src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author huffman
Mon Jul 05 09:14:51 2010 -0700 (2010-07-05)
changeset 37732 6432bf0d7191
parent 37673 f69f4b079275
child 38642 8fa437809c67
permissions -rw-r--r--
generalize type of is_interval to class euclidean_space
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(*  Title:      HOL/Library/Convex_Euclidean_Space.thy
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    Author:     Robert Himmelmann, TU Muenchen
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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 Topology_Euclidean_Space Convex
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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 basis_0[simp]:"(basis i::'a::euclidean_space) = 0 \<longleftrightarrow> i\<ge>DIM('a)"
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  using norm_basis[of i, where 'a='a] unfolding norm_eq_zero[where 'a='a,THEN sym] by auto
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lemma scaleR_2:
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  fixes x :: "'a::real_vector"
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  shows "scaleR 2 x = x + x"
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unfolding one_add_one_is_two [symmetric] scaleR_left_distrib by simp
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declare euclidean_simps[simp]
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lemma vector_choose_size: "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
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  apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"]) using DIM_positive[where 'a='a] by auto
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lemma setsum_delta_notmem: assumes "x\<notin>s"
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  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
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        "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
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        "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
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        "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
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  apply(rule_tac [!] setsum_cong2) using assms by auto
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lemma setsum_delta'':
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  fixes s::"'a::real_vector set" assumes "finite s"
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  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)"
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proof-
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  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)" by auto
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  show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
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qed
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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)" by auto
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lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
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  (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})"
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  using image_affinity_interval[of m 0 a b] by auto
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lemma dist_triangle_eq:
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  fixes x y z :: "'a::euclidean_space"
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  shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
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proof- have *:"x - y + (y - z) = x - z" by auto
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  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
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    by(auto simp add:norm_minus_commute) qed
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lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
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lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
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  unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
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lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
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  unfolding norm_eq_sqrt_inner by simp
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lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
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  unfolding norm_eq_sqrt_inner by simp
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subsection {* Affine set and affine hull.*}
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definition
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  affine :: "'a::real_vector set \<Rightarrow> bool" where
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  "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)"
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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)"
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unfolding affine_def by(metis eq_diff_eq')
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lemma affine_empty[intro]: "affine {}"
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  unfolding affine_def by auto
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lemma affine_sing[intro]: "affine {x}"
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  unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
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lemma affine_UNIV[intro]: "affine UNIV"
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  unfolding affine_def by auto
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lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
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  unfolding affine_def by auto 
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lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
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  unfolding affine_def by auto
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lemma affine_affine_hull: "affine(affine hull s)"
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  unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
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  unfolding mem_def by auto
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lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
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by (metis affine_affine_hull hull_same mem_def)
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lemma setsum_restrict_set'': assumes "finite A"
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  shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
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  unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
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subsection {* Some explicit formulations (from Lars Schewe). *}
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lemma affine: fixes V::"'a::real_vector set"
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  shows "affine V \<longleftrightarrow> (\<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)"
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unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
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defer apply(rule, rule, rule, rule, rule) proof-
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  fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
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    "\<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"
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  thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
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    using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]] and as(1-3) 
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    by(auto simp add: scaleR_left_distrib[THEN sym])
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next
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  fix s u 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"
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    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
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  def n \<equiv> "card s"
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  have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
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  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
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    assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
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    then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
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    thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
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      by(auto simp add: setsum_clauses(2))
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  next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
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      case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
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      assume IA:"\<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;
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               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" and
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        as:"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"
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           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
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      have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
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        assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
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        thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
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          less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
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      then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
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      have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
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      have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
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      have **:"setsum u (s - {x}) = 1 - u x"
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        using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
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      have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
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      have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
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        case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
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          assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
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          thus False using True by auto qed auto
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        thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
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        unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
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      next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
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        then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
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        thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
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          using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
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      hence "u x + (1 - u x) = 1 \<Longrightarrow> 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"
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        apply-apply(rule as(3)[rule_format]) 
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        unfolding  RealVector.scaleR_right.setsum using x(1) as(6) by auto
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      thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
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         apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
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         using `u x \<noteq> 1` by auto 
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    qed auto
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  next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
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    thus ?thesis using as(4,5) by simp
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  qed(insert `s\<noteq>{}` `finite s`, auto)
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qed
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lemma affine_hull_explicit:
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  "affine hull p = {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}"
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  apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
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  apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
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  fix x assume "x\<in>p" thus "\<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"
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    apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
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next
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  fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
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  thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
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next
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  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}" unfolding affine_def
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    apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
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    fix u v ::real assume uv:"u + v = 1"
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    fix x 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"
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    then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
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    fix y 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"
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    then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
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    have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
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    have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
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    show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
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      apply(rule_tac x="sx \<union> sy" in exI)
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      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)
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      unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
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      unfolding scaleR_scaleR[THEN sym] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
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      unfolding x y using x(1-3) y(1-3) uv by simp qed qed
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lemma affine_hull_finite:
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  assumes "finite s"
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  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
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  unfolding affine_hull_explicit and expand_set_eq and mem_Collect_eq apply (rule,rule)
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  apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
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  fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
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  thus "\<exists>sa u. finite sa \<and> \<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"
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    apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
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next
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  fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
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  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"
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  thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
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    unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
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subsection {* Stepping theorems and hence small special cases. *}
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lemma affine_hull_empty[simp]: "affine hull {} = {}"
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  apply(rule hull_unique) unfolding mem_def by auto
himmelma@33175
   209
himmelma@33175
   210
lemma affine_hull_finite_step:
himmelma@33175
   211
  fixes y :: "'a::real_vector"
himmelma@33175
   212
  shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
himmelma@33175
   213
  "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
himmelma@33175
   214
                (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
himmelma@33175
   215
proof-
himmelma@33175
   216
  show ?th1 by simp
himmelma@33175
   217
  assume ?as 
himmelma@33175
   218
  { assume ?lhs
himmelma@33175
   219
    then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
himmelma@33175
   220
    have ?rhs proof(cases "a\<in>s")
himmelma@33175
   221
      case True hence *:"insert a s = s" by auto
himmelma@33175
   222
      show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
himmelma@33175
   223
    next
himmelma@33175
   224
      case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
himmelma@33175
   225
    qed  } moreover
himmelma@33175
   226
  { assume ?rhs
himmelma@33175
   227
    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" by auto
himmelma@33175
   228
    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)" by auto
himmelma@33175
   229
    have ?lhs proof(cases "a\<in>s")
himmelma@33175
   230
      case True thus ?thesis
himmelma@33175
   231
        apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
himmelma@33175
   232
        unfolding setsum_clauses(2)[OF `?as`]  apply simp
himmelma@33175
   233
        unfolding scaleR_left_distrib and setsum_addf 
himmelma@33175
   234
        unfolding vu and * and scaleR_zero_left
himmelma@33175
   235
        by (auto simp add: setsum_delta[OF `?as`])
himmelma@33175
   236
    next
himmelma@33175
   237
      case False 
himmelma@33175
   238
      hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
himmelma@33175
   239
               "\<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
   240
      from False show ?thesis
himmelma@33175
   241
        apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
himmelma@33175
   242
        unfolding setsum_clauses(2)[OF `?as`] and * using vu
himmelma@33175
   243
        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)]
himmelma@33175
   244
        using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
himmelma@33175
   245
    qed }
himmelma@33175
   246
  ultimately show "?lhs = ?rhs" by blast
himmelma@33175
   247
qed
himmelma@33175
   248
himmelma@33175
   249
lemma affine_hull_2:
himmelma@33175
   250
  fixes a b :: "'a::real_vector"
himmelma@33175
   251
  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
himmelma@33175
   252
proof-
himmelma@33175
   253
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
himmelma@33175
   254
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
himmelma@33175
   255
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
himmelma@33175
   256
    using affine_hull_finite[of "{a,b}"] by auto
himmelma@33175
   257
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
himmelma@33175
   258
    by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
himmelma@33175
   259
  also have "\<dots> = ?rhs" unfolding * by auto
himmelma@33175
   260
  finally show ?thesis by auto
himmelma@33175
   261
qed
himmelma@33175
   262
himmelma@33175
   263
lemma affine_hull_3:
himmelma@33175
   264
  fixes a b c :: "'a::real_vector"
himmelma@33175
   265
  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}" (is "?lhs = ?rhs")
himmelma@33175
   266
proof-
himmelma@33175
   267
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
himmelma@33175
   268
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
himmelma@33175
   269
  show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
himmelma@33175
   270
    unfolding * apply auto
himmelma@33175
   271
    apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
himmelma@33175
   272
    apply(rule_tac x=u in exI) by(auto intro!: exI)
himmelma@33175
   273
qed
himmelma@33175
   274
himmelma@33175
   275
subsection {* Some relations between affine hull and subspaces. *}
himmelma@33175
   276
himmelma@33175
   277
lemma affine_hull_insert_subset_span:
hoelzl@37489
   278
  fixes a :: "'a::euclidean_space"
himmelma@33175
   279
  shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
hoelzl@37489
   280
  unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
himmelma@33175
   281
  apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
himmelma@33175
   282
  fix x t u 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"
himmelma@33175
   283
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
himmelma@33175
   284
  thus "\<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)"
himmelma@33175
   285
    apply(rule_tac x="x - a" in exI)
himmelma@33175
   286
    apply (rule conjI, simp)
himmelma@33175
   287
    apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
himmelma@33175
   288
    apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
himmelma@33175
   289
    apply (rule conjI) using as(1) apply simp
himmelma@33175
   290
    apply (erule conjI)
himmelma@33175
   291
    using as(1)
himmelma@33175
   292
    apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib setsum_subtractf scaleR_left.setsum[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
himmelma@33175
   293
    unfolding as by simp qed
himmelma@33175
   294
himmelma@33175
   295
lemma affine_hull_insert_span:
hoelzl@37489
   296
  fixes a :: "'a::euclidean_space"
himmelma@33175
   297
  assumes "a \<notin> s"
himmelma@33175
   298
  shows "affine hull (insert a s) =
himmelma@33175
   299
            {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
himmelma@33175
   300
  apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
himmelma@33175
   301
  unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
himmelma@33175
   302
  fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
hoelzl@37489
   303
  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" unfolding span_explicit by auto
himmelma@33175
   304
  def f \<equiv> "(\<lambda>x. x + a) ` t"
himmelma@33175
   305
  have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt 
himmelma@33175
   306
    by(auto simp add: setsum_reindex[unfolded inj_on_def])
himmelma@33175
   307
  have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
himmelma@33175
   308
  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"
himmelma@33175
   309
    apply(rule_tac x="insert a f" in exI)
himmelma@33175
   310
    apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
himmelma@33175
   311
    using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
hoelzl@35577
   312
    unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
hoelzl@35577
   313
    by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
himmelma@33175
   314
himmelma@33175
   315
lemma affine_hull_span:
hoelzl@37489
   316
  fixes a :: "'a::euclidean_space"
himmelma@33175
   317
  assumes "a \<in> s"
himmelma@33175
   318
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
himmelma@33175
   319
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
himmelma@33175
   320
himmelma@33175
   321
subsection {* Cones. *}
himmelma@33175
   322
himmelma@33175
   323
definition
himmelma@33175
   324
  cone :: "'a::real_vector set \<Rightarrow> bool" where
himmelma@33175
   325
  "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
himmelma@33175
   326
himmelma@33175
   327
lemma cone_empty[intro, simp]: "cone {}"
himmelma@33175
   328
  unfolding cone_def by auto
himmelma@33175
   329
himmelma@33175
   330
lemma cone_univ[intro, simp]: "cone UNIV"
himmelma@33175
   331
  unfolding cone_def by auto
himmelma@33175
   332
himmelma@33175
   333
lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
himmelma@33175
   334
  unfolding cone_def by auto
himmelma@33175
   335
himmelma@33175
   336
subsection {* Conic hull. *}
himmelma@33175
   337
himmelma@33175
   338
lemma cone_cone_hull: "cone (cone hull s)"
himmelma@33175
   339
  unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
himmelma@33175
   340
  by (auto simp add: mem_def)
himmelma@33175
   341
himmelma@33175
   342
lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
himmelma@33175
   343
  apply(rule hull_eq[unfolded mem_def])
himmelma@33175
   344
  using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
himmelma@33175
   345
himmelma@33175
   346
subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
himmelma@33175
   347
himmelma@33175
   348
definition
himmelma@33175
   349
  affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
himmelma@33175
   350
  "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
himmelma@33175
   351
himmelma@33175
   352
lemma affine_dependent_explicit:
himmelma@33175
   353
  "affine_dependent p \<longleftrightarrow>
himmelma@33175
   354
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
himmelma@33175
   355
    (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
himmelma@33175
   356
  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
himmelma@33175
   357
  apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
himmelma@33175
   358
proof-
himmelma@33175
   359
  fix x s u 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"
himmelma@33175
   360
  have "x\<notin>s" using as(1,4) by auto
himmelma@33175
   361
  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"
himmelma@33175
   362
    apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
himmelma@33175
   363
    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
himmelma@33175
   364
next
himmelma@33175
   365
  fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
himmelma@33175
   366
  have "s \<noteq> {v}" using as(3,6) by auto
himmelma@33175
   367
  thus "\<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" 
himmelma@33175
   368
    apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
himmelma@33175
   369
    unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
himmelma@33175
   370
qed
himmelma@33175
   371
himmelma@33175
   372
lemma affine_dependent_explicit_finite:
himmelma@33175
   373
  fixes s :: "'a::real_vector set" assumes "finite s"
himmelma@33175
   374
  shows "affine_dependent s \<longleftrightarrow> (\<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
   375
  (is "?lhs = ?rhs")
himmelma@33175
   376
proof
himmelma@33175
   377
  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))" by auto
himmelma@33175
   378
  assume ?lhs
himmelma@33175
   379
  then obtain t u v where "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
   380
    unfolding affine_dependent_explicit by auto
himmelma@33175
   381
  thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
himmelma@33175
   382
    apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
himmelma@33175
   383
    unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
himmelma@33175
   384
next
himmelma@33175
   385
  assume ?rhs
himmelma@33175
   386
  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
himmelma@33175
   387
  thus ?lhs unfolding affine_dependent_explicit using assms by auto
himmelma@33175
   388
qed
himmelma@33175
   389
himmelma@33175
   390
subsection {* A general lemma. *}
himmelma@33175
   391
himmelma@33175
   392
lemma convex_connected:
himmelma@33175
   393
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   394
  assumes "convex s" shows "connected s"
himmelma@33175
   395
proof-
himmelma@33175
   396
  { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
himmelma@33175
   397
    assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
himmelma@33175
   398
    then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
himmelma@33175
   399
    hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
himmelma@33175
   400
himmelma@33175
   401
    { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
himmelma@33175
   402
      { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
himmelma@33175
   403
          by (simp add: algebra_simps)
himmelma@33175
   404
        assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
himmelma@33175
   405
        hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
himmelma@33175
   406
          unfolding * and scaleR_right_diff_distrib[THEN sym]
himmelma@33175
   407
          unfolding less_divide_eq using n by auto  }
himmelma@33175
   408
      hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
himmelma@33175
   409
        apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
himmelma@33175
   410
        apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
himmelma@33175
   411
himmelma@33175
   412
    have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
himmelma@33175
   413
      apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
himmelma@33175
   414
      using * apply(simp add: dist_norm)
himmelma@33175
   415
      using as(1,2)[unfolded open_dist] apply simp
himmelma@33175
   416
      using as(1,2)[unfolded open_dist] apply simp
himmelma@33175
   417
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
himmelma@33175
   418
      using as(3) by auto
himmelma@33175
   419
    then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
himmelma@33175
   420
    hence False using as(4) 
himmelma@33175
   421
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
himmelma@33175
   422
      using x1(2) x2(2) by auto  }
himmelma@33175
   423
  thus ?thesis unfolding connected_def by auto
himmelma@33175
   424
qed
himmelma@33175
   425
himmelma@33175
   426
subsection {* One rather trivial consequence. *}
himmelma@33175
   427
hoelzl@34964
   428
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
himmelma@33175
   429
  by(simp add: convex_connected convex_UNIV)
himmelma@33175
   430
hoelzl@36623
   431
subsection {* Balls, being convex, are connected. *}
himmelma@33175
   432
hoelzl@37489
   433
lemma convex_box: fixes a::"'a::euclidean_space"
hoelzl@37489
   434
  assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
hoelzl@37489
   435
  shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
hoelzl@37489
   436
  using assms unfolding convex_def by(auto simp add:euclidean_simps)
himmelma@33175
   437
hoelzl@37489
   438
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
hoelzl@36623
   439
  by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
himmelma@33175
   440
himmelma@33175
   441
lemma convex_local_global_minimum:
himmelma@33175
   442
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   443
  assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
himmelma@33175
   444
  shows "\<forall>y\<in>s. f x \<le> f y"
himmelma@33175
   445
proof(rule ccontr)
himmelma@33175
   446
  have "x\<in>s" using assms(1,3) by auto
himmelma@33175
   447
  assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
himmelma@33175
   448
  then obtain y where "y\<in>s" and y:"f x > f y" by auto
himmelma@33175
   449
  hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
himmelma@33175
   450
himmelma@33175
   451
  then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
himmelma@33175
   452
    using real_lbound_gt_zero[of 1 "e / dist x y"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
himmelma@33175
   453
  hence "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
himmelma@33175
   454
    using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]] by auto
himmelma@33175
   455
  moreover
himmelma@33175
   456
  have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
himmelma@33175
   457
  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
himmelma@33175
   458
    using u unfolding pos_less_divide_eq[OF xy] by auto
himmelma@33175
   459
  hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
himmelma@33175
   460
  ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
himmelma@33175
   461
qed
himmelma@33175
   462
himmelma@33175
   463
lemma convex_ball:
himmelma@33175
   464
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   465
  shows "convex (ball x e)" 
himmelma@33175
   466
proof(auto simp add: convex_def)
himmelma@33175
   467
  fix y z assume yz:"dist x y < e" "dist x z < e"
himmelma@33175
   468
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   469
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
himmelma@33175
   470
    using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
hoelzl@36623
   471
  thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using convex_bound_lt[OF yz uv] by auto
himmelma@33175
   472
qed
himmelma@33175
   473
himmelma@33175
   474
lemma convex_cball:
himmelma@33175
   475
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   476
  shows "convex(cball x e)"
huffman@36362
   477
proof(auto simp add: convex_def Ball_def)
himmelma@33175
   478
  fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
himmelma@33175
   479
  fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   480
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
himmelma@33175
   481
    using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
hoelzl@36623
   482
  thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto 
himmelma@33175
   483
qed
himmelma@33175
   484
himmelma@33175
   485
lemma connected_ball:
himmelma@33175
   486
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   487
  shows "connected (ball x e)"
himmelma@33175
   488
  using convex_connected convex_ball by auto
himmelma@33175
   489
himmelma@33175
   490
lemma connected_cball:
himmelma@33175
   491
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   492
  shows "connected(cball x e)"
himmelma@33175
   493
  using convex_connected convex_cball by auto
himmelma@33175
   494
himmelma@33175
   495
subsection {* Convex hull. *}
himmelma@33175
   496
himmelma@33175
   497
lemma convex_convex_hull: "convex(convex hull s)"
himmelma@33175
   498
  unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
himmelma@33175
   499
  unfolding mem_def by auto
himmelma@33175
   500
haftmann@34064
   501
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
nipkow@36071
   502
by (metis convex_convex_hull hull_same mem_def)
himmelma@33175
   503
himmelma@33175
   504
lemma bounded_convex_hull:
himmelma@33175
   505
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   506
  assumes "bounded s" shows "bounded(convex hull s)"
himmelma@33175
   507
proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
himmelma@33175
   508
  show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
himmelma@33175
   509
    unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
himmelma@33175
   510
    unfolding subset_eq mem_cball dist_norm using B by auto qed
himmelma@33175
   511
himmelma@33175
   512
lemma finite_imp_bounded_convex_hull:
himmelma@33175
   513
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   514
  shows "finite s \<Longrightarrow> bounded(convex hull s)"
himmelma@33175
   515
  using bounded_convex_hull finite_imp_bounded by auto
himmelma@33175
   516
himmelma@33175
   517
subsection {* Stepping theorems for convex hulls of finite sets. *}
himmelma@33175
   518
himmelma@33175
   519
lemma convex_hull_empty[simp]: "convex hull {} = {}"
himmelma@33175
   520
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@33175
   521
himmelma@33175
   522
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
himmelma@33175
   523
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@33175
   524
himmelma@33175
   525
lemma convex_hull_insert:
himmelma@33175
   526
  fixes s :: "'a::real_vector set"
himmelma@33175
   527
  assumes "s \<noteq> {}"
himmelma@33175
   528
  shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
himmelma@33175
   529
                                    b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
himmelma@33175
   530
 apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
himmelma@33175
   531
 fix x assume x:"x = a \<or> x \<in> s"
himmelma@33175
   532
 thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
himmelma@33175
   533
   apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
himmelma@33175
   534
next
himmelma@33175
   535
  fix x assume "x\<in>?hull"
himmelma@33175
   536
  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" by auto
himmelma@33175
   537
  have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
himmelma@33175
   538
    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
himmelma@33175
   539
  thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
himmelma@33175
   540
    apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
himmelma@33175
   541
next
himmelma@33175
   542
  show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
himmelma@33175
   543
    fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
himmelma@33175
   544
    from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
himmelma@33175
   545
    from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
himmelma@33175
   546
    have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
himmelma@33175
   547
    have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
himmelma@33175
   548
    proof(cases "u * v1 + v * v2 = 0")
himmelma@33175
   549
      have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
nipkow@36071
   550
      case True hence **:"u * v1 = 0" "v * v2 = 0"
nipkow@36071
   551
        using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by arith+
himmelma@33175
   552
      hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
himmelma@33175
   553
      thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
himmelma@33175
   554
    next
himmelma@33175
   555
      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
himmelma@33175
   556
      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
himmelma@33175
   557
      also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
himmelma@33175
   558
      case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
himmelma@33175
   559
        apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
himmelma@33175
   560
        using as(1,2) obt1(1,2) obt2(1,2) by auto 
himmelma@33175
   561
      thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
himmelma@33175
   562
        apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
himmelma@33175
   563
        apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
himmelma@33175
   564
        unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
himmelma@33175
   565
        by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
himmelma@33175
   566
    qed note * = this
nipkow@36071
   567
    have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
nipkow@36071
   568
    have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
himmelma@33175
   569
    have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
himmelma@33175
   570
      apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
himmelma@33175
   571
    also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
himmelma@33175
   572
    finally 
himmelma@33175
   573
    show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
himmelma@33175
   574
      apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
himmelma@33175
   575
      using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
himmelma@33175
   576
  qed
himmelma@33175
   577
qed
himmelma@33175
   578
himmelma@33175
   579
himmelma@33175
   580
subsection {* Explicit expression for convex hull. *}
himmelma@33175
   581
himmelma@33175
   582
lemma convex_hull_indexed:
himmelma@33175
   583
  fixes s :: "'a::real_vector set"
himmelma@33175
   584
  shows "convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
himmelma@33175
   585
                            (setsum u {1..k} = 1) \<and>
himmelma@33175
   586
                            (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
himmelma@33175
   587
  apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
himmelma@33175
   588
  apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
himmelma@33175
   589
proof-
himmelma@33175
   590
  fix x assume "x\<in>s"
himmelma@33175
   591
  thus "x \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
himmelma@33175
   592
next
himmelma@33175
   593
  fix t assume as:"s \<subseteq> t" "convex t"
himmelma@33175
   594
  show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
himmelma@33175
   595
    fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
himmelma@33175
   596
    show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
himmelma@33175
   597
      using assm(1,2) as(1) by auto qed
himmelma@33175
   598
next
himmelma@33175
   599
  fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and xy:"x\<in>?hull" "y\<in>?hull"
himmelma@33175
   600
  from xy obtain k1 u1 x1 where 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" by auto
himmelma@33175
   601
  from xy obtain k2 u2 x2 where 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" by auto
himmelma@33175
   602
  have *:"\<And>P (x1::'a) x2 s1 s2 i.(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
   603
    "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
himmelma@33175
   604
    prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
himmelma@33175
   605
  have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
himmelma@33175
   606
  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
himmelma@33175
   607
    apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
himmelma@33175
   608
    apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
hoelzl@35577
   609
    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
himmelma@33175
   610
    unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
himmelma@33175
   611
    fix i assume i:"i \<in> {1..k1+k2}"
himmelma@33175
   612
    show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and> (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
himmelma@33175
   613
    proof(cases "i\<in>{1..k1}")
himmelma@33175
   614
      case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
himmelma@33175
   615
    next def j \<equiv> "i - k1"
himmelma@33175
   616
      case False with i have "j \<in> {1..k2}" unfolding j_def by auto
himmelma@33175
   617
      thus ?thesis unfolding j_def[symmetric] using False
himmelma@33175
   618
        using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
himmelma@33175
   619
  qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
himmelma@33175
   620
qed
himmelma@33175
   621
himmelma@33175
   622
lemma convex_hull_finite:
himmelma@33175
   623
  fixes s :: "'a::real_vector set"
himmelma@33175
   624
  assumes "finite s"
himmelma@33175
   625
  shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
himmelma@33175
   626
         setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
himmelma@33175
   627
proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
himmelma@33175
   628
  fix x assume "x\<in>s" thus " \<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" 
himmelma@33175
   629
    apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
himmelma@33175
   630
    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
himmelma@33175
   631
next
himmelma@33175
   632
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   633
  fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
himmelma@33175
   634
  fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
himmelma@33175
   635
  { fix x assume "x\<in>s"
himmelma@33175
   636
    hence "0 \<le> u * ux x + v * uy x" using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
himmelma@33175
   637
      by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
himmelma@33175
   638
  moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
himmelma@33175
   639
    unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
himmelma@33175
   640
  moreover 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)"
himmelma@33175
   641
    unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
himmelma@33175
   642
  ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<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)"
himmelma@33175
   643
    apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
himmelma@33175
   644
next
himmelma@33175
   645
  fix t assume t:"s \<subseteq> t" "convex t" 
himmelma@33175
   646
  fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
himmelma@33175
   647
  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
himmelma@33175
   648
    using assms and t(1) by auto
himmelma@33175
   649
qed
himmelma@33175
   650
himmelma@33175
   651
subsection {* Another formulation from Lars Schewe. *}
himmelma@33175
   652
himmelma@33175
   653
lemma setsum_constant_scaleR:
himmelma@33175
   654
  fixes y :: "'a::real_vector"
himmelma@33175
   655
  shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
himmelma@33175
   656
apply (cases "finite A")
himmelma@33175
   657
apply (induct set: finite)
himmelma@33175
   658
apply (simp_all add: algebra_simps)
himmelma@33175
   659
done
himmelma@33175
   660
himmelma@33175
   661
lemma convex_hull_explicit:
himmelma@33175
   662
  fixes p :: "'a::real_vector set"
himmelma@33175
   663
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
himmelma@33175
   664
             (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
himmelma@33175
   665
proof-
himmelma@33175
   666
  { fix x assume "x\<in>?lhs"
himmelma@33175
   667
    then obtain k u y where 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
   668
      unfolding convex_hull_indexed by auto
himmelma@33175
   669
himmelma@33175
   670
    have fin:"finite {1..k}" by auto
himmelma@33175
   671
    have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
himmelma@33175
   672
    { fix j assume "j\<in>{1..k}"
himmelma@33175
   673
      hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
himmelma@33175
   674
        using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
himmelma@33175
   675
        apply(rule setsum_nonneg) using obt(1) by auto } 
himmelma@33175
   676
    moreover
himmelma@33175
   677
    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
himmelma@33175
   678
      unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
himmelma@33175
   679
    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
himmelma@33175
   680
      using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
himmelma@33175
   681
      unfolding scaleR_left.setsum using obt(3) by auto
himmelma@33175
   682
    ultimately 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"
himmelma@33175
   683
      apply(rule_tac x="y ` {1..k}" in exI)
himmelma@33175
   684
      apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
himmelma@33175
   685
    hence "x\<in>?rhs" by auto  }
himmelma@33175
   686
  moreover
himmelma@33175
   687
  { fix y assume "y\<in>?rhs"
himmelma@33175
   688
    then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
himmelma@33175
   689
himmelma@33175
   690
    obtain f where f:"inj_on f {1..card s}" "f ` {1..card s} = s" using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
himmelma@33175
   691
    
himmelma@33175
   692
    { fix i::nat assume "i\<in>{1..card s}"
himmelma@33175
   693
      hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
himmelma@33175
   694
      hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
himmelma@33175
   695
    moreover have *:"finite {1..card s}" by auto
himmelma@33175
   696
    { fix y assume "y\<in>s"
himmelma@33175
   697
      then obtain i where "i\<in>{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"] by auto
himmelma@33175
   698
      hence "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}" apply auto using f(1)[unfolded inj_on_def] apply(erule_tac x=x in ballE) by auto
himmelma@33175
   699
      hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
himmelma@33175
   700
      hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
himmelma@33175
   701
            "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
himmelma@33175
   702
        by (auto simp add: setsum_constant_scaleR)   }
himmelma@33175
   703
himmelma@33175
   704
    hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
himmelma@33175
   705
      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
himmelma@33175
   706
      unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
himmelma@33175
   707
      using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
himmelma@33175
   708
    
himmelma@33175
   709
    ultimately 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> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
himmelma@33175
   710
      apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastsimp
himmelma@33175
   711
    hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
himmelma@33175
   712
  ultimately show ?thesis unfolding expand_set_eq by blast
himmelma@33175
   713
qed
himmelma@33175
   714
himmelma@33175
   715
subsection {* A stepping theorem for that expansion. *}
himmelma@33175
   716
himmelma@33175
   717
lemma convex_hull_finite_step:
himmelma@33175
   718
  fixes s :: "'a::real_vector set" assumes "finite s"
himmelma@33175
   719
  shows "(\<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)
himmelma@33175
   720
     \<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)" (is "?lhs = ?rhs")
himmelma@33175
   721
proof(rule, case_tac[!] "a\<in>s")
himmelma@33175
   722
  assume "a\<in>s" hence *:"insert a s = s" by auto
himmelma@33175
   723
  assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
himmelma@33175
   724
next
himmelma@33175
   725
  assume ?lhs then obtain u where 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" by auto
himmelma@33175
   726
  assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
himmelma@33175
   727
    apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
himmelma@33175
   728
next
himmelma@33175
   729
  assume "a\<in>s" hence *:"insert a s = s" by auto
himmelma@33175
   730
  have fin:"finite (insert a s)" using assms by auto
himmelma@33175
   731
  assume ?rhs 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" by auto
himmelma@33175
   732
  show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
himmelma@33175
   733
    unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
himmelma@33175
   734
next
himmelma@33175
   735
  assume ?rhs 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" by auto
himmelma@33175
   736
  moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
himmelma@33175
   737
    apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
himmelma@33175
   738
  ultimately show ?lhs apply(rule_tac x="\<lambda>x. if a = x then v else u x" in exI)  unfolding setsum_clauses(2)[OF assms] by auto
himmelma@33175
   739
qed
himmelma@33175
   740
himmelma@33175
   741
subsection {* Hence some special cases. *}
himmelma@33175
   742
himmelma@33175
   743
lemma convex_hull_2:
himmelma@33175
   744
  "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}"
himmelma@33175
   745
proof- have *:"\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" by auto have **:"finite {b}" by auto
himmelma@33175
   746
show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
himmelma@33175
   747
  apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
himmelma@33175
   748
  apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
himmelma@33175
   749
himmelma@33175
   750
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
himmelma@33175
   751
  unfolding convex_hull_2 unfolding Collect_def 
himmelma@33175
   752
proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
himmelma@33175
   753
  fix x 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) = (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
himmelma@33175
   754
    unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
himmelma@33175
   755
himmelma@33175
   756
lemma convex_hull_3:
himmelma@33175
   757
  "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}"
himmelma@33175
   758
proof-
himmelma@33175
   759
  have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
himmelma@33175
   760
  have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
hoelzl@37489
   761
         "\<And>x y z ::_::euclidean_space. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: field_simps)
himmelma@33175
   762
  show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
himmelma@33175
   763
    unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
himmelma@33175
   764
    apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
himmelma@33175
   765
    apply(rule_tac x="1 - v - w" in exI) apply simp apply(rule_tac x=v in exI) apply simp apply(rule_tac x="\<lambda>x. w" in exI) by simp qed
himmelma@33175
   766
himmelma@33175
   767
lemma convex_hull_3_alt:
himmelma@33175
   768
  "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}"
himmelma@33175
   769
proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
himmelma@33175
   770
  show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply (simp add: algebra_simps)
himmelma@33175
   771
    apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
himmelma@33175
   772
himmelma@33175
   773
subsection {* Relations among closure notions and corresponding hulls. *}
himmelma@33175
   774
himmelma@33175
   775
text {* TODO: Generalize linear algebra concepts defined in @{text
himmelma@33175
   776
Euclidean_Space.thy} so that we can generalize these lemmas. *}
himmelma@33175
   777
himmelma@33175
   778
lemma subspace_imp_affine:
hoelzl@37489
   779
  fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> affine s"
hoelzl@37489
   780
  unfolding subspace_def affine_def by auto
himmelma@33175
   781
himmelma@33175
   782
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
himmelma@33175
   783
  unfolding affine_def convex_def by auto
himmelma@33175
   784
himmelma@33175
   785
lemma subspace_imp_convex:
hoelzl@37489
   786
  fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> convex s"
himmelma@33175
   787
  using subspace_imp_affine affine_imp_convex by auto
himmelma@33175
   788
himmelma@33175
   789
lemma affine_hull_subset_span:
hoelzl@37489
   790
  fixes s :: "(_::euclidean_space) set" shows "(affine hull s) \<subseteq> (span s)"
nipkow@36071
   791
by (metis hull_minimal mem_def span_inc subspace_imp_affine subspace_span)
himmelma@33175
   792
himmelma@33175
   793
lemma convex_hull_subset_span:
hoelzl@37489
   794
  fixes s :: "(_::euclidean_space) set" shows "(convex hull s) \<subseteq> (span s)"
nipkow@36071
   795
by (metis hull_minimal mem_def span_inc subspace_imp_convex subspace_span)
himmelma@33175
   796
himmelma@33175
   797
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
nipkow@36071
   798
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset mem_def)
nipkow@36071
   799
himmelma@33175
   800
himmelma@33175
   801
lemma affine_dependent_imp_dependent:
hoelzl@37489
   802
  fixes s :: "(_::euclidean_space) set" shows "affine_dependent s \<Longrightarrow> dependent s"
himmelma@33175
   803
  unfolding affine_dependent_def dependent_def 
himmelma@33175
   804
  using affine_hull_subset_span by auto
himmelma@33175
   805
himmelma@33175
   806
lemma dependent_imp_affine_dependent:
hoelzl@37489
   807
  fixes s :: "(_::euclidean_space) set"
himmelma@33175
   808
  assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
himmelma@33175
   809
  shows "affine_dependent (insert a s)"
himmelma@33175
   810
proof-
hoelzl@37489
   811
  from assms(1)[unfolded dependent_explicit] obtain S u v 
himmelma@33175
   812
    where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
himmelma@33175
   813
  def t \<equiv> "(\<lambda>x. x + a) ` S"
himmelma@33175
   814
himmelma@33175
   815
  have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
himmelma@33175
   816
  have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
himmelma@33175
   817
  have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
himmelma@33175
   818
himmelma@33175
   819
  hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
himmelma@33175
   820
  moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
himmelma@33175
   821
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
himmelma@33175
   822
  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
himmelma@33175
   823
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
himmelma@33175
   824
  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"
himmelma@33175
   825
    apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
himmelma@33175
   826
  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)"
himmelma@33175
   827
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
himmelma@33175
   828
  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
himmelma@33175
   829
    unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
himmelma@33175
   830
    using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
himmelma@33175
   831
  hence "(\<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"
hoelzl@37489
   832
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *)
himmelma@33175
   833
  ultimately show ?thesis unfolding affine_dependent_explicit
himmelma@33175
   834
    apply(rule_tac x="insert a t" in exI) by auto 
himmelma@33175
   835
qed
himmelma@33175
   836
himmelma@33175
   837
lemma convex_cone:
himmelma@33175
   838
  "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)" (is "?lhs = ?rhs")
himmelma@33175
   839
proof-
himmelma@33175
   840
  { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
himmelma@33175
   841
    hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
himmelma@33175
   842
    hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
himmelma@33175
   843
      apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
himmelma@33175
   844
      apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
huffman@36362
   845
  thus ?thesis unfolding convex_def cone_def by blast
himmelma@33175
   846
qed
himmelma@33175
   847
hoelzl@37489
   848
lemma affine_dependent_biggerset: fixes s::"('a::euclidean_space) set"
hoelzl@37489
   849
  assumes "finite s" "card s \<ge> DIM('a) + 2"
himmelma@33175
   850
  shows "affine_dependent s"
himmelma@33175
   851
proof-
himmelma@33175
   852
  have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
himmelma@33175
   853
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
himmelma@33175
   854
  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
himmelma@33175
   855
    apply(rule card_image) unfolding inj_on_def by auto
hoelzl@37489
   856
  also have "\<dots> > DIM('a)" using assms(2)
himmelma@33175
   857
    unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
himmelma@33175
   858
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
himmelma@33175
   859
    apply(rule dependent_imp_affine_dependent)
himmelma@33175
   860
    apply(rule dependent_biggerset) by auto qed
himmelma@33175
   861
himmelma@33175
   862
lemma affine_dependent_biggerset_general:
hoelzl@37489
   863
  assumes "finite (s::('a::euclidean_space) set)" "card s \<ge> dim s + 2"
himmelma@33175
   864
  shows "affine_dependent s"
himmelma@33175
   865
proof-
himmelma@33175
   866
  from assms(2) have "s \<noteq> {}" by auto
himmelma@33175
   867
  then obtain a where "a\<in>s" by auto
himmelma@33175
   868
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
himmelma@33175
   869
  have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
himmelma@33175
   870
    apply(rule card_image) unfolding inj_on_def by auto
himmelma@33175
   871
  have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
himmelma@33175
   872
    apply(rule subset_le_dim) unfolding subset_eq
himmelma@33175
   873
    using `a\<in>s` by (auto simp add:span_superset span_sub)
himmelma@33175
   874
  also have "\<dots> < dim s + 1" by auto
himmelma@33175
   875
  also have "\<dots> \<le> card (s - {a})" using assms
himmelma@33175
   876
    using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
himmelma@33175
   877
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
himmelma@33175
   878
    apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
himmelma@33175
   879
himmelma@33175
   880
subsection {* Caratheodory's theorem. *}
himmelma@33175
   881
hoelzl@37489
   882
lemma convex_hull_caratheodory: fixes p::"('a::euclidean_space) set"
hoelzl@37489
   883
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
himmelma@33175
   884
  (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
himmelma@33175
   885
  unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
himmelma@33175
   886
proof(rule,rule)
himmelma@33175
   887
  fix y 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> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
   888
  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
   889
  then obtain N where "?P N" by auto
himmelma@33175
   890
  hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
himmelma@33175
   891
  then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
himmelma@33175
   892
  then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
himmelma@33175
   893
hoelzl@37489
   894
  have "card s \<le> DIM('a) + 1" proof(rule ccontr, simp only: not_le)
hoelzl@37489
   895
    assume "DIM('a) + 1 < card s"
himmelma@33175
   896
    hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
himmelma@33175
   897
    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
   898
      using affine_dependent_explicit_finite[OF obt(1)] by auto
himmelma@33175
   899
    def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
himmelma@33175
   900
    have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
himmelma@33175
   901
      assume as:"\<forall>x\<in>s. 0 \<le> w x"
himmelma@33175
   902
      hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
himmelma@33175
   903
      hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
himmelma@33175
   904
        using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
himmelma@33175
   905
      thus False using wv(1) by auto
himmelma@33175
   906
    qed hence "i\<noteq>{}" unfolding i_def by auto
himmelma@33175
   907
himmelma@33175
   908
    hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
himmelma@33175
   909
      using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
himmelma@33175
   910
    have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
himmelma@33175
   911
      fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
himmelma@33175
   912
      show"0 \<le> u v + t * w v" proof(cases "w v < 0")
himmelma@33175
   913
        case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
himmelma@33175
   914
          using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
himmelma@33175
   915
        case True hence "t \<le> u v / (- w v)" using `v\<in>s`
himmelma@33175
   916
          unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
himmelma@33175
   917
        thus ?thesis unfolding real_0_le_add_iff
himmelma@33175
   918
          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
himmelma@33175
   919
      qed qed
himmelma@33175
   920
himmelma@33175
   921
    obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
himmelma@33175
   922
      using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
himmelma@33175
   923
    hence a:"a\<in>s" "u a + t * w a = 0" by auto
hoelzl@37489
   924
    have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
hoelzl@37489
   925
      unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
himmelma@33175
   926
    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
himmelma@33175
   927
      unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
himmelma@33175
   928
    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" 
himmelma@33175
   929
      unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
hoelzl@37489
   930
      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
himmelma@33175
   931
    ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
hoelzl@37489
   932
      apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a
hoelzl@37489
   933
      by (auto simp add: * scaleR_left_distrib)
himmelma@33175
   934
    thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
hoelzl@37489
   935
  thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1
himmelma@33175
   936
    \<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" using obt by auto
himmelma@33175
   937
qed auto
himmelma@33175
   938
himmelma@33175
   939
lemma caratheodory:
hoelzl@37489
   940
 "convex hull p = {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
hoelzl@37489
   941
      card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
himmelma@33175
   942
  unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
himmelma@33175
   943
  fix x assume "x \<in> convex hull p"
hoelzl@37489
   944
  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
himmelma@33175
   945
     "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"unfolding convex_hull_caratheodory by auto
hoelzl@37489
   946
  thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
himmelma@33175
   947
    apply(rule_tac x=s in exI) using hull_subset[of s convex]
himmelma@33175
   948
  using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
himmelma@33175
   949
next
hoelzl@37489
   950
  fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
hoelzl@37489
   951
  then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s" by auto
himmelma@33175
   952
  thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
himmelma@33175
   953
qed
himmelma@33175
   954
himmelma@33175
   955
subsection {* Openness and compactness are preserved by convex hull operation. *}
himmelma@33175
   956
hoelzl@34964
   957
lemma open_convex_hull[intro]:
himmelma@33175
   958
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   959
  assumes "open s"
himmelma@33175
   960
  shows "open(convex hull s)"
huffman@36362
   961
  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
himmelma@33175
   962
proof(rule, rule) fix a
himmelma@33175
   963
  assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
himmelma@33175
   964
  then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
himmelma@33175
   965
himmelma@33175
   966
  from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
himmelma@33175
   967
    using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
himmelma@33175
   968
  have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
himmelma@33175
   969
himmelma@33175
   970
  show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
himmelma@33175
   971
    apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
himmelma@33175
   972
  proof-
himmelma@33175
   973
    show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
himmelma@33175
   974
      using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
himmelma@33175
   975
  next  fix y assume "y \<in> cball a (Min i)"
himmelma@33175
   976
    hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
himmelma@33175
   977
    { fix x assume "x\<in>t"
himmelma@33175
   978
      hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
himmelma@33175
   979
      hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
himmelma@33175
   980
      moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
huffman@36362
   981
      ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
himmelma@33175
   982
    moreover
himmelma@33175
   983
    have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
himmelma@33175
   984
    have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
himmelma@33175
   985
      unfolding setsum_reindex[OF *] o_def using obt(4) by auto
himmelma@33175
   986
    moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
himmelma@33175
   987
      unfolding setsum_reindex[OF *] o_def using obt(4,5)
himmelma@33175
   988
      by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
himmelma@33175
   989
    ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
himmelma@33175
   990
      apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
himmelma@33175
   991
      using obt(1, 3) by auto
himmelma@33175
   992
  qed
himmelma@33175
   993
qed
himmelma@33175
   994
himmelma@33175
   995
lemma compact_convex_combinations:
himmelma@33175
   996
  fixes s t :: "'a::real_normed_vector set"
himmelma@33175
   997
  assumes "compact s" "compact t"
himmelma@33175
   998
  shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
himmelma@33175
   999
proof-
himmelma@33175
  1000
  let ?X = "{0..1} \<times> s \<times> t"
himmelma@33175
  1001
  let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
himmelma@33175
  1002
  have *:"{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
himmelma@33175
  1003
    apply(rule set_ext) unfolding image_iff mem_Collect_eq
himmelma@33175
  1004
    apply rule apply auto
himmelma@33175
  1005
    apply (rule_tac x=u in rev_bexI, simp)
himmelma@33175
  1006
    apply (erule rev_bexI, erule rev_bexI, simp)
himmelma@33175
  1007
    by auto
himmelma@33175
  1008
  have "continuous_on ({0..1} \<times> s \<times> t)
himmelma@33175
  1009
     (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
himmelma@33175
  1010
    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
himmelma@33175
  1011
  thus ?thesis unfolding *
himmelma@33175
  1012
    apply (rule compact_continuous_image)
hoelzl@37489
  1013
    apply (intro compact_Times compact_interval assms)
himmelma@33175
  1014
    done
himmelma@33175
  1015
qed
himmelma@33175
  1016
hoelzl@37489
  1017
lemma compact_convex_hull: fixes s::"('a::euclidean_space) set"
himmelma@33175
  1018
  assumes "compact s"  shows "compact(convex hull s)"
himmelma@33175
  1019
proof(cases "s={}")
himmelma@33175
  1020
  case True thus ?thesis using compact_empty by simp
himmelma@33175
  1021
next
himmelma@33175
  1022
  case False then obtain w where "w\<in>s" by auto
himmelma@33175
  1023
  show ?thesis unfolding caratheodory[of s]
hoelzl@37489
  1024
  proof(induct ("DIM('a) + 1"))
himmelma@33175
  1025
    have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
huffman@36362
  1026
      using compact_empty by auto
himmelma@33175
  1027
    case 0 thus ?case unfolding * by simp
himmelma@33175
  1028
  next
himmelma@33175
  1029
    case (Suc n)
himmelma@33175
  1030
    show ?case proof(cases "n=0")
himmelma@33175
  1031
      case True have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
himmelma@33175
  1032
        unfolding expand_set_eq and mem_Collect_eq proof(rule, rule)
himmelma@33175
  1033
        fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1034
        then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
himmelma@33175
  1035
        show "x\<in>s" proof(cases "card t = 0")
huffman@36362
  1036
          case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
himmelma@33175
  1037
        next
himmelma@33175
  1038
          case False hence "card t = Suc 0" using t(3) `n=0` by auto
himmelma@33175
  1039
          then obtain a where "t = {a}" unfolding card_Suc_eq by auto
huffman@36362
  1040
          thus ?thesis using t(2,4) by simp
himmelma@33175
  1041
        qed
himmelma@33175
  1042
      next
himmelma@33175
  1043
        fix x assume "x\<in>s"
himmelma@33175
  1044
        thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1045
          apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
himmelma@33175
  1046
      qed thus ?thesis using assms by simp
himmelma@33175
  1047
    next
himmelma@33175
  1048
      case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
himmelma@33175
  1049
        { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
himmelma@33175
  1050
        0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
himmelma@33175
  1051
        unfolding expand_set_eq and mem_Collect_eq proof(rule,rule)
himmelma@33175
  1052
        fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
himmelma@33175
  1053
          0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
himmelma@33175
  1054
        then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
himmelma@33175
  1055
          "0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n"  "v \<in> convex hull t" by auto
himmelma@33175
  1056
        moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
himmelma@33175
  1057
          apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
himmelma@33175
  1058
          using obt(7) and hull_mono[of t "insert u t"] by auto
himmelma@33175
  1059
        ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1060
          apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
himmelma@33175
  1061
      next
himmelma@33175
  1062
        fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1063
        then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
himmelma@33175
  1064
        let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
himmelma@33175
  1065
          0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
himmelma@33175
  1066
        show ?P proof(cases "card t = Suc n")
himmelma@33175
  1067
          case False hence "card t \<le> n" using t(3) by auto
himmelma@33175
  1068
          thus ?P apply(rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) using `w\<in>s` and t
himmelma@33175
  1069
            by(auto intro!: exI[where x=t])
himmelma@33175
  1070
        next
himmelma@33175
  1071
          case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
himmelma@33175
  1072
          show ?P proof(cases "u={}")
himmelma@33175
  1073
            case True hence "x=a" using t(4)[unfolded au] by auto
himmelma@33175
  1074
            show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
huffman@36362
  1075
              using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
himmelma@33175
  1076
          next
himmelma@33175
  1077
            case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
himmelma@33175
  1078
              using t(4)[unfolded au convex_hull_insert[OF False]] by auto
himmelma@33175
  1079
            have *:"1 - vx = ux" using obt(3) by auto
himmelma@33175
  1080
            show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
himmelma@33175
  1081
              using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
himmelma@33175
  1082
              by(auto intro!: exI[where x=u])
himmelma@33175
  1083
          qed
himmelma@33175
  1084
        qed
himmelma@33175
  1085
      qed
himmelma@33175
  1086
      thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
himmelma@33175
  1087
    qed
huffman@36362
  1088
  qed
himmelma@33175
  1089
qed
himmelma@33175
  1090
himmelma@33175
  1091
lemma finite_imp_compact_convex_hull:
hoelzl@37489
  1092
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1093
  shows "finite s \<Longrightarrow> compact(convex hull s)"
nipkow@36071
  1094
by (metis compact_convex_hull finite_imp_compact)
himmelma@33175
  1095
himmelma@33175
  1096
subsection {* Extremal points of a simplex are some vertices. *}
himmelma@33175
  1097
himmelma@33175
  1098
lemma dist_increases_online:
himmelma@33175
  1099
  fixes a b d :: "'a::real_inner"
himmelma@33175
  1100
  assumes "d \<noteq> 0"
himmelma@33175
  1101
  shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
himmelma@33175
  1102
proof(cases "inner a d - inner b d > 0")
himmelma@33175
  1103
  case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
himmelma@33175
  1104
    apply(rule_tac add_pos_pos) using assms by auto
himmelma@33175
  1105
  thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
himmelma@33175
  1106
    by (simp add: algebra_simps inner_commute)
himmelma@33175
  1107
next
himmelma@33175
  1108
  case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
himmelma@33175
  1109
    apply(rule_tac add_pos_nonneg) using assms by auto
himmelma@33175
  1110
  thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
himmelma@33175
  1111
    by (simp add: algebra_simps inner_commute)
himmelma@33175
  1112
qed
himmelma@33175
  1113
himmelma@33175
  1114
lemma norm_increases_online:
himmelma@33175
  1115
  fixes d :: "'a::real_inner"
himmelma@33175
  1116
  shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
himmelma@33175
  1117
  using dist_increases_online[of d a 0] unfolding dist_norm by auto
himmelma@33175
  1118
himmelma@33175
  1119
lemma simplex_furthest_lt:
himmelma@33175
  1120
  fixes s::"'a::real_inner set" assumes "finite s"
himmelma@33175
  1121
  shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
himmelma@33175
  1122
proof(induct_tac rule: finite_induct[of s])
himmelma@33175
  1123
  fix x s assume as:"finite s" "x\<notin>s" "\<forall>x\<in>convex hull s. x \<notin> s \<longrightarrow> (\<exists>y\<in>convex hull s. norm (x - a) < norm (y - a))"
himmelma@33175
  1124
  show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow> (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
himmelma@33175
  1125
  proof(rule,rule,cases "s = {}")
himmelma@33175
  1126
    case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
himmelma@33175
  1127
    obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *\<^sub>R x + v *\<^sub>R b"
himmelma@33175
  1128
      using y(1)[unfolded convex_hull_insert[OF False]] by auto
himmelma@33175
  1129
    show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
himmelma@33175
  1130
    proof(cases "y\<in>convex hull s")
himmelma@33175
  1131
      case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
himmelma@33175
  1132
        using as(3)[THEN bspec[where x=y]] and y(2) by auto
himmelma@33175
  1133
      thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
himmelma@33175
  1134
    next
himmelma@33175
  1135
      case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
himmelma@33175
  1136
        assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
himmelma@33175
  1137
        thus ?thesis using False and obt(4) by auto
himmelma@33175
  1138
      next
himmelma@33175
  1139
        assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
himmelma@33175
  1140
        thus ?thesis using y(2) by auto
himmelma@33175
  1141
      next
himmelma@33175
  1142
        assume "u\<noteq>0" "v\<noteq>0"
himmelma@33175
  1143
        then obtain w where w:"w>0" "w<u" "w<v" using real_lbound_gt_zero[of u v] and obt(1,2) by auto
himmelma@33175
  1144
        have "x\<noteq>b" proof(rule ccontr) 
himmelma@33175
  1145
          assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
himmelma@33175
  1146
            using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
himmelma@33175
  1147
          thus False using obt(4) and False by simp qed
himmelma@33175
  1148
        hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
himmelma@33175
  1149
        show ?thesis using dist_increases_online[OF *, of a y]
himmelma@33175
  1150
        proof(erule_tac disjE)
himmelma@33175
  1151
          assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
himmelma@33175
  1152
          hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
himmelma@33175
  1153
            unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
himmelma@33175
  1154
          moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
himmelma@33175
  1155
            unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
himmelma@33175
  1156
            apply(rule_tac x="u + w" in exI) apply rule defer 
himmelma@33175
  1157
            apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
himmelma@33175
  1158
          ultimately show ?thesis by auto
himmelma@33175
  1159
        next
himmelma@33175
  1160
          assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
himmelma@33175
  1161
          hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
himmelma@33175
  1162
            unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
himmelma@33175
  1163
          moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
himmelma@33175
  1164
            unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
himmelma@33175
  1165
            apply(rule_tac x="u - w" in exI) apply rule defer 
himmelma@33175
  1166
            apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
himmelma@33175
  1167
          ultimately show ?thesis by auto
himmelma@33175
  1168
        qed
himmelma@33175
  1169
      qed auto
himmelma@33175
  1170
    qed
himmelma@33175
  1171
  qed auto
himmelma@33175
  1172
qed (auto simp add: assms)
himmelma@33175
  1173
himmelma@33175
  1174
lemma simplex_furthest_le:
hoelzl@37489
  1175
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1176
  assumes "finite s" "s \<noteq> {}"
himmelma@33175
  1177
  shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
himmelma@33175
  1178
proof-
himmelma@33175
  1179
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
himmelma@33175
  1180
  then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
himmelma@33175
  1181
    using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
himmelma@33175
  1182
    unfolding dist_commute[of a] unfolding dist_norm by auto
himmelma@33175
  1183
  thus ?thesis proof(cases "x\<in>s")
himmelma@33175
  1184
    case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
himmelma@33175
  1185
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
himmelma@33175
  1186
    thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
himmelma@33175
  1187
  qed auto
himmelma@33175
  1188
qed
himmelma@33175
  1189
himmelma@33175
  1190
lemma simplex_furthest_le_exists:
hoelzl@37489
  1191
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1192
  shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
himmelma@33175
  1193
  using simplex_furthest_le[of s] by (cases "s={}")auto
himmelma@33175
  1194
himmelma@33175
  1195
lemma simplex_extremal_le:
hoelzl@37489
  1196
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1197
  assumes "finite s" "s \<noteq> {}"
himmelma@33175
  1198
  shows "\<exists>u\<in>s. \<exists>v\<in>s. \<forall>x\<in>convex hull s. \<forall>y \<in> convex hull s. norm(x - y) \<le> norm(u - v)"
himmelma@33175
  1199
proof-
himmelma@33175
  1200
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
himmelma@33175
  1201
  then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
himmelma@33175
  1202
    "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
himmelma@33175
  1203
    using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
himmelma@33175
  1204
  thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
himmelma@33175
  1205
    assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
himmelma@33175
  1206
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
himmelma@33175
  1207
    thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
himmelma@33175
  1208
  next
himmelma@33175
  1209
    assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
himmelma@33175
  1210
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
himmelma@33175
  1211
    thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
himmelma@33175
  1212
      by (auto simp add: norm_minus_commute)
himmelma@33175
  1213
  qed auto
himmelma@33175
  1214
qed 
himmelma@33175
  1215
himmelma@33175
  1216
lemma simplex_extremal_le_exists:
hoelzl@37489
  1217
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1218
  shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
himmelma@33175
  1219
  \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
himmelma@33175
  1220
  using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
himmelma@33175
  1221
himmelma@33175
  1222
subsection {* Closest point of a convex set is unique, with a continuous projection. *}
himmelma@33175
  1223
himmelma@33175
  1224
definition
huffman@36337
  1225
  closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a" where
himmelma@33175
  1226
 "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
himmelma@33175
  1227
himmelma@33175
  1228
lemma closest_point_exists:
himmelma@33175
  1229
  assumes "closed s" "s \<noteq> {}"
himmelma@33175
  1230
  shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
himmelma@33175
  1231
  unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
himmelma@33175
  1232
  using distance_attains_inf[OF assms(1,2), of a] by auto
himmelma@33175
  1233
himmelma@33175
  1234
lemma closest_point_in_set:
himmelma@33175
  1235
  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
himmelma@33175
  1236
  by(meson closest_point_exists)
himmelma@33175
  1237
himmelma@33175
  1238
lemma closest_point_le:
himmelma@33175
  1239
  "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
himmelma@33175
  1240
  using closest_point_exists[of s] by auto
himmelma@33175
  1241
himmelma@33175
  1242
lemma closest_point_self:
himmelma@33175
  1243
  assumes "x \<in> s"  shows "closest_point s x = x"
himmelma@33175
  1244
  unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
himmelma@33175
  1245
  using assms by auto
himmelma@33175
  1246
himmelma@33175
  1247
lemma closest_point_refl:
himmelma@33175
  1248
 "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
himmelma@33175
  1249
  using closest_point_in_set[of s x] closest_point_self[of x s] by auto
himmelma@33175
  1250
huffman@36337
  1251
lemma closer_points_lemma:
himmelma@33175
  1252
  assumes "inner y z > 0"
himmelma@33175
  1253
  shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
himmelma@33175
  1254
proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
himmelma@33175
  1255
  thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
himmelma@33175
  1256
    fix v assume "0<v" "v \<le> inner y z / inner z z"
himmelma@33175
  1257
    thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
himmelma@33175
  1258
      by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
himmelma@33175
  1259
  qed(rule divide_pos_pos, auto) qed
himmelma@33175
  1260
himmelma@33175
  1261
lemma closer_point_lemma:
himmelma@33175
  1262
  assumes "inner (y - x) (z - x) > 0"
himmelma@33175
  1263
  shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
himmelma@33175
  1264
proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
himmelma@33175
  1265
    using closer_points_lemma[OF assms] by auto
himmelma@33175
  1266
  show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
himmelma@33175
  1267
    unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
himmelma@33175
  1268
himmelma@33175
  1269
lemma any_closest_point_dot:
himmelma@33175
  1270
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
himmelma@33175
  1271
  shows "inner (a - x) (y - x) \<le> 0"
himmelma@33175
  1272
proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
himmelma@33175
  1273
  then obtain u where u:"u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
himmelma@33175
  1274
  let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
himmelma@33175
  1275
  thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
himmelma@33175
  1276
himmelma@33175
  1277
lemma any_closest_point_unique:
huffman@36337
  1278
  fixes x :: "'a::real_inner"
himmelma@33175
  1279
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
himmelma@33175
  1280
  "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
himmelma@33175
  1281
  shows "x = y" using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
himmelma@33175
  1282
  unfolding norm_pths(1) and norm_le_square
himmelma@33175
  1283
  by (auto simp add: algebra_simps)
himmelma@33175
  1284
himmelma@33175
  1285
lemma closest_point_unique:
himmelma@33175
  1286
  assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
himmelma@33175
  1287
  shows "x = closest_point s a"
himmelma@33175
  1288
  using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
himmelma@33175
  1289
  using closest_point_exists[OF assms(2)] and assms(3) by auto
himmelma@33175
  1290
himmelma@33175
  1291
lemma closest_point_dot:
himmelma@33175
  1292
  assumes "convex s" "closed s" "x \<in> s"
himmelma@33175
  1293
  shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
himmelma@33175
  1294
  apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
himmelma@33175
  1295
  using closest_point_exists[OF assms(2)] and assms(3) by auto
himmelma@33175
  1296
himmelma@33175
  1297
lemma closest_point_lt:
himmelma@33175
  1298
  assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
himmelma@33175
  1299
  shows "dist a (closest_point s a) < dist a x"
himmelma@33175
  1300
  apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
himmelma@33175
  1301
  apply(rule closest_point_unique[OF assms(1-3), of a])
himmelma@33175
  1302
  using closest_point_le[OF assms(2), of _ a] by fastsimp
himmelma@33175
  1303
himmelma@33175
  1304
lemma closest_point_lipschitz:
himmelma@33175
  1305
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@33175
  1306
  shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
himmelma@33175
  1307
proof-
himmelma@33175
  1308
  have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
himmelma@33175
  1309
       "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
himmelma@33175
  1310
    apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
himmelma@33175
  1311
    using closest_point_exists[OF assms(2-3)] by auto
himmelma@33175
  1312
  thus ?thesis unfolding dist_norm and norm_le
himmelma@33175
  1313
    using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
himmelma@33175
  1314
    by (simp add: inner_add inner_diff inner_commute) qed
himmelma@33175
  1315
himmelma@33175
  1316
lemma continuous_at_closest_point:
himmelma@33175
  1317
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@33175
  1318
  shows "continuous (at x) (closest_point s)"
himmelma@33175
  1319
  unfolding continuous_at_eps_delta 
himmelma@33175
  1320
  using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
himmelma@33175
  1321
himmelma@33175
  1322
lemma continuous_on_closest_point:
himmelma@33175
  1323
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@33175
  1324
  shows "continuous_on t (closest_point s)"
nipkow@36071
  1325
by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
himmelma@33175
  1326
himmelma@33175
  1327
subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
himmelma@33175
  1328
himmelma@33175
  1329
lemma supporting_hyperplane_closed_point:
huffman@36337
  1330
  fixes z :: "'a::{real_inner,heine_borel}"
himmelma@33175
  1331
  assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
himmelma@33175
  1332
  shows "\<exists>a b. \<exists>y\<in>s. inner a z < b \<and> (inner a y = b) \<and> (\<forall>x\<in>s. inner a x \<ge> b)"
himmelma@33175
  1333
proof-
himmelma@33175
  1334
  from distance_attains_inf[OF assms(2-3)] obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x" by auto
himmelma@33175
  1335
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
himmelma@33175
  1336
    apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
himmelma@33175
  1337
    show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
himmelma@33175
  1338
      unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
himmelma@33175
  1339
  next
himmelma@33175
  1340
    fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
himmelma@33175
  1341
      using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
himmelma@33175
  1342
    assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
himmelma@33175
  1343
      "v>0" "v\<le>1" "dist (y + v *\<^sub>R (x - y)) z < dist y z" using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
himmelma@33175
  1344
    thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
himmelma@33175
  1345
  qed auto
himmelma@33175
  1346
qed
himmelma@33175
  1347
himmelma@33175
  1348
lemma separating_hyperplane_closed_point:
huffman@36337
  1349
  fixes z :: "'a::{real_inner,heine_borel}"
himmelma@33175
  1350
  assumes "convex s" "closed s" "z \<notin> s"
himmelma@33175
  1351
  shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
himmelma@33175
  1352
proof(cases "s={}")
himmelma@33175
  1353
  case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
himmelma@33175
  1354
    using less_le_trans[OF _ inner_ge_zero[of z]] by auto
himmelma@33175
  1355
next
himmelma@33175
  1356
  case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
himmelma@33175
  1357
    using distance_attains_inf[OF assms(2) False] by auto
himmelma@33175
  1358
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
himmelma@33175
  1359
    apply rule defer apply rule proof-
himmelma@33175
  1360
    fix x assume "x\<in>s"
himmelma@33175
  1361
    have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
himmelma@33175
  1362
      assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
himmelma@33175
  1363
      then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
himmelma@33175
  1364
      thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
himmelma@33175
  1365
        using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
himmelma@33175
  1366
        using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
himmelma@33175
  1367
    moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
himmelma@33175
  1368
    hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
himmelma@33175
  1369
    ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
himmelma@33175
  1370
      unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
himmelma@33175
  1371
  qed(insert `y\<in>s` `z\<notin>s`, auto)
himmelma@33175
  1372
qed
himmelma@33175
  1373
himmelma@33175
  1374
lemma separating_hyperplane_closed_0:
hoelzl@37489
  1375
  assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 \<notin> s"
himmelma@33175
  1376
  shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
hoelzl@37489
  1377
  proof(cases "s={}")
hoelzl@37489
  1378
  case True have "norm ((basis 0)::'a) = 1" by auto
hoelzl@37489
  1379
  hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
hoelzl@37489
  1380
    apply(subst norm_le_zero_iff[THEN sym]) by auto
hoelzl@37489
  1381
  thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
hoelzl@37489
  1382
    using True using DIM_positive[where 'a='a] by auto
himmelma@33175
  1383
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
himmelma@35542
  1384
    apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
himmelma@33175
  1385
himmelma@33175
  1386
subsection {* Now set-to-set for closed/compact sets. *}
himmelma@33175
  1387
himmelma@33175
  1388
lemma separating_hyperplane_closed_compact:
hoelzl@37489
  1389
  assumes "convex (s::('a::euclidean_space) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
himmelma@33175
  1390
  shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
himmelma@33175
  1391
proof(cases "s={}")
himmelma@33175
  1392
  case True
himmelma@33175
  1393
  obtain b where b:"b>0" "\<forall>x\<in>t. norm x \<le> b" using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
hoelzl@37489
  1394
  obtain z::"'a" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
himmelma@33175
  1395
  hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
himmelma@33175
  1396
  then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
himmelma@33175
  1397
    using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
himmelma@33175
  1398
  thus ?thesis using True by auto
himmelma@33175
  1399
next
himmelma@33175
  1400
  case False then obtain y where "y\<in>s" by auto
himmelma@33175
  1401
  obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
himmelma@33175
  1402
    using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
himmelma@33175
  1403
    using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
himmelma@33175
  1404
  hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
paulson@33270
  1405
  def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
himmelma@33175
  1406
  show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
himmelma@33175
  1407
    apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
himmelma@33175
  1408
    from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
himmelma@33175
  1409
      apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
paulson@33270
  1410
    hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
himmelma@33175
  1411
    fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
himmelma@33175
  1412
  next
himmelma@33175
  1413
    fix x assume "x\<in>s" 
paulson@33270
  1414
    hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
himmelma@33175
  1415
      using ab[THEN bspec[where x=x]] by auto
himmelma@33175
  1416
    thus "k + b / 2 < inner a x" using `0 < b` by auto
himmelma@33175
  1417
  qed
himmelma@33175
  1418
qed
himmelma@33175
  1419
himmelma@33175
  1420
lemma separating_hyperplane_compact_closed:
hoelzl@37489
  1421
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1422
  assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
himmelma@33175
  1423
  shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
himmelma@33175
  1424
proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
himmelma@33175
  1425
    using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
himmelma@33175
  1426
  thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
himmelma@33175
  1427
himmelma@33175
  1428
subsection {* General case without assuming closure and getting non-strict separation. *}
himmelma@33175
  1429
himmelma@33175
  1430
lemma separating_hyperplane_set_0:
hoelzl@37489
  1431
  assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
himmelma@33175
  1432
  shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
hoelzl@37489
  1433
proof- let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
himmelma@33175
  1434
  have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
himmelma@33175
  1435
    apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
himmelma@33175
  1436
    defer apply(rule,rule,erule conjE) proof-
himmelma@33175
  1437
    fix f assume as:"f \<subseteq> ?k ` s" "finite f"
himmelma@33175
  1438
    obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
himmelma@33175
  1439
    then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
himmelma@33175
  1440
      using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
himmelma@33175
  1441
      using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
himmelma@33175
  1442
      using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
himmelma@33175
  1443
    hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)" apply(rule_tac x="inverse(norm a) *\<^sub>R a" in exI)
himmelma@33175
  1444
       using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
himmelma@33175
  1445
       apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
himmelma@33175
  1446
       by(auto simp add: inner_commute elim!: ballE)
himmelma@33175
  1447
    thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
himmelma@33175
  1448
  qed(insert closed_halfspace_ge, auto)
himmelma@33175
  1449
  then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
himmelma@33175
  1450
  thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
himmelma@33175
  1451
himmelma@33175
  1452
lemma separating_hyperplane_sets:
hoelzl@37489
  1453
  assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
himmelma@33175
  1454
  shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
himmelma@33175
  1455
proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
paulson@33270
  1456
  obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x" 
paulson@33270
  1457
    using assms(3-5) by auto 
paulson@33270
  1458
  hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
paulson@33270
  1459
    by (force simp add: inner_diff)
paulson@33270
  1460
  thus ?thesis
paulson@33270
  1461
    apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
paulson@33270
  1462
    apply auto
paulson@33270
  1463
    apply (rule Sup[THEN isLubD2]) 
paulson@33270
  1464
    prefer 4
paulson@33270
  1465
    apply (rule Sup_least) 
paulson@33270
  1466
     using assms(3-5) apply (auto simp add: setle_def)
nipkow@36071
  1467
    apply metis
paulson@33270
  1468
    done
paulson@33270
  1469
qed
himmelma@33175
  1470
himmelma@33175
  1471
subsection {* More convexity generalities. *}
himmelma@33175
  1472
himmelma@33175
  1473
lemma convex_closure:
himmelma@33175
  1474
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1475
  assumes "convex s" shows "convex(closure s)"
himmelma@33175
  1476
  unfolding convex_def Ball_def closure_sequential
himmelma@33175
  1477
  apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
himmelma@33175
  1478
  apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
himmelma@33175
  1479
  apply(rule assms[unfolded convex_def, rule_format]) prefer 6
himmelma@33175
  1480
  apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
himmelma@33175
  1481
himmelma@33175
  1482
lemma convex_interior:
himmelma@33175
  1483
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1484
  assumes "convex s" shows "convex(interior s)"
himmelma@33175
  1485
  unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
himmelma@33175
  1486
  fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
himmelma@33175
  1487
  fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
himmelma@33175
  1488
  show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
himmelma@33175
  1489
    apply rule unfolding subset_eq defer apply rule proof-
himmelma@33175
  1490
    fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
himmelma@33175
  1491
    hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
himmelma@33175
  1492
      apply(rule_tac assms[unfolded convex_alt, rule_format])
himmelma@33175
  1493
      using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
himmelma@33175
  1494
    thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
himmelma@33175
  1495
hoelzl@34964
  1496
lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
himmelma@33175
  1497
  using hull_subset[of s convex] convex_hull_empty by auto
himmelma@33175
  1498
himmelma@33175
  1499
subsection {* Moving and scaling convex hulls. *}
himmelma@33175
  1500
himmelma@33175
  1501
lemma convex_hull_translation_lemma:
himmelma@33175
  1502
  "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
nipkow@36071
  1503
by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono mem_def)
himmelma@33175
  1504
himmelma@33175
  1505
lemma convex_hull_bilemma: fixes neg
himmelma@33175
  1506
  assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
himmelma@33175
  1507
  shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
himmelma@33175
  1508
  \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
himmelma@33175
  1509
  using assms by(metis subset_antisym) 
himmelma@33175
  1510
himmelma@33175
  1511
lemma convex_hull_translation:
himmelma@33175
  1512
  "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
himmelma@33175
  1513
  apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
himmelma@33175
  1514
himmelma@33175
  1515
lemma convex_hull_scaling_lemma:
himmelma@33175
  1516
 "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
nipkow@36071
  1517
by (metis convex_convex_hull convex_scaling hull_subset mem_def subset_hull subset_image_iff)
himmelma@33175
  1518
himmelma@33175
  1519
lemma convex_hull_scaling:
himmelma@33175
  1520
  "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
himmelma@33175
  1521
  apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
huffman@36362
  1522
  unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
himmelma@33175
  1523
himmelma@33175
  1524
lemma convex_hull_affinity:
himmelma@33175
  1525
  "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
nipkow@36071
  1526
by(simp only: image_image[THEN sym] convex_hull_scaling convex_hull_translation)
himmelma@33175
  1527
himmelma@33175
  1528
subsection {* Convex set as intersection of halfspaces. *}
himmelma@33175
  1529
himmelma@33175
  1530
lemma convex_halfspace_intersection:
hoelzl@37489
  1531
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1532
  assumes "closed s" "convex s"
himmelma@33175
  1533
  shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
himmelma@33175
  1534
  apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
himmelma@33175
  1535
  fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
himmelma@33175
  1536
  hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
himmelma@33175
  1537
  thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
himmelma@33175
  1538
    apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
himmelma@33175
  1539
qed auto
himmelma@33175
  1540
himmelma@33175
  1541
subsection {* Radon's theorem (from Lars Schewe). *}
himmelma@33175
  1542
himmelma@33175
  1543
lemma radon_ex_lemma:
himmelma@33175
  1544
  assumes "finite c" "affine_dependent c"
himmelma@33175
  1545
  shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
himmelma@33175
  1546
proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
himmelma@33175
  1547
  thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
himmelma@33175
  1548
    and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
himmelma@33175
  1549
himmelma@33175
  1550
lemma radon_s_lemma:
himmelma@33175
  1551
  assumes "finite s" "setsum f s = (0::real)"
himmelma@33175
  1552
  shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
himmelma@33175
  1553
proof- have *:"\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" by auto
himmelma@33175
  1554
  show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
himmelma@33175
  1555
    using assms(2) by assumption qed
himmelma@33175
  1556
himmelma@33175
  1557
lemma radon_v_lemma:
hoelzl@37489
  1558
  assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
himmelma@33175
  1559
  shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
himmelma@33175
  1560
proof-
himmelma@33175
  1561
  have *:"\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" using assms(3) by auto 
himmelma@33175
  1562
  show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
himmelma@33175
  1563
    using assms(2) by assumption qed
himmelma@33175
  1564
himmelma@33175
  1565
lemma radon_partition:
himmelma@33175
  1566
  assumes "finite c" "affine_dependent c"
himmelma@33175
  1567
  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
himmelma@33175
  1568
  obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
himmelma@33175
  1569
  have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
himmelma@33175
  1570
  def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
himmelma@33175
  1571
  have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
himmelma@33175
  1572
    case False hence "u v < 0" by auto
himmelma@33175
  1573
    thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
himmelma@33175
  1574
      case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
himmelma@33175
  1575
    next
himmelma@33175
  1576
      case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
himmelma@33175
  1577
      thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
himmelma@33175
  1578
  qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
himmelma@33175
  1579
huffman@36778
  1580
  hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
himmelma@33175
  1581
  moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
himmelma@33175
  1582
    "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
himmelma@33175
  1583
    using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
himmelma@33175
  1584
  hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
himmelma@33175
  1585
   "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)" 
himmelma@33175
  1586
    unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
himmelma@33175
  1587
  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
himmelma@33175
  1588
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
himmelma@33175
  1589
himmelma@33175
  1590
  ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
himmelma@33175
  1591
    apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
himmelma@33175
  1592
    using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
hoelzl@37489
  1593
    by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
himmelma@33175
  1594
  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
himmelma@33175
  1595
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
himmelma@33175
  1596
  hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
himmelma@33175
  1597
    apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
himmelma@33175
  1598
    using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
hoelzl@37489
  1599
    by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
himmelma@33175
  1600
  ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
himmelma@33175
  1601
qed
himmelma@33175
  1602
himmelma@33175
  1603
lemma radon: assumes "affine_dependent c"
himmelma@33175
  1604
  obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
himmelma@33175
  1605
proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
himmelma@33175
  1606
  hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
himmelma@33175
  1607
  from radon_partition[OF *] guess m .. then guess p ..
himmelma@33175
  1608
  thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
himmelma@33175
  1609
himmelma@33175
  1610
subsection {* Helly's theorem. *}
himmelma@33175
  1611
hoelzl@37489
  1612
lemma helly_induct: fixes f::"('a::euclidean_space) set set"
hoelzl@37489
  1613
  assumes "card f = n" "n \<ge> DIM('a) + 1"
hoelzl@37489
  1614
  "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
himmelma@33175
  1615
  shows "\<Inter> f \<noteq> {}"
hoelzl@33715
  1616
using assms proof(induct n arbitrary: f)
himmelma@33175
  1617
case (Suc n)
hoelzl@33715
  1618
have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
hoelzl@37489
  1619
show "\<Inter> f \<noteq> {}" apply(cases "n = DIM('a)") apply(rule Suc(5)[rule_format])
hoelzl@33715
  1620
  unfolding `card f = Suc n` proof-
hoelzl@37489
  1621
  assume ng:"n \<noteq> DIM('a)" hence "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" apply(rule_tac bchoice) unfolding ex_in_conv
hoelzl@33715
  1622
    apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
hoelzl@33715
  1623
    defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
himmelma@33175
  1624
  then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
himmelma@33175
  1625
  show ?thesis proof(cases "inj_on X f")
himmelma@33175
  1626
    case False then obtain s t where st:"s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t" unfolding inj_on_def by auto
himmelma@33175
  1627
    hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
himmelma@33175
  1628
    show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
himmelma@33175
  1629
      apply(rule, rule X[rule_format]) using X st by auto
himmelma@33175
  1630
  next case True then obtain m p where mp:"m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
himmelma@33175
  1631
      using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
hoelzl@33715
  1632
      unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
himmelma@33175
  1633
    have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
himmelma@33175
  1634
    then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
himmelma@33175
  1635
    hence "f \<union> (g \<union> h) = f" by auto
himmelma@33175
  1636
    hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
himmelma@33175
  1637
      unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
himmelma@33175
  1638
    have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
himmelma@33175
  1639
    have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
hoelzl@33715
  1640
      apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  unfolding mem_def unfolding subset_eq
himmelma@33175
  1641
      apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
himmelma@33175
  1642
      fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
himmelma@33175
  1643
      thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
himmelma@33175
  1644
      fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
himmelma@33175
  1645
      thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
himmelma@33175
  1646
    qed(auto)
himmelma@33175
  1647
    thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
huffman@37647
  1648
qed(auto) qed(auto)
himmelma@33175
  1649
hoelzl@37489
  1650
lemma helly: fixes f::"('a::euclidean_space) set set"
hoelzl@37489
  1651
  assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
hoelzl@37489
  1652
          "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
himmelma@33175
  1653
  shows "\<Inter> f \<noteq>{}"
hoelzl@33715
  1654
  apply(rule helly_induct) using assms by auto
himmelma@33175
  1655
himmelma@33175
  1656
subsection {* Convex hull is "preserved" by a linear function. *}
himmelma@33175
  1657
himmelma@33175
  1658
lemma convex_hull_linear_image:
himmelma@33175
  1659
  assumes "bounded_linear f"
himmelma@33175
  1660
  shows "f ` (convex hull s) = convex hull (f ` s)"
himmelma@33175
  1661
  apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
himmelma@33175
  1662
  apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
himmelma@33175
  1663
  apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
himmelma@33175
  1664
proof-
himmelma@33175
  1665
  interpret f: bounded_linear f by fact
himmelma@33175
  1666
  show "convex {x. f x \<in> convex hull f ` s}" 
himmelma@33175
  1667
  unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
himmelma@33175
  1668
  interpret f: bounded_linear f by fact
himmelma@33175
  1669
  show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
himmelma@33175
  1670
    unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
himmelma@33175
  1671
qed auto
himmelma@33175
  1672
himmelma@33175
  1673
lemma in_convex_hull_linear_image:
himmelma@33175
  1674
  assumes "bounded_linear f" "x \<in> convex hull s"
himmelma@33175
  1675
  shows "(f x) \<in> convex hull (f ` s)"
himmelma@33175
  1676
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
himmelma@33175
  1677
himmelma@33175
  1678
subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
himmelma@33175
  1679
himmelma@33175
  1680
lemma compact_frontier_line_lemma:
hoelzl@37489
  1681
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1682
  assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
himmelma@33175
  1683
  obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
himmelma@33175
  1684
proof-
himmelma@33175
  1685
  obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
himmelma@33175
  1686
  let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
huffman@36431
  1687
  have A:"?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
huffman@36431
  1688
    by auto
hoelzl@37489
  1689
  have *:"\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
himmelma@33175
  1690
  have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
himmelma@33175
  1691
    apply(rule, rule continuous_vmul)
hoelzl@37489
  1692
    apply(rule continuous_at_id) by(rule compact_interval)
hoelzl@37489
  1693
  moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}" apply(rule *[OF _ assms(2)])
himmelma@33175
  1694
    unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
himmelma@33175
  1695
  ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
himmelma@33175
  1696
    "y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
himmelma@33175
  1697
himmelma@33175
  1698
  have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
himmelma@33175
  1699
  { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
himmelma@33175
  1700
    hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
himmelma@33175
  1701
      using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
himmelma@33175
  1702
    hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
himmelma@33175
  1703
      apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
himmelma@33175
  1704
      using as(1) `u\<ge>0` by(auto simp add:field_simps) 
himmelma@33175
  1705
    hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
himmelma@33175
  1706
  } note u_max = this
himmelma@33175
  1707
himmelma@33175
  1708
  have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[THEN sym]
himmelma@33175
  1709
    prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
himmelma@33175
  1710
    fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
himmelma@33175
  1711
    hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
himmelma@33175
  1712
    thus False using u_max[OF _ as] by auto
himmelma@33175
  1713
  qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
nipkow@36071
  1714
  thus ?thesis by(metis that[of u] u_max obt(1))
nipkow@36071
  1715
qed
himmelma@33175
  1716
himmelma@33175
  1717
lemma starlike_compact_projective:
hoelzl@37489
  1718
  assumes "compact s" "cball (0::'a::euclidean_space) 1 \<subseteq> s "
himmelma@33175
  1719
  "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
hoelzl@37489
  1720
  shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
himmelma@33175
  1721
proof-
himmelma@33175
  1722
  have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
hoelzl@37489
  1723
  def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
himmelma@33175
  1724
  have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
himmelma@33175
  1725
    using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
himmelma@33175
  1726
  have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
himmelma@33175
  1727
himmelma@33175
  1728
  have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
himmelma@33175
  1729
    apply rule unfolding pi_def
himmelma@33175
  1730
    apply (rule continuous_mul)
himmelma@33175
  1731
    apply (rule continuous_at_inv[unfolded o_def])
himmelma@33175
  1732
    apply (rule continuous_at_norm)
himmelma@33175
  1733
    apply simp
himmelma@33175
  1734
    apply (rule continuous_at_id)
himmelma@33175
  1735
    done
hoelzl@37489
  1736
  def sphere \<equiv> "{x::'a. norm x = 1}"
himmelma@33175
  1737
  have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x" unfolding pi_def sphere_def by auto
himmelma@33175
  1738
himmelma@33175
  1739
  have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
himmelma@33175
  1740
  have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
himmelma@33175
  1741
    fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
himmelma@33175
  1742
    hence "x\<noteq>0" using `0\<notin>frontier s` by auto
himmelma@33175
  1743
    obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
himmelma@33175
  1744
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
himmelma@33175
  1745
    have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
himmelma@33175
  1746
      assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
himmelma@33175
  1747
      assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
himmelma@33175
  1748
        using v and x and fs unfolding inverse_less_1_iff by auto qed
himmelma@33175
  1749
    show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
himmelma@33175
  1750
      assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
himmelma@33175
  1751
        using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
himmelma@33175
  1752
himmelma@33175
  1753
  have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
himmelma@33175
  1754
    apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
himmelma@33175
  1755
    apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_ext,rule) 
himmelma@33175
  1756
    unfolding inj_on_def prefer 3 apply(rule,rule,rule)
himmelma@33175
  1757
  proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
himmelma@33175
  1758
    thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
himmelma@33175
  1759
  next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
himmelma@33175
  1760
    then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
himmelma@33175
  1761
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
himmelma@33175
  1762
    thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
himmelma@33175
  1763
  next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
himmelma@33175
  1764
    hence xys:"x\<in>s" "y\<in>s" using fs by auto
himmelma@33175
  1765
    from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
himmelma@33175
  1766
    from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
himmelma@33175
  1767
    from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto 
himmelma@33175
  1768
    have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
himmelma@33175
  1769
      unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
himmelma@33175
  1770
    hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
himmelma@33175
  1771
      using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
himmelma@33175
  1772
      using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
himmelma@33175
  1773
      using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
himmelma@33175
  1774
    thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
himmelma@33175
  1775
  qed(insert `0 \<notin> frontier s`, auto)
himmelma@33175
  1776
  then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
himmelma@33175
  1777
    "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
himmelma@33175
  1778
  
himmelma@33175
  1779
  have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
himmelma@33175
  1780
    apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
himmelma@33175
  1781
hoelzl@37489
  1782
  { fix x assume as:"x \<in> cball (0::'a) 1"
himmelma@33175
  1783
    have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
himmelma@33175
  1784
      case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
himmelma@33175
  1785
      thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
himmelma@33175
  1786
        apply(rule_tac fs[unfolded subset_eq, rule_format])
himmelma@33175
  1787
        unfolding surf(5)[THEN sym] by auto
himmelma@33175
  1788
    next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
himmelma@33175
  1789
        unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
himmelma@33175
  1790
himmelma@33175
  1791
  { fix x assume "x\<in>s"
himmelma@33175
  1792
    hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
himmelma@33175
  1793
      case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
himmelma@33175
  1794
    next let ?a = "inverse (norm (surf (pi x)))"
himmelma@33175
  1795
      case False hence invn:"inverse (norm x) \<noteq> 0" by auto
himmelma@33175
  1796
      from False have pix:"pi x\<in>sphere" using pi(1) by auto
himmelma@33175
  1797
      hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
himmelma@33175
  1798
      hence **:"norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
himmelma@33175
  1799
      hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
himmelma@33175
  1800
        apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
himmelma@33175
  1801
      have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
himmelma@33175
  1802
      hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
himmelma@33175
  1803
        unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
himmelma@33175
  1804
      moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))" 
himmelma@33175
  1805
        unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
himmelma@33175
  1806
      moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
himmelma@33175
  1807
      hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
himmelma@33175
  1808
        using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
himmelma@33175
  1809
        using False `x\<in>s` by(auto simp add:field_simps)
himmelma@33175
  1810
      ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
himmelma@33175
  1811
        apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
himmelma@33175
  1812
        unfolding pi(2)[OF `?a > 0`] by auto
himmelma@33175
  1813
    qed } note hom2 = this
himmelma@33175
  1814
himmelma@33175
  1815
  show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
himmelma@33175
  1816
    apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
himmelma@33175
  1817
    prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
hoelzl@37489
  1818
    fix x::"'a" assume as:"x \<in> cball 0 1"
himmelma@33175
  1819
    thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
himmelma@33175
  1820
      case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
himmelma@33175
  1821
        using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
hoelzl@37489
  1822
    next obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
hoelzl@37489
  1823
      hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer 
hoelzl@37489
  1824
        apply(erule_tac x="basis 0" in ballE)
hoelzl@37489
  1825
        unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
hoelzl@37489
  1826
        by(auto simp add:norm_basis[unfolded One_nat_def])
himmelma@33175
  1827
      case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
himmelma@33175
  1828
        apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
huffman@36586
  1829
        unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
hoelzl@37489
  1830
        fix e and x::"'a" assume as:"norm x < e / B" "0 < norm x" "0<e"
himmelma@33175
  1831
        hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
himmelma@33175
  1832
        hence "norm (surf (pi x)) \<le> B" using B fs by auto
himmelma@33175
  1833
        hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
himmelma@33175
  1834
        also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
himmelma@33175
  1835
        also have "\<dots> = e" using `B>0` by auto
himmelma@33175
  1836
        finally show "norm x * norm (surf (pi x)) < e" by assumption
himmelma@33175
  1837
      qed(insert `B>0`, auto) qed
himmelma@33175
  1838
  next { fix x assume as:"surf (pi x) = 0"
himmelma@33175
  1839
      have "x = 0" proof(rule ccontr)
himmelma@33175
  1840
        assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
himmelma@33175
  1841
        hence "surf (pi x) \<in> frontier s" using surf(5) by auto
himmelma@33175
  1842
        thus False using `0\<notin>frontier s` unfolding as by simp qed
himmelma@33175
  1843
    } note surf_0 = this
himmelma@33175
  1844
    show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
himmelma@33175
  1845
      fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
himmelma@33175
  1846
      thus "x=y" proof(cases "x=0 \<or> y=0") 
himmelma@33175
  1847
        case True thus ?thesis using as by(auto elim: surf_0) next
himmelma@33175
  1848
        case False
himmelma@33175
  1849
        hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
himmelma@33175
  1850
          using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
himmelma@33175
  1851
        moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
himmelma@33175
  1852
        ultimately have *:"pi x = pi y" using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]] by auto 
himmelma@33175
  1853
        moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
himmelma@33175
  1854
        ultimately show ?thesis using injpi by auto qed qed
himmelma@33175
  1855
  qed auto qed
himmelma@33175
  1856
hoelzl@37489
  1857
lemma homeomorphic_convex_compact_lemma: fixes s::"('a::euclidean_space) set"
himmelma@33175
  1858
  assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
hoelzl@37489
  1859
  shows "s homeomorphic (cball (0::'a) 1)"
himmelma@33175
  1860
  apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
himmelma@33175
  1861
  fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
himmelma@33175
  1862
  hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
himmelma@33175
  1863
    apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
himmelma@33175
  1864
    unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
himmelma@33175
  1865
    fix y assume "dist (u *\<^sub>R x) y < 1 - u"
himmelma@33175
  1866
    hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
himmelma@33175
  1867
      using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
himmelma@33175
  1868
      unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
himmelma@33175
  1869
      apply (rule mult_left_le_imp_le[of "1 - u"])
hoelzl@36844
  1870
      unfolding mult_assoc[symmetric] using `u<1` by auto
himmelma@33175
  1871
    thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
himmelma@33175
  1872
      using as unfolding scaleR_scaleR by auto qed auto
himmelma@33175
  1873
  thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
himmelma@33175
  1874
hoelzl@37489
  1875
lemma homeomorphic_convex_compact_cball: fixes e::real and s::"('a::euclidean_space) set"
himmelma@33175
  1876
  assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
hoelzl@37489
  1877
  shows "s homeomorphic (cball (b::'a) e)"
himmelma@33175
  1878
proof- obtain a where "a\<in>interior s" using assms(3) by auto
himmelma@33175
  1879
  then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
hoelzl@37489
  1880
  let ?d = "inverse d" and ?n = "0::'a"
himmelma@33175
  1881
  have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
himmelma@33175
  1882
    apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
himmelma@33175
  1883
    apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
himmelma@33175
  1884
    by(auto simp add: mult_right_le_one_le)
himmelma@33175
  1885
  hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
himmelma@33175
  1886
    using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
himmelma@33175
  1887
    using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
himmelma@33175
  1888
  thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
himmelma@33175
  1889
    apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
himmelma@33175
  1890
    using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
himmelma@33175
  1891
hoelzl@37489
  1892
lemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) set"
himmelma@33175
  1893
  assumes "convex s" "compact s" "interior s \<noteq> {}"
himmelma@33175
  1894
          "convex t" "compact t" "interior t \<noteq> {}"
himmelma@33175
  1895
  shows "s homeomorphic t"
himmelma@33175
  1896
  using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
himmelma@33175
  1897
himmelma@33175
  1898
subsection {* Epigraphs of convex functions. *}
himmelma@33175
  1899
huffman@36338
  1900
definition "epigraph s (f::_ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
huffman@36338
  1901
huffman@36338
  1902
lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
himmelma@33175
  1903
hoelzl@34964
  1904
(** This might break sooner or later. In fact it did already once. **)
himmelma@33175
  1905
lemma convex_epigraph: 
himmelma@33175
  1906
  "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
huffman@36338
  1907
  unfolding convex_def convex_on_def
huffman@36338
  1908
  unfolding Ball_def split_paired_All epigraph_def
huffman@36338
  1909
  unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
hoelzl@34964
  1910
  apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
hoelzl@34964
  1911
  apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
huffman@36338
  1912
  apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)
himmelma@33175
  1913
nipkow@36071
  1914
lemma convex_epigraphI:
nipkow@36071
  1915
  "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex(epigraph s f)"
nipkow@36071
  1916
unfolding convex_epigraph by auto
nipkow@36071
  1917
nipkow@36071
  1918
lemma convex_epigraph_convex:
nipkow@36071
  1919
  "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
nipkow@36071
  1920
by(simp add: convex_epigraph)
himmelma@33175
  1921
himmelma@33175
  1922
subsection {* Use this to derive general bound property of convex function. *}
himmelma@33175
  1923
himmelma@33175
  1924
lemma convex_on:
himmelma@33175
  1925
  assumes "convex s"
himmelma@33175
  1926
  shows "convex_on s f \<longleftrightarrow> (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
himmelma@33175
  1927
   f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
himmelma@33175
  1928
  unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
huffman@36338
  1929
  unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
huffman@36338
  1930
  apply safe
huffman@36338
  1931
  apply (drule_tac x=k in spec)
huffman@36338
  1932
  apply (drule_tac x=u in spec)
huffman@36338
  1933
  apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
huffman@36338
  1934
  apply simp
huffman@36338
  1935
  using assms[unfolded convex] apply simp
huffman@36778
  1936
  apply(rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
huffman@36338
  1937
  defer apply(rule setsum_mono) apply(erule_tac x=i in allE) unfolding real_scaleR_def
hoelzl@34964
  1938
  apply(rule mult_left_mono)using assms[unfolded convex] by auto
himmelma@33175
  1939
huffman@36338
  1940
himmelma@33175
  1941
subsection {* Convexity of general and special intervals. *}
himmelma@33175
  1942
huffman@37732
  1943
lemma convexI: (* TODO: move to Library/Convex.thy *)
huffman@37732
  1944
  assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
huffman@37732
  1945
  shows "convex s"
huffman@37732
  1946
using assms unfolding convex_def by fast
huffman@37732
  1947
himmelma@33175
  1948
lemma is_interval_convex:
huffman@37732
  1949
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1950
  assumes "is_interval s" shows "convex s"
huffman@37732
  1951
proof (rule convexI)
himmelma@33175
  1952
  fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
himmelma@33175
  1953
  hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
himmelma@33175
  1954
  { fix a b assume "\<not> b \<le> u * a + v * b"
himmelma@33175
  1955
    hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
himmelma@33175
  1956
    hence "a < b" unfolding * using as(4) *(2) apply(rule_tac mult_left_less_imp_less[of "1 - v"]) by(auto simp add: field_simps)
himmelma@33175
  1957
    hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
himmelma@33175
  1958
  } moreover
himmelma@33175
  1959
  { fix a b assume "\<not> u * a + v * b \<le> a"
himmelma@33175
  1960
    hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
haftmann@36350
  1961
    hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
himmelma@33175
  1962
    hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
himmelma@33175
  1963
  ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
hoelzl@37489
  1964
    using as(3-) DIM_positive[where 'a='a] by(auto simp add:euclidean_simps) qed
himmelma@33175
  1965
himmelma@33175
  1966
lemma is_interval_connected:
huffman@37732
  1967
  fixes s :: "('a::euclidean_space) set"
himmelma@33175
  1968
  shows "is_interval s \<Longrightarrow> connected s"
himmelma@33175
  1969
  using is_interval_convex convex_connected by auto
himmelma@33175
  1970
hoelzl@37489
  1971
lemma convex_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"
himmelma@33175
  1972
  apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
himmelma@33175
  1973
huffman@36431
  1974
(* FIXME: rewrite these lemmas without using vec1
himmelma@33175
  1975
subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
himmelma@33175
  1976
himmelma@33175
  1977
lemma is_interval_1:
himmelma@33175
  1978
  "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b \<longrightarrow> x \<in> s)"
hoelzl@34964
  1979
  unfolding is_interval_def forall_1 by auto
himmelma@33175
  1980
himmelma@33175
  1981
lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
himmelma@33175
  1982
  apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
himmelma@33175
  1983
  apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
himmelma@33175
  1984
  fix a b x assume as:"connected s" "a \<in> s" "b \<in> s" "dest_vec1 a \<le> dest_vec1 x" "dest_vec1 x \<le> dest_vec1 b" "x\<notin>s"
himmelma@33175
  1985
  hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
himmelma@33175
  1986
  let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
himmelma@33175
  1987
  { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
huffman@36362
  1988
    using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
hoelzl@34964
  1989
  moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
himmelma@33175
  1990
  hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
himmelma@33175
  1991
  ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
himmelma@33175
  1992
    apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
nipkow@36071
  1993
    apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
nipkow@36071
  1994
    by(auto simp add: field_simps) qed
himmelma@33175
  1995
himmelma@33175
  1996
lemma is_interval_convex_1:
himmelma@33175
  1997
  "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
nipkow@36071
  1998
by(metis is_interval_convex convex_connected is_interval_connected_1)
himmelma@33175
  1999
himmelma@33175
  2000
lemma convex_connected_1:
himmelma@33175
  2001
  "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
nipkow@36071
  2002
by(metis is_interval_convex convex_connected is_interval_connected_1)
huffman@36431
  2003
*)
himmelma@33175
  2004
subsection {* Another intermediate value theorem formulation. *}
himmelma@33175
  2005
huffman@37673
  2006
lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
hoelzl@37489
  2007
  assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
hoelzl@37489
  2008
  shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
himmelma@33175
  2009
proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
hoelzl@37489
  2010
    using assms(1) by auto
himmelma@33175
  2011
  thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
huffman@36431
  2012
    using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
himmelma@33175
  2013
    using assms by(auto intro!: imageI) qed
himmelma@33175
  2014
huffman@37673
  2015
lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
huffman@36431
  2016
  shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
hoelzl@37489
  2017
   \<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
nipkow@36071
  2018
by(rule ivt_increasing_component_on_1)
nipkow@36071
  2019
  (auto simp add: continuous_at_imp_continuous_on)
himmelma@33175
  2020
huffman@37673
  2021
lemma ivt_decreasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
hoelzl@37489
  2022
  assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
hoelzl@37489
  2023
  shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
hoelzl@37489
  2024
  apply(subst neg_equal_iff_equal[THEN sym])
hoelzl@37489
  2025
  using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"] using assms using continuous_on_neg
hoelzl@37489
  2026
  by (auto simp add:euclidean_simps)
himmelma@33175
  2027
huffman@37673
  2028
lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
huffman@36431
  2029
  shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
hoelzl@37489
  2030
    \<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
nipkow@36071
  2031
by(rule ivt_decreasing_component_on_1)
nipkow@36071
  2032
  (auto simp: continuous_at_imp_continuous_on)
himmelma@33175
  2033
himmelma@33175
  2034
subsection {* A bound within a convex hull, and so an interval. *}
himmelma@33175
  2035
himmelma@33175
  2036
lemma convex_on_convex_hull_bound:
himmelma@33175
  2037
  assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
himmelma@33175
  2038
  shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
himmelma@33175
  2039
  fix x assume "x\<in>convex hull s"
himmelma@33175
  2040
  then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
himmelma@33175
  2041
    unfolding convex_hull_indexed mem_Collect_eq by auto
himmelma@33175
  2042
  have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
himmelma@33175
  2043
    unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
himmelma@33175
  2044
    using assms(2) obt(1) by auto
himmelma@33175
  2045
  thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
himmelma@33175
  2046
    unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
himmelma@33175
  2047
himmelma@33175
  2048
lemma unit_interval_convex_hull:
hoelzl@37489
  2049
  "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
hoelzl@37489
  2050
  (is "?int = convex hull ?points")
hoelzl@37489
  2051
proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
hoelzl@37489
  2052
  { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n" 
himmelma@33175
  2053
  hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
hoelzl@37489
  2054
    case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
himmelma@33175
  2055
    thus "x\<in>convex hull ?points" using 01 by auto
himmelma@33175
  2056
  next
hoelzl@37489
  2057
    case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
hoelzl@37489
  2058
      case True hence "x = 0" apply(subst euclidean_eq) by auto
himmelma@33175
  2059
      thus "x\<in>convex hull ?points" using 01 by auto
himmelma@33175
  2060
    next
hoelzl@37489
  2061
      case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
hoelzl@37489
  2062
      have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
hoelzl@37489
  2063
      then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
hoelzl@37489
  2064
      have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
himmelma@33175
  2065
        unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
himmelma@33175
  2066
        defer apply(rule_tac x=j in bexI) using i' by auto
hoelzl@37489
  2067
      have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
hoelzl@37489
  2068
        using i'(2-) `x$$i \<noteq> 0` by auto
hoelzl@37489
  2069
      show ?thesis proof(cases "x$$i=1")
hoelzl@37489
  2070
        case True have "\<forall>j\<in>{i. i<DIM('a) \<and> x$$i \<noteq> 0}. x$$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
hoelzl@37489
  2071
        proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
hoelzl@37489
  2072
          hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
hoelzl@37489
  2073
          hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto 
hoelzl@37489
  2074
          hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
hoelzl@37489
  2075
          thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
himmelma@33175
  2076
        thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
huffman@36362
  2077
          by auto
hoelzl@37489
  2078
      next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
hoelzl@37489
  2079
        case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
hoelzl@37489
  2080
          apply(subst euclidean_eq) by(auto simp add: field_simps euclidean_simps)
hoelzl@37489
  2081
        { fix j assume j:"j<DIM('a)"
hoelzl@37489
  2082
          have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
himmelma@33175
  2083
            apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
hoelzl@37489
  2084
            using Suc(2)[unfolded mem_interval, rule_format, of j] using j
hoelzl@37489
  2085
            by(auto simp add:field_simps euclidean_simps)
himmelma@33175
  2086
          hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
hoelzl@37489
  2087
        moreover have "i\<in>{j. j<DIM('a) \<and> x$$j \<noteq> 0} - {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}"
hoelzl@37489
  2088
          using i01 using i'(3) by auto
hoelzl@37489
  2089
        hence "{j. j<DIM('a) \<and> x$$j \<noteq> 0} \<noteq> {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}" using i'(3) by blast
hoelzl@37489
  2090
        hence **:"{j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<subset> {j. j<DIM('a) \<and> x$$j \<noteq> 0}" apply - apply rule 
hoelzl@37489
  2091
          by( auto simp add:euclidean_simps)
hoelzl@37489
  2092
        have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
hoelzl@37489
  2093
          using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
himmelma@33175
  2094
        ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
himmelma@33175
  2095
          apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
huffman@36362
  2096
          unfolding mem_interval using i01 Suc(3) by auto
himmelma@33175
  2097
      qed qed qed } note * = this
hoelzl@37489
  2098
  have **:"DIM('a) = card {..<DIM('a)}" by auto
himmelma@33175
  2099
  show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
hoelzl@37489
  2100
    apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **) 
hoelzl@37489
  2101
    apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
himmelma@33175
  2102
    unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
hoelzl@37489
  2103
    by(auto simp add: mem_def[of _ convex]) qed
himmelma@33175
  2104
himmelma@33175
  2105
subsection {* And this is a finite set of vertices. *}
himmelma@33175
  2106
hoelzl@37489
  2107
lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
hoelzl@37489
  2108
  apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
hoelzl@37489
  2109
  apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
himmelma@33175
  2110
  prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
hoelzl@37489
  2111
  fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
hoelzl@37489
  2112
  show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
hoelzl@37489
  2113
    apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
hoelzl@37489
  2114
    using as apply(subst euclidean_eq) by auto qed auto
himmelma@33175
  2115
himmelma@33175
  2116
subsection {* Hence any cube (could do any nonempty interval). *}
himmelma@33175
  2117
himmelma@33175
  2118
lemma cube_convex_hull:
hoelzl@37489
  2119
  assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
hoelzl@37489
  2120
  "finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
hoelzl@37489
  2121
  let ?d = "(\<chi>\<chi> i. d)::'a"
hoelzl@37489
  2122
  have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_ext, rule)
himmelma@33175
  2123
    unfolding image_iff defer apply(erule bexE) proof-
himmelma@33175
  2124
    fix y assume as:"y\<in>{x - ?d .. x + ?d}"
hoelzl@37489
  2125
    { fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
hoelzl@37489
  2126
        using as[unfolded mem_interval, THEN spec[where x=i]] i
hoelzl@37489
  2127
        by(auto simp add:euclidean_simps)
hoelzl@37489
  2128
      hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
himmelma@33175
  2129
        apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
huffman@36362
  2130
        using assms by(auto simp add: field_simps)
hoelzl@37489
  2131
      hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
hoelzl@37489
  2132
            "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
hoelzl@37489
  2133
    hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
hoelzl@37489
  2134
      by(auto simp add: euclidean_simps field_simps)
hoelzl@37489
  2135
    thus "\<exists>z\<in>{0..\<chi>\<chi> i.1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI) 
hoelzl@37489
  2136
      using assms by auto
himmelma@33175
  2137
  next
hoelzl@37489
  2138
    fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z" 
hoelzl@37489
  2139
    have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
hoelzl@37489
  2140
      using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
himmelma@33175
  2141
      apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
hoelzl@37489
  2142
      using assms by auto
himmelma@33175
  2143
    thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
hoelzl@37489
  2144
      apply(erule_tac x=i in allE) using assms by(auto simp add: euclidean_simps) qed
hoelzl@37489
  2145
  obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
himmelma@33175
  2146
  thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
himmelma@33175
  2147
himmelma@33175
  2148
subsection {* Bounded convex function on open set is continuous. *}
himmelma@33175
  2149
himmelma@33175
  2150
lemma convex_on_bounded_continuous:
huffman@36338
  2151
  fixes s :: "('a::real_normed_vector) set"
himmelma@33175
  2152
  assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
himmelma@33175
  2153
  shows "continuous_on s f"