53302
   731
    apply (rule_tac x=u in exI)
wenzelm@53302
   732
    apply force
wenzelm@49529
   733
    done
himmelma@33175
   734
qed
himmelma@33175
   735
hoelzl@40377
   736
lemma mem_affine:
wenzelm@53333
   737
  assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
wenzelm@53347
   738
  shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
hoelzl@40377
   739
  using assms affine_def[of S] by auto
hoelzl@40377
   740
hoelzl@40377
   741
lemma mem_affine_3:
wenzelm@53333
   742
  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
wenzelm@53347
   743
  shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
wenzelm@49529
   744
proof -
wenzelm@53347
   745
  have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
wenzelm@49529
   746
    using affine_hull_3[of x y z] assms by auto
wenzelm@49529
   747
  moreover
wenzelm@53347
   748
  have "affine hull {x, y, z} \<subseteq> affine hull S"
wenzelm@49529
   749
    using hull_mono[of "{x, y, z}" "S"] assms by auto
wenzelm@49529
   750
  moreover
wenzelm@53347
   751
  have "affine hull S = S"
wenzelm@53347
   752
    using assms affine_hull_eq[of S] by auto
wenzelm@49531
   753
  ultimately show ?thesis by auto
hoelzl@40377
   754
qed
hoelzl@40377
   755
hoelzl@40377
   756
lemma mem_affine_3_minus:
wenzelm@53333
   757
  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
wenzelm@53333
   758
  shows "x + v *\<^sub>R (y-z) \<in> S"
wenzelm@53333
   759
  using mem_affine_3[of S x y z 1 v "-v"] assms
wenzelm@53333
   760
  by (simp add: algebra_simps)
hoelzl@40377
   761
hoelzl@40377
   762
huffman@44467
   763
subsubsection {* Some relations between affine hull and subspaces *}
himmelma@33175
   764
himmelma@33175
   765
lemma affine_hull_insert_subset_span:
wenzelm@49529
   766
  "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
wenzelm@49529
   767
  unfolding subset_eq Ball_def
wenzelm@49529
   768
  unfolding affine_hull_explicit span_explicit mem_Collect_eq
wenzelm@50804
   769
  apply (rule, rule)
wenzelm@50804
   770
  apply (erule exE)+
wenzelm@50804
   771
  apply (erule conjE)+
wenzelm@49529
   772
proof -
wenzelm@49529
   773
  fix x t u
wenzelm@49529
   774
  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
   775
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}"
wenzelm@53333
   776
    using as(3) by auto
wenzelm@49529
   777
  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
   778
    apply (rule_tac x="x - a" in exI)
himmelma@33175
   779
    apply (rule conjI, simp)
wenzelm@49529
   780
    apply (rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
wenzelm@49529
   781
    apply (rule_tac x="\<lambda>x. u (x + a)" in exI)
himmelma@33175
   782
    apply (rule conjI) using as(1) apply simp
himmelma@33175
   783
    apply (erule conjI)
himmelma@33175
   784
    using as(1)
wenzelm@49529
   785
    apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib
wenzelm@49530
   786
      setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)
wenzelm@49529
   787
    unfolding as
wenzelm@49529
   788
    apply simp
wenzelm@49529
   789
    done
wenzelm@49529
   790
qed
himmelma@33175
   791
himmelma@33175
   792
lemma affine_hull_insert_span:
himmelma@33175
   793
  assumes "a \<notin> s"
wenzelm@49529
   794
  shows "affine hull (insert a s) = {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
wenzelm@49529
   795
  apply (rule, rule affine_hull_insert_subset_span)
wenzelm@49529
   796
  unfolding subset_eq Ball_def
wenzelm@49529
   797
  unfolding affine_hull_explicit and mem_Collect_eq
wenzelm@49529
   798
proof (rule, rule, erule exE, erule conjE)
wenzelm@49531
   799
  fix y v
wenzelm@49529
   800
  assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
wenzelm@53339
   801
  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
   802
    unfolding span_explicit by auto
himmelma@33175
   803
  def f \<equiv> "(\<lambda>x. x + a) ` t"
wenzelm@53333
   804
  have f: "finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a"
wenzelm@49529
   805
    unfolding f_def using obt by (auto simp add: setsum_reindex[unfolded inj_on_def])
wenzelm@53333
   806
  have *: "f \<inter> {a} = {}" "f \<inter> - {a} = f"
wenzelm@53333
   807
    using f(2) assms by auto
himmelma@33175
   808
  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
   809
    apply (rule_tac x = "insert a f" in exI)
wenzelm@49529
   810
    apply (rule_tac x = "\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
wenzelm@53339
   811
    using assms and f
wenzelm@53339
   812
    unfolding setsum_clauses(2)[OF f(1)] and if_smult
hoelzl@35577
   813
    unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
wenzelm@49529
   814
    apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)
wenzelm@49529
   815
    done
wenzelm@49529
   816
qed
himmelma@33175
   817
himmelma@33175
   818
lemma affine_hull_span:
himmelma@33175
   819
  assumes "a \<in> s"
himmelma@33175
   820
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
himmelma@33175
   821
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
himmelma@33175
   822
wenzelm@49529
   823
huffman@44467
   824
subsubsection {* Parallel affine sets *}
hoelzl@40377
   825
wenzelm@53347
   826
definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
wenzelm@53339
   827
  where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
hoelzl@40377
   828
hoelzl@40377
   829
lemma affine_parallel_expl_aux:
wenzelm@49529
   830
  fixes S T :: "'a::real_vector set"
wenzelm@53339
   831
  assumes "\<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
wenzelm@53339
   832
  shows "T = (\<lambda>x. a + x) ` S"
wenzelm@49529
   833
proof -
wenzelm@53302
   834
  {
wenzelm@53302
   835
    fix x
wenzelm@53339
   836
    assume "x \<in> T"
wenzelm@53339
   837
    then have "( - a) + x \<in> S"
wenzelm@53339
   838
      using assms by auto
wenzelm@53339
   839
    then have "x \<in> ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   840
      using imageI[of "-a+x" S "(\<lambda>x. a+x)"] by auto
wenzelm@53302
   841
  }
wenzelm@53339
   842
  moreover have "T \<ge> (\<lambda>x. a + x) ` S"
wenzelm@53333
   843
    using assms by auto
wenzelm@49529
   844
  ultimately show ?thesis by auto
wenzelm@49529
   845
qed
wenzelm@49529
   846
wenzelm@53339
   847
lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
wenzelm@49529
   848
  unfolding affine_parallel_def
wenzelm@49529
   849
  using affine_parallel_expl_aux[of S _ T] by auto
wenzelm@49529
   850
wenzelm@49529
   851
lemma affine_parallel_reflex: "affine_parallel S S"
wenzelm@53302
   852
  unfolding affine_parallel_def
wenzelm@53302
   853
  apply (rule exI[of _ "0"])
wenzelm@53302
   854
  apply auto
wenzelm@53302
   855
  done
hoelzl@40377
   856
hoelzl@40377
   857
lemma affine_parallel_commut:
wenzelm@49529
   858
  assumes "affine_parallel A B"
wenzelm@49529
   859
  shows "affine_parallel B A"
wenzelm@49529
   860
proof -
haftmann@54230
   861
  from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
wenzelm@49529
   862
    unfolding affine_parallel_def by auto
haftmann@54230
   863
  have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
haftmann@54230
   864
  from B show ?thesis
wenzelm@53333
   865
    using translation_galois [of B a A]
wenzelm@53333
   866
    unfolding affine_parallel_def by auto
hoelzl@40377
   867
qed
hoelzl@40377
   868
hoelzl@40377
   869
lemma affine_parallel_assoc:
wenzelm@53339
   870
  assumes "affine_parallel A B"
wenzelm@53339
   871
    and "affine_parallel B C"
wenzelm@49531
   872
  shows "affine_parallel A C"
wenzelm@49529
   873
proof -
wenzelm@53333
   874
  from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
wenzelm@49531
   875
    unfolding affine_parallel_def by auto
wenzelm@49531
   876
  moreover
wenzelm@53333
   877
  from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
wenzelm@49529
   878
    unfolding affine_parallel_def by auto
wenzelm@49529
   879
  ultimately show ?thesis
wenzelm@49529
   880
    using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
hoelzl@40377
   881
qed
hoelzl@40377
   882
hoelzl@40377
   883
lemma affine_translation_aux:
hoelzl@40377
   884
  fixes a :: "'a::real_vector"
wenzelm@53333
   885
  assumes "affine ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   886
  shows "affine S"
wenzelm@53302
   887
proof -
wenzelm@53302
   888
  {
wenzelm@53302
   889
    fix x y u v
wenzelm@53333
   890
    assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
wenzelm@53333
   891
    then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
wenzelm@53333
   892
      by auto
wenzelm@53339
   893
    then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
wenzelm@49529
   894
      using xy assms unfolding affine_def by auto
wenzelm@53339
   895
    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
   896
      by (simp add: algebra_simps)
wenzelm@53339
   897
    also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
wenzelm@53339
   898
      using `u + v = 1` by auto
wenzelm@53339
   899
    ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
wenzelm@53333
   900
      using h1 by auto
wenzelm@49529
   901
    then have "u *\<^sub>R x + v *\<^sub>R y : S" by auto
wenzelm@49529
   902
  }
wenzelm@49529
   903
  then show ?thesis unfolding affine_def by auto
hoelzl@40377
   904
qed
hoelzl@40377
   905
hoelzl@40377
   906
lemma affine_translation:
hoelzl@40377
   907
  fixes a :: "'a::real_vector"
wenzelm@53339
   908
  shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
wenzelm@49529
   909
proof -
wenzelm@53339
   910
  have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
wenzelm@53339
   911
    using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
wenzelm@49529
   912
    using translation_assoc[of "-a" a S] by auto
wenzelm@49529
   913
  then show ?thesis using affine_translation_aux by auto
hoelzl@40377
   914
qed
hoelzl@40377
   915
hoelzl@40377
   916
lemma parallel_is_affine:
wenzelm@49529
   917
  fixes S T :: "'a::real_vector set"
wenzelm@49529
   918
  assumes "affine S" "affine_parallel S T"
wenzelm@49529
   919
  shows "affine T"
wenzelm@49529
   920
proof -
wenzelm@53339
   921
  from assms obtain a where "T = (\<lambda>x. a + x) ` S"
wenzelm@49531
   922
    unfolding affine_parallel_def by auto
wenzelm@53339
   923
  then show ?thesis
wenzelm@53339
   924
    using affine_translation assms by auto
hoelzl@40377
   925
qed
hoelzl@40377
   926
huffman@44361
   927
lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
hoelzl@40377
   928
  unfolding subspace_def affine_def by auto
hoelzl@40377
   929
wenzelm@49529
   930
huffman@44467
   931
subsubsection {* Subspace parallel to an affine set *}
hoelzl@40377
   932
wenzelm@53339
   933
lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
wenzelm@49529
   934
proof -
wenzelm@53333
   935
  have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
wenzelm@49529
   936
    using subspace_imp_affine[of S] subspace_0 by auto
wenzelm@53302
   937
  {
wenzelm@53333
   938
    assume assm: "affine S \<and> 0 \<in> S"
wenzelm@53302
   939
    {
wenzelm@53302
   940
      fix c :: real
wenzelm@54465
   941
      fix x
wenzelm@54465
   942
      assume x: "x \<in> S"
wenzelm@49529
   943
      have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
wenzelm@49529
   944
      moreover
wenzelm@53339
   945
      have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
wenzelm@54465
   946
        using affine_alt[of S] assm x by auto
wenzelm@53333
   947
      ultimately have "c *\<^sub>R x \<in> S" by auto
wenzelm@49529
   948
    }
wenzelm@53333
   949
    then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
wenzelm@49529
   950
wenzelm@53302
   951
    {
wenzelm@53302
   952
      fix x y
wenzelm@54465
   953
      assume xy: "x \<in> S" "y \<in> S"
wenzelm@49529
   954
      def u == "(1 :: real)/2"
wenzelm@53302
   955
      have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
wenzelm@53302
   956
        by auto
wenzelm@49529
   957
      moreover
wenzelm@53302
   958
      have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
wenzelm@53302
   959
        by (simp add: algebra_simps)
wenzelm@49529
   960
      moreover
wenzelm@54465
   961
      have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
wenzelm@54465
   962
        using affine_alt[of S] assm xy by auto
wenzelm@49529
   963
      ultimately
wenzelm@53333
   964
      have "(1/2) *\<^sub>R (x+y) \<in> S"
wenzelm@53302
   965
        using u_def by auto
wenzelm@49529
   966
      moreover
wenzelm@54465
   967
      have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
wenzelm@53302
   968
        by auto
wenzelm@49529
   969
      ultimately
wenzelm@54465
   970
      have "x + y \<in> S"
wenzelm@53302
   971
        using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
wenzelm@49529
   972
    }
wenzelm@53302
   973
    then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
wenzelm@53302
   974
      by auto
wenzelm@53302
   975
    then have "subspace S"
wenzelm@53302
   976
      using h1 assm unfolding subspace_def by auto
wenzelm@49529
   977
  }
wenzelm@49529
   978
  then show ?thesis using h0 by metis
hoelzl@40377
   979
qed
hoelzl@40377
   980
hoelzl@40377
   981
lemma affine_diffs_subspace:
wenzelm@53333
   982
  assumes "affine S" "a \<in> S"
wenzelm@53302
   983
  shows "subspace ((\<lambda>x. (-a)+x) ` S)"
wenzelm@49529
   984
proof -
haftmann@54230
   985
  have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
wenzelm@53302
   986
  have "affine ((\<lambda>x. (-a)+x) ` S)"
wenzelm@49531
   987
    using  affine_translation assms by auto
wenzelm@53302
   988
  moreover have "0 : ((\<lambda>x. (-a)+x) ` S)"
wenzelm@53333
   989
    using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
wenzelm@49531
   990
  ultimately show ?thesis using subspace_affine by auto
hoelzl@40377
   991
qed
hoelzl@40377
   992
hoelzl@40377
   993
lemma parallel_subspace_explicit:
wenzelm@54465
   994
  assumes "affine S"
wenzelm@54465
   995
    and "a \<in> S"
wenzelm@54465
   996
  assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
wenzelm@54465
   997
  shows "subspace L \<and> affine_parallel S L"
wenzelm@49529
   998
proof -
haftmann@54230
   999
  from assms have "L = plus (- a) ` S" by auto
haftmann@54230
  1000
  then have par: "affine_parallel S L"
wenzelm@54465
  1001
    unfolding affine_parallel_def ..
wenzelm@49531
  1002
  then have "affine L" using assms parallel_is_affine by auto
wenzelm@53302
  1003
  moreover have "0 \<in> L"
haftmann@54230
  1004
    using assms by auto
wenzelm@53302
  1005
  ultimately show ?thesis
wenzelm@53302
  1006
    using subspace_affine par by auto
hoelzl@40377
  1007
qed
hoelzl@40377
  1008
hoelzl@40377
  1009
lemma parallel_subspace_aux:
wenzelm@53302
  1010
  assumes "subspace A"
wenzelm@53302
  1011
    and "subspace B"
wenzelm@53302
  1012
    and "affine_parallel A B"
wenzelm@53302
  1013
  shows "A \<supseteq> B"
wenzelm@49529
  1014
proof -
wenzelm@54465
  1015
  from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
wenzelm@49529
  1016
    using affine_parallel_expl[of A B] by auto
wenzelm@53302
  1017
  then have "-a \<in> A"
wenzelm@53302
  1018
    using assms subspace_0[of B] by auto
wenzelm@53302
  1019
  then have "a \<in> A"
wenzelm@53302
  1020
    using assms subspace_neg[of A "-a"] by auto
wenzelm@53302
  1021
  then show ?thesis
wenzelm@54465
  1022
    using assms a unfolding subspace_def by auto
hoelzl@40377
  1023
qed
hoelzl@40377
  1024
hoelzl@40377
  1025
lemma parallel_subspace:
wenzelm@53302
  1026
  assumes "subspace A"
wenzelm@53302
  1027
    and "subspace B"
wenzelm@53302
  1028
    and "affine_parallel A B"
wenzelm@49529
  1029
  shows "A = B"
wenzelm@49529
  1030
proof
wenzelm@53302
  1031
  show "A \<supseteq> B"
wenzelm@49529
  1032
    using assms parallel_subspace_aux by auto
wenzelm@53302
  1033
  show "A \<subseteq> B"
wenzelm@49529
  1034
    using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
hoelzl@40377
  1035
qed
hoelzl@40377
  1036
hoelzl@40377
  1037
lemma affine_parallel_subspace:
wenzelm@53302
  1038
  assumes "affine S" "S \<noteq> {}"
wenzelm@53339
  1039
  shows "\<exists>!L. subspace L \<and> affine_parallel S L"
wenzelm@49529
  1040
proof -
wenzelm@53339
  1041
  have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
wenzelm@49531
  1042
    using assms parallel_subspace_explicit by auto
wenzelm@53302
  1043
  {
wenzelm@53302
  1044
    fix L1 L2
wenzelm@53339
  1045
    assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
wenzelm@49529
  1046
    then have "affine_parallel L1 L2"
wenzelm@49529
  1047
      using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
wenzelm@49529
  1048
    then have "L1 = L2"
wenzelm@49529
  1049
      using ass parallel_subspace by auto
wenzelm@49529
  1050
  }
wenzelm@49529
  1051
  then show ?thesis using ex by auto
wenzelm@49529
  1052
qed
wenzelm@49529
  1053
hoelzl@40377
  1054
huffman@44467
  1055
subsection {* Cones *}
himmelma@33175
  1056
wenzelm@49529
  1057
definition cone :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@53339
  1058
  where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
himmelma@33175
  1059
himmelma@33175
  1060
lemma cone_empty[intro, simp]: "cone {}"
himmelma@33175
  1061
  unfolding cone_def by auto
himmelma@33175
  1062
himmelma@33175
  1063
lemma cone_univ[intro, simp]: "cone UNIV"
himmelma@33175
  1064
  unfolding cone_def by auto
himmelma@33175
  1065
wenzelm@53339
  1066
lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
himmelma@33175
  1067
  unfolding cone_def by auto
himmelma@33175
  1068
wenzelm@49529
  1069
huffman@44467
  1070
subsubsection {* Conic hull *}
himmelma@33175
  1071
himmelma@33175
  1072
lemma cone_cone_hull: "cone (cone hull s)"
huffman@44170
  1073
  unfolding hull_def by auto
himmelma@33175
  1074
wenzelm@53302
  1075
lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
wenzelm@49529
  1076
  apply (rule hull_eq)
wenzelm@53302
  1077
  using cone_Inter
wenzelm@53302
  1078
  unfolding subset_eq
wenzelm@53302
  1079
  apply auto
wenzelm@49529
  1080
  done
himmelma@33175
  1081
hoelzl@40377
  1082
lemma mem_cone:
wenzelm@53302
  1083
  assumes "cone S" "x \<in> S" "c \<ge> 0"
hoelzl@40377
  1084
  shows "c *\<^sub>R x : S"
hoelzl@40377
  1085
  using assms cone_def[of S] by auto
hoelzl@40377
  1086
hoelzl@40377
  1087
lemma cone_contains_0:
wenzelm@49529
  1088
  assumes "cone S"
wenzelm@53302
  1089
  shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
wenzelm@49529
  1090
proof -
wenzelm@53302
  1091
  {
wenzelm@53302
  1092
    assume "S \<noteq> {}"
wenzelm@53302
  1093
    then obtain a where "a \<in> S" by auto
wenzelm@53302
  1094
    then have "0 \<in> S"
wenzelm@53302
  1095
      using assms mem_cone[of S a 0] by auto
wenzelm@49529
  1096
  }
wenzelm@49529
  1097
  then show ?thesis by auto
hoelzl@40377
  1098
qed
hoelzl@40377
  1099
huffman@44361
  1100
lemma cone_0: "cone {0}"
wenzelm@49529
  1101
  unfolding cone_def by auto
hoelzl@40377
  1102
wenzelm@53302
  1103
lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (Union f)"
hoelzl@40377
  1104
  unfolding cone_def by blast
hoelzl@40377
  1105
hoelzl@40377
  1106
lemma cone_iff:
wenzelm@53347
  1107
  assumes "S \<noteq> {}"
wenzelm@53347
  1108
  shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
wenzelm@49529
  1109
proof -
wenzelm@53302
  1110
  {
wenzelm@53302
  1111
    assume "cone S"
wenzelm@53302
  1112
    {
wenzelm@53347
  1113
      fix c :: real
wenzelm@53347
  1114
      assume "c > 0"
wenzelm@53302
  1115
      {
wenzelm@53302
  1116
        fix x
wenzelm@53347
  1117
        assume "x \<in> S"
wenzelm@53347
  1118
        then have "x \<in> (op *\<^sub>R c) ` S"
wenzelm@49529
  1119
          unfolding image_def
wenzelm@49529
  1120
          using `cone S` `c>0` mem_cone[of S x "1/c"]
wenzelm@54465
  1121
            exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
wenzelm@53347
  1122
          by auto
wenzelm@49529
  1123
      }
wenzelm@49529
  1124
      moreover
wenzelm@53302
  1125
      {
wenzelm@53302
  1126
        fix x
wenzelm@53347
  1127
        assume "x \<in> (op *\<^sub>R c) ` S"
wenzelm@53347
  1128
        then have "x \<in> S"
wenzelm@53347
  1129
          using `cone S` `c > 0`
wenzelm@53347
  1130
          unfolding cone_def image_def `c > 0` by auto
wenzelm@49529
  1131
      }
wenzelm@53302
  1132
      ultimately have "(op *\<^sub>R c) ` S = S" by auto
hoelzl@40377
  1133
    }
wenzelm@53339
  1134
    then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
wenzelm@49529
  1135
      using `cone S` cone_contains_0[of S] assms by auto
wenzelm@49529
  1136
  }
wenzelm@49529
  1137
  moreover
wenzelm@53302
  1138
  {
wenzelm@53339
  1139
    assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
wenzelm@53302
  1140
    {
wenzelm@53302
  1141
      fix x
wenzelm@53302
  1142
      assume "x \<in> S"
wenzelm@53347
  1143
      fix c1 :: real
wenzelm@53347
  1144
      assume "c1 \<ge> 0"
wenzelm@53347
  1145
      then have "c1 = 0 \<or> c1 > 0" by auto
wenzelm@53347
  1146
      then have "c1 *\<^sub>R x \<in> S" using a `x \<in> S` by auto
wenzelm@49529
  1147
    }
wenzelm@49529
  1148
    then have "cone S" unfolding cone_def by auto
hoelzl@40377
  1149
  }
wenzelm@49529
  1150
  ultimately show ?thesis by blast
wenzelm@49529
  1151
qed
wenzelm@49529
  1152
wenzelm@49529
  1153
lemma cone_hull_empty: "cone hull {} = {}"
wenzelm@49529
  1154
  by (metis cone_empty cone_hull_eq)
wenzelm@49529
  1155
wenzelm@53302
  1156
lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
wenzelm@49529
  1157
  by (metis bot_least cone_hull_empty hull_subset xtrans(5))
wenzelm@49529
  1158
wenzelm@53302
  1159
lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
wenzelm@49529
  1160
  using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
wenzelm@49529
  1161
  by auto
hoelzl@40377
  1162
hoelzl@40377
  1163
lemma mem_cone_hull:
wenzelm@53347
  1164
  assumes "x \<in> S" "c \<ge> 0"
wenzelm@53302
  1165
  shows "c *\<^sub>R x \<in> cone hull S"
wenzelm@49529
  1166
  by (metis assms cone_cone_hull hull_inc mem_cone)
wenzelm@49529
  1167
wenzelm@53339
  1168
lemma cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
wenzelm@53339
  1169
  (is "?lhs = ?rhs")
wenzelm@49529
  1170
proof -
wenzelm@53302
  1171
  {
wenzelm@53302
  1172
    fix x
wenzelm@53302
  1173
    assume "x \<in> ?rhs"
wenzelm@54465
  1174
    then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
wenzelm@49529
  1175
      by auto
wenzelm@53347
  1176
    fix c :: real
wenzelm@53347
  1177
    assume c: "c \<ge> 0"
wenzelm@53339
  1178
    then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
wenzelm@54465
  1179
      using x by (simp add: algebra_simps)
wenzelm@49529
  1180
    moreover
wenzelm@53302
  1181
    have "c * cx \<ge> 0"
wenzelm@54465
  1182
      using c x using mult_nonneg_nonneg by auto
wenzelm@49529
  1183
    ultimately
wenzelm@54465
  1184
    have "c *\<^sub>R x \<in> ?rhs" using x by auto
wenzelm@53302
  1185
  }
wenzelm@53347
  1186
  then have "cone ?rhs"
wenzelm@53347
  1187
    unfolding cone_def by auto
wenzelm@53347
  1188
  then have "?rhs \<in> Collect cone"
wenzelm@53347
  1189
    unfolding mem_Collect_eq by auto
wenzelm@53302
  1190
  {
wenzelm@53302
  1191
    fix x
wenzelm@53302
  1192
    assume "x \<in> S"
wenzelm@53302
  1193
    then have "1 *\<^sub>R x \<in> ?rhs"
wenzelm@49531
  1194
      apply auto
wenzelm@53347
  1195
      apply (rule_tac x = 1 in exI)
wenzelm@49529
  1196
      apply auto
wenzelm@49529
  1197
      done
wenzelm@53302
  1198
    then have "x \<in> ?rhs" by auto
wenzelm@53347
  1199
  }
wenzelm@53347
  1200
  then have "S \<subseteq> ?rhs" by auto
wenzelm@53302
  1201
  then have "?lhs \<subseteq> ?rhs"
wenzelm@53302
  1202
    using `?rhs \<in> Collect cone` hull_minimal[of S "?rhs" "cone"] by auto
wenzelm@49529
  1203
  moreover
wenzelm@53302
  1204
  {
wenzelm@53302
  1205
    fix x
wenzelm@53302
  1206
    assume "x \<in> ?rhs"
wenzelm@54465
  1207
    then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
wenzelm@53339
  1208
      by auto
wenzelm@53339
  1209
    then have "xx \<in> cone hull S"
wenzelm@53339
  1210
      using hull_subset[of S] by auto
wenzelm@53302
  1211
    then have "x \<in> ?lhs"
wenzelm@54465
  1212
      using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
wenzelm@49529
  1213
  }
wenzelm@49529
  1214
  ultimately show ?thesis by auto
hoelzl@40377
  1215
qed
hoelzl@40377
  1216
hoelzl@40377
  1217
lemma cone_closure:
wenzelm@53347
  1218
  fixes S :: "'a::real_normed_vector set"
wenzelm@49529
  1219
  assumes "cone S"
wenzelm@49529
  1220
  shows "cone (closure S)"
wenzelm@49529
  1221
proof (cases "S = {}")
wenzelm@49529
  1222
  case True
wenzelm@49529
  1223
  then show ?thesis by auto
wenzelm@49529
  1224
next
wenzelm@49529
  1225
  case False
wenzelm@53339
  1226
  then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
wenzelm@49529
  1227
    using cone_iff[of S] assms by auto
wenzelm@53339
  1228
  then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` closure S = closure S)"
wenzelm@49529
  1229
    using closure_subset by (auto simp add: closure_scaleR)
wenzelm@53339
  1230
  then show ?thesis
wenzelm@53339
  1231
    using cone_iff[of "closure S"] by auto
wenzelm@49529
  1232
qed
wenzelm@49529
  1233
hoelzl@40377
  1234
huffman@44467
  1235
subsection {* Affine dependence and consequential theorems (from Lars Schewe) *}
himmelma@33175
  1236
wenzelm@49529
  1237
definition affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@53339
  1238
  where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
himmelma@33175
  1239
himmelma@33175
  1240
lemma affine_dependent_explicit:
himmelma@33175
  1241
  "affine_dependent p \<longleftrightarrow>
himmelma@33175
  1242
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
wenzelm@53347
  1243
      (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
wenzelm@49529
  1244
  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq
wenzelm@49529
  1245
  apply rule
wenzelm@49529
  1246
  apply (erule bexE, erule exE, erule exE)
wenzelm@49529
  1247
  apply (erule conjE)+
wenzelm@49529
  1248
  defer
wenzelm@49529
  1249
  apply (erule exE, erule exE)
wenzelm@49529
  1250
  apply (erule conjE)+
wenzelm@49529
  1251
  apply (erule bexE)
wenzelm@49529
  1252
proof -
wenzelm@49529
  1253
  fix x s u
wenzelm@49529
  1254
  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
  1255
  have "x \<notin> s" using as(1,4) by auto
himmelma@33175
  1256
  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
  1257
    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
  1258
    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as
wenzelm@53339
  1259
    using as
wenzelm@53339
  1260
    apply auto
wenzelm@49529
  1261
    done
himmelma@33175
  1262
next
wenzelm@49529
  1263
  fix s u v
wenzelm@53302
  1264
  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"
wenzelm@53339
  1265
  have "s \<noteq> {v}"
wenzelm@53339
  1266
    using as(3,6) by auto
wenzelm@49529
  1267
  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
  1268
    apply (rule_tac x=v in bexI)
wenzelm@53302
  1269
    apply (rule_tac x="s - {v}" in exI)
wenzelm@53302
  1270
    apply (rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
wenzelm@49530
  1271
    unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
wenzelm@49530
  1272
    unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]
wenzelm@53302
  1273
    using as
wenzelm@53302
  1274
    apply auto
wenzelm@49529
  1275
    done
himmelma@33175
  1276
qed
himmelma@33175
  1277
himmelma@33175
  1278
lemma affine_dependent_explicit_finite:
wenzelm@49529
  1279
  fixes s :: "'a::real_vector set"
wenzelm@49529
  1280
  assumes "finite s"
wenzelm@53302
  1281
  shows "affine_dependent s \<longleftrightarrow>
wenzelm@53302
  1282
    (\<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
  1283
  (is "?lhs = ?rhs")
himmelma@33175
  1284
proof
wenzelm@53347
  1285
  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
  1286
    by auto
himmelma@33175
  1287
  assume ?lhs
wenzelm@49529
  1288
  then obtain t u v where
wenzelm@53339
  1289
    "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
  1290
    unfolding affine_dependent_explicit by auto
wenzelm@49529
  1291
  then show ?rhs
wenzelm@49529
  1292
    apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
wenzelm@49530
  1293
    apply auto unfolding * and setsum_restrict_set[OF assms, symmetric]
wenzelm@49529
  1294
    unfolding Int_absorb1[OF `t\<subseteq>s`]
wenzelm@49529
  1295
    apply auto
wenzelm@49529
  1296
    done
himmelma@33175
  1297
next
himmelma@33175
  1298
  assume ?rhs
wenzelm@53339
  1299
  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"
wenzelm@53339
  1300
    by auto
wenzelm@49529
  1301
  then show ?lhs unfolding affine_dependent_explicit
wenzelm@49529
  1302
    using assms by auto
wenzelm@49529
  1303
qed
wenzelm@49529
  1304
himmelma@33175
  1305
huffman@44465
  1306
subsection {* Connectedness of convex sets *}
huffman@44465
  1307
hoelzl@51480
  1308
lemma connectedD:
hoelzl@51480
  1309
  "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
  1310
  by (metis connected_def)
himmelma@33175
  1311
himmelma@33175
  1312
lemma convex_connected:
himmelma@33175
  1313
  fixes s :: "'a::real_normed_vector set"
wenzelm@53302
  1314
  assumes "convex s"
wenzelm@53302
  1315
  shows "connected s"
hoelzl@51480
  1316
proof (rule connectedI)
hoelzl@51480
  1317
  fix A B
hoelzl@51480
  1318
  assume "open A" "open B" "A \<inter> B \<inter> s = {}" "s \<subseteq> A \<union> B"
hoelzl@51480
  1319
  moreover
hoelzl@51480
  1320
  assume "A \<inter> s \<noteq> {}" "B \<inter> s \<noteq> {}"
hoelzl@51480
  1321
  then obtain a b where a: "a \<in> A" "a \<in> s" and b: "b \<in> B" "b \<in> s" by auto
hoelzl@51480
  1322
  def f \<equiv> "\<lambda>u. u *\<^sub>R a + (1 - u) *\<^sub>R b"
hoelzl@51480
  1323
  then have "continuous_on {0 .. 1} f"
hoelzl@51480
  1324
    by (auto intro!: continuous_on_intros)
hoelzl@51480
  1325
  then have "connected (f ` {0 .. 1})"
hoelzl@51480
  1326
    by (auto intro!: connected_continuous_image)
hoelzl@51480
  1327
  note connectedD[OF this, of A B]
hoelzl@51480
  1328
  moreover have "a \<in> A \<inter> f ` {0 .. 1}"
hoelzl@51480
  1329
    using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
hoelzl@51480
  1330
  moreover have "b \<in> B \<inter> f ` {0 .. 1}"
hoelzl@51480
  1331
    using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
hoelzl@51480
  1332
  moreover have "f ` {0 .. 1} \<subseteq> s"
hoelzl@51480
  1333
    using `convex s` a b unfolding convex_def f_def by auto
hoelzl@51480
  1334
  ultimately show False by auto
himmelma@33175
  1335
qed
himmelma@33175
  1336
huffman@44467
  1337
text {* One rather trivial consequence. *}
himmelma@33175
  1338
hoelzl@34964
  1339
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
himmelma@33175
  1340
  by(simp add: convex_connected convex_UNIV)
himmelma@33175
  1341
huffman@44467
  1342
text {* Balls, being convex, are connected. *}
himmelma@33175
  1343
wenzelm@53302
  1344
lemma convex_box:
wenzelm@53347
  1345
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
hoelzl@50526
  1346
  shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
hoelzl@50526
  1347
  using assms unfolding convex_def
hoelzl@50526
  1348
  by (auto simp: inner_add_left)
hoelzl@50526
  1349
hoelzl@50526
  1350
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
hoelzl@36623
  1351
  by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
himmelma@33175
  1352
himmelma@33175
  1353
lemma convex_local_global_minimum:
himmelma@33175
  1354
  fixes s :: "'a::real_normed_vector set"
wenzelm@53347
  1355
  assumes "e > 0"
wenzelm@53347
  1356
    and "convex_on s f"
wenzelm@53347
  1357
    and "ball x e \<subseteq> s"
wenzelm@53347
  1358
    and "\<forall>y\<in>ball x e. f x \<le> f y"
himmelma@33175
  1359
  shows "\<forall>y\<in>s. f x \<le> f y"
wenzelm@53302
  1360
proof (rule ccontr)
wenzelm@53302
  1361
  have "x \<in> s" using assms(1,3) by auto
wenzelm@53302
  1362
  assume "\<not> ?thesis"
wenzelm@53302
  1363
  then obtain y where "y\<in>s" and y: "f x > f y" by auto
wenzelm@53347
  1364
  then have xy: "0 < dist x y"
wenzelm@53347
  1365
    by (auto simp add: dist_nz[symmetric])
wenzelm@53347
  1366
wenzelm@53347
  1367
  then obtain u where "0 < u" "u \<le> 1" and u: "u < e / dist x y"
wenzelm@50804
  1368
    using real_lbound_gt_zero[of 1 "e / dist x y"]
wenzelm@53347
  1369
    using xy `e>0` and divide_pos_pos[of e "dist x y"]
wenzelm@53347
  1370
    by auto
wenzelm@53302
  1371
  then have "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y"
wenzelm@53302
  1372
    using `x\<in>s` `y\<in>s`
wenzelm@53302
  1373
    using assms(2)[unfolded convex_on_def,
wenzelm@53302
  1374
      THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
wenzelm@50804
  1375
    by auto
himmelma@33175
  1376
  moreover
wenzelm@50804
  1377
  have *: "x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)"
wenzelm@50804
  1378
    by (simp add: algebra_simps)
wenzelm@50804
  1379
  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e"
wenzelm@53302
  1380
    unfolding mem_ball dist_norm
wenzelm@53302
  1381
    unfolding * and norm_scaleR and abs_of_pos[OF `0<u`]
wenzelm@50804
  1382
    unfolding dist_norm[symmetric]
wenzelm@53302
  1383
    using u
wenzelm@53302
  1384
    unfolding pos_less_divide_eq[OF xy]
wenzelm@53302
  1385
    by auto
wenzelm@53302
  1386
  then have "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)"
wenzelm@53302
  1387
    using assms(4) by auto
wenzelm@50804
  1388
  ultimately show False
wenzelm@53302
  1389
    using mult_strict_left_mono[OF y `u>0`]
wenzelm@53302
  1390
    unfolding left_diff_distrib
wenzelm@53302
  1391
    by auto
himmelma@33175
  1392
qed
himmelma@33175
  1393
himmelma@33175
  1394
lemma convex_ball:
himmelma@33175
  1395
  fixes x :: "'a::real_normed_vector"
wenzelm@49531
  1396
  shows "convex (ball x e)"
wenzelm@50804
  1397
proof (auto simp add: convex_def)
wenzelm@50804
  1398
  fix y z
wenzelm@50804
  1399
  assume yz: "dist x y < e" "dist x z < e"
wenzelm@50804
  1400
  fix u v :: real
wenzelm@50804
  1401
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@50804
  1402
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
wenzelm@50804
  1403
    using uv yz
huffman@53620
  1404
    using convex_on_dist [of "ball x e" x, unfolded convex_on_def,
wenzelm@53302
  1405
      THEN bspec[where x=y], THEN bspec[where x=z]]
wenzelm@50804
  1406
    by auto
wenzelm@50804
  1407
  then show "dist x (u *\<^sub>R y + v *\<^sub>R z) < e"
wenzelm@50804
  1408
    using convex_bound_lt[OF yz uv] by auto
himmelma@33175
  1409
qed
himmelma@33175
  1410
himmelma@33175
  1411
lemma convex_cball:
himmelma@33175
  1412
  fixes x :: "'a::real_normed_vector"
wenzelm@53347
  1413
  shows "convex (cball x e)"
wenzelm@53347
  1414
proof -
wenzelm@53347
  1415
  {
wenzelm@53347
  1416
    fix y z
wenzelm@53347
  1417
    assume yz: "dist x y \<le> e" "dist x z \<le> e"
wenzelm@53347
  1418
    fix u v :: real
wenzelm@53347
  1419
    assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@53347
  1420
    have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
wenzelm@53347
  1421
      using uv yz
huffman@53620
  1422
      using convex_on_dist [of "cball x e" x, unfolded convex_on_def,
wenzelm@53347
  1423
        THEN bspec[where x=y], THEN bspec[where x=z]]
wenzelm@53347
  1424
      by auto
wenzelm@53347
  1425
    then have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e"
wenzelm@53347
  1426
      using convex_bound_le[OF yz uv] by auto
wenzelm@53347
  1427
  }
wenzelm@53347
  1428
  then show ?thesis by (auto simp add: convex_def Ball_def)
himmelma@33175
  1429
qed
himmelma@33175
  1430
himmelma@33175
  1431
lemma connected_ball:
himmelma@33175
  1432
  fixes x :: "'a::real_normed_vector"
himmelma@33175
  1433
  shows "connected (ball x e)"
himmelma@33175
  1434
  using convex_connected convex_ball by auto
himmelma@33175
  1435
himmelma@33175
  1436
lemma connected_cball:
himmelma@33175
  1437
  fixes x :: "'a::real_normed_vector"
wenzelm@53302
  1438
  shows "connected (cball x e)"
himmelma@33175
  1439
  using convex_connected convex_cball by auto
himmelma@33175
  1440
wenzelm@50804
  1441
huffman@44467
  1442
subsection {* Convex hull *}
himmelma@33175
  1443
wenzelm@53302
  1444
lemma convex_convex_hull: "convex (convex hull s)"
wenzelm@53302
  1445
  unfolding hull_def
wenzelm@53302
  1446
  using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
huffman@44170
  1447
  by auto
himmelma@33175
  1448
haftmann@34064
  1449
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
wenzelm@50804
  1450
  by (metis convex_convex_hull hull_same)
himmelma@33175
  1451
himmelma@33175
  1452
lemma bounded_convex_hull:
himmelma@33175
  1453
  fixes s :: "'a::real_normed_vector set"
wenzelm@53347
  1454
  assumes "bounded s"
wenzelm@53347
  1455
  shows "bounded (convex hull s)"
wenzelm@50804
  1456
proof -
wenzelm@50804
  1457
  from assms obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
wenzelm@50804
  1458
    unfolding bounded_iff by auto
wenzelm@50804
  1459
  show ?thesis
wenzelm@50804
  1460
    apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
huffman@44170
  1461
    unfolding subset_hull[of convex, OF convex_cball]
wenzelm@53302
  1462
    unfolding subset_eq mem_cball dist_norm using B
wenzelm@53302
  1463
    apply auto
wenzelm@50804
  1464
    done
wenzelm@50804
  1465
qed
himmelma@33175
  1466
himmelma@33175
  1467
lemma finite_imp_bounded_convex_hull:
himmelma@33175
  1468
  fixes s :: "'a::real_normed_vector set"
wenzelm@53302
  1469
  shows "finite s \<Longrightarrow> bounded (convex hull s)"
wenzelm@53302
  1470
  using bounded_convex_hull finite_imp_bounded
wenzelm@53302
  1471
  by auto
himmelma@33175
  1472
wenzelm@50804
  1473
huffman@44467
  1474
subsubsection {* Convex hull is "preserved" by a linear function *}
hoelzl@40377
  1475
hoelzl@40377
  1476
lemma convex_hull_linear_image:
huffman@53620
  1477
  assumes f: "linear f"
hoelzl@40377
  1478
  shows "f ` (convex hull s) = convex hull (f ` s)"
huffman@53620
  1479
proof
huffman@53620
  1480
  show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
huffman@53620
  1481
    by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
huffman@53620
  1482
  show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
huffman@53620
  1483
  proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
huffman@53620
  1484
    show "s \<subseteq> f -` (convex hull (f ` s))"
huffman@53620
  1485
      by (fast intro: hull_inc)
huffman@53620
  1486
    show "convex (f -` (convex hull (f ` s)))"
huffman@53620
  1487
      by (intro convex_linear_vimage [OF f] convex_convex_hull)
huffman@53620
  1488
  qed
huffman@53620
  1489
qed
hoelzl@40377
  1490
hoelzl@40377
  1491
lemma in_convex_hull_linear_image:
huffman@53620
  1492
  assumes "linear f"
wenzelm@53347
  1493
    and "x \<in> convex hull s"
wenzelm@53339
  1494
  shows "f x \<in> convex hull (f ` s)"
wenzelm@50804
  1495
  using convex_hull_linear_image[OF assms(1)] assms(2) by auto
wenzelm@50804
  1496
huffman@53620
  1497
lemma convex_hull_Times:
huffman@53620
  1498
  "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
huffman@53620
  1499
proof
huffman@53620
  1500
  show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
huffman@53620
  1501
    by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
huffman@53620
  1502
  have "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)"
huffman@53620
  1503
  proof (intro hull_induct)
huffman@53620
  1504
    fix x y assume "x \<in> s" and "y \<in> t"
huffman@53620
  1505
    then show "(x, y) \<in> convex hull (s \<times> t)"
huffman@53620
  1506
      by (simp add: hull_inc)
huffman@53620
  1507
  next
huffman@53620
  1508
    fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
huffman@53620
  1509
    have "convex ?S"
huffman@53620
  1510
      by (intro convex_linear_vimage convex_translation convex_convex_hull,
huffman@53620
  1511
        simp add: linear_iff)
huffman@53620
  1512
    also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
huffman@53620
  1513
      by (auto simp add: uminus_add_conv_diff image_def Bex_def)
huffman@53620
  1514
    finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
huffman@53620
  1515
  next
huffman@53620
  1516
    show "convex {x. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)}"
huffman@53620
  1517
    proof (unfold Collect_ball_eq, rule convex_INT [rule_format])
huffman@53620
  1518
      fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
huffman@53620
  1519
      have "convex ?S"
huffman@53620
  1520
      by (intro convex_linear_vimage convex_translation convex_convex_hull,
huffman@53620
  1521
        simp add: linear_iff)
huffman@53620
  1522
      also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
huffman@53620
  1523
        by (auto simp add: uminus_add_conv_diff image_def Bex_def)
huffman@53620
  1524
      finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
huffman@53620
  1525
    qed
huffman@53620
  1526
  qed
huffman@53620
  1527
  then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
huffman@53620
  1528
    unfolding subset_eq split_paired_Ball_Sigma .
huffman@53620
  1529
qed
huffman@53620
  1530
hoelzl@40377
  1531
huffman@44467
  1532
subsubsection {* Stepping theorems for convex hulls of finite sets *}
himmelma@33175
  1533
himmelma@33175
  1534
lemma convex_hull_empty[simp]: "convex hull {} = {}"
wenzelm@50804
  1535
  by (rule hull_unique) auto
himmelma@33175
  1536
himmelma@33175
  1537
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
wenzelm@50804
  1538
  by (rule hull_unique) auto
himmelma@33175
  1539
himmelma@33175
  1540
lemma convex_hull_insert:
himmelma@33175
  1541
  fixes s :: "'a::real_vector set"
himmelma@33175
  1542
  assumes "s \<noteq> {}"
wenzelm@50804
  1543
  shows "convex hull (insert a s) =
wenzelm@50804
  1544
    {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@53347
  1545
  (is "_ = ?hull")
wenzelm@50804
  1546
  apply (rule, rule hull_minimal, rule)
wenzelm@50804
  1547
  unfolding insert_iff
wenzelm@50804
  1548
  prefer 3
wenzelm@50804
  1549
  apply rule
wenzelm@50804
  1550
proof -
wenzelm@50804
  1551
  fix x
wenzelm@50804
  1552
  assume x: "x = a \<or> x \<in> s"
wenzelm@50804
  1553
  then show "x \<in> ?hull"
wenzelm@50804
  1554
    apply rule
wenzelm@50804
  1555
    unfolding mem_Collect_eq
wenzelm@50804
  1556
    apply (rule_tac x=1 in exI)
wenzelm@50804
  1557
    defer
wenzelm@50804
  1558
    apply (rule_tac x=0 in exI)
wenzelm@50804
  1559
    using assms hull_subset[of s convex]
wenzelm@50804
  1560
    apply auto
wenzelm@50804
  1561
    done
himmelma@33175
  1562
next
wenzelm@50804
  1563
  fix x
wenzelm@50804
  1564
  assume "x \<in> ?hull"
wenzelm@50804
  1565
  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
  1566
    by auto
wenzelm@53339
  1567
  have "a \<in> convex hull insert a s" "b \<in> convex hull insert a s"
wenzelm@50804
  1568
    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4)
wenzelm@50804
  1569
    by auto
wenzelm@50804
  1570
  then show "x \<in> convex hull insert a s"
huffman@53676
  1571
    unfolding obt(5) using obt(1-3)
huffman@53676
  1572
    by (rule convexD [OF convex_convex_hull])
himmelma@33175
  1573
next
wenzelm@50804
  1574
  show "convex ?hull"
huffman@53676
  1575
  proof (rule convexI)
wenzelm@50804
  1576
    fix x y u v
wenzelm@50804
  1577
    assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
wenzelm@53339
  1578
    from as(4) obtain u1 v1 b1 where
wenzelm@53339
  1579
      obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
wenzelm@53339
  1580
      by auto
wenzelm@53339
  1581
    from as(5) obtain u2 v2 b2 where
wenzelm@53339
  1582
      obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
wenzelm@53339
  1583
      by auto
wenzelm@50804
  1584
    have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
wenzelm@50804
  1585
      by (auto simp add: algebra_simps)
wenzelm@50804
  1586
    have **: "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y =
wenzelm@50804
  1587
      (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
wenzelm@50804
  1588
    proof (cases "u * v1 + v * v2 = 0")
wenzelm@50804
  1589
      case True
wenzelm@50804
  1590
      have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
wenzelm@50804
  1591
        by (auto simp add: algebra_simps)
wenzelm@50804
  1592
      from True have ***: "u * v1 = 0" "v * v2 = 0"
wenzelm@53302
  1593
        using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`]
wenzelm@53302
  1594
        by arith+
wenzelm@50804
  1595
      then have "u * u1 + v * u2 = 1"
wenzelm@50804
  1596
        using as(3) obt1(3) obt2(3) by auto
wenzelm@50804
  1597
      then show ?thesis
wenzelm@50804
  1598
        unfolding obt1(5) obt2(5) *
wenzelm@50804
  1599
        using assms hull_subset[of s convex]
wenzelm@50804
  1600
        by (auto simp add: *** scaleR_right_distrib)
himmelma@33175
  1601
    next
wenzelm@50804
  1602
      case False
wenzelm@50804
  1603
      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
wenzelm@50804
  1604
        using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
wenzelm@50804
  1605
      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
wenzelm@50804
  1606
        using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
wenzelm@50804
  1607
      also have "\<dots> = u * v1 + v * v2"
wenzelm@50804
  1608
        by simp
wenzelm@50804
  1609
      finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
wenzelm@50804
  1610
      have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
wenzelm@50804
  1611
        apply (rule add_nonneg_nonneg)
wenzelm@50804
  1612
        prefer 4
wenzelm@50804
  1613
        apply (rule add_nonneg_nonneg)
wenzelm@50804
  1614
        apply (rule_tac [!] mult_nonneg_nonneg)
wenzelm@53302
  1615
        using as(1,2) obt1(1,2) obt2(1,2)
wenzelm@53302
  1616
        apply auto
wenzelm@50804
  1617
        done
wenzelm@50804
  1618
      then show ?thesis
wenzelm@50804
  1619
        unfolding obt1(5) obt2(5)
wenzelm@50804
  1620
        unfolding * and **
wenzelm@50804
  1621
        using False
wenzelm@53339
  1622
        apply (rule_tac
wenzelm@53339
  1623
          x = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI)
wenzelm@50804
  1624
        defer
huffman@53676
  1625
        apply (rule convexD [OF convex_convex_hull])
wenzelm@50804
  1626
        using obt1(4) obt2(4)
wenzelm@49530
  1627
        unfolding add_divide_distrib[symmetric] and zero_le_divide_iff
wenzelm@50804
  1628
        apply (auto simp add: scaleR_left_distrib scaleR_right_distrib)
wenzelm@50804
  1629
        done
wenzelm@50804
  1630
    qed
wenzelm@50804
  1631
    have u1: "u1 \<le> 1"
wenzelm@50804
  1632
      unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
wenzelm@50804
  1633
    have u2: "u2 \<le> 1"
wenzelm@50804
  1634
      unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
wenzelm@53339
  1635
    have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
wenzelm@50804
  1636
      apply (rule add_mono)
wenzelm@50804
  1637
      apply (rule_tac [!] mult_right_mono)
wenzelm@50804
  1638
      using as(1,2) obt1(1,2) obt2(1,2)
wenzelm@50804
  1639
      apply auto
wenzelm@50804
  1640
      done
wenzelm@50804
  1641
    also have "\<dots> \<le> 1"
wenzelm@50804
  1642
      unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
wenzelm@50804
  1643
    finally show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
wenzelm@50804
  1644
      unfolding mem_Collect_eq
wenzelm@50804
  1645
      apply (rule_tac x="u * u1 + v * u2" in exI)
wenzelm@50804
  1646
      apply (rule conjI)
wenzelm@50804
  1647
      defer
wenzelm@50804
  1648
      apply (rule_tac x="1 - u * u1 - v * u2" in exI)
wenzelm@50804
  1649
      unfolding Bex_def
wenzelm@50804
  1650
      using as(1,2) obt1(1,2) obt2(1,2) **
wenzelm@50804
  1651
      apply (auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
wenzelm@50804
  1652
      done
himmelma@33175
  1653
  qed
himmelma@33175
  1654
qed
himmelma@33175
  1655
himmelma@33175
  1656
huffman@44467
  1657
subsubsection {* Explicit expression for convex hull *}
himmelma@33175
  1658
himmelma@33175
  1659
lemma convex_hull_indexed:
himmelma@33175
  1660
  fixes s :: "'a::real_vector set"
wenzelm@50804
  1661
  shows "convex hull s =
wenzelm@53347
  1662
    {y. \<exists>k u x.
wenzelm@53347
  1663
      (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
wenzelm@53347
  1664
      (setsum u {1..k} = 1) \<and> (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}"
wenzelm@53339
  1665
  (is "?xyz = ?hull")
wenzelm@50804
  1666
  apply (rule hull_unique)
wenzelm@50804
  1667
  apply rule
wenzelm@50804
  1668
  defer
huffman@53676
  1669
  apply (rule convexI)
wenzelm@50804
  1670
proof -
wenzelm@50804
  1671
  fix x
wenzelm@50804
  1672
  assume "x\<in>s"
wenzelm@50804
  1673
  then show "x \<in> ?hull"
wenzelm@50804
  1674
    unfolding mem_Collect_eq
wenzelm@50804
  1675
    apply (rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI)
wenzelm@50804
  1676
    apply auto
wenzelm@50804
  1677
    done
himmelma@33175
  1678
next
wenzelm@50804
  1679
  fix t
wenzelm@50804
  1680
  assume as: "s \<subseteq> t" "convex t"
wenzelm@50804
  1681
  show "?hull \<subseteq> t"
wenzelm@50804
  1682
    apply rule
wenzelm@50804
  1683
    unfolding mem_Collect_eq
wenzelm@53302
  1684
    apply (elim exE conjE)
wenzelm@50804
  1685
  proof -
wenzelm@50804
  1686
    fix x k u y
wenzelm@50804
  1687
    assume assm:
wenzelm@50804
  1688
      "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s"
wenzelm@50804
  1689
      "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
wenzelm@50804
  1690
    show "x\<in>t"
wenzelm@50804
  1691
      unfolding assm(3) [symmetric]
wenzelm@50804
  1692
      apply (rule as(2)[unfolded convex, rule_format])
wenzelm@50804
  1693
      using assm(1,2) as(1) apply auto
wenzelm@50804
  1694
      done
wenzelm@50804
  1695
  qed
himmelma@33175
  1696
next
wenzelm@50804
  1697
  fix x y u v
wenzelm@53347
  1698
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
wenzelm@53347
  1699
  assume xy: "x \<in> ?hull" "y \<in> ?hull"
wenzelm@50804
  1700
  from xy obtain k1 u1 x1 where
wenzelm@53339
  1701
    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
  1702
    by auto
wenzelm@50804
  1703
  from xy obtain k2 u2 x2 where
wenzelm@53339
  1704
    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
  1705
    by auto
wenzelm@50804
  1706
  have *: "\<And>P (x1::'a) x2 s1 s2 i.
wenzelm@50804
  1707
    (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
  1708
    "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
wenzelm@50804
  1709
    prefer 3
wenzelm@50804
  1710
    apply (rule, rule)
wenzelm@50804
  1711
    unfolding image_iff
wenzelm@50804
  1712
    apply (rule_tac x = "x - k1" in bexI)
wenzelm@50804
  1713
    apply (auto simp add: not_le)
wenzelm@50804
  1714
    done
wenzelm@50804
  1715
  have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
wenzelm@50804
  1716
    unfolding inj_on_def by auto
wenzelm@50804
  1717
  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
wenzelm@50804
  1718
    apply rule
wenzelm@50804
  1719
    apply (rule_tac x="k1 + k2" in exI)
wenzelm@50804
  1720
    apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
wenzelm@50804
  1721
    apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI)
wenzelm@50804
  1722
    apply (rule, rule)
wenzelm@50804
  1723
    defer
wenzelm@50804
  1724
    apply rule
wenzelm@50804
  1725
    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
wenzelm@50804
  1726
      setsum_reindex[OF inj] and o_def Collect_mem_eq
wenzelm@50804
  1727
    unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]
wenzelm@50804
  1728
  proof -
wenzelm@50804
  1729
    fix i
wenzelm@50804
  1730
    assume i: "i \<in> {1..k1+k2}"
wenzelm@50804
  1731
    show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and>
wenzelm@50804
  1732
      (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
wenzelm@50804
  1733
    proof (cases "i\<in>{1..k1}")
wenzelm@50804
  1734
      case True
wenzelm@50804
  1735
      then show ?thesis
wenzelm@53347
  1736
        using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]]
wenzelm@53347
  1737
        by auto
wenzelm@50804
  1738
    next
wenzelm@50804
  1739
      case False
wenzelm@50804
  1740
      def j \<equiv> "i - k1"
wenzelm@53347
  1741
      from i False have "j \<in> {1..k2}"
wenzelm@53347
  1742
        unfolding j_def by auto
wenzelm@50804
  1743
      then show ?thesis
wenzelm@50804
  1744
        unfolding j_def[symmetric]
wenzelm@50804
  1745
        using False
wenzelm@50804
  1746
        using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]]
wenzelm@50804
  1747
        apply auto
wenzelm@50804
  1748
        done
wenzelm@50804
  1749
    qed
wenzelm@50804
  1750
  qed (auto simp add: not_le x(2,3) y(2,3) uv(3))
himmelma@33175
  1751
qed
himmelma@33175
  1752
himmelma@33175
  1753
lemma convex_hull_finite:
himmelma@33175
  1754
  fixes s :: "'a::real_vector set"
himmelma@33175
  1755
  assumes "finite s"
himmelma@33175
  1756
  shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
wenzelm@53339
  1757
    setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
wenzelm@53339
  1758
  (is "?HULL = ?set")
wenzelm@50804
  1759
proof (rule hull_unique, auto simp add: convex_def[of ?set])
wenzelm@50804
  1760
  fix x
wenzelm@50804
  1761
  assume "x \<in> s"
wenzelm@50804
  1762
  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
  1763
    apply (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI)
wenzelm@50804
  1764
    apply auto
wenzelm@50804
  1765
    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms]
wenzelm@50804
  1766
    apply auto
wenzelm@50804
  1767
    done
himmelma@33175
  1768
next
wenzelm@50804
  1769
  fix u v :: real
wenzelm@50804
  1770
  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@50804
  1771
  fix ux assume ux: "\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
wenzelm@50804
  1772
  fix uy assume uy: "\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
wenzelm@53339
  1773
  {
wenzelm@53339
  1774
    fix x
wenzelm@50804
  1775
    assume "x\<in>s"
wenzelm@50804
  1776
    then have "0 \<le> u * ux x + v * uy x"
wenzelm@50804
  1777
      using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
wenzelm@50804
  1778
      apply auto
wenzelm@50804
  1779
      apply (metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))
wenzelm@50804
  1780
      done
wenzelm@50804
  1781
  }
wenzelm@50804
  1782
  moreover
wenzelm@50804
  1783
  have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
wenzelm@53302
  1784
    unfolding setsum_addf and setsum_right_distrib[symmetric] and ux(2) uy(2)
wenzelm@53302
  1785
    using uv(3) by auto
wenzelm@50804
  1786
  moreover
wenzelm@50804
  1787
  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@53339
  1788
    unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[symmetric]
wenzelm@53339
  1789
      and scaleR_right.setsum [symmetric]
wenzelm@50804
  1790
    by auto
wenzelm@50804
  1791
  ultimately
wenzelm@50804
  1792
  show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and>
wenzelm@50804
  1793
      (\<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
  1794
    apply (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI)
wenzelm@50804
  1795
    apply auto
wenzelm@50804
  1796
    done
himmelma@33175
  1797
next
wenzelm@50804
  1798
  fix t
wenzelm@50804
  1799
  assume t: "s \<subseteq> t" "convex t"
wenzelm@50804
  1800
  fix u
wenzelm@50804
  1801
  assume u: "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
wenzelm@50804
  1802
  then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t"
wenzelm@50804
  1803
    using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
himmelma@33175
  1804
    using assms and t(1) by auto
himmelma@33175
  1805
qed
himmelma@33175
  1806
wenzelm@50804
  1807
huffman@44467
  1808
subsubsection {* Another formulation from Lars Schewe *}
himmelma@33175
  1809
himmelma@33175
  1810
lemma setsum_constant_scaleR:
himmelma@33175
  1811
  fixes y :: "'a::real_vector"
himmelma@33175
  1812
  shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
wenzelm@50804
  1813
  apply (cases "finite A")
wenzelm@50804
  1814
  apply (induct set: finite)
wenzelm@50804
  1815
  apply (simp_all add: algebra_simps)
wenzelm@50804
  1816
  done
himmelma@33175
  1817
himmelma@33175
  1818
lemma convex_hull_explicit:
himmelma@33175
  1819
  fixes p :: "'a::real_vector set"
wenzelm@53347
  1820
  shows "convex hull p =
wenzelm@53347
  1821
    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
wenzelm@53339
  1822
  (is "?lhs = ?rhs")
wenzelm@50804
  1823
proof -
wenzelm@53302
  1824
  {
wenzelm@53302
  1825
    fix x
wenzelm@53302
  1826
    assume "x\<in>?lhs"
wenzelm@50804
  1827
    then obtain k u y where
wenzelm@50804
  1828
        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
  1829
      unfolding convex_hull_indexed by auto
himmelma@33175
  1830
wenzelm@50804
  1831
    have fin: "finite {1..k}" by auto
wenzelm@50804
  1832
    have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
wenzelm@53302
  1833
    {
wenzelm@53302
  1834
      fix j
wenzelm@50804
  1835
      assume "j\<in>{1..k}"
wenzelm@50804
  1836
      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
  1837
        using obt(1)[THEN bspec[where x=j]] and obt(2)
wenzelm@50804
  1838
        apply simp
wenzelm@50804
  1839
        apply (rule setsum_nonneg)
wenzelm@50804
  1840
        using obt(1)
wenzelm@50804
  1841
        apply auto
wenzelm@50804
  1842
        done
wenzelm@50804
  1843
    }
himmelma@33175
  1844
    moreover
wenzelm@49531
  1845
    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
wenzelm@49530
  1846
      unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
himmelma@33175
  1847
    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
wenzelm@49530
  1848
      using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
himmelma@33175
  1849
      unfolding scaleR_left.setsum using obt(3) by auto
wenzelm@50804
  1850
    ultimately
wenzelm@50804
  1851
    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
  1852
      apply (rule_tac x="y ` {1..k}" in exI)
wenzelm@50804
  1853
      apply (rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI)
wenzelm@50804
  1854
      apply auto
wenzelm@50804
  1855
      done
wenzelm@50804
  1856
    then have "x\<in>?rhs" by auto
wenzelm@50804
  1857
  }
himmelma@33175
  1858
  moreover
wenzelm@53302
  1859
  {
wenzelm@53302
  1860
    fix y
wenzelm@53302
  1861
    assume "y\<in>?rhs"
wenzelm@50804
  1862
    then obtain s u where
wenzelm@53339
  1863
      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"
wenzelm@53339
  1864
      by auto
wenzelm@50804
  1865
wenzelm@50804
  1866
    obtain f where f: "inj_on f {1..card s}" "f ` {1..card s} = s"
wenzelm@50804
  1867
      using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
wenzelm@50804
  1868
wenzelm@53302
  1869
    {
wenzelm@53302
  1870
      fix i :: nat
wenzelm@50804
  1871
      assume "i\<in>{1..card s}"
wenzelm@50804
  1872
      then have "f i \<in> s"
wenzelm@50804
  1873
        apply (subst f(2)[symmetric])
wenzelm@50804
  1874
        apply auto
wenzelm@50804
  1875
        done
wenzelm@50804
  1876
      then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
wenzelm@50804
  1877
    }
wenzelm@53347
  1878
    moreover have *: "finite {1..card s}" by auto
wenzelm@53302
  1879
    {
wenzelm@53302
  1880
      fix y
wenzelm@50804
  1881
      assume "y\<in>s"
wenzelm@53302
  1882
      then obtain i where "i\<in>{1..card s}" "f i = y"
wenzelm@53302
  1883
        using f using image_iff[of y f "{1..card s}"]
wenzelm@50804
  1884
        by auto
wenzelm@50804
  1885
      then have "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}"
wenzelm@50804
  1886
        apply auto
wenzelm@50804
  1887
        using f(1)[unfolded inj_on_def]
wenzelm@50804
  1888
        apply(erule_tac x=x in ballE)
wenzelm@50804
  1889
        apply auto
wenzelm@50804
  1890
        done
wenzelm@50804
  1891
      then have "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
wenzelm@50804
  1892
      then have "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
wenzelm@50804
  1893
          "(\<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
  1894
        by (auto simp add: setsum_constant_scaleR)
wenzelm@50804
  1895
    }
wenzelm@50804
  1896
    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@53339
  1897
      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
wenzelm@53339
  1898
        and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
wenzelm@53339
  1899
      unfolding f
wenzelm@53339
  1900
      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
  1901
      using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u]
wenzelm@53302
  1902
      unfolding obt(4,5)
wenzelm@53302
  1903
      by auto
wenzelm@50804
  1904
    ultimately
wenzelm@50804
  1905
    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
  1906
        (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
wenzelm@50804
  1907
      apply (rule_tac x="card s" in exI)
wenzelm@50804
  1908
      apply (rule_tac x="u \<circ> f" in exI)
wenzelm@50804
  1909
      apply (rule_tac x=f in exI)
wenzelm@50804
  1910
      apply fastforce
wenzelm@50804
  1911
      done
wenzelm@53302
  1912
    then have "y \<in> ?lhs"
wenzelm@53302
  1913
      unfolding convex_hull_indexed by auto
wenzelm@50804
  1914
  }
wenzelm@53302
  1915
  ultimately show ?thesis
wenzelm@53302
  1916
    unfolding set_eq_iff by blast
himmelma@33175
  1917
qed
himmelma@33175
  1918
wenzelm@50804
  1919
huffman@44467
  1920
subsubsection {* A stepping theorem for that expansion *}
himmelma@33175
  1921
himmelma@33175
  1922
lemma convex_hull_finite_step:
wenzelm@50804
  1923
  fixes s :: "'a::real_vector set"
wenzelm@50804
  1924
  assumes "finite s"
wenzelm@53302
  1925
  shows
wenzelm@53302
  1926
    "(\<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
  1927
      \<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
  1928
  (is "?lhs = ?rhs")
wenzelm@50804
  1929
proof (rule, case_tac[!] "a\<in>s")
wenzelm@53302
  1930
  assume "a \<in> s"
wenzelm@53339
  1931
  then have *: "insert a s = s" by auto
wenzelm@50804
  1932
  assume ?lhs
wenzelm@50804
  1933
  then show ?rhs
wenzelm@50804
  1934
    unfolding *
wenzelm@50804
  1935
    apply (rule_tac x=0 in exI)
wenzelm@50804
  1936
    apply auto
wenzelm@50804
  1937
    done
himmelma@33175
  1938
next
wenzelm@50804
  1939
  assume ?lhs
wenzelm@53302
  1940
  then obtain u where
wenzelm@53302
  1941
      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
  1942
    by auto
wenzelm@50804
  1943
  assume "a \<notin> s"
wenzelm@50804
  1944
  then show ?rhs
wenzelm@50804
  1945
    apply (rule_tac x="u a" in exI)
wenzelm@50804
  1946
    using u(1)[THEN bspec[where x=a]]
wenzelm@50804
  1947
    apply simp
wenzelm@50804
  1948
    apply (rule_tac x=u in exI)
wenzelm@50804
  1949
    using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s`
wenzelm@50804
  1950
    apply auto
wenzelm@50804
  1951
    done
himmelma@33175
  1952
next
wenzelm@50804
  1953
  assume "a \<in> s"
wenzelm@50804
  1954
  then have *: "insert a s = s" by auto
wenzelm@50804
  1955
  have fin: "finite (insert a s)" using assms by auto
wenzelm@50804
  1956
  assume ?rhs
wenzelm@50804
  1957
  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
  1958
    by auto
wenzelm@50804
  1959
  show ?lhs
wenzelm@50804
  1960
    apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
wenzelm@50804
  1961
    unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
wenzelm@50804
  1962
    unfolding setsum_clauses(2)[OF assms]
wenzelm@50804
  1963
    using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s`
wenzelm@50804
  1964
    apply auto
wenzelm@50804
  1965
    done
himmelma@33175
  1966
next
wenzelm@50804
  1967
  assume ?rhs
wenzelm@53339
  1968
  then obtain v u where
wenzelm@53339
  1969
    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
  1970
    by auto
wenzelm@50804
  1971
  moreover
wenzelm@50804
  1972
  assume "a \<notin> s"
wenzelm@50804
  1973
  moreover
wenzelm@53302
  1974
  have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s"
wenzelm@53302
  1975
    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
  1976
    apply (rule_tac setsum_cong2)
wenzelm@50804
  1977
    defer
wenzelm@50804
  1978
    apply (rule_tac setsum_cong2)
wenzelm@50804
  1979
    using `a \<notin> s`
wenzelm@50804
  1980
    apply auto
wenzelm@50804
  1981
    done
wenzelm@50804
  1982
  ultimately show ?lhs
wenzelm@50804
  1983
    apply (rule_tac x="\<lambda>x. if a = x then v else u x" in exI)
wenzelm@50804
  1984
    unfolding setsum_clauses(2)[OF assms]
wenzelm@50804
  1985
    apply auto
wenzelm@50804
  1986
    done
wenzelm@50804
  1987
qed
wenzelm@50804
  1988
himmelma@33175
  1989
huffman@44467
  1990
subsubsection {* Hence some special cases *}
himmelma@33175
  1991
himmelma@33175
  1992
lemma convex_hull_2:
himmelma@33175
  1993
  "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
  1994
proof -
wenzelm@53302
  1995
  have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
wenzelm@53302
  1996
    by auto
wenzelm@53302
  1997
  have **: "finite {b}" by auto
wenzelm@53302
  1998
  show ?thesis
wenzelm@53302
  1999
    apply (simp add: convex_hull_finite)
wenzelm@53302
  2000
    unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
wenzelm@53302
  2001
    apply auto
wenzelm@53302
  2002
    apply (rule_tac x=v in exI)
wenzelm@53302
  2003
    apply (rule_tac x="1 - v" in exI)
wenzelm@53302
  2004
    apply simp
wenzelm@53302
  2005
    apply (rule_tac x=u in exI)
wenzelm@53302
  2006
    apply simp
wenzelm@53302
  2007
    apply (rule_tac x="\<lambda>x. v" in exI)
wenzelm@53302
  2008
    apply simp
wenzelm@53302
  2009
    done
wenzelm@53302
  2010
qed
himmelma@33175
  2011
himmelma@33175
  2012
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
  2013
  unfolding convex_hull_2
wenzelm@53302
  2014
proof (rule Collect_cong)
wenzelm@53302
  2015
  have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
wenzelm@53302
  2016
    by auto
wenzelm@53302
  2017
  fix x
wenzelm@53302
  2018
  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
  2019
    (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
wenzelm@53302
  2020
    unfolding *
wenzelm@53302
  2021
    apply auto
wenzelm@53302
  2022
    apply (rule_tac[!] x=u in exI)
wenzelm@53302
  2023
    apply (auto simp add: algebra_simps)
wenzelm@53302
  2024
    done
wenzelm@53302
  2025
qed
himmelma@33175
  2026
himmelma@33175
  2027
lemma convex_hull_3:
himmelma@33175
  2028
  "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
  2029
proof -
wenzelm@53302
  2030
  have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
wenzelm@53302
  2031
    by auto
wenzelm@53302
  2032
  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
huffman@44361
  2033
    by (auto simp add: field_simps)
wenzelm@53302
  2034
  show ?thesis
wenzelm@53302
  2035
    unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
wenzelm@53302
  2036
    unfolding convex_hull_finite_step[OF fin(3)]
wenzelm@53302
  2037
    apply (rule Collect_cong)
wenzelm@53302
  2038
    apply simp
wenzelm@53302
  2039
    apply auto
wenzelm@53302
  2040
    apply (rule_tac x=va in exI)
wenzelm@53302
  2041
    apply (rule_tac x="u c" in exI)
wenzelm@53302
  2042
    apply simp
wenzelm@53302
  2043
    apply (rule_tac x="1 - v - w" in exI)
wenzelm@53302
  2044
    apply simp
wenzelm@53302
  2045
    apply (rule_tac x=v in exI)
wenzelm@53302
  2046
    apply simp
wenzelm@53302
  2047
    apply (rule_tac x="\<lambda>x. w" in exI)
wenzelm@53302
  2048
    apply simp
wenzelm@53302
  2049
    done
wenzelm@53302
  2050
qed
himmelma@33175
  2051
himmelma@33175
  2052
lemma convex_hull_3_alt:
himmelma@33175
  2053
  "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
  2054
proof -
wenzelm@53302
  2055
  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
wenzelm@53302
  2056
    by auto
wenzelm@53302
  2057
  show ?thesis
wenzelm@53302
  2058
    unfolding convex_hull_3
wenzelm@53302
  2059
    apply (auto simp add: *)
wenzelm@53302
  2060
    apply (rule_tac x=v in exI)
wenzelm@53302
  2061
    apply (rule_tac x=w in exI)
wenzelm@53302
  2062
    apply (simp add: algebra_simps)
wenzelm@53302
  2063
    apply (rule_tac x=u in exI)
wenzelm@53302
  2064
    apply (rule_tac x=v in exI)
wenzelm@53302
  2065
    apply (simp add: algebra_simps)
wenzelm@53302
  2066
    done
wenzelm@53302
  2067
qed
wenzelm@53302
  2068
himmelma@33175
  2069
huffman@44467
  2070
subsection {* Relations among closure notions and corresponding hulls *}
himmelma@33175
  2071
himmelma@33175
  2072
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
himmelma@33175
  2073
  unfolding affine_def convex_def by auto
himmelma@33175
  2074
huffman@44361
  2075
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
himmelma@33175
  2076
  using subspace_imp_affine affine_imp_convex by auto
himmelma@33175
  2077
huffman@44361
  2078
lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
wenzelm@53302
  2079
  by (metis hull_minimal span_inc subspace_imp_affine subspace_span)
himmelma@33175
  2080
huffman@44361
  2081
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
wenzelm@53302
  2082
  by (metis hull_minimal span_inc subspace_imp_convex subspace_span)
himmelma@33175
  2083
himmelma@33175
  2084
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
wenzelm@53302
  2085
  by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
wenzelm@53302
  2086
wenzelm@53302
  2087
wenzelm@53302
  2088
lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
wenzelm@49531
  2089
  unfolding affine_dependent_def dependent_def
himmelma@33175
  2090
  using affine_hull_subset_span by auto
himmelma@33175
  2091
himmelma@33175
  2092
lemma dependent_imp_affine_dependent:
wenzelm@53302
  2093
  assumes "dependent {x - a| x . x \<in> s}"
wenzelm@53302
  2094
    and "a \<notin> s"
himmelma@33175
  2095
  shows "affine_dependent (insert a s)"
wenzelm@53302
  2096
proof -
wenzelm@49531
  2097
  from assms(1)[unfolded dependent_explicit] obtain S u v
wenzelm@53347
  2098
    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"
wenzelm@53347
  2099
    by auto
himmelma@33175
  2100
  def t \<equiv> "(\<lambda>x. x + a) ` S"
himmelma@33175
  2101
wenzelm@53347
  2102
  have inj: "inj_on (\<lambda>x. x + a) S"
wenzelm@53302
  2103
    unfolding inj_on_def by auto
wenzelm@53302
  2104
  have "0 \<notin> S"
wenzelm@53302
  2105
    using obt(2) assms(2) unfolding subset_eq by auto
wenzelm@53347
  2106
  have fin: "finite t" and "t \<subseteq> s"
wenzelm@53302
  2107
    unfolding t_def using obt(1,2) by auto
wenzelm@53302
  2108
  then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
wenzelm@53302
  2109
    by auto
wenzelm@53302
  2110
  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
  2111
    apply (rule setsum_cong2)
wenzelm@53302
  2112
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2113
    apply auto
wenzelm@53302
  2114
    done
himmelma@33175
  2115
  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
  2116
    unfolding setsum_clauses(2)[OF fin]
wenzelm@53302
  2117
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2118
    apply auto
wenzelm@53302
  2119
    unfolding *
wenzelm@53302
  2120
    apply auto
wenzelm@53302
  2121
    done
himmelma@33175
  2122
  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
  2123
    apply (rule_tac x="v + a" in bexI)
wenzelm@53302
  2124
    using obt(3,4) and `0\<notin>S`
wenzelm@53302
  2125
    unfolding t_def
wenzelm@53302
  2126
    apply auto
wenzelm@53302
  2127
    done
wenzelm@53302
  2128
  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
  2129
    apply (rule setsum_cong2)
wenzelm@53302
  2130
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2131
    apply auto
wenzelm@53302
  2132
    done
wenzelm@49531
  2133
  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
wenzelm@53302
  2134
    unfolding scaleR_left.setsum
wenzelm@53302
  2135
    unfolding t_def and setsum_reindex[OF inj] and o_def
wenzelm@53302
  2136
    using obt(5)
wenzelm@53302
  2137
    by (auto simp add: setsum_addf scaleR_right_distrib)
wenzelm@53302
  2138
  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
  2139
    unfolding setsum_clauses(2)[OF fin]
wenzelm@53302
  2140
    using `a\<notin>s` `t\<subseteq>s`
wenzelm@53302
  2141
    by (auto simp add: *)
wenzelm@53302
  2142
  ultimately show ?thesis
wenzelm@53302
  2143
    unfolding affine_dependent_explicit
wenzelm@53302
  2144
    apply (rule_tac x="insert a t" in exI)
wenzelm@53302
  2145
    apply auto
wenzelm@53302
  2146
    done
himmelma@33175
  2147
qed
himmelma@33175
  2148
himmelma@33175
  2149
lemma convex_cone:
wenzelm@53302
  2150
  "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
  2151
  (is "?lhs = ?rhs")
wenzelm@53302
  2152
proof -
wenzelm@53302
  2153
  {
wenzelm@53302
  2154
    fix x y
wenzelm@53302
  2155
    assume "x\<in>s" "y\<in>s" and ?lhs
wenzelm@53302
  2156
    then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
wenzelm@53302
  2157
      unfolding cone_def by auto
wenzelm@53302
  2158
    then have "x + y \<in> s"
wenzelm@53302
  2159
      using `?lhs`[unfolded convex_def, THEN conjunct1]
wenzelm@53302
  2160
      apply (erule_tac x="2*\<^sub>R x" in ballE)
wenzelm@53302
  2161
      apply (erule_tac x="2*\<^sub>R y" in ballE)
wenzelm@53302
  2162
      apply (erule_tac x="1/2" in allE)
wenzelm@53302
  2163
      apply simp
wenzelm@53302
  2164
      apply (erule_tac x="1/2" in allE)
wenzelm@53302
  2165
      apply auto
wenzelm@53302
  2166
      done
wenzelm@53302
  2167
  }
wenzelm@53302
  2168
  then show ?thesis
wenzelm@53302
  2169
    unfolding convex_def cone_def by blast
wenzelm@53302
  2170
qed
wenzelm@53302
  2171
wenzelm@53302
  2172
lemma affine_dependent_biggerset:
wenzelm@53347
  2173
  fixes s :: "'a::euclidean_space set"
hoelzl@37489
  2174
  assumes "finite s" "card s \<ge> DIM('a) + 2"
himmelma@33175
  2175
  shows "affine_dependent s"
wenzelm@53302
  2176
proof -
wenzelm@53302
  2177
  have "s \<noteq> {}" using assms by auto
wenzelm@53302
  2178
  then obtain a where "a\<in>s" by auto
wenzelm@53302
  2179
  have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
wenzelm@53302
  2180
    by auto
wenzelm@53302
  2181
  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
wenzelm@53302
  2182
    unfolding *
wenzelm@53302
  2183
    apply (rule card_image)
wenzelm@53302
  2184
    unfolding inj_on_def
wenzelm@53302
  2185
    apply auto
wenzelm@53302
  2186
    done
hoelzl@37489
  2187
  also have "\<dots> > DIM('a)" using assms(2)
himmelma@33175
  2188
    unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
wenzelm@53302
  2189
  finally show ?thesis
wenzelm@53302
  2190
    apply (subst insert_Diff[OF `a\<in>s`, symmetric])
wenzelm@53302
  2191
    apply (rule dependent_imp_affine_dependent)
wenzelm@53302
  2192
    apply (rule dependent_biggerset)
wenzelm@53302
  2193
    apply auto
wenzelm@53302
  2194
    done
wenzelm@53302
  2195
qed
himmelma@33175
  2196
himmelma@33175
  2197
lemma affine_dependent_biggerset_general:
wenzelm@53347
  2198
  assumes "finite (s :: 'a::euclidean_space set)"
wenzelm@53347
  2199
    and "card s \<ge> dim s + 2"
himmelma@33175
  2200
  shows "affine_dependent s"
wenzelm@53302
  2201
proof -
himmelma@33175
  2202
  from assms(2) have "s \<noteq> {}" by auto
himmelma@33175
  2203
  then obtain a where "a\<in>s" by auto
wenzelm@53302
  2204
  have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
wenzelm@53302
  2205
    by auto
wenzelm@53302
  2206
  have **: "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
wenzelm@53302
  2207
    unfolding *
wenzelm@53302
  2208
    apply (rule card_image)
wenzelm@53302
  2209
    unfolding inj_on_def
wenzelm@53302
  2210
    apply auto
wenzelm@53302
  2211
    done
himmelma@33175
  2212
  have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
wenzelm@53302
  2213
    apply (rule subset_le_dim)
wenzelm@53302
  2214
    unfolding subset_eq
wenzelm@53302
  2215
    using `a\<in>s`
wenzelm@53302
  2216
    apply (auto simp add:span_superset span_sub)
wenzelm@53302
  2217
    done
himmelma@33175
  2218
  also have "\<dots> < dim s + 1" by auto
wenzelm@53302
  2219
  also have "\<dots> \<le> card (s - {a})"
wenzelm@53302
  2220
    using assms
wenzelm@53302
  2221
    using card_Diff_singleton[OF assms(1) `a\<in>s`]
wenzelm@53302
  2222
    by auto
wenzelm@53302
  2223
  finally show ?thesis
wenzelm@53302
  2224
    apply (subst insert_Diff[OF `a\<in>s`, symmetric])
wenzelm@53302
  2225
    apply (rule dependent_imp_affine_dependent)
wenzelm@53302
  2226
    apply (rule dependent_biggerset_general)
wenzelm@53302
  2227
    unfolding **
wenzelm@53302
  2228
    apply auto
wenzelm@53302
  2229
    done
wenzelm@53302
  2230
qed
wenzelm@53302
  2231
himmelma@33175
  2232
himmelma@33175
  2233
subsection {* Caratheodory's theorem. *}
himmelma@33175
  2234
wenzelm@53302
  2235
lemma convex_hull_caratheodory:
wenzelm@53302
  2236
  fixes p :: "('a::euclidean_space) set"
wenzelm@53302
  2237
  shows "convex hull p =
wenzelm@53302
  2238
    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
wenzelm@53302
  2239
      (\<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
  2240
  unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
wenzelm@53302
  2241
proof (rule, rule)
wenzelm@53302
  2242
  fix y
wenzelm@53302
  2243
  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
  2244
    setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
  2245
  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
  2246
  then obtain N where "?P N" by auto
wenzelm@53302
  2247
  then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
wenzelm@53302
  2248
    apply (rule_tac ex_least_nat_le)
wenzelm@53302
  2249
    apply auto
wenzelm@53302
  2250
    done
wenzelm@53302
  2251
  then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
wenzelm@53302
  2252
    by blast
wenzelm@53302
  2253
  then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
wenzelm@53302
  2254
    "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
wenzelm@53302
  2255
wenzelm@53302
  2256
  have "card s \<le> DIM('a) + 1"
wenzelm@53302
  2257
  proof (rule ccontr, simp only: not_le)
hoelzl@37489
  2258
    assume "DIM('a) + 1 < card s"
wenzelm@53302
  2259
    then have "affine_dependent s"
wenzelm@53302
  2260
      using affine_dependent_biggerset[OF obt(1)] by auto
wenzelm@53302
  2261
    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
  2262
      using affine_dependent_explicit_finite[OF obt(1)] by auto
wenzelm@53302
  2263
    def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
wenzelm@53302
  2264
    def t \<equiv> "Min i"
wenzelm@53302
  2265
    have "\<exists>x\<in>s. w x < 0"
wenzelm@53302
  2266
    proof (rule ccontr, simp add: not_less)
himmelma@33175
  2267
      assume as:"\<forall>x\<in>s. 0 \<le> w x"
wenzelm@53302
  2268
      then have "setsum w (s - {v}) \<ge> 0"
wenzelm@53302
  2269
        apply (rule_tac setsum_nonneg)
wenzelm@53302
  2270
        apply auto
wenzelm@53302
  2271
        done
wenzelm@53302
  2272
      then have "setsum w s > 0"
wenzelm@53302
  2273
        unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
wenzelm@53302
  2274
        using as[THEN bspec[where x=v]] and `v\<in>s`
wenzelm@53302
  2275
        using `w v \<noteq> 0`
wenzelm@53302
  2276
        by auto
wenzelm@53302
  2277
      then show False using wv(1) by auto
wenzelm@53302
  2278
    qed
wenzelm@53302
  2279
    then have "i \<noteq> {}" unfolding i_def by auto
wenzelm@53302
  2280
wenzelm@53302
  2281
    then have "t \<ge> 0"
wenzelm@53302
  2282
      using Min_ge_iff[of i 0 ] and obt(1)
wenzelm@53302
  2283
      unfolding t_def i_def
wenzelm@53302
  2284
      using obt(4)[unfolded le_less]
wenzelm@53302
  2285
      apply auto
wenzelm@53302
  2286
      unfolding divide_le_0_iff
wenzelm@53302
  2287
      apply auto
wenzelm@53302
  2288
      done
wenzelm@53302
  2289
    have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
wenzelm@53302
  2290
    proof
wenzelm@53302
  2291
      fix v
wenzelm@53302
  2292
      assume "v \<in> s"
wenzelm@53302
  2293
      then have v: "0 \<le> u v"
wenzelm@53302
  2294
        using obt(4)[THEN bspec[where x=v]] by auto
wenzelm@53302
  2295
      show "0 \<le> u v + t * w v"
wenzelm@53302
  2296
      proof (cases "w v < 0")
wenzelm@53302
  2297
        case False
wenzelm@53302
  2298
        then show ?thesis
wenzelm@53302
  2299
          apply (rule_tac add_nonneg_nonneg)
wenzelm@53302
  2300
          using v
wenzelm@53302
  2301
          apply simp
wenzelm@53302
  2302
          apply (rule mult_nonneg_nonneg)
wenzelm@53302
  2303
          using `t\<ge>0`
wenzelm@53302
  2304
          apply auto
wenzelm@53302
  2305
          done
wenzelm@53302
  2306
      next
wenzelm@53302
  2307
        case True
wenzelm@53302
  2308
        then have "t \<le> u v / (- w v)"
wenzelm@53302
  2309
          using `v\<in>s`
wenzelm@53302
  2310
          unfolding t_def i_def
wenzelm@53302
  2311
          apply (rule_tac Min_le)
wenzelm@53302
  2312
          using obt(1)
wenzelm@53302
  2313
          apply auto
wenzelm@53302
  2314
          done
wenzelm@53302
  2315
        then show ?thesis
wenzelm@53302
  2316
          unfolding real_0_le_add_iff
wenzelm@53302
  2317
          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
wenzelm@53302
  2318
          by auto
wenzelm@53302
  2319
      qed
wenzelm@53302
  2320
    qed
wenzelm@53302
  2321
wenzelm@53302
  2322
    obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
himmelma@33175
  2323
      using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
wenzelm@53302
  2324
    then have a: "a \<in> s" "u a + t * w a = 0" by auto
wenzelm@53302
  2325
    have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
wenzelm@49531
  2326
      unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto
himmelma@33175
  2327
    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
wenzelm@49530
  2328
      unfolding setsum_addf wv(1) setsum_right_distrib[symmetric] obt(5) by auto
wenzelm@49531
  2329
    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
  2330
      unfolding setsum_addf obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
hoelzl@37489
  2331
      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
wenzelm@53302
  2332
    ultimately have "?P (n - 1)"
wenzelm@53302
  2333
      apply (rule_tac x="(s - {a})" in exI)
wenzelm@53302
  2334
      apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
wenzelm@53302
  2335
      using obt(1-3) and t and a
wenzelm@53302
  2336
      apply (auto simp add: * scaleR_left_distrib)
wenzelm@53302
  2337
      done
wenzelm@53302
  2338
    then show False
wenzelm@53302
  2339
      using smallest[THEN spec[where x="n - 1"]] by auto
wenzelm@53302
  2340
  qed
wenzelm@53302
  2341
  then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
wenzelm@53347
  2342
      (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
wenzelm@53347
  2343
    using obt by auto
himmelma@33175
  2344
qed auto
himmelma@33175
  2345
himmelma@33175
  2346
lemma caratheodory:
wenzelm@53302
  2347
  "convex hull p =
wenzelm@53302
  2348
    {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
hoelzl@37489
  2349
      card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
wenzelm@53302
  2350
  unfolding set_eq_iff
wenzelm@53302
  2351
  apply rule
wenzelm@53302
  2352
  apply rule
wenzelm@53302
  2353
  unfolding mem_Collect_eq
wenzelm@53302
  2354
proof -
wenzelm@53302
  2355
  fix x
wenzelm@53302
  2356
  assume "x \<in> convex hull p"
hoelzl@37489
  2357
  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
wenzelm@53302
  2358
    "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
wenzelm@53302
  2359
    unfolding convex_hull_caratheodory by auto
wenzelm@53302
  2360
  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
  2361
    apply (rule_tac x=s in exI)
wenzelm@53302
  2362
    using hull_subset[of s convex]
wenzelm@53347
  2363
    using convex_convex_hull[unfolded convex_explicit, of s,
wenzelm@53347
  2364
      THEN spec[where x=s], THEN spec[where x=u]]
wenzelm@53302
  2365
    apply auto
wenzelm@53302
  2366
    done
himmelma@33175
  2367
next
wenzelm@53302
  2368
  fix x
wenzelm@53302
  2369
  assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
wenzelm@53302
  2370
  then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
wenzelm@53302
  2371
    by auto
wenzelm@53302
  2372
  then show "x \<in> convex hull p"
wenzelm@53302
  2373
    using hull_mono[OF `s\<subseteq>p`] by auto
himmelma@33175
  2374
qed
himmelma@33175
  2375
hoelzl@40377
  2376
hoelzl@40377
  2377
subsection {* Some Properties of Affine Dependent Sets *}
hoelzl@40377
  2378
wenzelm@53347
  2379
lemma affine_independent_empty: "\<not> affine_dependent {}"
hoelzl@40377
  2380
  by (simp add: affine_dependent_def)
hoelzl@40377
  2381
wenzelm@53302
  2382
lemma affine_independent_sing: "\<not> affine_dependent {a}"
wenzelm@53302
  2383
  by (simp add: affine_dependent_def)
wenzelm@53302
  2384
wenzelm@53302
  2385
lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2386
proof -
wenzelm@53302
  2387
  have "affine ((\<lambda>x. a + x) ` (affine hull S))"
wenzelm@53302
  2388
    using affine_translation affine_affine_hull by auto
wenzelm@53347
  2389
  moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2390
    using hull_subset[of S] by auto
wenzelm@53347
  2391
  ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2392
    by (metis hull_minimal)
wenzelm@53302
  2393
  have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
haftmann@54230
  2394
    using affine_translation affine_affine_hull by (auto simp del: uminus_add_conv_diff)
wenzelm@53347
  2395
  moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
wenzelm@53302
  2396
    using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
wenzelm@53347
  2397
  moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
wenzelm@53302
  2398
    using translation_assoc[of "-a" a] by auto
wenzelm@53302
  2399
  ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
wenzelm@53302
  2400
    by (metis hull_minimal)
wenzelm@53302
  2401
  then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
wenzelm@53302
  2402
    by auto
wenzelm@54465
  2403
  then show ?thesis using h1 by auto
hoelzl@40377
  2404
qed
hoelzl@40377
  2405
hoelzl@40377
  2406
lemma affine_dependent_translation:
hoelzl@40377
  2407
  assumes "affine_dependent S"
wenzelm@53339
  2408
  shows "affine_dependent ((\<lambda>x. a + x) ` S)"
wenzelm@53302
  2409
proof -
wenzelm@54465
  2410
  obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
wenzelm@53302
  2411
    using assms affine_dependent_def by auto
wenzelm@53302
  2412
  have "op + a ` (S - {x}) = op + a ` S - {a + x}"
wenzelm@53302
  2413
    by auto
wenzelm@53347
  2414
  then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
wenzelm@54465
  2415
    using affine_hull_translation[of a "S - {x}"] x by auto
wenzelm@53347
  2416
  moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
wenzelm@54465
  2417
    using x by auto
wenzelm@53302
  2418
  ultimately show ?thesis
wenzelm@53302
  2419
    unfolding affine_dependent_def by auto
hoelzl@40377
  2420
qed
hoelzl@40377
  2421
hoelzl@40377
  2422
lemma affine_dependent_translation_eq:
wenzelm@54465
  2423
  "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
wenzelm@53302
  2424
proof -
wenzelm@53302
  2425
  {
wenzelm@53339
  2426
    assume "affine_dependent ((\<lambda>x. a + x) ` S)"
wenzelm@53302
  2427
    then have "affine_dependent S"
wenzelm@53339
  2428
      using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
wenzelm@53302
  2429
      by auto
wenzelm@53302
  2430
  }
wenzelm@53302
  2431
  then show ?thesis
wenzelm@53302
  2432
    using affine_dependent_translation by auto
hoelzl@40377
  2433
qed
hoelzl@40377
  2434
hoelzl@40377
  2435
lemma affine_hull_0_dependent:
wenzelm@53339
  2436
  assumes "0 \<in> affine hull S"
hoelzl@40377
  2437
  shows "dependent S"
wenzelm@53302
  2438
proof -
wenzelm@54465
  2439
  obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
wenzelm@53302
  2440
    using assms affine_hull_explicit[of S] by auto
wenzelm@53339
  2441
  then have "\<exists>v\<in>s. u v \<noteq> 0"
wenzelm@53302
  2442
    using setsum_not_0[of "u" "s"] by auto
wenzelm@53339
  2443
  then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
wenzelm@54465
  2444
    using s_u by auto
wenzelm@53302
  2445
  then show ?thesis
wenzelm@53302
  2446
    unfolding dependent_explicit[of S] by auto
hoelzl@40377
  2447
qed
hoelzl@40377
  2448
hoelzl@40377
  2449
lemma affine_dependent_imp_dependent2:
hoelzl@40377
  2450
  assumes "affine_dependent (insert 0 S)"
hoelzl@40377
  2451
  shows "dependent S"
wenzelm@53302
  2452
proof -
wenzelm@54465
  2453
  obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
wenzelm@53302
  2454
    using affine_dependent_def[of "(insert 0 S)"] assms by blast
wenzelm@53302
  2455
  then have "x \<in> span (insert 0 S - {x})"
wenzelm@53302
  2456
    using affine_hull_subset_span by auto
wenzelm@53302
  2457
  moreover have "span (insert 0 S - {x}) = span (S - {x})"
wenzelm@53302
  2458
    using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
wenzelm@53302
  2459
  ultimately have "x \<in> span (S - {x})" by auto
wenzelm@53302
  2460
  then have "x \<noteq> 0 \<Longrightarrow> dependent S"
wenzelm@54465
  2461
    using x dependent_def by auto