src/HOL/Library/Convex_Euclidean_Space.thy
author huffman
Thu Jun 11 09:03:24 2009 -0700 (2009-06-11)
changeset 31563 ded2364d14d4
parent 31561 a5e168fd2bb9
child 31565 da5a5589418e
permissions -rw-r--r--
cleaned up some proofs
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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
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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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declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
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declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
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declare dot_ladd[simp] dot_radd[simp] dot_lsub[simp] dot_rsub[simp]
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declare dot_lmult[simp] dot_rmult[simp] dot_lneg[simp] dot_rneg[simp]
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declare UNIV_1[simp]
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term "(x::real^'n \<Rightarrow> real) 0"
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lemma dim1in[intro]:"Suc 0 \<in> {1::nat .. CARD(1)}" by auto
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lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_less_eq_def Cart_lambda_beta dest_vec1_def basis_component vector_uminus_component
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lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul continuous_const continuous_sub continuous_at_id continuous_within_id
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lemmas continuous_on_intros = continuous_on_add continuous_on_const continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg continuous_on_sub
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  uniformly_continuous_on_add uniformly_continuous_on_const uniformly_continuous_on_id uniformly_continuous_on_compose uniformly_continuous_on_cmul uniformly_continuous_on_neg uniformly_continuous_on_sub
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lemma dest_vec1_simps[simp]: fixes a::"real^1"
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  shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
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  "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
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  by(auto simp add:vector_component_simps all_1 Cart_eq)
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lemma nequals0I:"x\<in>A \<Longrightarrow> A \<noteq> {}" by auto
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lemma norm_not_0:"(x::real^'n::finite)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
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lemma vector_unminus_smult[simp]: "(-1::real) *s x = -x"
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  unfolding vector_sneg_minus1 by simp
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  (* TODO: move to Euclidean_Space.thy *)
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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'': fixes s::"(real^'n) set" assumes "finite s"
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  shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *s x) = (if y\<in>s then (f y) *s 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)) *s x = (if x=y then (f x) *s x else 0)" by auto
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  show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *s x"] by auto
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qed
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lemma not_disjointI:"x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A \<inter> B \<noteq> {}" by blast
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lemma if_smult:"(if P then x else (y::real)) *s v = (if P then x *s v else y *s v)" by auto
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lemma mem_interval_1: fixes x :: "real^1" shows
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 "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
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 "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
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by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def all_1)
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lemma image_smult_interval:"(\<lambda>x. m *s (x::real^'n::finite)) ` {a..b} =
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  (if {a..b} = {} then {} else if 0 \<le> m then {m *s a..m *s b} else {m *s b..m *s a})"
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  using image_affinity_interval[of m 0 a b] by auto
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lemma dest_vec1_inverval:
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  "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
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  "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
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  "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
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  "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
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  apply(rule_tac [!] equalityI)
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  unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
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  apply(rule_tac [!] allI)apply(rule_tac [!] impI)
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  apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
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  apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
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  by (auto simp add: vector_less_def vector_less_eq_def all_1 dest_vec1_def
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    vec1_dest_vec1[unfolded dest_vec1_def One_nat_def])
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lemma dest_vec1_setsum: assumes "finite S"
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  shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
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  using dest_vec1_sum[OF assms] by auto
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lemma dist_triangle_eq:"dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *s (y - z) = norm (y - z) *s (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_eqI:"x = y \<Longrightarrow> norm x = norm y" by auto 
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lemma norm_minus_eqI:"(x::real^'n::finite) = - 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 dimindex_ge_1:"CARD(_::finite) \<ge> 1"
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  using one_le_card_finite by auto
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lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1" 
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  by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff) 
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lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
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subsection {* Affine set and affine hull.*}
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definition "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v::real. u + v = 1 \<longrightarrow> (u *s x + v *s 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) *s x + u *s y \<in> s)"
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proof- have *:"\<And>u v ::real. u + v = 1 \<longleftrightarrow> v = 1 - u" by auto
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  { fix x y assume "x\<in>s" "y\<in>s"
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    hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *s x + u *s y \<in> s)" apply auto 
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      apply(erule_tac[!] x="1 - u" in allE) unfolding * by auto  }
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  thus ?thesis unfolding affine_def by auto qed
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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: vector_sadd_rdistrib[THEN sym]) 
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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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proof-
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  { fix f assume "f \<subseteq> affine"
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    hence "affine (\<Inter>f)" using affine_Inter[of f] unfolding subset_eq mem_def by auto  }
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  thus ?thesis using hull_eq[unfolded mem_def, of affine s] by auto
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qed
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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::"(real^'n) 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) *s 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 *s x) \<in> V"
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  thus "u *s x + v *s 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: vector_sadd_rdistrib[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 *s x + v *s 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 *s 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::"(real^'n) set" and u::"real^'n\<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 *s x + v *s y \<in> V; finite s;
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               s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> V" and
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	as:"Suc n \<equiv> card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s 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) *s 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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      thus "(\<Sum>x\<in>s. u x *s x) \<in> V" unfolding vector_smult_assoc[THEN sym] and setsum_cmul
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 	 apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
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	 using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *s (\<Sum>xa\<in>s - {x}. u xa *s xa)"], 
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	 THEN spec[where x="u x"], THEN spec[where x="1 - u x"]] and rev_subsetD[OF `x\<in>s` `s\<subseteq>V`] and `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) *s 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 *s 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 *s 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 *s v) = y}" unfolding affine_def
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   219
    apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
himmelma@31276
   220
    fix u v ::real assume uv:"u + v = 1"
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   221
    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 *s v) = x"
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   222
    then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *s v) = x" by auto
himmelma@31276
   223
    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 *s v) = y"
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   224
    then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *s v) = y" by auto
himmelma@31276
   225
    have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
himmelma@31276
   226
    have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
himmelma@31276
   227
    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 *s v) = u *s x + v *s y"
himmelma@31276
   228
      apply(rule_tac x="sx \<union> sy" in exI)
himmelma@31276
   229
      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)
himmelma@31276
   230
      unfolding vector_sadd_rdistrib setsum_addf if_smult vector_smult_lzero  ** setsum_restrict_set[OF xy, THEN sym]
himmelma@31276
   231
      unfolding vector_smult_assoc[THEN sym] setsum_cmul and setsum_right_distrib[THEN sym]
himmelma@31276
   232
      unfolding x y using x(1-3) y(1-3) uv by simp qed qed
himmelma@31276
   233
himmelma@31276
   234
lemma affine_hull_finite:
himmelma@31276
   235
  assumes "finite s"
himmelma@31276
   236
  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
himmelma@31276
   237
  unfolding affine_hull_explicit and expand_set_eq and mem_Collect_eq apply (rule,rule)
himmelma@31276
   238
  apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
himmelma@31276
   239
  fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"
himmelma@31276
   240
  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 *s v) = x"
himmelma@31276
   241
    apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
himmelma@31276
   242
next
himmelma@31276
   243
  fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
himmelma@31276
   244
  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *s v) = x"
himmelma@31276
   245
  thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
himmelma@31276
   246
    unfolding if_smult vector_smult_lzero and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
himmelma@31276
   247
himmelma@31276
   248
subsection {* Stepping theorems and hence small special cases. *}
himmelma@31276
   249
himmelma@31276
   250
lemma affine_hull_empty[simp]: "affine hull {} = {}"
himmelma@31276
   251
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@31276
   252
himmelma@31276
   253
lemma affine_hull_finite_step:
himmelma@31276
   254
  shows "(\<exists>u::real^'n=>real. setsum u {} = w \<and> setsum (\<lambda>x. u x *s x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
himmelma@31276
   255
  "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *s x) (insert a s) = y) \<longleftrightarrow>
himmelma@31276
   256
                (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *s x) s = y - v *s a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
himmelma@31276
   257
proof-
himmelma@31276
   258
  show ?th1 by simp
himmelma@31276
   259
  assume ?as 
himmelma@31276
   260
  { assume ?lhs
himmelma@31276
   261
    then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *s x) = y" by auto
himmelma@31276
   262
    have ?rhs proof(cases "a\<in>s")
himmelma@31276
   263
      case True hence *:"insert a s = s" by auto
himmelma@31276
   264
      show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
himmelma@31276
   265
    next
himmelma@31276
   266
      case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
himmelma@31276
   267
    qed  } moreover
himmelma@31276
   268
  { assume ?rhs
himmelma@31276
   269
    then obtain v u where vu:"setsum u s = w - v"  "(\<Sum>x\<in>s. u x *s x) = y - v *s a" by auto
himmelma@31276
   270
    have *:"\<And>x M. (if x = a then v else M) *s x = (if x = a then v *s x else M *s x)" by auto
himmelma@31276
   271
    have ?lhs proof(cases "a\<in>s")
himmelma@31276
   272
      case True thus ?thesis
himmelma@31276
   273
	apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
himmelma@31276
   274
	unfolding setsum_clauses(2)[OF `?as`]  apply simp
himmelma@31276
   275
	unfolding vector_sadd_rdistrib and setsum_addf 
huffman@31445
   276
	unfolding vu and * and vector_smult_lzero
himmelma@31276
   277
	by (auto simp add: setsum_delta[OF `?as`])
himmelma@31276
   278
    next
himmelma@31276
   279
      case False 
himmelma@31276
   280
      hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
himmelma@31276
   281
               "\<And>x. x \<in> s \<Longrightarrow> u x *s x = (if x = a then v *s x else u x *s x)" by auto
himmelma@31276
   282
      from False show ?thesis
himmelma@31276
   283
	apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
himmelma@31276
   284
	unfolding setsum_clauses(2)[OF `?as`] and * using vu
himmelma@31276
   285
	using setsum_cong2[of s "\<lambda>x. u x *s x" "\<lambda>x. if x = a then v *s x else u x *s x", OF **(2)]
himmelma@31276
   286
	using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
himmelma@31276
   287
    qed }
himmelma@31276
   288
  ultimately show "?lhs = ?rhs" by blast
himmelma@31276
   289
qed
himmelma@31276
   290
himmelma@31276
   291
lemma affine_hull_2: "affine hull {a,b::real^'n} = {u *s a + v *s b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
himmelma@31276
   292
proof-
himmelma@31276
   293
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
himmelma@31276
   294
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real^'n)" by auto
himmelma@31276
   295
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *s v) = y}"
himmelma@31276
   296
    using affine_hull_finite[of "{a,b}"] by auto
himmelma@31276
   297
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *s b = y - v *s a}"
himmelma@31276
   298
    by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
himmelma@31276
   299
  also have "\<dots> = ?rhs" unfolding * by auto
himmelma@31276
   300
  finally show ?thesis by auto
himmelma@31276
   301
qed
himmelma@31276
   302
himmelma@31276
   303
lemma affine_hull_3: "affine hull {a,b,c::real^'n} = { u *s a + v *s b + w *s c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
himmelma@31276
   304
proof-
himmelma@31276
   305
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
himmelma@31276
   306
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real^'n)" by auto
himmelma@31276
   307
  show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
himmelma@31276
   308
    unfolding * apply auto
himmelma@31276
   309
    apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
himmelma@31276
   310
    apply(rule_tac x=u in exI) by(auto intro!: exI)
himmelma@31276
   311
qed
himmelma@31276
   312
himmelma@31276
   313
subsection {* Some relations between affine hull and subspaces. *}
himmelma@31276
   314
himmelma@31276
   315
lemma affine_hull_insert_subset_span:
himmelma@31276
   316
  "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
himmelma@31276
   317
  unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
himmelma@31276
   318
  apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
himmelma@31276
   319
  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 *s v) = x"
himmelma@31276
   320
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
himmelma@31276
   321
  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 *s v) = v)"
himmelma@31276
   322
    apply(rule_tac x="x - a" in exI) apply rule defer apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
himmelma@31276
   323
    apply(rule_tac x="\<lambda>x. u (x + a)" in exI) using as(1)
himmelma@31276
   324
    apply(simp add: setsum_reindex[unfolded inj_on_def] setsum_subtractf setsum_diff1 setsum_vmul[THEN sym])
himmelma@31276
   325
    unfolding as by simp_all qed
himmelma@31276
   326
himmelma@31276
   327
lemma affine_hull_insert_span:
himmelma@31276
   328
  assumes "a \<notin> s"
himmelma@31276
   329
  shows "affine hull (insert a s) =
himmelma@31276
   330
            {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
himmelma@31276
   331
  apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
himmelma@31276
   332
  unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
himmelma@31276
   333
  fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
himmelma@31276
   334
  then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *s v) = y" unfolding span_explicit by auto
himmelma@31276
   335
  def f \<equiv> "(\<lambda>x. x + a) ` t"
himmelma@31276
   336
  have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *s (v - a)) = y - a" unfolding f_def using obt 
himmelma@31276
   337
    by(auto simp add: setsum_reindex[unfolded inj_on_def])
himmelma@31276
   338
  have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
himmelma@31276
   339
  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 *s v) = y"
himmelma@31276
   340
    apply(rule_tac x="insert a f" in exI)
himmelma@31276
   341
    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@31276
   342
    using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
himmelma@31276
   343
    unfolding setsum_cases[OF f(1), of "{a}", unfolded singleton_iff] and *
himmelma@31276
   344
    by (auto simp add: setsum_subtractf setsum_vmul field_simps) qed
himmelma@31276
   345
himmelma@31276
   346
lemma affine_hull_span:
himmelma@31276
   347
  assumes "a \<in> s"
himmelma@31276
   348
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
himmelma@31276
   349
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
himmelma@31276
   350
himmelma@31276
   351
subsection {* Convexity. *}
himmelma@31276
   352
himmelma@31276
   353
definition "convex (s::(real^'n) set) \<longleftrightarrow>
himmelma@31276
   354
        (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. (u + v = 1) \<longrightarrow> (u *s x + v *s y) \<in> s)"
himmelma@31276
   355
himmelma@31276
   356
lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *s x + u *s y) \<in> s)"
himmelma@31276
   357
proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
himmelma@31276
   358
  show ?thesis unfolding convex_def apply auto
himmelma@31276
   359
    apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
himmelma@31276
   360
    by (auto simp add: *) qed
himmelma@31276
   361
himmelma@31276
   362
lemma mem_convex:
himmelma@31276
   363
  assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
himmelma@31276
   364
  shows "((1 - u) *s a + u *s b) \<in> s"
himmelma@31276
   365
  using assms unfolding convex_alt by auto
himmelma@31276
   366
himmelma@31276
   367
lemma convex_empty[intro]: "convex {}"
himmelma@31276
   368
  unfolding convex_def by simp
himmelma@31276
   369
himmelma@31276
   370
lemma convex_singleton[intro]: "convex {a}"
himmelma@31276
   371
  unfolding convex_def by (auto simp add:vector_sadd_rdistrib[THEN sym])
himmelma@31276
   372
himmelma@31276
   373
lemma convex_UNIV[intro]: "convex UNIV"
himmelma@31276
   374
  unfolding convex_def by auto
himmelma@31276
   375
himmelma@31276
   376
lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
himmelma@31276
   377
  unfolding convex_def by auto
himmelma@31276
   378
himmelma@31276
   379
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
himmelma@31276
   380
  unfolding convex_def by auto
himmelma@31276
   381
himmelma@31276
   382
lemma convex_halfspace_le: "convex {x. a \<bullet> x \<le> b}"
himmelma@31276
   383
  unfolding convex_def apply auto
himmelma@31276
   384
  unfolding dot_radd dot_rmult by (metis real_convex_bound_le) 
himmelma@31276
   385
himmelma@31276
   386
lemma convex_halfspace_ge: "convex {x. a \<bullet> x \<ge> b}"
himmelma@31276
   387
proof- have *:"{x. a \<bullet> x \<ge> b} = {x. -a \<bullet> x \<le> -b}" by auto
himmelma@31276
   388
  show ?thesis apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
himmelma@31276
   389
himmelma@31276
   390
lemma convex_hyperplane: "convex {x. a \<bullet> x = b}"
himmelma@31276
   391
proof-
himmelma@31276
   392
  have *:"{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}" by auto
himmelma@31276
   393
  show ?thesis unfolding * apply(rule convex_Int)
himmelma@31276
   394
    using convex_halfspace_le convex_halfspace_ge by auto
himmelma@31276
   395
qed
himmelma@31276
   396
himmelma@31276
   397
lemma convex_halfspace_lt: "convex {x. a \<bullet> x < b}"
himmelma@31276
   398
  unfolding convex_def by(auto simp add: real_convex_bound_lt dot_radd dot_rmult)
himmelma@31276
   399
himmelma@31276
   400
lemma convex_halfspace_gt: "convex {x. a \<bullet> x > b}"
himmelma@31276
   401
   using convex_halfspace_lt[of "-a" "-b"] by(auto simp add: dot_lneg neg_less_iff_less)
himmelma@31276
   402
himmelma@31276
   403
lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
himmelma@31276
   404
  unfolding convex_def apply auto apply(erule_tac x=i in allE)+
himmelma@31276
   405
  apply(rule add_nonneg_nonneg) by(auto simp add: mult_nonneg_nonneg)
himmelma@31276
   406
himmelma@31276
   407
subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
himmelma@31276
   408
himmelma@31276
   409
lemma convex: "convex s \<longleftrightarrow>
himmelma@31276
   410
  (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
himmelma@31276
   411
           \<longrightarrow> setsum (\<lambda>i. u i *s x i) {1..k} \<in> s)"
himmelma@31276
   412
  unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
himmelma@31276
   413
  fix x y u v assume as:"\<forall>(k::nat) u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *s x i) \<in> s"
himmelma@31276
   414
    "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
himmelma@31276
   415
  show "u *s x + v *s y \<in> s" using as(1)[THEN spec[where x=2], THEN spec[where x="\<lambda>n. if n=1 then u else v"], THEN spec[where x="\<lambda>n. if n=1 then x else y"]] and as(2-)
himmelma@31276
   416
    by (auto simp add: setsum_head_Suc) 
himmelma@31276
   417
next
himmelma@31276
   418
  fix k u x assume as:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s" 
himmelma@31276
   419
  show "(\<forall>i::nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *s x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
himmelma@31276
   420
  case (Suc k) show ?case proof(cases "u (Suc k) = 1")
himmelma@31276
   421
    case True hence "(\<Sum>i = Suc 0..k. u i *s x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
himmelma@31276
   422
      fix i assume i:"i \<in> {Suc 0..k}" "u i *s x i \<noteq> 0"
himmelma@31276
   423
      hence ui:"u i \<noteq> 0" by auto
himmelma@31276
   424
      hence "setsum (\<lambda>k. if k=i then u i else 0) {1 .. k} \<le> setsum u {1 .. k}" apply(rule_tac setsum_mono) using Suc(2) by auto
himmelma@31276
   425
      hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta) 
himmelma@31276
   426
      hence "setsum u {1 .. k} > 0"  using ui apply(rule_tac less_le_trans[of _ "u i"]) using Suc(2)[THEN spec[where x=i]] and i(1) by auto
himmelma@31276
   427
      thus False using Suc(3) unfolding setsum_cl_ivl_Suc and True by simp qed
himmelma@31276
   428
    thus ?thesis unfolding setsum_cl_ivl_Suc using True and Suc(2) by auto
himmelma@31276
   429
  next
himmelma@31276
   430
    have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
himmelma@31276
   431
    have **:"u (Suc k) \<le> 1" apply(rule ccontr) unfolding not_le using Suc(3) using setsum_nonneg[of "{1..k}" u] using Suc(2) by auto
himmelma@31276
   432
    have ***:"\<And>i k. (u i / (1 - u (Suc k))) *s x i = (inverse (1 - u (Suc k))) *s (u i *s x i)" unfolding real_divide_def by auto
himmelma@31276
   433
    case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
himmelma@31276
   434
    have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
himmelma@31276
   435
      apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
himmelma@31276
   436
    hence "(1 - u (Suc k)) *s (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) + u (Suc k) *s x (Suc k) \<in> s"
himmelma@31276
   437
      apply(rule as[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using Suc(2)[THEN spec[where x="Suc k"]] and ** by auto
himmelma@31276
   438
    thus ?thesis unfolding setsum_cl_ivl_Suc and *** and setsum_cmul using nn by auto qed qed auto qed
himmelma@31276
   439
himmelma@31276
   440
himmelma@31276
   441
lemma convex_explicit: "convex (s::(real^'n) set) \<longleftrightarrow>
himmelma@31276
   442
  (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *s x) t \<in> s)"
himmelma@31276
   443
  unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
himmelma@31276
   444
  fix x y u v assume as:"\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
himmelma@31276
   445
  show "u *s x + v *s y \<in> s" proof(cases "x=y")
himmelma@31276
   446
    case True show ?thesis unfolding True and vector_sadd_rdistrib[THEN sym] using as(3,6) by auto next
himmelma@31276
   447
    case False thus ?thesis using as(1)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>z. if z=x then u else v"]] and as(2-) by auto qed
himmelma@31276
   448
next 
himmelma@31276
   449
  fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s" "finite (t::(real^'n) set)"
himmelma@31276
   450
  (*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
himmelma@31276
   451
  from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" apply(induct_tac t rule:finite_induct)
himmelma@31276
   452
    prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
himmelma@31276
   453
    fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *s x) \<in> s"
himmelma@31276
   454
    assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
himmelma@31276
   455
    show "(\<Sum>x\<in>insert x f. u x *s x) \<in> s" proof(cases "u x = 1")
himmelma@31276
   456
      case True hence "setsum (\<lambda>x. u x *s x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
himmelma@31276
   457
	fix y assume y:"y \<in> f" "u y *s y \<noteq> 0"
himmelma@31276
   458
	hence uy:"u y \<noteq> 0" by auto
himmelma@31276
   459
	hence "setsum (\<lambda>k. if k=y then u y else 0) f \<le> setsum u f" apply(rule_tac setsum_mono) using as(4) by auto
himmelma@31276
   460
	hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta) 
himmelma@31276
   461
	hence "setsum u f > 0" using uy apply(rule_tac less_le_trans[of _ "u y"]) using as(4) and y(1) by auto
himmelma@31276
   462
	thus False using as(2,5) unfolding setsum_clauses(2)[OF as(1)] and True by auto qed
himmelma@31276
   463
      thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2,3) unfolding True by auto
himmelma@31276
   464
    next
himmelma@31276
   465
      have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
himmelma@31276
   466
      have **:"u x \<le> 1" apply(rule ccontr) unfolding not_le using as(5)[unfolded setsum_clauses(2)[OF as(1)]] and as(2)
himmelma@31276
   467
	using setsum_nonneg[of f u] and as(4) by auto
himmelma@31276
   468
      case False hence "inverse (1 - u x) *s (\<Sum>x\<in>f. u x *s x) \<in> s" unfolding setsum_cmul[THEN sym] and vector_smult_assoc
himmelma@31276
   469
	apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
himmelma@31276
   470
	unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
himmelma@31276
   471
      hence "u x *s x + (1 - u x) *s ((inverse (1 - u x)) *s setsum (\<lambda>x. u x *s x) f) \<in>s" 
himmelma@31276
   472
	apply(rule_tac asm(1)[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using as and ** False by auto 
himmelma@31276
   473
      thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
himmelma@31276
   474
  qed auto thus "t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" by auto
himmelma@31276
   475
qed
himmelma@31276
   476
himmelma@31276
   477
lemma convex_finite: assumes "finite s"
himmelma@31276
   478
  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
himmelma@31276
   479
                      \<longrightarrow> setsum (\<lambda>x. u x *s x) s \<in> s)"
himmelma@31276
   480
  unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
himmelma@31276
   481
  fix t u assume as:"\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
himmelma@31276
   482
  have *:"s \<inter> t = t" using as(3) by auto
himmelma@31276
   483
  show "(\<Sum>x\<in>t. u x *s x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
himmelma@31276
   484
    unfolding if_smult and setsum_cases[OF assms] and * using as(2-) by auto
himmelma@31276
   485
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
himmelma@31276
   486
himmelma@31276
   487
subsection {* Cones. *}
himmelma@31276
   488
himmelma@31276
   489
definition "cone (s::(real^'n) set) \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *s x) \<in> s)"
himmelma@31276
   490
himmelma@31276
   491
lemma cone_empty[intro, simp]: "cone {}"
himmelma@31276
   492
  unfolding cone_def by auto
himmelma@31276
   493
himmelma@31276
   494
lemma cone_univ[intro, simp]: "cone UNIV"
himmelma@31276
   495
  unfolding cone_def by auto
himmelma@31276
   496
himmelma@31276
   497
lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
himmelma@31276
   498
  unfolding cone_def by auto
himmelma@31276
   499
himmelma@31276
   500
subsection {* Conic hull. *}
himmelma@31276
   501
himmelma@31276
   502
lemma cone_cone_hull: "cone (cone hull s)"
himmelma@31276
   503
  unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
himmelma@31276
   504
  by (auto simp add: mem_def)
himmelma@31276
   505
himmelma@31276
   506
lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
himmelma@31276
   507
  apply(rule hull_eq[unfolded mem_def])
himmelma@31276
   508
  using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
himmelma@31276
   509
himmelma@31276
   510
subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
himmelma@31276
   511
himmelma@31276
   512
definition "affine_dependent (s::(real^'n) set) \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
himmelma@31276
   513
himmelma@31276
   514
lemma affine_dependent_explicit:
himmelma@31276
   515
  "affine_dependent p \<longleftrightarrow>
himmelma@31276
   516
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
himmelma@31276
   517
    (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) s = 0)"
himmelma@31276
   518
  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
himmelma@31276
   519
  apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
himmelma@31276
   520
proof-
himmelma@31276
   521
  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 *s v) = x"
himmelma@31276
   522
  have "x\<notin>s" using as(1,4) by auto
himmelma@31276
   523
  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 *s v) = 0"
himmelma@31276
   524
    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@31276
   525
    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@31276
   526
next
himmelma@31276
   527
  fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *s v) = 0" "v \<in> s" "u v \<noteq> 0"
himmelma@31276
   528
  have "s \<noteq> {v}" using as(3,6) by auto
himmelma@31276
   529
  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 *s v) = x" 
himmelma@31276
   530
    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@31518
   531
    unfolding vector_smult_assoc[THEN sym] and setsum_cmul unfolding setsum_right_distrib[THEN sym] and setsum_diff1_ring[OF as(1,5)] using as by auto
himmelma@31276
   532
qed
himmelma@31276
   533
himmelma@31276
   534
lemma affine_dependent_explicit_finite:
himmelma@31276
   535
  assumes "finite (s::(real^'n) set)"
himmelma@31276
   536
  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 *s v) s = 0)"
himmelma@31276
   537
  (is "?lhs = ?rhs")
himmelma@31276
   538
proof
himmelma@31276
   539
  have *:"\<And>vt u v. (if vt then u v else 0) *s v = (if vt then (u v) *s v else (0::real^'n))" by auto
himmelma@31276
   540
  assume ?lhs
himmelma@31276
   541
  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 *s v) = 0"
himmelma@31276
   542
    unfolding affine_dependent_explicit by auto
himmelma@31276
   543
  thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
himmelma@31276
   544
    apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
himmelma@31276
   545
    unfolding Int_absorb2[OF `t\<subseteq>s`, unfolded Int_commute] by auto
himmelma@31276
   546
next
himmelma@31276
   547
  assume ?rhs
himmelma@31276
   548
  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *s v) = 0" by auto
himmelma@31276
   549
  thus ?lhs unfolding affine_dependent_explicit using assms by auto
himmelma@31276
   550
qed
himmelma@31276
   551
himmelma@31276
   552
subsection {* A general lemma. *}
himmelma@31276
   553
himmelma@31276
   554
lemma convex_connected:
himmelma@31276
   555
  assumes "convex s" shows "connected s"
himmelma@31276
   556
proof-
himmelma@31276
   557
  { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
himmelma@31276
   558
    assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
himmelma@31276
   559
    then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
himmelma@31276
   560
    hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
himmelma@31276
   561
himmelma@31276
   562
    { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
himmelma@31276
   563
      { fix y have *:"(1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2) = (y - x) *s x1 - (y - x) *s x2"
himmelma@31276
   564
	  by(simp add: ring_simps vector_sadd_rdistrib vector_sub_rdistrib)
himmelma@31276
   565
	assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
himmelma@31276
   566
	hence "norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
himmelma@31276
   567
	  unfolding * and vector_ssub_ldistrib[THEN sym] and norm_mul 
himmelma@31276
   568
	  unfolding less_divide_eq using n by auto  }
himmelma@31276
   569
      hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
himmelma@31276
   570
	apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
himmelma@31276
   571
	apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
himmelma@31276
   572
himmelma@31276
   573
    have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e2"
huffman@31285
   574
      apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
huffman@31289
   575
      using * apply(simp add: dist_norm)
huffman@31418
   576
      using as(1,2)[unfolded open_dist] apply simp
huffman@31418
   577
      using as(1,2)[unfolded open_dist] apply simp
himmelma@31276
   578
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
himmelma@31276
   579
      using as(3) by auto
himmelma@31276
   580
    then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *s x1 + x *s x2 \<notin> e1"  "(1 - x) *s x1 + x *s x2 \<notin> e2" by auto
himmelma@31276
   581
    hence False using as(4) 
himmelma@31276
   582
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
himmelma@31276
   583
      using x1(2) x2(2) by auto  }
himmelma@31276
   584
  thus ?thesis unfolding connected_def by auto
himmelma@31276
   585
qed
himmelma@31276
   586
himmelma@31276
   587
subsection {* One rather trivial consequence. *}
himmelma@31276
   588
huffman@31345
   589
lemma connected_UNIV: "connected (UNIV :: (real ^ _) set)"
himmelma@31276
   590
  by(simp add: convex_connected convex_UNIV)
himmelma@31276
   591
himmelma@31276
   592
subsection {* Convex functions into the reals. *}
himmelma@31276
   593
himmelma@31276
   594
definition "convex_on s (f::real^'n \<Rightarrow> real) = 
himmelma@31276
   595
  (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *s x + v *s y) \<le> u * f x + v * f y)"
himmelma@31276
   596
himmelma@31276
   597
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
himmelma@31276
   598
  unfolding convex_on_def by auto
himmelma@31276
   599
himmelma@31276
   600
lemma convex_add:
himmelma@31276
   601
  assumes "convex_on s f" "convex_on s g"
himmelma@31276
   602
  shows "convex_on s (\<lambda>x. f x + g x)"
himmelma@31276
   603
proof-
himmelma@31276
   604
  { fix x y assume "x\<in>s" "y\<in>s" moreover
himmelma@31276
   605
    fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   606
    ultimately have "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
himmelma@31276
   607
      using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
himmelma@31276
   608
      using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
himmelma@31276
   609
      apply - apply(rule add_mono) by auto
himmelma@31276
   610
    hence "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps)  }
himmelma@31276
   611
  thus ?thesis unfolding convex_on_def by auto 
himmelma@31276
   612
qed
himmelma@31276
   613
himmelma@31276
   614
lemma convex_cmul: 
himmelma@31276
   615
  assumes "0 \<le> (c::real)" "convex_on s f"
himmelma@31276
   616
  shows "convex_on s (\<lambda>x. c * f x)"
himmelma@31276
   617
proof-
himmelma@31276
   618
  have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: ring_simps)
himmelma@31276
   619
  show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
himmelma@31276
   620
qed
himmelma@31276
   621
himmelma@31276
   622
lemma convex_lower:
himmelma@31276
   623
  assumes "convex_on s f"  "x\<in>s"  "y \<in> s"  "0 \<le> u"  "0 \<le> v"  "u + v = 1"
himmelma@31276
   624
  shows "f (u *s x + v *s y) \<le> max (f x) (f y)"
himmelma@31276
   625
proof-
himmelma@31276
   626
  let ?m = "max (f x) (f y)"
himmelma@31276
   627
  have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" apply(rule add_mono) 
himmelma@31276
   628
    using assms(4,5) by(auto simp add: mult_mono1)
himmelma@31276
   629
  also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
himmelma@31276
   630
  finally show ?thesis using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
himmelma@31276
   631
    using assms(2-6) by auto 
himmelma@31276
   632
qed
himmelma@31276
   633
himmelma@31276
   634
lemma convex_local_global_minimum:
himmelma@31276
   635
  assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
himmelma@31276
   636
  shows "\<forall>y\<in>s. f x \<le> f y"
himmelma@31276
   637
proof(rule ccontr)
himmelma@31276
   638
  have "x\<in>s" using assms(1,3) by auto
himmelma@31276
   639
  assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
himmelma@31276
   640
  then obtain y where "y\<in>s" and y:"f x > f y" by auto
himmelma@31276
   641
  hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
himmelma@31276
   642
himmelma@31276
   643
  then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
himmelma@31276
   644
    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@31276
   645
  hence "f ((1-u) *s x + u *s y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
himmelma@31276
   646
    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@31276
   647
  moreover
himmelma@31276
   648
  have *:"x - ((1 - u) *s x + u *s y) = u *s (x - y)" by (simp add: vector_ssub_ldistrib vector_sub_rdistrib)
huffman@31289
   649
  have "(1 - u) *s x + u *s y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_mul and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
himmelma@31276
   650
    using u unfolding pos_less_divide_eq[OF xy] by auto
himmelma@31276
   651
  hence "f x \<le> f ((1 - u) *s x + u *s y)" using assms(4) by auto
himmelma@31276
   652
  ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
himmelma@31276
   653
qed
himmelma@31276
   654
himmelma@31276
   655
lemma convex_distance: "convex_on s (\<lambda>x. dist a x)"
huffman@31289
   656
proof(auto simp add: convex_on_def dist_norm)
himmelma@31276
   657
  fix x y assume "x\<in>s" "y\<in>s"
himmelma@31276
   658
  fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   659
  have "a = u *s a + v *s a" unfolding vector_sadd_rdistrib[THEN sym] and `u+v=1` by simp
himmelma@31276
   660
  hence *:"a - (u *s x + v *s y) = (u *s (a - x)) + (v *s (a - y))" by auto
himmelma@31276
   661
  show "norm (a - (u *s x + v *s y)) \<le> u * norm (a - x) + v * norm (a - y)"
himmelma@31276
   662
    unfolding * using norm_triangle_ineq[of "u *s (a - x)" "v *s (a - y)"] unfolding norm_mul
himmelma@31276
   663
    using `0 \<le> u` `0 \<le> v` by auto
himmelma@31276
   664
qed
himmelma@31276
   665
himmelma@31276
   666
subsection {* Arithmetic operations on sets preserve convexity. *}
himmelma@31276
   667
himmelma@31276
   668
lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *s x) ` s)"
himmelma@31276
   669
  unfolding convex_def and image_iff apply auto
himmelma@31276
   670
  apply (rule_tac x="u *s x+v *s y" in bexI) by (auto simp add: field_simps)
himmelma@31276
   671
himmelma@31276
   672
lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
himmelma@31276
   673
  unfolding convex_def and image_iff apply auto
himmelma@31276
   674
  apply (rule_tac x="u *s x+v *s y" in bexI) by auto
himmelma@31276
   675
himmelma@31276
   676
lemma convex_sums:
himmelma@31276
   677
  assumes "convex s" "convex t"
himmelma@31276
   678
  shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
himmelma@31276
   679
proof(auto simp add: convex_def image_iff)
himmelma@31276
   680
  fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
himmelma@31276
   681
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   682
  show "\<exists>x y. u *s xa + u *s ya + (v *s xb + v *s yb) = x + y \<and> x \<in> s \<and> y \<in> t"
himmelma@31276
   683
    apply(rule_tac x="u *s xa + v *s xb" in exI) apply(rule_tac x="u *s ya + v *s yb" in exI)
himmelma@31276
   684
    using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
himmelma@31276
   685
    using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
himmelma@31276
   686
    using uv xy by auto
himmelma@31276
   687
qed
himmelma@31276
   688
himmelma@31276
   689
lemma convex_differences: 
himmelma@31276
   690
  assumes "convex s" "convex t"
himmelma@31276
   691
  shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
himmelma@31276
   692
proof-
himmelma@31276
   693
  have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}" unfolding image_iff apply auto
himmelma@31276
   694
    apply(rule_tac x=xa in exI) apply(rule_tac x="-y" in exI) apply simp
himmelma@31276
   695
    apply(rule_tac x=xa in exI) apply(rule_tac x=xb in exI) by simp
himmelma@31276
   696
  thus ?thesis using convex_sums[OF assms(1)  convex_negations[OF assms(2)]] by auto
himmelma@31276
   697
qed
himmelma@31276
   698
himmelma@31276
   699
lemma convex_translation: assumes "convex s" shows "convex ((\<lambda>x. a + x) ` s)"
himmelma@31276
   700
proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
himmelma@31276
   701
  thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
himmelma@31276
   702
himmelma@31276
   703
lemma convex_affinity: assumes "convex (s::(real^'n) set)" shows "convex ((\<lambda>x. a + c *s x) ` s)"
himmelma@31276
   704
proof- have "(\<lambda>x. a + c *s x) ` s = op + a ` op *s c ` s" by auto
himmelma@31276
   705
  thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
himmelma@31276
   706
himmelma@31276
   707
lemma convex_linear_image: assumes c:"convex s" and l:"linear f" shows "convex(f ` s)"
himmelma@31276
   708
proof(auto simp add: convex_def)
himmelma@31276
   709
  fix x y assume xy:"x \<in> s" "y \<in> s"
himmelma@31276
   710
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   711
  show "u *s f x + v *s f y \<in> f ` s" unfolding image_iff
himmelma@31276
   712
    apply(rule_tac x="u *s x + v *s y" in bexI)
himmelma@31276
   713
    unfolding linear_add[OF l] linear_cmul[OF l] 
himmelma@31276
   714
    using c[unfolded convex_def] xy uv by auto
himmelma@31276
   715
qed
himmelma@31276
   716
himmelma@31276
   717
subsection {* Balls, being convex, are connected. *}
himmelma@31276
   718
himmelma@31276
   719
lemma convex_ball: "convex (ball x e)" 
himmelma@31276
   720
proof(auto simp add: convex_def)
himmelma@31276
   721
  fix y z assume yz:"dist x y < e" "dist x z < e"
himmelma@31276
   722
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   723
  have "dist x (u *s y + v *s z) \<le> u * dist x y + v * dist x z" using uv yz
himmelma@31276
   724
    using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
himmelma@31276
   725
  thus "dist x (u *s y + v *s z) < e" using real_convex_bound_lt[OF yz uv] by auto 
himmelma@31276
   726
qed
himmelma@31276
   727
himmelma@31276
   728
lemma convex_cball: "convex(cball x e)"
himmelma@31276
   729
proof(auto simp add: convex_def Ball_def mem_cball)
himmelma@31276
   730
  fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
himmelma@31276
   731
  fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   732
  have "dist x (u *s y + v *s z) \<le> u * dist x y + v * dist x z" using uv yz
himmelma@31276
   733
    using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
himmelma@31276
   734
  thus "dist x (u *s y + v *s z) \<le> e" using real_convex_bound_le[OF yz uv] by auto 
himmelma@31276
   735
qed
himmelma@31276
   736
huffman@31345
   737
lemma connected_ball: "connected(ball (x::real^_) e)" (* FIXME: generalize *)
himmelma@31276
   738
  using convex_connected convex_ball by auto
himmelma@31276
   739
huffman@31345
   740
lemma connected_cball: "connected(cball (x::real^_) e)" (* FIXME: generalize *)
himmelma@31276
   741
  using convex_connected convex_cball by auto
himmelma@31276
   742
himmelma@31276
   743
subsection {* Convex hull. *}
himmelma@31276
   744
himmelma@31276
   745
lemma convex_convex_hull: "convex(convex hull s)"
himmelma@31276
   746
  unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
himmelma@31276
   747
  unfolding mem_def by auto
himmelma@31276
   748
himmelma@31276
   749
lemma convex_hull_eq: "(convex hull s = s) \<longleftrightarrow> convex s" apply(rule hull_eq[unfolded mem_def])
himmelma@31276
   750
  using convex_Inter[unfolded Ball_def mem_def] by auto
himmelma@31276
   751
himmelma@31276
   752
lemma bounded_convex_hull: assumes "bounded s" shows "bounded(convex hull s)"
huffman@31533
   753
proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
himmelma@31276
   754
  show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
himmelma@31276
   755
    unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
huffman@31289
   756
    unfolding subset_eq mem_cball dist_norm using B by auto qed
himmelma@31276
   757
himmelma@31276
   758
lemma finite_imp_bounded_convex_hull:
himmelma@31276
   759
  "finite s \<Longrightarrow> bounded(convex hull s)"
himmelma@31276
   760
  using bounded_convex_hull finite_imp_bounded by auto
himmelma@31276
   761
himmelma@31276
   762
subsection {* Stepping theorems for convex hulls of finite sets. *}
himmelma@31276
   763
himmelma@31276
   764
lemma convex_hull_empty[simp]: "convex hull {} = {}"
himmelma@31276
   765
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@31276
   766
himmelma@31276
   767
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
himmelma@31276
   768
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@31276
   769
himmelma@31276
   770
lemma convex_hull_insert:
himmelma@31276
   771
  assumes "s \<noteq> {}"
himmelma@31276
   772
  shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
himmelma@31276
   773
                                    b \<in> (convex hull s) \<and> (x = u *s a + v *s b)}" (is "?xyz = ?hull")
himmelma@31276
   774
 apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
himmelma@31276
   775
 fix x assume x:"x = a \<or> x \<in> s"
himmelma@31276
   776
 thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
himmelma@31276
   777
   apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
himmelma@31276
   778
next
himmelma@31276
   779
  fix x assume "x\<in>?hull"
himmelma@31276
   780
  then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *s a + v *s b" by auto
himmelma@31276
   781
  have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
himmelma@31276
   782
    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
himmelma@31276
   783
  thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
himmelma@31276
   784
    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@31276
   785
next
himmelma@31276
   786
  show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
himmelma@31276
   787
    fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
himmelma@31276
   788
    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 *s a + v1 *s b1" by auto
himmelma@31276
   789
    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 *s a + v2 *s b2" by auto
himmelma@31276
   790
    have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
himmelma@31276
   791
    have "\<exists>b \<in> convex hull s. u *s x + v *s y = (u * u1) *s a + (v * u2) *s a + (b - (u * u1) *s b - (v * u2) *s b)"
himmelma@31276
   792
    proof(cases "u * v1 + v * v2 = 0")
himmelma@31276
   793
      have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
himmelma@31276
   794
      case True hence **:"u * v1 = 0" "v * v2 = 0" apply- apply(rule_tac [!] ccontr)
himmelma@31276
   795
	using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by auto
himmelma@31276
   796
      hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
himmelma@31276
   797
      thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: **) 
himmelma@31276
   798
    next
himmelma@31276
   799
      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@31276
   800
      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@31276
   801
      also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
himmelma@31276
   802
      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@31276
   803
	apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
himmelma@31276
   804
	using as(1,2) obt1(1,2) obt2(1,2) by auto 
himmelma@31276
   805
      thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
himmelma@31276
   806
	apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *s b1 + ((v * v2) / (u * v1 + v * v2)) *s b2" in bexI) defer
himmelma@31276
   807
	apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
himmelma@31276
   808
	unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff by auto
himmelma@31276
   809
    qed note * = this
himmelma@31276
   810
    have u1:"u1 \<le> 1" apply(rule ccontr) unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
himmelma@31276
   811
    have u2:"u2 \<le> 1" apply(rule ccontr) unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
himmelma@31276
   812
    have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
himmelma@31276
   813
      apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
himmelma@31276
   814
    also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
himmelma@31276
   815
    finally 
himmelma@31276
   816
    show "u *s x + v *s y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
himmelma@31276
   817
      apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
himmelma@31276
   818
      using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add:field_simps)
himmelma@31276
   819
  qed
himmelma@31276
   820
qed
himmelma@31276
   821
himmelma@31276
   822
himmelma@31276
   823
subsection {* Explicit expression for convex hull. *}
himmelma@31276
   824
himmelma@31276
   825
lemma convex_hull_indexed:
himmelma@31276
   826
  "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@31276
   827
                            (setsum u {1..k} = 1) \<and>
himmelma@31276
   828
                            (setsum (\<lambda>i. u i *s x i) {1..k} = y)}" (is "?xyz = ?hull")
himmelma@31276
   829
  apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
himmelma@31276
   830
  apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
himmelma@31276
   831
proof-
himmelma@31276
   832
  fix x assume "x\<in>s"
himmelma@31276
   833
  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@31276
   834
next
himmelma@31276
   835
  fix t assume as:"s \<subseteq> t" "convex t"
himmelma@31276
   836
  show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
himmelma@31276
   837
    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 *s y i) = x"
himmelma@31276
   838
    show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
himmelma@31276
   839
      using assm(1,2) as(1) by auto qed
himmelma@31276
   840
next
himmelma@31276
   841
  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@31276
   842
  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 *s x1 i) = x" by auto
himmelma@31276
   843
  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 *s x2 i) = y" by auto
himmelma@31276
   844
  have *:"\<And>P x1 x2 s1 s2 i.(if P i then s1 else s2) *s (if P i then x1 else x2) = (if P i then s1 *s x1 else s2 *s x2)"
himmelma@31276
   845
    "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
himmelma@31276
   846
    prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
himmelma@31276
   847
  have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
himmelma@31276
   848
  show "u *s x + v *s y \<in> ?hull" apply(rule)
himmelma@31276
   849
    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@31276
   850
    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)
himmelma@31276
   851
    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def
himmelma@31276
   852
    unfolding vector_smult_assoc[THEN sym] setsum_cmul setsum_right_distrib[THEN sym] proof-
himmelma@31276
   853
    fix i assume i:"i \<in> {1..k1+k2}"
himmelma@31276
   854
    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@31276
   855
    proof(cases "i\<in>{1..k1}")
himmelma@31276
   856
      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@31276
   857
    next def j \<equiv> "i - k1"
himmelma@31276
   858
      case False with i have "j \<in> {1..k2}" unfolding j_def by auto
himmelma@31276
   859
      thus ?thesis unfolding j_def[symmetric] using False
himmelma@31276
   860
	using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
himmelma@31276
   861
  qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
himmelma@31276
   862
qed
himmelma@31276
   863
himmelma@31276
   864
lemma convex_hull_finite:
himmelma@31276
   865
  assumes "finite (s::(real^'n)set)"
himmelma@31276
   866
  shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
himmelma@31276
   867
         setsum u s = 1 \<and> setsum (\<lambda>x. u x *s x) s = y}" (is "?HULL = ?set")
himmelma@31276
   868
proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
himmelma@31276
   869
  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 *s x) = x" 
himmelma@31276
   870
    apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
himmelma@31276
   871
    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
himmelma@31276
   872
next
himmelma@31276
   873
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@31276
   874
  fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
himmelma@31276
   875
  fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
himmelma@31276
   876
  { fix x assume "x\<in>s"
himmelma@31276
   877
    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@31276
   878
      by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
himmelma@31276
   879
  moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
himmelma@31276
   880
    unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
himmelma@31276
   881
  moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
himmelma@31276
   882
    unfolding vector_sadd_rdistrib and setsum_addf and vector_smult_assoc[THEN sym] and setsum_cmul by auto
himmelma@31276
   883
  ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
himmelma@31276
   884
    apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
himmelma@31276
   885
next
himmelma@31276
   886
  fix t assume t:"s \<subseteq> t" "convex t" 
himmelma@31276
   887
  fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
himmelma@31276
   888
  thus "(\<Sum>x\<in>s. u x *s x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
himmelma@31276
   889
    using assms and t(1) by auto
himmelma@31276
   890
qed
himmelma@31276
   891
himmelma@31276
   892
subsection {* Another formulation from Lars Schewe. *}
himmelma@31276
   893
himmelma@31276
   894
lemma convex_hull_explicit:
himmelma@31276
   895
  "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
himmelma@31276
   896
             (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}" (is "?lhs = ?rhs")
himmelma@31276
   897
proof-
himmelma@31276
   898
  { fix x assume "x\<in>?lhs"
himmelma@31276
   899
    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 *s y i) = x"
himmelma@31276
   900
      unfolding convex_hull_indexed by auto
himmelma@31276
   901
himmelma@31276
   902
    have fin:"finite {1..k}" by auto
himmelma@31276
   903
    have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
himmelma@31276
   904
    { fix j assume "j\<in>{1..k}"
himmelma@31276
   905
      hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
himmelma@31276
   906
	using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
himmelma@31276
   907
	apply(rule setsum_nonneg) using obt(1) by auto } 
himmelma@31276
   908
    moreover
himmelma@31276
   909
    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
himmelma@31276
   910
      unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
himmelma@31276
   911
    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *s v) = x"
himmelma@31276
   912
      using setsum_image_gen[OF fin, of "\<lambda>i. u i *s y i" y, THEN sym]
himmelma@31276
   913
      unfolding setsum_vmul[OF fin']  using obt(3) by auto
himmelma@31276
   914
    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 *s v) = x"
himmelma@31276
   915
      apply(rule_tac x="y ` {1..k}" in exI)
himmelma@31276
   916
      apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
himmelma@31276
   917
    hence "x\<in>?rhs" by auto  }
himmelma@31276
   918
  moreover
himmelma@31276
   919
  { fix y assume "y\<in>?rhs"
himmelma@31276
   920
    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 *s v) = y" by auto
himmelma@31276
   921
himmelma@31276
   922
    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@31276
   923
    
himmelma@31276
   924
    { fix i::nat assume "i\<in>{1..card s}"
himmelma@31276
   925
      hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
himmelma@31276
   926
      hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
himmelma@31276
   927
    moreover have *:"finite {1..card s}" by auto
himmelma@31276
   928
    { fix y assume "y\<in>s"
himmelma@31276
   929
      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@31276
   930
      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@31276
   931
      hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
himmelma@31276
   932
      hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y" "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *s f x) = u y *s y" by auto   }
himmelma@31276
   933
himmelma@31276
   934
    hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *s f i) = y"
himmelma@31276
   935
      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *s f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
himmelma@31276
   936
      unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *s f x)" "\<lambda>v. u v *s v"]
himmelma@31276
   937
      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@31276
   938
    
himmelma@31276
   939
    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 *s x i) = y"
himmelma@31276
   940
      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@31276
   941
    hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
himmelma@31276
   942
  ultimately show ?thesis unfolding expand_set_eq by blast
himmelma@31276
   943
qed
himmelma@31276
   944
himmelma@31276
   945
subsection {* A stepping theorem for that expansion. *}
himmelma@31276
   946
himmelma@31276
   947
lemma convex_hull_finite_step:
himmelma@31276
   948
  assumes "finite (s::(real^'n) set)"
himmelma@31276
   949
  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 *s x) (insert a s) = y)
himmelma@31276
   950
     \<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 *s x) s = y - v *s a)" (is "?lhs = ?rhs")
himmelma@31276
   951
proof(rule, case_tac[!] "a\<in>s")
himmelma@31276
   952
  assume "a\<in>s" hence *:"insert a s = s" by auto
himmelma@31276
   953
  assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
himmelma@31276
   954
next
himmelma@31276
   955
  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 *s x) = y" by auto
himmelma@31276
   956
  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@31276
   957
    apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
himmelma@31276
   958
next
himmelma@31276
   959
  assume "a\<in>s" hence *:"insert a s = s" by auto
himmelma@31276
   960
  have fin:"finite (insert a s)" using assms by auto
himmelma@31276
   961
  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 *s x) = y - v *s a" by auto
himmelma@31276
   962
  show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding vector_sadd_rdistrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
himmelma@31276
   963
    unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
himmelma@31276
   964
next
himmelma@31276
   965
  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 *s x) = y - v *s a" by auto
himmelma@31276
   966
  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) *s x) = (\<Sum>x\<in>s. u x *s x)"
himmelma@31276
   967
    apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
himmelma@31276
   968
  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@31276
   969
qed
himmelma@31276
   970
himmelma@31276
   971
subsection {* Hence some special cases. *}
himmelma@31276
   972
himmelma@31276
   973
lemma convex_hull_2:
himmelma@31276
   974
  "convex hull {a,b} = {u *s a + v *s b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
himmelma@31276
   975
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@31276
   976
show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
himmelma@31276
   977
  apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
himmelma@31276
   978
  apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
himmelma@31276
   979
himmelma@31276
   980
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *s (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
himmelma@31276
   981
  unfolding convex_hull_2 unfolding Collect_def 
himmelma@31276
   982
proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
himmelma@31276
   983
  fix x show "(\<exists>v u. x = v *s a + u *s b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *s (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
himmelma@31276
   984
    unfolding * apply auto apply(rule_tac[!] x=u in exI) by auto qed
himmelma@31276
   985
himmelma@31276
   986
lemma convex_hull_3:
himmelma@31276
   987
  "convex hull {a::real^'n,b,c} = { u *s a + v *s b + w *s c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
himmelma@31276
   988
proof-
himmelma@31276
   989
  have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
himmelma@31276
   990
  have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
himmelma@31276
   991
         "\<And>x y z ::real^'n. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: ring_simps)
himmelma@31276
   992
  show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
himmelma@31276
   993
    unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
himmelma@31276
   994
    apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
himmelma@31276
   995
    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@31276
   996
himmelma@31276
   997
lemma convex_hull_3_alt:
himmelma@31276
   998
  "convex hull {a,b,c} = {a + u *s (b - a) + v *s (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
himmelma@31276
   999
proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
himmelma@31276
  1000
  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
himmelma@31276
  1001
    apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by simp qed
himmelma@31276
  1002
himmelma@31276
  1003
subsection {* Relations among closure notions and corresponding hulls. *}
himmelma@31276
  1004
himmelma@31276
  1005
lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
himmelma@31276
  1006
  unfolding subspace_def affine_def by auto
himmelma@31276
  1007
himmelma@31276
  1008
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
himmelma@31276
  1009
  unfolding affine_def convex_def by auto
himmelma@31276
  1010
himmelma@31276
  1011
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
himmelma@31276
  1012
  using subspace_imp_affine affine_imp_convex by auto
himmelma@31276
  1013
himmelma@31276
  1014
lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
himmelma@31276
  1015
  unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
himmelma@31276
  1016
  using subspace_imp_affine  by auto
himmelma@31276
  1017
himmelma@31276
  1018
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
himmelma@31276
  1019
  unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
himmelma@31276
  1020
  using subspace_imp_convex by auto
himmelma@31276
  1021
himmelma@31276
  1022
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
himmelma@31276
  1023
  unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
himmelma@31276
  1024
  using affine_imp_convex by auto
himmelma@31276
  1025
himmelma@31276
  1026
lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
himmelma@31276
  1027
  unfolding affine_dependent_def dependent_def 
himmelma@31276
  1028
  using affine_hull_subset_span by auto
himmelma@31276
  1029
himmelma@31276
  1030
lemma dependent_imp_affine_dependent:
himmelma@31276
  1031
  assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
himmelma@31276
  1032
  shows "affine_dependent (insert a s)"
himmelma@31276
  1033
proof-
himmelma@31276
  1034
  from assms(1)[unfolded dependent_explicit] obtain S u v 
himmelma@31276
  1035
    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 *s v) = 0" by auto
himmelma@31276
  1036
  def t \<equiv> "(\<lambda>x. x + a) ` S"
himmelma@31276
  1037
himmelma@31276
  1038
  have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
himmelma@31276
  1039
  have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
himmelma@31276
  1040
  have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
himmelma@31276
  1041
himmelma@31276
  1042
  hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
himmelma@31276
  1043
  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@31276
  1044
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
himmelma@31276
  1045
  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
himmelma@31276
  1046
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
himmelma@31276
  1047
  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@31276
  1048
    apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
himmelma@31276
  1049
  moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *s x) = (\<Sum>x\<in>t. Q x *s x)"
himmelma@31276
  1050
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
himmelma@31276
  1051
  have "(\<Sum>x\<in>t. u (x - a)) *s a = (\<Sum>v\<in>t. u (v - a) *s v)" 
himmelma@31276
  1052
    unfolding setsum_vmul[OF fin(1)] unfolding t_def and setsum_reindex[OF inj] and o_def
himmelma@31276
  1053
    using obt(5) by (auto simp add: setsum_addf)
himmelma@31276
  1054
  hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *s v) = 0"
himmelma@31276
  1055
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *  vector_smult_lneg) 
himmelma@31276
  1056
  ultimately show ?thesis unfolding affine_dependent_explicit
himmelma@31276
  1057
    apply(rule_tac x="insert a t" in exI) by auto 
himmelma@31276
  1058
qed
himmelma@31276
  1059
himmelma@31276
  1060
lemma convex_cone:
himmelma@31276
  1061
  "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 *s x) \<in> s)" (is "?lhs = ?rhs")
himmelma@31276
  1062
proof-
himmelma@31276
  1063
  { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
himmelma@31276
  1064
    hence "2 *s x \<in>s" "2 *s y \<in> s" unfolding cone_def by auto
himmelma@31276
  1065
    hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
himmelma@31276
  1066
      apply(erule_tac x="2*s x" in ballE) apply(erule_tac x="2*s y" in ballE)
himmelma@31276
  1067
      apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
himmelma@31276
  1068
  thus ?thesis unfolding convex_def cone_def by blast
himmelma@31276
  1069
qed
himmelma@31276
  1070
himmelma@31276
  1071
lemma affine_dependent_biggerset: fixes s::"(real^'n::finite) set"
himmelma@31276
  1072
  assumes "finite s" "card s \<ge> CARD('n) + 2"
himmelma@31276
  1073
  shows "affine_dependent s"
himmelma@31276
  1074
proof-
himmelma@31276
  1075
  have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
himmelma@31276
  1076
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
himmelma@31276
  1077
  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
himmelma@31276
  1078
    apply(rule card_image) unfolding inj_on_def by auto
himmelma@31276
  1079
  also have "\<dots> > CARD('n)" using assms(2)
himmelma@31276
  1080
    unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
himmelma@31276
  1081
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
himmelma@31276
  1082
    apply(rule dependent_imp_affine_dependent)
himmelma@31276
  1083
    apply(rule dependent_biggerset) by auto qed
himmelma@31276
  1084
himmelma@31276
  1085
lemma affine_dependent_biggerset_general:
himmelma@31276
  1086
  assumes "finite (s::(real^'n::finite) set)" "card s \<ge> dim s + 2"
himmelma@31276
  1087
  shows "affine_dependent s"
himmelma@31276
  1088
proof-
himmelma@31276
  1089
  from assms(2) have "s \<noteq> {}" by auto
himmelma@31276
  1090
  then obtain a where "a\<in>s" by auto
himmelma@31276
  1091
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
himmelma@31276
  1092
  have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
himmelma@31276
  1093
    apply(rule card_image) unfolding inj_on_def by auto
himmelma@31276
  1094
  have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
himmelma@31276
  1095
    apply(rule subset_le_dim) unfolding subset_eq
himmelma@31276
  1096
    using `a\<in>s` by (auto simp add:span_superset span_sub)
himmelma@31276
  1097
  also have "\<dots> < dim s + 1" by auto
himmelma@31276
  1098
  also have "\<dots> \<le> card (s - {a})" using assms
himmelma@31276
  1099
    using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
himmelma@31276
  1100
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
himmelma@31276
  1101
    apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
himmelma@31276
  1102
himmelma@31276
  1103
subsection {* Caratheodory's theorem. *}
himmelma@31276
  1104
himmelma@31276
  1105
lemma convex_hull_caratheodory: fixes p::"(real^'n::finite) set"
himmelma@31276
  1106
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and>
himmelma@31276
  1107
  (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
himmelma@31276
  1108
  unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
himmelma@31276
  1109
proof(rule,rule)
himmelma@31276
  1110
  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 *s v) = y"
himmelma@31276
  1111
  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 *s v) = y"
himmelma@31276
  1112
  then obtain N where "?P N" by auto
himmelma@31276
  1113
  hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
himmelma@31276
  1114
  then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
himmelma@31276
  1115
  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 *s v) = y" by auto
himmelma@31276
  1116
himmelma@31276
  1117
  have "card s \<le> CARD('n) + 1" proof(rule ccontr, simp only: not_le)
himmelma@31276
  1118
    assume "CARD('n) + 1 < card s"
himmelma@31276
  1119
    hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
himmelma@31276
  1120
    then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *s v) = 0"
himmelma@31276
  1121
      using affine_dependent_explicit_finite[OF obt(1)] by auto
himmelma@31276
  1122
    def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
himmelma@31276
  1123
    have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
himmelma@31276
  1124
      assume as:"\<forall>x\<in>s. 0 \<le> w x"
himmelma@31276
  1125
      hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
himmelma@31276
  1126
      hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
himmelma@31276
  1127
	using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
himmelma@31276
  1128
      thus False using wv(1) by auto
himmelma@31276
  1129
    qed hence "i\<noteq>{}" unfolding i_def by auto
himmelma@31276
  1130
himmelma@31276
  1131
    hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
himmelma@31276
  1132
      using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
himmelma@31276
  1133
    have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
himmelma@31276
  1134
      fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
himmelma@31276
  1135
      show"0 \<le> u v + t * w v" proof(cases "w v < 0")
himmelma@31276
  1136
	case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
himmelma@31276
  1137
	  using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
himmelma@31276
  1138
	case True hence "t \<le> u v / (- w v)" using `v\<in>s`
himmelma@31276
  1139
	  unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
himmelma@31276
  1140
	thus ?thesis unfolding real_0_le_add_iff
himmelma@31276
  1141
	  using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
himmelma@31276
  1142
      qed qed
himmelma@31276
  1143
himmelma@31276
  1144
    obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
himmelma@31276
  1145
      using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
himmelma@31276
  1146
    hence a:"a\<in>s" "u a + t * w a = 0" by auto
himmelma@31276
  1147
    have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'a::ring)" unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
himmelma@31276
  1148
    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
himmelma@31276
  1149
      unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
himmelma@31276
  1150
    moreover have "(\<Sum>v\<in>s. u v *s v + (t * w v) *s v) - (u a *s a + (t * w a) *s a) = y" 
himmelma@31276
  1151
      unfolding setsum_addf obt(6) vector_smult_assoc[THEN sym] setsum_cmul wv(4)
huffman@31445
  1152
      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]]
huffman@31445
  1153
      by (simp add: vector_smult_lneg)
himmelma@31276
  1154
    ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
himmelma@31276
  1155
      apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a by (auto simp add: *)
himmelma@31276
  1156
    thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
himmelma@31276
  1157
  thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1
himmelma@31276
  1158
    \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y" using obt by auto
himmelma@31276
  1159
qed auto
himmelma@31276
  1160
himmelma@31276
  1161
lemma caratheodory:
himmelma@31276
  1162
 "convex hull p = {x::real^'n::finite. \<exists>s. finite s \<and> s \<subseteq> p \<and>
himmelma@31276
  1163
      card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s}"
himmelma@31276
  1164
  unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
himmelma@31276
  1165
  fix x assume "x \<in> convex hull p"
himmelma@31276
  1166
  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1"
himmelma@31276
  1167
     "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"unfolding convex_hull_caratheodory by auto
himmelma@31276
  1168
  thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
himmelma@31276
  1169
    apply(rule_tac x=s in exI) using hull_subset[of s convex]
himmelma@31276
  1170
  using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
himmelma@31276
  1171
next
himmelma@31276
  1172
  fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
himmelma@31276
  1173
  then obtain s where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1" "x \<in> convex hull s" by auto
himmelma@31276
  1174
  thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
himmelma@31276
  1175
qed
himmelma@31276
  1176
himmelma@31276
  1177
subsection {* Openness and compactness are preserved by convex hull operation. *}
himmelma@31276
  1178
himmelma@31276
  1179
lemma open_convex_hull:
himmelma@31276
  1180
  assumes "open s"
himmelma@31276
  1181
  shows "open(convex hull s)"
himmelma@31276
  1182
  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10) 
himmelma@31276
  1183
proof(rule, rule) fix a
himmelma@31276
  1184
  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 *s v) = a"
himmelma@31276
  1185
  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 *s v) = a" by auto
himmelma@31276
  1186
himmelma@31276
  1187
  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@31276
  1188
    using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
himmelma@31276
  1189
  have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
himmelma@31276
  1190
himmelma@31276
  1191
  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 *s v) = y}"
himmelma@31276
  1192
    apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
himmelma@31276
  1193
  proof-
himmelma@31276
  1194
    show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
himmelma@31276
  1195
      using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
himmelma@31276
  1196
  next  fix y assume "y \<in> cball a (Min i)"
huffman@31289
  1197
    hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
himmelma@31276
  1198
    { fix x assume "x\<in>t"
himmelma@31276
  1199
      hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
huffman@31289
  1200
      hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
himmelma@31276
  1201
      moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
himmelma@31276
  1202
      ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by auto }
himmelma@31276
  1203
    moreover
himmelma@31276
  1204
    have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
himmelma@31276
  1205
    have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
himmelma@31276
  1206
      unfolding setsum_reindex[OF *] o_def using obt(4) by auto
himmelma@31276
  1207
    moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *s v) = y"
himmelma@31276
  1208
      unfolding setsum_reindex[OF *] o_def using obt(4,5)
himmelma@31276
  1209
      by (simp add: setsum_addf setsum_subtractf setsum_vmul[OF obt(1), THEN sym]) 
himmelma@31276
  1210
    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 *s v) = y"
himmelma@31276
  1211
      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@31276
  1212
      using obt(1, 3) by auto
himmelma@31276
  1213
  qed
himmelma@31276
  1214
qed
himmelma@31276
  1215
huffman@31558
  1216
lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
huffman@31558
  1217
unfolding open_vector_def all_1
huffman@31558
  1218
by (auto simp add: dest_vec1_def)
huffman@31558
  1219
huffman@31558
  1220
lemma tendsto_dest_vec1: "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
huffman@31558
  1221
  unfolding tendsto_def
huffman@31558
  1222
  apply clarify
huffman@31558
  1223
  apply (drule_tac x="dest_vec1 -` S" in spec)
huffman@31558
  1224
  apply (simp add: open_dest_vec1_vimage)
huffman@31558
  1225
  done
huffman@31558
  1226
huffman@31558
  1227
lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
huffman@31558
  1228
  unfolding continuous_def by (rule tendsto_dest_vec1)
himmelma@31276
  1229
huffman@31561
  1230
lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
huffman@31561
  1231
by (induct x) simp
huffman@31561
  1232
huffman@31561
  1233
(* TODO: move *)
huffman@31561
  1234
lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
huffman@31561
  1235
unfolding compact_def
huffman@31561
  1236
apply clarify
huffman@31561
  1237
apply (drule_tac x="fst \<circ> f" in spec)
huffman@31561
  1238
apply (drule mp, simp add: mem_Times_iff)
huffman@31561
  1239
apply (clarify, rename_tac l1 r1)
huffman@31561
  1240
apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
huffman@31561
  1241
apply (drule mp, simp add: mem_Times_iff)
huffman@31561
  1242
apply (clarify, rename_tac l2 r2)
huffman@31561
  1243
apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
huffman@31561
  1244
apply (rule_tac x="r1 \<circ> r2" in exI)
huffman@31561
  1245
apply (rule conjI, simp add: subseq_def)
huffman@31561
  1246
apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
huffman@31561
  1247
apply (drule (1) tendsto_Pair) back
huffman@31561
  1248
apply (simp add: o_def)
huffman@31561
  1249
done
huffman@31561
  1250
huffman@31561
  1251
(* TODO: move *)
huffman@31561
  1252
lemma compact_real_interval:
huffman@31561
  1253
  fixes a b :: real shows "compact {a..b}"
huffman@31561
  1254
proof -
huffman@31561
  1255
  have "continuous_on {vec1 a .. vec1 b} dest_vec1"
huffman@31561
  1256
    unfolding continuous_on
huffman@31561
  1257
    by (simp add: tendsto_dest_vec1 Lim_at_within Lim_ident_at)
huffman@31561
  1258
  moreover have "compact {vec1 a .. vec1 b}" by (rule compact_interval)
huffman@31561
  1259
  ultimately have "compact (dest_vec1 ` {vec1 a .. vec1 b})"
huffman@31561
  1260
    by (rule compact_continuous_image)
huffman@31561
  1261
  also have "dest_vec1 ` {vec1 a .. vec1 b} = {a..b}"
huffman@31561
  1262
    by (auto simp add: image_def Bex_def exists_vec1)
huffman@31561
  1263
  finally show ?thesis .
huffman@31561
  1264
qed
huffman@31561
  1265
himmelma@31276
  1266
lemma compact_convex_combinations:
huffman@31558
  1267
  fixes s t :: "(real ^ 'n::finite) set"
himmelma@31276
  1268
  assumes "compact s" "compact t"
himmelma@31276
  1269
  shows "compact { (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
himmelma@31276
  1270
proof-
huffman@31561
  1271
  let ?X = "{0..1} \<times> s \<times> t"
huffman@31561
  1272
  let ?h = "(\<lambda>z. (1 - fst z) *s fst (snd z) + fst z *s snd (snd z))"
himmelma@31276
  1273
  have *:"{ (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
huffman@31561
  1274
    apply(rule set_ext) unfolding image_iff mem_Collect_eq
huffman@31561
  1275
    apply rule apply auto
huffman@31561
  1276
    apply (rule_tac x=u in rev_bexI, simp)
huffman@31561
  1277
    apply (erule rev_bexI, erule rev_bexI, simp)
huffman@31561
  1278
    by auto
huffman@31561
  1279
  { fix u::"real" fix x y assume as:"0 \<le> u" "u \<le> 1" "x \<in> s" "y \<in> t"
huffman@31561
  1280
    hence "continuous (at (u, x, y))
huffman@31561
  1281
           (\<lambda>z. fst (snd z) - fst z *s fst (snd z) + fst z *s snd (snd z))"
huffman@31561
  1282
      apply (auto intro!: continuous_add continuous_sub continuous_mul)
huffman@31561
  1283
      unfolding continuous_at
huffman@31561
  1284
      by (safe intro!: tendsto_fst tendsto_snd Lim_at_id [unfolded id_def])
huffman@31561
  1285
  }
huffman@31561
  1286
  hence "continuous_on ({0..1} \<times> s \<times> t)
huffman@31561
  1287
     (\<lambda>z. (1 - fst z) *s fst (snd z) + fst z *s snd (snd z))"
huffman@31561
  1288
    apply(rule_tac continuous_at_imp_continuous_on) by auto
himmelma@31276
  1289
 thus ?thesis unfolding * apply(rule compact_continuous_image)
huffman@31561
  1290
    defer apply(rule compact_Times) defer apply(rule compact_Times)
huffman@31561
  1291
    using compact_real_interval assms by auto
himmelma@31276
  1292
qed
himmelma@31276
  1293
himmelma@31276
  1294
lemma compact_convex_hull: fixes s::"(real^'n::finite) set"
himmelma@31276
  1295
  assumes "compact s"  shows "compact(convex hull s)"
himmelma@31276
  1296
proof(cases "s={}")
himmelma@31276
  1297
  case True thus ?thesis using compact_empty by simp
himmelma@31276
  1298
next
himmelma@31276
  1299
  case False then obtain w where "w\<in>s" by auto
himmelma@31276
  1300
  show ?thesis unfolding caratheodory[of s]
himmelma@31276
  1301
  proof(induct "CARD('n) + 1")
himmelma@31276
  1302
    have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
himmelma@31276
  1303
      using compact_empty by (auto simp add: convex_hull_empty)
himmelma@31276
  1304
    case 0 thus ?case unfolding * by simp
himmelma@31276
  1305
  next
himmelma@31276
  1306
    case (Suc n)
himmelma@31276
  1307
    show ?case proof(cases "n=0")
himmelma@31276
  1308
      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@31276
  1309
	unfolding expand_set_eq and mem_Collect_eq proof(rule, rule)
himmelma@31276
  1310
	fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@31276
  1311
	then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
himmelma@31276
  1312
	show "x\<in>s" proof(cases "card t = 0")
himmelma@31276
  1313
	  case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by(simp add: convex_hull_empty)
himmelma@31276
  1314
	next
himmelma@31276
  1315
	  case False hence "card t = Suc 0" using t(3) `n=0` by auto
himmelma@31276
  1316
	  then obtain a where "t = {a}" unfolding card_Suc_eq by auto
himmelma@31276
  1317
	  thus ?thesis using t(2,4) by (simp add: convex_hull_singleton)
himmelma@31276
  1318
	qed
himmelma@31276
  1319
      next
himmelma@31276
  1320
	fix x assume "x\<in>s"
himmelma@31276
  1321
	thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@31276
  1322
	  apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
himmelma@31276
  1323
      qed thus ?thesis using assms by simp
himmelma@31276
  1324
    next
himmelma@31276
  1325
      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@31276
  1326
	{ (1 - u) *s x + u *s y | x y u. 
himmelma@31276
  1327
	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@31276
  1328
	unfolding expand_set_eq and mem_Collect_eq proof(rule,rule)
himmelma@31276
  1329
	fix x assume "\<exists>u v c. x = (1 - c) *s u + c *s v \<and>
himmelma@31276
  1330
          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@31276
  1331
	then obtain u v c t where obt:"x = (1 - c) *s u + c *s v"
himmelma@31276
  1332
          "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@31276
  1333
	moreover have "(1 - c) *s u + c *s v \<in> convex hull insert u t"
himmelma@31276
  1334
	  apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
himmelma@31276
  1335
	  using obt(7) and hull_mono[of t "insert u t"] by auto
himmelma@31276
  1336
	ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@31276
  1337
	  apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
himmelma@31276
  1338
      next
himmelma@31276
  1339
	fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@31276
  1340
	then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
himmelma@31276
  1341
	let ?P = "\<exists>u v c. x = (1 - c) *s u + c *s v \<and>
himmelma@31276
  1342
          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@31276
  1343
	show ?P proof(cases "card t = Suc n")
himmelma@31276
  1344
	  case False hence "card t \<le> n" using t(3) by auto
himmelma@31276
  1345
	  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@31276
  1346
	    by(auto intro!: exI[where x=t])
himmelma@31276
  1347
	next
himmelma@31276
  1348
	  case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
himmelma@31276
  1349
	  show ?P proof(cases "u={}")
himmelma@31276
  1350
	    case True hence "x=a" using t(4)[unfolded au] by auto
himmelma@31276
  1351
	    show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
himmelma@31276
  1352
	      using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
himmelma@31276
  1353
	  next
himmelma@31276
  1354
	    case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *s a + vx *s b"
himmelma@31276
  1355
	      using t(4)[unfolded au convex_hull_insert[OF False]] by auto
himmelma@31276
  1356
	    have *:"1 - vx = ux" using obt(3) by auto
himmelma@31276
  1357
	    show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
himmelma@31276
  1358
	      using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
himmelma@31276
  1359
	      by(auto intro!: exI[where x=u])
himmelma@31276
  1360
	  qed
himmelma@31276
  1361
	qed
himmelma@31276
  1362
      qed
himmelma@31276
  1363
      thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
himmelma@31276
  1364
    qed
himmelma@31276
  1365
  qed 
himmelma@31276
  1366
qed
himmelma@31276
  1367
himmelma@31276
  1368
lemma finite_imp_compact_convex_hull:
himmelma@31276
  1369
 "finite s \<Longrightarrow> compact(convex hull s)"
himmelma@31276
  1370
  apply(drule finite_imp_compact, drule compact_convex_hull) by assumption
himmelma@31276
  1371
himmelma@31276
  1372
subsection {* Extremal points of a simplex are some vertices. *}
himmelma@31276
  1373
huffman@31285
  1374
lemma dist_increases_online:
huffman@31285
  1375
  fixes a b d :: "real ^ 'n::finite"
huffman@31285
  1376
  assumes "d \<noteq> 0"
himmelma@31276
  1377
  shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
himmelma@31276
  1378
proof(cases "a \<bullet> d - b \<bullet> d > 0")
himmelma@31276
  1379
  case True hence "0 < d \<bullet> d + (a \<bullet> d * 2 - b \<bullet> d * 2)" 
himmelma@31276
  1380
    apply(rule_tac add_pos_pos) using assms by auto
huffman@31289
  1381
  thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and real_vector_norm_def and real_sqrt_less_iff
himmelma@31276
  1382
    by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
himmelma@31276
  1383
next
himmelma@31276
  1384
  case False hence "0 < d \<bullet> d + (b \<bullet> d * 2 - a \<bullet> d * 2)" 
himmelma@31276
  1385
    apply(rule_tac add_pos_nonneg) using assms by auto
huffman@31289
  1386
  thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and real_vector_norm_def and real_sqrt_less_iff
himmelma@31276
  1387
    by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
himmelma@31276
  1388
qed
himmelma@31276
  1389
himmelma@31276
  1390
lemma norm_increases_online:
himmelma@31276
  1391
 "(d::real^'n::finite) \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
huffman@31289
  1392
  using dist_increases_online[of d a 0] unfolding dist_norm by auto
himmelma@31276
  1393
himmelma@31276
  1394
lemma simplex_furthest_lt:
himmelma@31276
  1395
  fixes s::"(real^'n::finite) set" assumes "finite s"
himmelma@31276
  1396
  shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
himmelma@31276
  1397
proof(induct_tac rule: finite_induct[of s])
himmelma@31276
  1398
  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@31276
  1399
  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@31276
  1400
  proof(rule,rule,cases "s = {}")
himmelma@31276
  1401
    case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
himmelma@31276
  1402
    obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *s x + v *s b"
himmelma@31276
  1403
      using y(1)[unfolded convex_hull_insert[OF False]] by auto
himmelma@31276
  1404
    show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
himmelma@31276
  1405
    proof(cases "y\<in>convex hull s")
himmelma@31276
  1406
      case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
himmelma@31276
  1407
	using as(3)[THEN bspec[where x=y]] and y(2) by auto
himmelma@31276
  1408
      thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
himmelma@31276
  1409
    next
himmelma@31276
  1410
      case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
himmelma@31276
  1411
	assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
himmelma@31276
  1412
	thus ?thesis using False and obt(4) by auto
himmelma@31276
  1413
      next
himmelma@31276
  1414
	assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
himmelma@31276
  1415
	thus ?thesis using y(2) by auto
himmelma@31276
  1416
      next
himmelma@31276
  1417
	assume "u\<noteq>0" "v\<noteq>0"
himmelma@31276
  1418
	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@31276
  1419
	have "x\<noteq>b" proof(rule ccontr) 
himmelma@31276
  1420
	  assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
himmelma@31276
  1421
	    using obt(3) by(auto simp add: vector_sadd_rdistrib[THEN sym])
himmelma@31276
  1422
	  thus False using obt(4) and False by simp qed
himmelma@31276
  1423
	hence *:"w *s (x - b) \<noteq> 0" using w(1) by auto
himmelma@31276
  1424
	show ?thesis using dist_increases_online[OF *, of a y]
himmelma@31276
  1425
 	proof(erule_tac disjE)
himmelma@31276
  1426
	  assume "dist a y < dist a (y + w *s (x - b))"
himmelma@31276
  1427
	  hence "norm (y - a) < norm ((u + w) *s x + (v - w) *s b - a)"
huffman@31289
  1428
	    unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: ring_simps)
himmelma@31276
  1429
	  moreover have "(u + w) *s x + (v - w) *s b \<in> convex hull insert x s"
himmelma@31276
  1430
	    unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
himmelma@31276
  1431
	    apply(rule_tac x="u + w" in exI) apply rule defer 
himmelma@31276
  1432
	    apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
himmelma@31276
  1433
	  ultimately show ?thesis by auto
himmelma@31276
  1434
	next
himmelma@31276
  1435
	  assume "dist a y < dist a (y - w *s (x - b))"
himmelma@31276
  1436
	  hence "norm (y - a) < norm ((u - w) *s x + (v + w) *s b - a)"
huffman@31289
  1437
	    unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: ring_simps)
himmelma@31276
  1438
	  moreover have "(u - w) *s x + (v + w) *s b \<in> convex hull insert x s"
himmelma@31276
  1439
	    unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
himmelma@31276
  1440
	    apply(rule_tac x="u - w" in exI) apply rule defer 
himmelma@31276
  1441
	    apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
himmelma@31276
  1442
	  ultimately show ?thesis by auto
himmelma@31276
  1443
	qed
himmelma@31276
  1444
      qed auto
himmelma@31276
  1445
    qed
himmelma@31276
  1446
  qed auto
himmelma@31276
  1447
qed (auto simp add: assms)
himmelma@31276
  1448
himmelma@31276
  1449
lemma simplex_furthest_le:
himmelma@31276
  1450
  assumes "finite s" "s \<noteq> {}"
himmelma@31276
  1451
  shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
himmelma@31276
  1452
proof-
himmelma@31276
  1453
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
himmelma@31276
  1454
  then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
himmelma@31276
  1455
    using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
huffman@31289
  1456
    unfolding dist_commute[of a] unfolding dist_norm by auto
himmelma@31276
  1457
  thus ?thesis proof(cases "x\<in>s")
himmelma@31276
  1458
    case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
himmelma@31276
  1459
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
himmelma@31276
  1460
    thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
himmelma@31276
  1461
  qed auto
himmelma@31276
  1462
qed
himmelma@31276
  1463
himmelma@31276
  1464
lemma simplex_furthest_le_exists:
himmelma@31276
  1465
  "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
himmelma@31276
  1466
  using simplex_furthest_le[of s] by (cases "s={}")auto
himmelma@31276
  1467
himmelma@31276
  1468
lemma simplex_extremal_le:
himmelma@31276
  1469
  assumes "finite s" "s \<noteq> {}"
himmelma@31276
  1470
  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@31276
  1471
proof-
himmelma@31276
  1472
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
himmelma@31276
  1473
  then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
himmelma@31276
  1474
    "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
himmelma@31276
  1475
    using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
himmelma@31276
  1476
  thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
himmelma@31276
  1477
    assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
himmelma@31276
  1478
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
himmelma@31276
  1479
    thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
himmelma@31276
  1480
  next
himmelma@31276
  1481
    assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
himmelma@31276
  1482
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
himmelma@31276
  1483
    thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
himmelma@31276
  1484
      by (auto simp add: norm_minus_commute)
himmelma@31276
  1485
  qed auto
himmelma@31276
  1486
qed 
himmelma@31276
  1487
himmelma@31276
  1488
lemma simplex_extremal_le_exists:
himmelma@31276
  1489
  "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
himmelma@31276
  1490
  \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
himmelma@31276
  1491
  using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
himmelma@31276
  1492
himmelma@31276
  1493
subsection {* Closest point of a convex set is unique, with a continuous projection. *}
himmelma@31276
  1494
huffman@31289
  1495
definition
huffman@31289
  1496
  closest_point :: "(real ^ 'n::finite) set \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 'n" where
himmelma@31276
  1497
 "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
himmelma@31276
  1498
himmelma@31276
  1499
lemma closest_point_exists:
himmelma@31276
  1500
  assumes "closed s" "s \<noteq> {}"
himmelma@31276
  1501
  shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
himmelma@31276
  1502
  unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
himmelma@31276
  1503
  using distance_attains_inf[OF assms(1,2), of a] by auto
himmelma@31276
  1504
himmelma@31276
  1505
lemma closest_point_in_set:
himmelma@31276
  1506
  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
himmelma@31276
  1507
  by(meson closest_point_exists)
himmelma@31276
  1508
himmelma@31276
  1509
lemma closest_point_le:
himmelma@31276
  1510
  "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
himmelma@31276
  1511
  using closest_point_exists[of s] by auto
himmelma@31276
  1512
himmelma@31276
  1513
lemma closest_point_self:
himmelma@31276
  1514
  assumes "x \<in> s"  shows "closest_point s x = x"
himmelma@31276
  1515
  unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
himmelma@31276
  1516
  using assms by auto
himmelma@31276
  1517
himmelma@31276
  1518
lemma closest_point_refl:
himmelma@31276
  1519
 "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
himmelma@31276
  1520
  using closest_point_in_set[of s x] closest_point_self[of x s] by auto
himmelma@31276
  1521
himmelma@31276
  1522
lemma closer_points_lemma: fixes y::"real^'n::finite"
himmelma@31276
  1523
  assumes "y \<bullet> z > 0"
himmelma@31276
  1524
  shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *s z - y) < norm y"
himmelma@31276
  1525
proof- have z:"z \<bullet> z > 0" unfolding dot_pos_lt using assms by auto
himmelma@31276
  1526
  thus ?thesis using assms apply(rule_tac x="(y \<bullet> z) / (z \<bullet> z)" in exI) apply(rule) defer proof(rule+)
himmelma@31276
  1527
    fix v assume "0<v" "v \<le> y \<bullet> z / (z \<bullet> z)"
himmelma@31276
  1528
    thus "norm (v *s z - y) < norm y" unfolding norm_lt using z and assms
himmelma@31276
  1529
      by (simp add: field_simps dot_sym  mult_strict_left_mono[OF _ `0<v`])
himmelma@31276
  1530
  qed(rule divide_pos_pos, auto) qed
himmelma@31276
  1531
himmelma@31276
  1532
lemma closer_point_lemma:
huffman@31285
  1533
  fixes x y z :: "real ^ 'n::finite"
himmelma@31276
  1534
  assumes "(y - x) \<bullet> (z - x) > 0"
himmelma@31276
  1535
  shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *s (z - x)) y < dist x y"
himmelma@31276
  1536
proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *s (z - x) - (y - x)) < norm (y - x)"
himmelma@31276
  1537
    using closer_points_lemma[OF assms] by auto
himmelma@31276
  1538
  show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
huffman@31289
  1539
    unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
himmelma@31276
  1540
himmelma@31276
  1541
lemma any_closest_point_dot:
himmelma@31276
  1542
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
himmelma@31276
  1543
  shows "(a - x) \<bullet> (y - x) \<le> 0"
himmelma@31276
  1544
proof(rule ccontr) assume "\<not> (a - x) \<bullet> (y - x) \<le> 0"
himmelma@31276
  1545
  then obtain u where u:"u>0" "u\<le>1" "dist (x + u *s (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
himmelma@31276
  1546
  let ?z = "(1 - u) *s x + u *s y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
huffman@31285
  1547
  thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute field_simps) qed
himmelma@31276
  1548
himmelma@31276
  1549
lemma any_closest_point_unique:
himmelma@31276
  1550
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
himmelma@31276
  1551
  "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
himmelma@31276
  1552
  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@31276
  1553
  unfolding norm_pths(1) and norm_le_square by auto
himmelma@31276
  1554
himmelma@31276
  1555
lemma closest_point_unique:
himmelma@31276
  1556
  assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
himmelma@31276
  1557
  shows "x = closest_point s a"
himmelma@31276
  1558
  using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
himmelma@31276
  1559
  using closest_point_exists[OF assms(2)] and assms(3) by auto
himmelma@31276
  1560
himmelma@31276
  1561
lemma closest_point_dot:
himmelma@31276
  1562
  assumes "convex s" "closed s" "x \<in> s"
himmelma@31276
  1563
  shows "(a - closest_point s a) \<bullet> (x - closest_point s a) \<le> 0"
himmelma@31276
  1564
  apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
himmelma@31276
  1565
  using closest_point_exists[OF assms(2)] and assms(3) by auto
himmelma@31276
  1566
himmelma@31276
  1567
lemma closest_point_lt:
himmelma@31276
  1568
  assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
himmelma@31276
  1569
  shows "dist a (closest_point s a) < dist a x"
himmelma@31276
  1570
  apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
himmelma@31276
  1571
  apply(rule closest_point_unique[OF assms(1-3), of a])
himmelma@31276
  1572
  using closest_point_le[OF assms(2), of _ a] by fastsimp
himmelma@31276
  1573
himmelma@31276
  1574
lemma closest_point_lipschitz:
himmelma@31276
  1575
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@31276
  1576
  shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
himmelma@31276
  1577
proof-
himmelma@31276
  1578
  have "(x - closest_point s x) \<bullet> (closest_point s y - closest_point s x) \<le> 0"
himmelma@31276
  1579
       "(y - closest_point s y) \<bullet> (closest_point s x - closest_point s y) \<le> 0"
himmelma@31276
  1580
    apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
himmelma@31276
  1581
    using closest_point_exists[OF assms(2-3)] by auto
huffman@31289
  1582
  thus ?thesis unfolding dist_norm and norm_le
himmelma@31276
  1583
    using dot_pos_le[of "(x - closest_point s x) - (y - closest_point s y)"]
himmelma@31276
  1584
    by (auto simp add: dot_sym dot_ladd dot_radd) qed
himmelma@31276
  1585
himmelma@31276
  1586
lemma continuous_at_closest_point:
himmelma@31276
  1587
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@31276
  1588
  shows "continuous (at x) (closest_point s)"
himmelma@31276
  1589
  unfolding continuous_at_eps_delta 
himmelma@31276
  1590
  using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
himmelma@31276
  1591
himmelma@31276
  1592
lemma continuous_on_closest_point:
himmelma@31276
  1593
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@31276
  1594
  shows "continuous_on t (closest_point s)"
himmelma@31276
  1595
  apply(rule continuous_at_imp_continuous_on) using continuous_at_closest_point[OF assms] by auto
himmelma@31276
  1596
himmelma@31276
  1597
subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
himmelma@31276
  1598
himmelma@31276
  1599
lemma supporting_hyperplane_closed_point:
himmelma@31276
  1600
  assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
himmelma@31276
  1601
  shows "\<exists>a b. \<exists>y\<in>s. a \<bullet> z < b \<and> (a \<bullet> y = b) \<and> (\<forall>x\<in>s. a \<bullet> x \<ge> b)"
himmelma@31276
  1602
proof-
himmelma@31276
  1603
  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@31276
  1604
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="(y - z) \<bullet> y" in exI, rule_tac x=y in bexI)
himmelma@31276
  1605
    apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
himmelma@31276
  1606
    show "(y - z) \<bullet> z < (y - z) \<bullet> y" apply(subst diff_less_iff(1)[THEN sym])
himmelma@31276
  1607
      unfolding dot_rsub[THEN sym] and dot_pos_lt using `y\<in>s` `z\<notin>s` by auto
himmelma@31276
  1608
  next
himmelma@31276
  1609
    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) *s y + u *s x)"
himmelma@31276
  1610
      using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
himmelma@31276
  1611
    assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x" then obtain v where
himmelma@31276
  1612
      "v>0" "v\<le>1" "dist (y + v *s (x - y)) z < dist y z" using closer_point_lemma[of z y x] by auto
huffman@31285
  1613
    thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute field_simps)
himmelma@31276
  1614
  qed auto
himmelma@31276
  1615
qed
himmelma@31276
  1616
himmelma@31276
  1617
lemma separating_hyperplane_closed_point:
himmelma@31276
  1618
  assumes "convex s" "closed s" "z \<notin> s"
himmelma@31276
  1619
  shows "\<exists>a b. a \<bullet> z < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
himmelma@31276
  1620
proof(cases "s={}")
himmelma@31276
  1621
  case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
himmelma@31276
  1622
    using less_le_trans[OF _ dot_pos_le[of z]] by auto
himmelma@31276
  1623
next
himmelma@31276
  1624
  case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
himmelma@31276
  1625
    using distance_attains_inf[OF assms(2) False] by auto
himmelma@31276
  1626
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="(y - z) \<bullet> z + (norm(y - z))\<twosuperior> / 2" in exI)
himmelma@31276
  1627
    apply rule defer apply rule proof-
himmelma@31276
  1628
    fix x assume "x\<in>s"
himmelma@31276
  1629
    have "\<not> 0 < (z - y) \<bullet> (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
himmelma@31276
  1630
      assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *s (x - y)) z < dist y z"
himmelma@31276
  1631
      then obtain u where "u>0" "u\<le>1" "dist (y + u *s (x - y)) z < dist y z" by auto
himmelma@31276
  1632
      thus False using y[THEN bspec[where x="y + u *s (x - y)"]]
himmelma@31276
  1633
	using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
huffman@31285
  1634
	using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute field_simps) qed
himmelma@31276
  1635
    moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
himmelma@31276
  1636
    hence "0 < (y - z) \<bullet> (y - z)" unfolding norm_pow_2 by simp
himmelma@31276
  1637
    ultimately show "(y - z) \<bullet> z + (norm (y - z))\<twosuperior> / 2 < (y - z) \<bullet> x"
himmelma@31276
  1638
      unfolding norm_pow_2 and dlo_simps(3) by (auto simp add: field_simps dot_sym)
himmelma@31276
  1639
  qed(insert `y\<in>s` `z\<notin>s`, auto)
himmelma@31276
  1640
qed
himmelma@31276
  1641
himmelma@31276
  1642
lemma separating_hyperplane_closed_0:
himmelma@31276
  1643
  assumes "convex (s::(real^'n::finite) set)" "closed s" "0 \<notin> s"
himmelma@31276
  1644
  shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
himmelma@31276
  1645
  proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
himmelma@31276
  1646
  case True have "norm ((basis a)::real^'n::finite) = 1" 
himmelma@31276
  1647
    using norm_basis and dimindex_ge_1 by auto
himmelma@31276
  1648
  thus ?thesis apply(rule_tac x="basis a" in exI, rule_tac x=1 in exI) using True by auto
himmelma@31276
  1649
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
himmelma@31276
  1650
    apply - apply(erule exE)+ unfolding dot_rzero apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
himmelma@31276
  1651
himmelma@31276
  1652
subsection {* Now set-to-set for closed/compact sets. *}
himmelma@31276
  1653
himmelma@31276
  1654
lemma separating_hyperplane_closed_compact:
himmelma@31276
  1655
  assumes "convex (s::(real^'n::finite) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
himmelma@31276
  1656
  shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
himmelma@31276
  1657
proof(cases "s={}")
himmelma@31276
  1658
  case True
himmelma@31276
  1659
  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
himmelma@31276
  1660
  obtain z::"real^'n" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
himmelma@31276
  1661
  hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
himmelma@31276
  1662
  then obtain a b where ab:"a \<bullet> z < b" "\<forall>x\<in>t. b < a \<bullet> x"
himmelma@31276
  1663
    using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
himmelma@31276
  1664
  thus ?thesis using True by auto
himmelma@31276
  1665
next
himmelma@31276
  1666
  case False then obtain y where "y\<in>s" by auto
himmelma@31276
  1667
  obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < a \<bullet> x"
himmelma@31276
  1668
    using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
himmelma@31276
  1669
    using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
himmelma@31276
  1670
  hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + a \<bullet> y < a \<bullet> x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by auto
himmelma@31276
  1671
  def k \<equiv> "rsup ((\<lambda>x. a \<bullet> x) ` t)"
himmelma@31276
  1672
  show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
himmelma@31276
  1673
    apply(rule,rule) defer apply(rule) unfolding dot_lneg and neg_less_iff_less proof-
himmelma@31276
  1674
    from ab have "((\<lambda>x. a \<bullet> x) ` t) *<= (a \<bullet> y - b)"
himmelma@31276
  1675
      apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
himmelma@31276
  1676
    hence k:"isLub UNIV ((\<lambda>x. a \<bullet> x) ` t) k" unfolding k_def apply(rule_tac rsup) using assms(5) by auto
himmelma@31276
  1677
    fix x assume "x\<in>t" thus "a \<bullet> x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "a \<bullet> x"] by auto
himmelma@31276
  1678
  next
himmelma@31276
  1679
    fix x assume "x\<in>s" 
himmelma@31276
  1680
    hence "k \<le> a \<bullet> x - b" unfolding k_def apply(rule_tac rsup_le) using assms(5)
himmelma@31276
  1681
      unfolding setle_def
himmelma@31276
  1682
      using ab[THEN bspec[where x=x]] by auto
himmelma@31276
  1683
    thus "k + b / 2 < a \<bullet> x" using `0 < b` by auto
himmelma@31276
  1684
  qed
himmelma@31276
  1685
qed
himmelma@31276
  1686
himmelma@31276
  1687
lemma separating_hyperplane_compact_closed:
himmelma@31276
  1688
  assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
himmelma@31276
  1689
  shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
himmelma@31276
  1690
proof- obtain a b where "(\<forall>x\<in>t. a \<bullet> x < b) \<and> (\<forall>x\<in>s. b < a \<bullet> x)"
himmelma@31276
  1691
    using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
himmelma@31276
  1692
  thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
himmelma@31276
  1693
himmelma@31276
  1694
subsection {* General case without assuming closure and getting non-strict separation. *}
himmelma@31276
  1695
himmelma@31276
  1696
lemma separating_hyperplane_set_0:
himmelma@31276
  1697
  assumes "convex s" "(0::real^'n::finite) \<notin> s"
himmelma@31276
  1698
  shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> a \<bullet> x)"
himmelma@31276
  1699
proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> c \<bullet> x}"
himmelma@31276
  1700
  have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
himmelma@31276
  1701
    apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
himmelma@31276
  1702
    defer apply(rule,rule,erule conjE) proof-
himmelma@31276
  1703
    fix f assume as:"f \<subseteq> ?k ` s" "finite f"
himmelma@31518
  1704
    obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
himmelma@31276
  1705
    then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < a \<bullet> x"
himmelma@31276
  1706
      using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
himmelma@31276
  1707
      using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
himmelma@31276
  1708
      using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
himmelma@31276
  1709
    hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> y \<bullet> x)" apply(rule_tac x="inverse(norm a) *s a" in exI)
himmelma@31276
  1710
       using hull_subset[of c convex] unfolding subset_eq and dot_rmult
himmelma@31276
  1711
       apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
himmelma@31276
  1712
       by(auto simp add: dot_sym elim!: ballE) 
huffman@31289
  1713
    thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
himmelma@31276
  1714
  qed(insert closed_halfspace_ge, auto)
huffman@31289
  1715
  then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
himmelma@31276
  1716
  thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: dot_sym) qed
himmelma@31276
  1717
himmelma@31276
  1718
lemma separating_hyperplane_sets:
himmelma@31276
  1719
  assumes "convex s" "convex (t::(real^'n::finite) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
himmelma@31276
  1720
  shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. a \<bullet> x \<le> b) \<and> (\<forall>x\<in>t. a \<bullet> x \<ge> b)"
himmelma@31276
  1721
proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
himmelma@31276
  1722
  obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> a \<bullet> x"  using assms(3-5) by auto 
himmelma@31276
  1723
  hence "\<forall>x\<in>t. \<forall>y\<in>s. a \<bullet> y \<le> a \<bullet> x" apply- apply(rule, rule) apply(erule_tac x="x - y" in ballE) by auto
himmelma@31276
  1724
  thus ?thesis apply(rule_tac x=a in exI, rule_tac x="rsup ((\<lambda>x. a \<bullet> x) ` s)" in exI) using `a\<noteq>0`
himmelma@31276
  1725
    apply(rule) apply(rule,rule) apply(rule rsup[THEN isLubD2]) prefer 4 apply(rule,rule rsup_le) unfolding setle_def
himmelma@31276
  1726
    prefer 4 using assms(3-5) by blast+ qed
himmelma@31276
  1727
himmelma@31276
  1728
subsection {* More convexity generalities. *}
himmelma@31276
  1729
himmelma@31276
  1730
lemma convex_closure: assumes "convex s" shows "convex(closure s)"
himmelma@31276
  1731
  unfolding convex_def Ball_def closure_sequential
himmelma@31276
  1732
  apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
himmelma@31276
  1733
  apply(rule_tac x="\<lambda>n. u *s xb n + v *s xc n" in exI) apply(rule,rule)
himmelma@31276
  1734
  apply(rule assms[unfolded convex_def, rule_format]) prefer 6
himmelma@31276
  1735
  apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
himmelma@31276
  1736
himmelma@31276
  1737
lemma convex_interior: assumes "convex s" shows "convex(interior s)"
himmelma@31276
  1738
  unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
himmelma@31276
  1739
  fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
himmelma@31276
  1740
  fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
himmelma@31276
  1741
  show "\<exists>e>0. ball ((1 - u) *s x + u *s y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
himmelma@31276
  1742
    apply rule unfolding subset_eq defer apply rule proof-
himmelma@31276
  1743
    fix z assume "z \<in> ball ((1 - u) *s x + u *s y) (min d e)"
himmelma@31276
  1744
    hence "(1- u) *s (z - u *s (y - x)) + u *s (z + (1 - u) *s (y - x)) \<in> s"
himmelma@31276
  1745
      apply(rule_tac assms[unfolded convex_alt, rule_format])
huffman@31289
  1746
      using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: ring_simps)
himmelma@31276
  1747
    thus "z \<in> s" using u by (auto simp add: ring_simps) qed(insert u ed(3-4), auto) qed
himmelma@31276
  1748
himmelma@31276
  1749
lemma convex_hull_eq_empty: "convex hull s = {} \<longleftrightarrow> s = {}"
himmelma@31276
  1750
  using hull_subset[of s convex] convex_hull_empty by auto
himmelma@31276
  1751
himmelma@31276
  1752
subsection {* Moving and scaling convex hulls. *}
himmelma@31276
  1753
himmelma@31276
  1754
lemma convex_hull_translation_lemma:
himmelma@31276
  1755
  "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
himmelma@31276
  1756
  apply(rule hull_minimal, rule image_mono, rule hull_subset) unfolding mem_def
himmelma@31276
  1757
  using convex_translation[OF convex_convex_hull, of a s] by assumption
himmelma@31276
  1758
himmelma@31276
  1759
lemma convex_hull_bilemma: fixes neg
himmelma@31276
  1760
  assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
himmelma@31276
  1761
  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@31276
  1762
  \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
himmelma@31276
  1763
  using assms by(metis subset_antisym) 
himmelma@31276
  1764
himmelma@31276
  1765
lemma convex_hull_translation:
himmelma@31276
  1766
  "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
himmelma@31276
  1767
  apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
himmelma@31276
  1768
himmelma@31276
  1769
lemma convex_hull_scaling_lemma:
himmelma@31276
  1770
 "(convex hull ((\<lambda>x. c *s x) ` s)) \<subseteq> (\<lambda>x. c *s x) ` (convex hull s)"
himmelma@31276
  1771
  apply(rule hull_minimal, rule image_mono, rule hull_subset)
himmelma@31276
  1772
  unfolding mem_def by(rule convex_scaling, rule convex_convex_hull)
himmelma@31276
  1773
himmelma@31276
  1774
lemma convex_hull_scaling:
himmelma@31276
  1775
  "convex hull ((\<lambda>x. c *s x) ` s) = (\<lambda>x. c *s x) ` (convex hull s)"
himmelma@31276
  1776
  apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
himmelma@31276
  1777
  unfolding image_image vector_smult_assoc by(auto simp add:image_constant_conv convex_hull_eq_empty)
himmelma@31276
  1778
himmelma@31276
  1779
lemma convex_hull_affinity:
himmelma@31276
  1780
  "convex hull ((\<lambda>x. a + c *s x) ` s) = (\<lambda>x. a + c *s x) ` (convex hull s)"
himmelma@31276
  1781
  unfolding image_image[THEN sym] convex_hull_scaling convex_hull_translation  ..
himmelma@31276
  1782
himmelma@31276
  1783
subsection {* Convex set as intersection of halfspaces. *}
himmelma@31276
  1784
himmelma@31276
  1785
lemma convex_halfspace_intersection:
himmelma@31276
  1786
  assumes "closed s" "convex s"
himmelma@31276
  1787
  shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. a \<bullet> x \<le> b})}"
himmelma@31276
  1788
  apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
himmelma@31276
  1789
  fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. a \<bullet> x \<le> b}) \<longrightarrow> x \<in> xa"
himmelma@31276
  1790
  hence "\<forall>a b. s \<subseteq> {x. a \<bullet> x \<le> b} \<longrightarrow> x \<in> {x. a \<bullet> x \<le> b}" by blast
himmelma@31276
  1791
  thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
himmelma@31276
  1792
    apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
himmelma@31276
  1793
qed auto
himmelma@31276
  1794
himmelma@31276
  1795
subsection {* Radon's theorem (from Lars Schewe). *}
himmelma@31276
  1796
himmelma@31276
  1797
lemma radon_ex_lemma:
himmelma@31276
  1798
  assumes "finite c" "affine_dependent c"
himmelma@31276
  1799
  shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) c = 0"
himmelma@31276
  1800
proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
himmelma@31276
  1801
  thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult vector_smult_lzero
himmelma@31276
  1802
    and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
himmelma@31276
  1803
himmelma@31276
  1804
lemma radon_s_lemma:
himmelma@31276
  1805
  assumes "finite s" "setsum f s = (0::real)"
himmelma@31276
  1806
  shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
himmelma@31276
  1807
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@31276
  1808
  show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
himmelma@31276
  1809
    using assms(2) by assumption qed
himmelma@31276
  1810
himmelma@31276
  1811
lemma radon_v_lemma:
himmelma@31276
  1812
  assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::real^'n)"
himmelma@31276
  1813
  shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
himmelma@31276
  1814
proof-
himmelma@31276
  1815
  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@31276
  1816
  show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
himmelma@31276
  1817
    using assms(2) by assumption qed
himmelma@31276
  1818
himmelma@31276
  1819
lemma radon_partition:
himmelma@31276
  1820
  assumes "finite c" "affine_dependent c"
himmelma@31276
  1821
  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
himmelma@31276
  1822
  obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *s v) = 0" using radon_ex_lemma[OF assms] by auto
himmelma@31276
  1823
  have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
himmelma@31276
  1824
  def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *s setsum (\<lambda>x. u x *s x) {x\<in>c. u x > 0}"
himmelma@31276
  1825
  have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
himmelma@31276
  1826
    case False hence "u v < 0" by auto
himmelma@31276
  1827
    thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
himmelma@31276
  1828
      case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
himmelma@31276
  1829
    next
himmelma@31276
  1830
      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@31276
  1831
      thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
himmelma@31276
  1832
  qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
himmelma@31276
  1833
himmelma@31276
  1834
  hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding real_less_def apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
himmelma@31276
  1835
  moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
himmelma@31276
  1836
    "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *s x) = (\<Sum>x\<in>c. u x *s x)"
himmelma@31276
  1837
    using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
himmelma@31276
  1838
  hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
himmelma@31276
  1839
   "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *s x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *s x)" 
himmelma@31276
  1840
    unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
himmelma@31276
  1841
  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@31276
  1842
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
himmelma@31276
  1843
himmelma@31276
  1844
  ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
himmelma@31276
  1845
    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@31276
  1846
    using assms(1) unfolding vector_smult_assoc[THEN sym] setsum_cmul and z_def
himmelma@31276
  1847
    by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
himmelma@31276
  1848
  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@31276
  1849
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
himmelma@31276
  1850
  hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
himmelma@31276
  1851
    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@31276
  1852
    using assms(1) unfolding vector_smult_assoc[THEN sym] setsum_cmul and z_def using *
himmelma@31276
  1853
    by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
himmelma@31276
  1854
  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@31276
  1855
qed
himmelma@31276
  1856
himmelma@31276
  1857
lemma radon: assumes "affine_dependent c"
himmelma@31276
  1858
  obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
himmelma@31276
  1859
proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
himmelma@31276
  1860
  hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
himmelma@31276
  1861
  from radon_partition[OF *] guess m .. then guess p ..
himmelma@31276
  1862
  thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
himmelma@31276
  1863
himmelma@31276
  1864
subsection {* Helly's theorem. *}
himmelma@31276
  1865
himmelma@31276
  1866
lemma helly_induct: fixes f::"(real^'n::finite) set set"
himmelma@31276
  1867
  assumes "f hassize n" "n \<ge> CARD('n) + 1"
himmelma@31276
  1868
  "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
himmelma@31276
  1869
  shows "\<Inter> f \<noteq> {}"
himmelma@31276
  1870
  using assms unfolding hassize_def apply(erule_tac conjE) proof(induct n arbitrary: f)
himmelma@31276
  1871
case (Suc n)
himmelma@31276
  1872
show "\<Inter> f \<noteq> {}" apply(cases "n = CARD('n)") apply(rule Suc(4)[rule_format])
himmelma@31276
  1873
  unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6) proof-
himmelma@31276
  1874
  assume ng:"n \<noteq> CARD('n)" hence "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" apply(rule_tac bchoice) unfolding ex_in_conv
himmelma@31276
  1875
    apply(rule, rule Suc(1)[rule_format])  unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6)
himmelma@31276
  1876
    defer apply(rule Suc(3)[rule_format]) defer apply(rule Suc(4)[rule_format]) using Suc(2,5) by auto
himmelma@31276
  1877
  then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
himmelma@31276
  1878
  show ?thesis proof(cases "inj_on X f")
himmelma@31276
  1879
    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@31276
  1880
    hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
himmelma@31276
  1881
    show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
himmelma@31276
  1882
      apply(rule, rule X[rule_format]) using X st by auto
himmelma@31276
  1883
  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@31276
  1884
      using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
himmelma@31276
  1885
      unfolding card_image[OF True] and Suc(6) using Suc(2,5) and ng by auto
himmelma@31276
  1886
    have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
himmelma@31276
  1887
    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@31276
  1888
    hence "f \<union> (g \<union> h) = f" by auto
himmelma@31518
  1889
    hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
himmelma@31276
  1890
      unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
himmelma@31276
  1891
    have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
himmelma@31276
  1892
    have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
himmelma@31276
  1893
      apply(rule_tac [!] hull_minimal) using Suc(3) gh(3-4)  unfolding mem_def unfolding subset_eq
himmelma@31276
  1894
      apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
himmelma@31276
  1895
      fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
himmelma@31276
  1896
      thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
himmelma@31276
  1897
      fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
himmelma@31276
  1898
      thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
himmelma@31276
  1899
    qed(auto)
himmelma@31276
  1900
    thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
himmelma@31276
  1901
qed(insert dimindex_ge_1, auto) qed(auto)
himmelma@31276
  1902
himmelma@31276
  1903
lemma helly: fixes f::"(real^'n::finite) set set"
himmelma@31276
  1904
  assumes "finite f" "card f \<ge> CARD('n) + 1" "\<forall>s\<in>f. convex s"
himmelma@31276
  1905
          "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
himmelma@31276
  1906
  shows "\<Inter> f \<noteq>{}"
himmelma@31276
  1907
  apply(rule helly_induct) unfolding hassize_def using assms by auto
himmelma@31276
  1908
himmelma@31276
  1909
subsection {* Convex hull is "preserved" by a linear function. *}
himmelma@31276
  1910
himmelma@31276
  1911
lemma convex_hull_linear_image:
himmelma@31276
  1912
  assumes "linear f"
himmelma@31276
  1913
  shows "f ` (convex hull s) = convex hull (f ` s)"
himmelma@31276
  1914
  apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
himmelma@31276
  1915
  apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
himmelma@31276
  1916
  apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
himmelma@31276
  1917
proof- show "convex {x. f x \<in> convex hull f ` s}" 
himmelma@31276
  1918
  unfolding convex_def by(auto simp add: linear_cmul[OF assms]  linear_add[OF assms]
himmelma@31276
  1919
    convex_convex_hull[unfolded convex_def, rule_format]) next
himmelma@31276
  1920
  show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
himmelma@31276
  1921
    unfolding convex_def by (auto simp add: linear_cmul[OF assms, THEN sym]  linear_add[OF assms, THEN sym])
himmelma@31276
  1922
qed auto
himmelma@31276
  1923
himmelma@31276
  1924
lemma in_convex_hull_linear_image:
himmelma@31276
  1925
  assumes "linear f" "x \<in> convex hull s" shows "(f x) \<in> convex hull (f ` s)"
himmelma@31276
  1926
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
himmelma@31276
  1927
himmelma@31276
  1928
subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
himmelma@31276
  1929
himmelma@31276
  1930
lemma compact_frontier_line_lemma:
huffman@31401
  1931
  fixes s :: "(real ^ _) set"
himmelma@31276
  1932
  assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
himmelma@31276
  1933
  obtains u where "0 \<le> u" "(u *s x) \<in> frontier s" "\<forall>v>u. (v *s x) \<notin> s"
himmelma@31276
  1934
proof-
himmelma@31276
  1935
  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@31276
  1936
  let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *s x)}"
himmelma@31276
  1937
  have A:"?A = (\<lambda>u. dest_vec1 u *s x) ` {0 .. vec1 (b / norm x)}"
himmelma@31276
  1938
    unfolding image_image[of "\<lambda>u. u *s x" "\<lambda>x. dest_vec1 x", THEN sym]
himmelma@31276
  1939
    unfolding dest_vec1_inverval vec1_dest_vec1 by auto
himmelma@31276
  1940
  have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
huffman@31558
  1941
    apply(rule, rule continuous_vmul)
huffman@31558
  1942
    apply (rule continuous_dest_vec1)
huffman@31558
  1943
    apply(rule continuous_at_id) by(rule compact_interval)
himmelma@31276
  1944
  moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *s x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
himmelma@31276
  1945
    unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
himmelma@31276
  1946
  ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *s x"
himmelma@31276
  1947
    "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@31276
  1948
himmelma@31276
  1949
  have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
himmelma@31276
  1950
  { fix v assume as:"v > u" "v *s x \<in> s"
himmelma@31276
  1951
    hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
himmelma@31276
  1952
      using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] and norm_mul by auto
himmelma@31276
  1953
    hence "norm (v *s x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
himmelma@31276
  1954
      apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
himmelma@31276
  1955
      using as(1) `u\<ge>0` by(auto simp add:field_simps) 
himmelma@31276
  1956
    hence False unfolding obt(3) unfolding norm_mul using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
himmelma@31276
  1957
  } note u_max = this
himmelma@31276
  1958
himmelma@31276
  1959
  have "u *s x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *s x" in bexI) unfolding obt(3)[THEN sym]
himmelma@31276
  1960
    prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *s x" in exI) apply(rule, rule) proof-
himmelma@31276
  1961
    fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *s x \<in> s"
himmelma@31276
  1962
    hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
himmelma@31276
  1963
    thus False using u_max[OF _ as] by auto
huffman@31289
  1964
  qed(insert `y\<in>s`, auto simp add: dist_norm obt(3))
himmelma@31276
  1965
  thus ?thesis apply(rule_tac that[of u]) apply(rule obt(1), assumption)
himmelma@31276
  1966
    apply(rule,rule,rule ccontr) apply(rule u_max) by auto qed
himmelma@31276
  1967
himmelma@31276
  1968
lemma starlike_compact_projective:
himmelma@31276
  1969
  assumes "compact s" "cball (0::real^'n::finite) 1 \<subseteq> s "
himmelma@31276
  1970
  "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *s x) \<in> (s - frontier s )"
himmelma@31276
  1971
  shows "s homeomorphic (cball (0::real^'n::finite) 1)"
himmelma@31276
  1972
proof-
himmelma@31276
  1973
  have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
himmelma@31276
  1974
  def pi \<equiv> "\<lambda>x::real^'n. inverse (norm x) *s x"
himmelma@31276
  1975
  have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
himmelma@31276
  1976
    using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
himmelma@31276
  1977
  have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
himmelma@31276
  1978
himmelma@31276
  1979
  have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
huffman@31558
  1980
    apply rule unfolding pi_def
huffman@31558
  1981
    apply (rule continuous_mul)
huffman@31558
  1982
    apply (rule continuous_at_inv[unfolded o_def])
huffman@31558
  1983
    apply (rule continuous_at_norm)
huffman@31558
  1984
    apply simp
huffman@31558
  1985
    apply (rule continuous_at_id)
huffman@31558
  1986
    done
himmelma@31276
  1987
  def sphere \<equiv> "{x::real^'n. norm x = 1}"
himmelma@31276
  1988
  have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *s x) = pi x" unfolding pi_def sphere_def by auto
himmelma@31276
  1989
himmelma@31276
  1990
  have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
himmelma@31276
  1991
  have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *s x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
himmelma@31276
  1992
    fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
himmelma@31276
  1993
    hence "x\<noteq>0" using `0\<notin>frontier s` by auto
himmelma@31276
  1994
    obtain v where v:"0 \<le> v" "v *s x \<in> frontier s" "\<forall>w>v. w *s x \<notin> s"
himmelma@31276
  1995
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
himmelma@31276
  1996
    have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
himmelma@31276
  1997
      assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
himmelma@31276
  1998
      assume "v>1" thus False using assms(3)[THEN bspec[where x="v *s x"], THEN spec[where x="inverse v"]]
himmelma@31276
  1999
	using v and x and fs unfolding inverse_less_1_iff by auto qed
himmelma@31276
  2000
    show "u *s x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
himmelma@31276
  2001
      assume "u\<le>1" thus "u *s x \<in> s" apply(cases "u=1")
himmelma@31276
  2002
	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@31276
  2003
himmelma@31276
  2004
  have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
himmelma@31276
  2005
    apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
himmelma@31276
  2006
    apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_ext,rule) 
himmelma@31276
  2007
    unfolding inj_on_def prefer 3 apply(rule,rule,rule)
himmelma@31276
  2008
  proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
himmelma@31276
  2009
    thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
himmelma@31276
  2010
  next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
himmelma@31276
  2011
    then obtain u where "0 \<le> u" "u *s x \<in> frontier s" "\<forall>v>u. v *s x \<notin> s"
himmelma@31276
  2012
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
himmelma@31276
  2013
    thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *s x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
himmelma@31276
  2014
  next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
himmelma@31276
  2015
    hence xys:"x\<in>s" "y\<in>s" using fs by auto
himmelma@31276
  2016
    from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
himmelma@31276
  2017
    from nor have x:"x = norm x *s ((inverse (norm y)) *s y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
himmelma@31276
  2018
    from nor have y:"y = norm y *s ((inverse (norm x)) *s x)" unfolding as(3)[unfolded pi_def] by auto 
himmelma@31276
  2019
    have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
himmelma@31276
  2020
      unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
himmelma@31276
  2021
    hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
himmelma@31276
  2022
      using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
himmelma@31276
  2023
      using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
himmelma@31276
  2024
      using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
himmelma@31276
  2025
    thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
himmelma@31276
  2026
  qed(insert `0 \<notin> frontier s`, auto)
himmelma@31276
  2027
  then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
himmelma@31276
  2028
    "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
himmelma@31276
  2029
  
himmelma@31276
  2030
  have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
himmelma@31276
  2031
    apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
himmelma@31276
  2032
himmelma@31276
  2033
  { fix x assume as:"x \<in> cball (0::real^'n) 1"
himmelma@31276
  2034
    have "norm x *s surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
huffman@31289
  2035
      case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
himmelma@31276
  2036
      thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
himmelma@31276
  2037
	apply(rule_tac fs[unfolded subset_eq, rule_format])
himmelma@31276
  2038
	unfolding surf(5)[THEN sym] by auto
himmelma@31276
  2039
    next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
himmelma@31276
  2040
	unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
himmelma@31276
  2041
himmelma@31276
  2042
  { fix x assume "x\<in>s"
himmelma@31276
  2043
    hence "x \<in> (\<lambda>x. norm x *s surf (pi x)) ` cball 0 1" proof(cases "x=0")
himmelma@31276
  2044
      case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
himmelma@31276
  2045
    next let ?a = "inverse (norm (surf (pi x)))"
himmelma@31276
  2046
      case False hence invn:"inverse (norm x) \<noteq> 0" by auto
himmelma@31276
  2047
      from False have pix:"pi x\<in>sphere" using pi(1) by auto
himmelma@31276
  2048
      hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
himmelma@31276
  2049
      hence **:"norm x *s (?a *s surf (pi x)) = x" apply(rule_tac vector_mul_lcancel_imp[OF invn]) unfolding pi_def by auto
himmelma@31276
  2050
      hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
himmelma@31276
  2051
	apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
himmelma@31276
  2052
      have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
himmelma@31276
  2053
      hence "norm x = norm ((?a * norm x) *s surf (pi x))"
himmelma@31276
  2054
	unfolding norm_mul abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
himmelma@31276
  2055
      moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *s surf (pi x))" 
himmelma@31276
  2056
	unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
himmelma@31276
  2057
      moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
huffman@31289
  2058
      hence "dist 0 (inverse (norm (surf (pi x))) *s x) \<le> 1" unfolding dist_norm
himmelma@31276
  2059
	using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
himmelma@31276
  2060
	using False `x\<in>s` by(auto simp add:field_simps)
himmelma@31276
  2061
      ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *s x" in bexI)
himmelma@31276
  2062
	apply(subst injpi[THEN sym]) unfolding norm_mul abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
himmelma@31276
  2063
	unfolding pi(2)[OF `?a > 0`] by auto
himmelma@31276
  2064
    qed } note hom2 = this
himmelma@31276
  2065
himmelma@31276
  2066
  show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *s surf (pi x)"])
himmelma@31276
  2067
    apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
himmelma@31276
  2068
    prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
himmelma@31276
  2069
    fix x::"real^'n" assume as:"x \<in> cball 0 1"
himmelma@31276
  2070
    thus "continuous (at x) (\<lambda>x. norm x *s surf (pi x))" proof(cases "x=0")
huffman@31558
  2071
      case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
himmelma@31276
  2072
	using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
himmelma@31276
  2073
    next guess a using UNIV_witness[where 'a = 'n] ..
huffman@31533
  2074
      obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
himmelma@31276
  2075
      hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis a" in ballE) defer apply(erule_tac x="basis a" in ballE)
huffman@31289
  2076
	unfolding Ball_def mem_cball dist_norm by (auto simp add: norm_basis[unfolded One_nat_def])
himmelma@31276
  2077
      case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
himmelma@31276
  2078
	apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
huffman@31289
  2079
	unfolding norm_0 vector_smult_lzero dist_norm diff_0_right norm_mul abs_norm_cancel proof-
himmelma@31276
  2080
	fix e and x::"real^'n" assume as:"norm x < e / B" "0 < norm x" "0<e"
himmelma@31276
  2081
	hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
himmelma@31276
  2082
	hence "norm (surf (pi x)) \<le> B" using B fs by auto
himmelma@31276
  2083
	hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
himmelma@31276
  2084
	also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
himmelma@31276
  2085
	also have "\<dots> = e" using `B>0` by auto
himmelma@31276
  2086
	finally show "norm x * norm (surf (pi x)) < e" by assumption
himmelma@31276
  2087
      qed(insert `B>0`, auto) qed
himmelma@31276
  2088
  next { fix x assume as:"surf (pi x) = 0"
himmelma@31276
  2089
      have "x = 0" proof(rule ccontr)
himmelma@31276
  2090
	assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
himmelma@31276
  2091
	hence "surf (pi x) \<in> frontier s" using surf(5) by auto
himmelma@31276
  2092
	thus False using `0\<notin>frontier s` unfolding as by simp qed
himmelma@31276
  2093
    } note surf_0 = this
himmelma@31276
  2094
    show "inj_on (\<lambda>x. norm x *s surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
himmelma@31276
  2095
      fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *s surf (pi x) = norm y *s surf (pi y)"
himmelma@31276
  2096
      thus "x=y" proof(cases "x=0 \<or> y=0") 
himmelma@31276
  2097
	case True thus ?thesis using as by(auto elim: surf_0) next
himmelma@31276
  2098
	case False
himmelma@31276
  2099
	hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
himmelma@31276
  2100
	  using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
himmelma@31276
  2101
	moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
himmelma@31276
  2102
	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@31276
  2103
	moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
himmelma@31276
  2104
	ultimately show ?thesis using injpi by auto qed qed
himmelma@31276
  2105
  qed auto qed
himmelma@31276
  2106
himmelma@31276
  2107
lemma homeomorphic_convex_compact_lemma: fixes s::"(real^'n::finite) set"
himmelma@31276
  2108
  assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
himmelma@31276
  2109
  shows "s homeomorphic (cball (0::real^'n) 1)"
himmelma@31276
  2110
  apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
himmelma@31276
  2111
  fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
himmelma@31276
  2112
  hence "u *s x \<in> interior s" unfolding interior_def mem_Collect_eq
himmelma@31276
  2113
    apply(rule_tac x="ball (u *s x) (1 - u)" in exI) apply(rule, rule open_ball)
himmelma@31276
  2114
    unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
himmelma@31276
  2115
    fix y assume "dist (u *s x) y < 1 - u"
himmelma@31276
  2116
    hence "inverse (1 - u) *s (y - u *s x) \<in> s"
huffman@31289
  2117
      using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
himmelma@31276
  2118
      unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_mul      
himmelma@31276
  2119
      apply (rule mult_left_le_imp_le[of "1 - u"])
himmelma@31276
  2120
      unfolding class_semiring.mul_a using `u<1` by auto
himmelma@31276
  2121
    thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *s (y - u *s x)" x "1 - u" u]
himmelma@31276
  2122
      using as unfolding vector_smult_assoc by auto qed auto
himmelma@31276
  2123
  thus "u *s x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
himmelma@31276
  2124
himmelma@31276
  2125
lemma homeomorphic_convex_compact_cball: fixes e::real and s::"(real^'n::finite) set"
himmelma@31276
  2126
  assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
himmelma@31276
  2127
  shows "s homeomorphic (cball (b::real^'n::finite) e)"
himmelma@31276
  2128
proof- obtain a where "a\<in>interior s" using assms(3) by auto
himmelma@31276
  2129
  then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
himmelma@31276
  2130
  let ?d = "inverse d" and ?n = "0::real^'n"
himmelma@31276
  2131
  have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *s (x - a)) ` s"
himmelma@31276
  2132
    apply(rule, rule_tac x="d *s x + a" in image_eqI) defer
huffman@31289
  2133
    apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
himmelma@31276
  2134
    by(auto simp add: mult_right_le_one_le)
himmelma@31276
  2135
  hence "(\<lambda>x. inverse d *s (x - a)) ` s homeomorphic cball ?n 1"
himmelma@31276
  2136
    using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *s -a + ?d *s x) ` s", OF convex_affinity compact_affinity]
himmelma@31276
  2137
    using assms(1,2) by(auto simp add: uminus_add_conv_diff)
himmelma@31276
  2138
  thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
himmelma@31276
  2139
    apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *s -a"]])
himmelma@31276
  2140
    using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff) qed
himmelma@31276
  2141
himmelma@31276
  2142
lemma homeomorphic_convex_compact: fixes s::"(real^'n::finite) set" and t::"(real^'n) set"
himmelma@31276
  2143
  assumes "convex s" "compact s" "interior s \<noteq> {}"
himmelma@31276
  2144
          "convex t" "compact t" "interior t \<noteq> {}"
himmelma@31276
  2145
  shows "s homeomorphic t"
himmelma@31276
  2146
  using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
himmelma@31276
  2147
himmelma@31276
  2148
subsection {* Epigraphs of convex functions. *}
himmelma@31276
  2149
himmelma@31276
  2150
definition "epigraph s (f::real^'n \<Rightarrow> real) = {xy. fstcart xy \<in> s \<and> f(fstcart xy) \<le> dest_vec1 (sndcart xy)}"
himmelma@31276
  2151
himmelma@31276
  2152
lemma mem_epigraph: "(pastecart x (vec1 y)) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
himmelma@31276
  2153
himmelma@31276
  2154
lemma convex_epigraph: 
himmelma@31276
  2155
  "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
himmelma@31276
  2156
  unfolding convex_def convex_on_def unfolding Ball_def forall_pastecart epigraph_def
himmelma@31276
  2157
  unfolding mem_Collect_eq fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
himmelma@31276
  2158
  unfolding Ball_def[symmetric] unfolding dest_vec1_add dest_vec1_cmul
himmelma@31276
  2159
  apply(subst forall_dest_vec1[THEN sym])+ by(meson real_le_refl real_le_trans add_mono mult_left_mono) 
himmelma@31276
  2160
himmelma@31276
  2161
lemma convex_epigraphI: assumes "convex_on s f" "convex s"
himmelma@31276
  2162
  shows "convex(epigraph s f)" using assms unfolding convex_epigraph by auto
himmelma@31276
  2163
himmelma@31276
  2164
lemma convex_epigraph_convex: "convex s \<Longrightarrow> (convex_on s f \<longleftrightarrow> convex(epigraph s f))"
himmelma@31276
  2165
  using convex_epigraph by auto
himmelma@31276
  2166
himmelma@31276
  2167
subsection {* Use this to derive general bound property of convex function. *}
himmelma@31276
  2168
himmelma@31276
  2169
lemma forall_of_pastecart:
himmelma@31276
  2170
  "(\<forall>p. P (\<lambda>x. fstcart (p x)) (\<lambda>x. sndcart (p x))) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
himmelma@31276
  2171
  apply(erule_tac x="\<lambda>a. pastecart (x a) (y a)" in allE) unfolding o_def by auto
himmelma@31276
  2172
himmelma@31276
  2173
lemma forall_of_pastecart':
himmelma@31276
  2174
  "(\<forall>p. P (fstcart p) (sndcart p)) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
himmelma@31276
  2175
  apply(erule_tac x="pastecart x y" in allE) unfolding o_def by auto
himmelma@31276
  2176
himmelma@31276
  2177
lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
himmelma@31276
  2178
  apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto 
himmelma@31276
  2179
himmelma@31276
  2180
lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
himmelma@31276
  2181
  apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule 
himmelma@31276
  2182
  apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto
himmelma@31276
  2183
himmelma@31276
  2184
lemma convex_on:
himmelma@31276
  2185
  assumes "convex s"
himmelma@31276
  2186
  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@31276
  2187
   f (setsum (\<lambda>i. u i *s x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
himmelma@31276
  2188
  unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
himmelma@31276
  2189
  unfolding sndcart_setsum[OF finite_atLeastAtMost] fstcart_setsum[OF finite_atLeastAtMost] dest_vec1_setsum[OF finite_atLeastAtMost]
himmelma@31276
  2190
  unfolding fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
himmelma@31276
  2191
  unfolding dest_vec1_add dest_vec1_cmul apply(subst forall_of_pastecart)+ apply(subst forall_of_dest_vec1)+ apply rule
himmelma@31276
  2192
  using assms[unfolded convex] apply simp apply(rule,rule,rule)
himmelma@31276
  2193
  apply(erule_tac x=k in allE, erule_tac x=u in allE, erule_tac x=x in allE) apply rule apply rule apply rule defer
himmelma@31276
  2194
  apply(rule_tac j="\<Sum>i = 1..k. u i * f (x i)" in real_le_trans)
himmelma@31276
  2195
  defer apply(rule setsum_mono) apply(erule conjE)+ apply(erule_tac x=i in allE)apply(rule mult_left_mono)
himmelma@31276
  2196
  using assms[unfolded convex] by auto
himmelma@31276
  2197
himmelma@31276
  2198
subsection {* Convexity of general and special intervals. *}
himmelma@31276
  2199
himmelma@31281
  2200
lemma is_interval_convex: assumes "is_interval s" shows "convex s"
himmelma@31276
  2201
  unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
himmelma@31276
  2202
  fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
himmelma@31276
  2203
  hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
himmelma@31276
  2204
  { fix a b assume "\<not> b \<le> u * a + v * b"
himmelma@31276
  2205
    hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
himmelma@31276
  2206
    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@31276
  2207
    hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
himmelma@31276
  2208
  } moreover
himmelma@31276
  2209
  { fix a b assume "\<not> u * a + v * b \<le> a"
himmelma@31276
  2210
    hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
himmelma@31276
  2211
    hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: ring_simps)
himmelma@31276
  2212
    hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
himmelma@31281
  2213
  ultimately show "u *s x + v *s y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
himmelma@31276
  2214
    using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
himmelma@31276
  2215
huffman@31345
  2216
lemma is_interval_connected:
huffman@31345
  2217
  fixes s :: "(real ^ _) set"
huffman@31345
  2218
  shows "is_interval s \<Longrightarrow> connected s"
himmelma@31276
  2219
  using is_interval_convex convex_connected by auto
himmelma@31276
  2220
himmelma@31276
  2221
lemma convex_interval: "convex {a .. b}" "convex {a<..<b::real^'n::finite}"
himmelma@31281
  2222
  apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
himmelma@31276
  2223
berghofe@31360
  2224
subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
himmelma@31276
  2225
himmelma@31276
  2226
lemma is_interval_1:
himmelma@31281
  2227
  "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)"
himmelma@31281
  2228
  unfolding is_interval_def dest_vec1_def forall_1 by auto
himmelma@31281
  2229
himmelma@31281
  2230
lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
himmelma@31276
  2231
  apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
himmelma@31276
  2232
  apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
himmelma@31276
  2233
  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@31276
  2234
  hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
himmelma@31276
  2235
  let ?halfl = "{z. basis 1 \<bullet> z < dest_vec1 x} " and ?halfr = "{z. basis 1 \<bullet> z > dest_vec1 x} "
himmelma@31276
  2236
  { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
himmelma@31276
  2237
    using as(6) `y\<in>s` by (auto simp add: basis_component field_simps dest_vec1_eq[unfolded dest_vec1_def One_nat_def] dest_vec1_def) }
himmelma@31276
  2238
  moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: basis_component field_simps dest_vec1_def) 
himmelma@31276
  2239
  hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
himmelma@31276
  2240
  ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
himmelma@31276
  2241
    apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
himmelma@31276
  2242
    apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt) apply(rule, rule, rule ccontr)
himmelma@31276
  2243
    by(auto simp add: basis_component field_simps) qed 
himmelma@31276
  2244
himmelma@31276
  2245
lemma is_interval_convex_1:
himmelma@31281
  2246
  "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
himmelma@31276
  2247
  using is_interval_convex convex_connected is_interval_connected_1 by auto
himmelma@31276
  2248
himmelma@31276
  2249
lemma convex_connected_1:
himmelma@31276
  2250
  "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
himmelma@31276
  2251
  using is_interval_convex convex_connected is_interval_connected_1 by auto
himmelma@31276
  2252
himmelma@31276
  2253
subsection {* Another intermediate value theorem formulation. *}
himmelma@31276
  2254
himmelma@31276
  2255
lemma ivt_increasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
himmelma@31276
  2256
  assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f a)$k \<le> y" "y \<le> (f b)$k"
himmelma@31276
  2257
  shows "\<exists>x\<in>{a..b}. (f x)$k = y"
himmelma@31276
  2258
proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
himmelma@31276
  2259
    using assms(1) by(auto simp add: vector_less_eq_def dest_vec1_def)
himmelma@31276
  2260
  thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
himmelma@31276
  2261
    using connected_continuous_image[OF assms(2) convex_connected[OF convex_interval(1)]]
himmelma@31276
  2262
    using assms by(auto intro!: imageI) qed
himmelma@31276
  2263
himmelma@31276
  2264
lemma ivt_increasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
himmelma@31276
  2265
  assumes "dest_vec1 a \<le> dest_vec1 b"
himmelma@31276
  2266
  "\<forall>x\<in>{a .. b}. continuous (at x) f" "f a$k \<le> y" "y \<le> f b$k"
himmelma@31276
  2267
  shows "\<exists>x\<in>{a..b}. (f x)$k = y"
himmelma@31276
  2268
  apply(rule ivt_increasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
himmelma@31276
  2269
himmelma@31276
  2270
lemma ivt_decreasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
himmelma@31276
  2271
  assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f b)$k \<le> y" "y \<le> (f a)$k"
himmelma@31276
  2272
  shows "\<exists>x\<in>{a..b}. (f x)$k = y"
himmelma@31276
  2273
  apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
himmelma@31276
  2274
  apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
himmelma@31276
  2275
  by(auto simp add:vector_uminus_component)
himmelma@31276
  2276
himmelma@31276
  2277
lemma ivt_decreasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
himmelma@31276
  2278
  assumes "dest_vec1 a \<le> dest_vec1 b" "\<forall>x\<in>{a .. b}. continuous (at x) f" "f b$k \<le> y" "y \<le> f a$k"
himmelma@31276
  2279
  shows "\<exists>x\<in>{a..b}. (f x)$k = y"
himmelma@31276
  2280
  apply(rule ivt_decreasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
himmelma@31276
  2281
himmelma@31276
  2282
subsection {* A bound within a convex hull, and so an interval. *}
himmelma@31276
  2283
himmelma@31276
  2284
lemma convex_on_convex_hull_bound:
himmelma@31276
  2285
  assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
himmelma@31276
  2286
  shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
himmelma@31276
  2287
  fix x assume "x\<in>convex hull s"
himmelma@31276
  2288
  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 *s v i) = x"
himmelma@31276
  2289
    unfolding convex_hull_indexed mem_Collect_eq by auto
himmelma@31276
  2290
  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@31276
  2291
    unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
himmelma@31276
  2292
    using assms(2) obt(1) by auto
himmelma@31276
  2293
  thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
himmelma@31276
  2294
    unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
himmelma@31276
  2295
himmelma@31276
  2296
lemma unit_interval_convex_hull:
himmelma@31276
  2297
  "{0::real^'n::finite .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
himmelma@31276
  2298
proof- have 01:"{0,1} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
himmelma@31276
  2299
  { fix n x assume "x\<in>{0::real^'n .. 1}" "n \<le> CARD('n)" "card {i. x$i \<noteq> 0} \<le> n" 
himmelma@31276
  2300
  hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
himmelma@31276
  2301
    case 0 hence "x = 0" apply(subst Cart_eq) apply rule by auto
himmelma@31276
  2302
    thus "x\<in>convex hull ?points" using 01 by auto
himmelma@31276
  2303
  next