src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author hoelzl
Thu Mar 04 17:09:44 2010 +0100 (2010-03-04)
changeset 35577 43b93e294522
parent 35542 8f97d8caabfd
child 36071 c8ae8e56d42e
permissions -rw-r--r--
Generalized setsum_cases
himmelma@33175
     1
(*  Title:      HOL/Library/Convex_Euclidean_Space.thy
himmelma@33175
     2
    Author:     Robert Himmelmann, TU Muenchen
himmelma@33175
     3
*)
himmelma@33175
     4
himmelma@33175
     5
header {* Convex sets, functions and related things. *}
himmelma@33175
     6
himmelma@33175
     7
theory Convex_Euclidean_Space
himmelma@33175
     8
imports Topology_Euclidean_Space
himmelma@33175
     9
begin
himmelma@33175
    10
himmelma@33175
    11
himmelma@33175
    12
(* ------------------------------------------------------------------------- *)
himmelma@33175
    13
(* To be moved elsewhere                                                     *)
himmelma@33175
    14
(* ------------------------------------------------------------------------- *)
himmelma@33175
    15
himmelma@33175
    16
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
himmelma@33175
    17
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
himmelma@33175
    18
declare UNIV_1[simp]
himmelma@33175
    19
hoelzl@34964
    20
(*lemma dim1in[intro]:"Suc 0 \<in> {1::nat .. CARD(1)}" by auto*)
hoelzl@34964
    21
hoelzl@34964
    22
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component
himmelma@33175
    23
himmelma@33175
    24
lemma dest_vec1_simps[simp]: fixes a::"real^1"
himmelma@33175
    25
  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"*)
himmelma@33175
    26
  "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
hoelzl@34964
    27
  by(auto simp add:vector_component_simps forall_1 Cart_eq)
himmelma@33175
    28
hoelzl@34291
    29
lemma norm_not_0:"(x::real^'n)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
himmelma@33175
    30
himmelma@33175
    31
lemma setsum_delta_notmem: assumes "x\<notin>s"
himmelma@33175
    32
  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
himmelma@33175
    33
        "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
himmelma@33175
    34
        "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
himmelma@33175
    35
        "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
himmelma@33175
    36
  apply(rule_tac [!] setsum_cong2) using assms by auto
himmelma@33175
    37
himmelma@33175
    38
lemma setsum_delta'':
himmelma@33175
    39
  fixes s::"'a::real_vector set" assumes "finite s"
himmelma@33175
    40
  shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
himmelma@33175
    41
proof-
himmelma@33175
    42
  have *:"\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" by auto
himmelma@33175
    43
  show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
himmelma@33175
    44
qed
himmelma@33175
    45
himmelma@33175
    46
lemma not_disjointI:"x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A \<inter> B \<noteq> {}" by blast
himmelma@33175
    47
himmelma@33175
    48
lemma if_smult:"(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" by auto
himmelma@33175
    49
himmelma@33175
    50
lemma mem_interval_1: fixes x :: "real^1" shows
himmelma@33175
    51
 "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
himmelma@33175
    52
 "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
hoelzl@34964
    53
by(simp_all add: Cart_eq vector_less_def vector_le_def forall_1)
himmelma@33175
    54
hoelzl@34291
    55
lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n)) ` {a..b} =
himmelma@33175
    56
  (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
himmelma@33175
    57
  using image_affinity_interval[of m 0 a b] by auto
himmelma@33175
    58
himmelma@33175
    59
lemma dest_vec1_inverval:
himmelma@33175
    60
  "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
himmelma@33175
    61
  "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
himmelma@33175
    62
  "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
himmelma@33175
    63
  "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
himmelma@33175
    64
  apply(rule_tac [!] equalityI)
himmelma@33175
    65
  unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
himmelma@33175
    66
  apply(rule_tac [!] allI)apply(rule_tac [!] impI)
himmelma@33175
    67
  apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
himmelma@33175
    68
  apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
hoelzl@34964
    69
  by (auto simp add: vector_less_def vector_le_def forall_1
hoelzl@34964
    70
    vec1_dest_vec1[unfolded One_nat_def])
himmelma@33175
    71
himmelma@33175
    72
lemma dest_vec1_setsum: assumes "finite S"
himmelma@33175
    73
  shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
himmelma@33175
    74
  using dest_vec1_sum[OF assms] by auto
himmelma@33175
    75
himmelma@33175
    76
lemma dist_triangle_eq:
himmelma@33175
    77
  fixes x y z :: "real ^ _"
himmelma@33175
    78
  shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
himmelma@33175
    79
proof- have *:"x - y + (y - z) = x - z" by auto
himmelma@33175
    80
  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded smult_conv_scaleR *]
himmelma@33175
    81
    by(auto simp add:norm_minus_commute) qed
himmelma@33175
    82
himmelma@33175
    83
lemma norm_eqI:"x = y \<Longrightarrow> norm x = norm y" by auto 
hoelzl@34291
    84
lemma norm_minus_eqI:"(x::real^'n) = - y \<Longrightarrow> norm x = norm y" by auto
himmelma@33175
    85
himmelma@33175
    86
lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
himmelma@33175
    87
  unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
himmelma@33175
    88
himmelma@33175
    89
lemma dimindex_ge_1:"CARD(_::finite) \<ge> 1"
himmelma@33175
    90
  using one_le_card_finite by auto
himmelma@33175
    91
himmelma@33175
    92
lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1" 
himmelma@33175
    93
  by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff) 
himmelma@33175
    94
himmelma@33175
    95
lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
himmelma@33175
    96
himmelma@33175
    97
subsection {* Affine set and affine hull.*}
himmelma@33175
    98
himmelma@33175
    99
definition
himmelma@33175
   100
  affine :: "'a::real_vector set \<Rightarrow> bool" where
himmelma@33175
   101
  "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
himmelma@33175
   102
himmelma@33175
   103
lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
himmelma@33175
   104
proof- have *:"\<And>u v ::real. u + v = 1 \<longleftrightarrow> v = 1 - u" by auto
himmelma@33175
   105
  { fix x y assume "x\<in>s" "y\<in>s"
himmelma@33175
   106
    hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)" apply auto 
himmelma@33175
   107
      apply(erule_tac[!] x="1 - u" in allE) unfolding * by auto  }
himmelma@33175
   108
  thus ?thesis unfolding affine_def by auto qed
himmelma@33175
   109
himmelma@33175
   110
lemma affine_empty[intro]: "affine {}"
himmelma@33175
   111
  unfolding affine_def by auto
himmelma@33175
   112
himmelma@33175
   113
lemma affine_sing[intro]: "affine {x}"
himmelma@33175
   114
  unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
himmelma@33175
   115
himmelma@33175
   116
lemma affine_UNIV[intro]: "affine UNIV"
himmelma@33175
   117
  unfolding affine_def by auto
himmelma@33175
   118
himmelma@33175
   119
lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
himmelma@33175
   120
  unfolding affine_def by auto 
himmelma@33175
   121
himmelma@33175
   122
lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
himmelma@33175
   123
  unfolding affine_def by auto
himmelma@33175
   124
himmelma@33175
   125
lemma affine_affine_hull: "affine(affine hull s)"
himmelma@33175
   126
  unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
himmelma@33175
   127
  unfolding mem_def by auto
himmelma@33175
   128
himmelma@33175
   129
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
himmelma@33175
   130
proof-
himmelma@33175
   131
  { fix f assume "f \<subseteq> affine"
himmelma@33175
   132
    hence "affine (\<Inter>f)" using affine_Inter[of f] unfolding subset_eq mem_def by auto  }
himmelma@33175
   133
  thus ?thesis using hull_eq[unfolded mem_def, of affine s] by auto
himmelma@33175
   134
qed
himmelma@33175
   135
himmelma@33175
   136
lemma setsum_restrict_set'': assumes "finite A"
himmelma@33175
   137
  shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
himmelma@33175
   138
  unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
himmelma@33175
   139
himmelma@33175
   140
subsection {* Some explicit formulations (from Lars Schewe). *}
himmelma@33175
   141
himmelma@33175
   142
lemma affine: fixes V::"'a::real_vector set"
himmelma@33175
   143
  shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
himmelma@33175
   144
unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
himmelma@33175
   145
defer apply(rule, rule, rule, rule, rule) proof-
himmelma@33175
   146
  fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
himmelma@33175
   147
    "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
himmelma@33175
   148
  thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
himmelma@33175
   149
    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) 
himmelma@33175
   150
    by(auto simp add: scaleR_left_distrib[THEN sym])
himmelma@33175
   151
next
himmelma@33175
   152
  fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
himmelma@33175
   153
    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
himmelma@33175
   154
  def n \<equiv> "card s"
himmelma@33175
   155
  have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
himmelma@33175
   156
  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
himmelma@33175
   157
    assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
himmelma@33175
   158
    then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
himmelma@33175
   159
    thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
himmelma@33175
   160
      by(auto simp add: setsum_clauses(2))
himmelma@33175
   161
  next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
himmelma@33175
   162
      case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
himmelma@33175
   163
      assume IA:"\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
berghofe@34915
   164
               s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
berghofe@34915
   165
        as:"Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
himmelma@33175
   166
           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
himmelma@33175
   167
      have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
himmelma@33175
   168
        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
himmelma@33175
   169
        thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
himmelma@33175
   170
          less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
himmelma@33175
   171
      then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
himmelma@33175
   172
himmelma@33175
   173
      have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
himmelma@33175
   174
      have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
himmelma@33175
   175
      have **:"setsum u (s - {x}) = 1 - u x"
himmelma@33175
   176
        using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
himmelma@33175
   177
      have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
himmelma@33175
   178
      have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
himmelma@33175
   179
        case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
himmelma@33175
   180
          assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
himmelma@33175
   181
          thus False using True by auto qed auto
himmelma@33175
   182
        thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
himmelma@33175
   183
        unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
himmelma@33175
   184
      next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
himmelma@33175
   185
        then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
himmelma@33175
   186
        thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
himmelma@33175
   187
          using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
himmelma@33175
   188
      thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
himmelma@33175
   189
         apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
himmelma@33175
   190
         using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *\<^sub>R (\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa)"], 
himmelma@33175
   191
         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
himmelma@33175
   192
    qed auto
himmelma@33175
   193
  next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
himmelma@33175
   194
    thus ?thesis using as(4,5) by simp
himmelma@33175
   195
  qed(insert `s\<noteq>{}` `finite s`, auto)
himmelma@33175
   196
qed
himmelma@33175
   197
himmelma@33175
   198
lemma affine_hull_explicit:
himmelma@33175
   199
  "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
himmelma@33175
   200
  apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
himmelma@33175
   201
  apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
himmelma@33175
   202
  fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
himmelma@33175
   203
    apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
himmelma@33175
   204
next
himmelma@33175
   205
  fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
himmelma@33175
   206
  thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
himmelma@33175
   207
next
himmelma@33175
   208
  show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}" unfolding affine_def
himmelma@33175
   209
    apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
himmelma@33175
   210
    fix u v ::real assume uv:"u + v = 1"
himmelma@33175
   211
    fix x assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
himmelma@33175
   212
    then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
himmelma@33175
   213
    fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
   214
    then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
himmelma@33175
   215
    have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
himmelma@33175
   216
    have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
himmelma@33175
   217
    show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
himmelma@33175
   218
      apply(rule_tac x="sx \<union> sy" in exI)
himmelma@33175
   219
      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@33175
   220
      unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
himmelma@33175
   221
      unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
himmelma@33175
   222
      unfolding x y using x(1-3) y(1-3) uv by simp qed qed
himmelma@33175
   223
himmelma@33175
   224
lemma affine_hull_finite:
himmelma@33175
   225
  assumes "finite s"
himmelma@33175
   226
  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
himmelma@33175
   227
  unfolding affine_hull_explicit and expand_set_eq and mem_Collect_eq apply (rule,rule)
himmelma@33175
   228
  apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
himmelma@33175
   229
  fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
himmelma@33175
   230
  thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
himmelma@33175
   231
    apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
himmelma@33175
   232
next
himmelma@33175
   233
  fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
himmelma@33175
   234
  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
himmelma@33175
   235
  thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
himmelma@33175
   236
    unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
himmelma@33175
   237
himmelma@33175
   238
subsection {* Stepping theorems and hence small special cases. *}
himmelma@33175
   239
himmelma@33175
   240
lemma affine_hull_empty[simp]: "affine hull {} = {}"
himmelma@33175
   241
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@33175
   242
himmelma@33175
   243
lemma affine_hull_finite_step:
himmelma@33175
   244
  fixes y :: "'a::real_vector"
himmelma@33175
   245
  shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
himmelma@33175
   246
  "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
himmelma@33175
   247
                (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
himmelma@33175
   248
proof-
himmelma@33175
   249
  show ?th1 by simp
himmelma@33175
   250
  assume ?as 
himmelma@33175
   251
  { assume ?lhs
himmelma@33175
   252
    then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
himmelma@33175
   253
    have ?rhs proof(cases "a\<in>s")
himmelma@33175
   254
      case True hence *:"insert a s = s" by auto
himmelma@33175
   255
      show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
himmelma@33175
   256
    next
himmelma@33175
   257
      case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
himmelma@33175
   258
    qed  } moreover
himmelma@33175
   259
  { assume ?rhs
himmelma@33175
   260
    then obtain v u where vu:"setsum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
himmelma@33175
   261
    have *:"\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" by auto
himmelma@33175
   262
    have ?lhs proof(cases "a\<in>s")
himmelma@33175
   263
      case True thus ?thesis
himmelma@33175
   264
        apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
himmelma@33175
   265
        unfolding setsum_clauses(2)[OF `?as`]  apply simp
himmelma@33175
   266
        unfolding scaleR_left_distrib and setsum_addf 
himmelma@33175
   267
        unfolding vu and * and scaleR_zero_left
himmelma@33175
   268
        by (auto simp add: setsum_delta[OF `?as`])
himmelma@33175
   269
    next
himmelma@33175
   270
      case False 
himmelma@33175
   271
      hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
himmelma@33175
   272
               "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
himmelma@33175
   273
      from False show ?thesis
himmelma@33175
   274
        apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
himmelma@33175
   275
        unfolding setsum_clauses(2)[OF `?as`] and * using vu
himmelma@33175
   276
        using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
himmelma@33175
   277
        using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
himmelma@33175
   278
    qed }
himmelma@33175
   279
  ultimately show "?lhs = ?rhs" by blast
himmelma@33175
   280
qed
himmelma@33175
   281
himmelma@33175
   282
lemma affine_hull_2:
himmelma@33175
   283
  fixes a b :: "'a::real_vector"
himmelma@33175
   284
  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
himmelma@33175
   285
proof-
himmelma@33175
   286
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
himmelma@33175
   287
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
himmelma@33175
   288
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
himmelma@33175
   289
    using affine_hull_finite[of "{a,b}"] by auto
himmelma@33175
   290
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
himmelma@33175
   291
    by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
himmelma@33175
   292
  also have "\<dots> = ?rhs" unfolding * by auto
himmelma@33175
   293
  finally show ?thesis by auto
himmelma@33175
   294
qed
himmelma@33175
   295
himmelma@33175
   296
lemma affine_hull_3:
himmelma@33175
   297
  fixes a b c :: "'a::real_vector"
himmelma@33175
   298
  shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
himmelma@33175
   299
proof-
himmelma@33175
   300
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
himmelma@33175
   301
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
himmelma@33175
   302
  show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
himmelma@33175
   303
    unfolding * apply auto
himmelma@33175
   304
    apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
himmelma@33175
   305
    apply(rule_tac x=u in exI) by(auto intro!: exI)
himmelma@33175
   306
qed
himmelma@33175
   307
himmelma@33175
   308
subsection {* Some relations between affine hull and subspaces. *}
himmelma@33175
   309
himmelma@33175
   310
lemma affine_hull_insert_subset_span:
himmelma@33175
   311
  fixes a :: "real ^ _"
himmelma@33175
   312
  shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
himmelma@33175
   313
  unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq smult_conv_scaleR
himmelma@33175
   314
  apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
himmelma@33175
   315
  fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
himmelma@33175
   316
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
himmelma@33175
   317
  thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
himmelma@33175
   318
    apply(rule_tac x="x - a" in exI)
himmelma@33175
   319
    apply (rule conjI, simp)
himmelma@33175
   320
    apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
himmelma@33175
   321
    apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
himmelma@33175
   322
    apply (rule conjI) using as(1) apply simp
himmelma@33175
   323
    apply (erule conjI)
himmelma@33175
   324
    using as(1)
himmelma@33175
   325
    apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib setsum_subtractf scaleR_left.setsum[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
himmelma@33175
   326
    unfolding as by simp qed
himmelma@33175
   327
himmelma@33175
   328
lemma affine_hull_insert_span:
himmelma@33175
   329
  fixes a :: "real ^ _"
himmelma@33175
   330
  assumes "a \<notin> s"
himmelma@33175
   331
  shows "affine hull (insert a s) =
himmelma@33175
   332
            {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
himmelma@33175
   333
  apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
himmelma@33175
   334
  unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
himmelma@33175
   335
  fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
himmelma@33175
   336
  then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y" unfolding span_explicit smult_conv_scaleR by auto
himmelma@33175
   337
  def f \<equiv> "(\<lambda>x. x + a) ` t"
himmelma@33175
   338
  have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt 
himmelma@33175
   339
    by(auto simp add: setsum_reindex[unfolded inj_on_def])
himmelma@33175
   340
  have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
himmelma@33175
   341
  show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
himmelma@33175
   342
    apply(rule_tac x="insert a f" in exI)
himmelma@33175
   343
    apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
himmelma@33175
   344
    using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
hoelzl@35577
   345
    unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
hoelzl@35577
   346
    by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
himmelma@33175
   347
himmelma@33175
   348
lemma affine_hull_span:
himmelma@33175
   349
  fixes a :: "real ^ _"
himmelma@33175
   350
  assumes "a \<in> s"
himmelma@33175
   351
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
himmelma@33175
   352
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
himmelma@33175
   353
himmelma@33175
   354
subsection {* Convexity. *}
himmelma@33175
   355
himmelma@33175
   356
definition
himmelma@33175
   357
  convex :: "'a::real_vector set \<Rightarrow> bool" where
himmelma@33175
   358
  "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
himmelma@33175
   359
himmelma@33175
   360
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) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
himmelma@33175
   361
proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
himmelma@33175
   362
  show ?thesis unfolding convex_def apply auto
himmelma@33175
   363
    apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
himmelma@33175
   364
    by (auto simp add: *) qed
himmelma@33175
   365
himmelma@33175
   366
lemma mem_convex:
himmelma@33175
   367
  assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
himmelma@33175
   368
  shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
himmelma@33175
   369
  using assms unfolding convex_alt by auto
himmelma@33175
   370
hoelzl@33714
   371
lemma convex_vec1:"convex (vec1 ` s) = convex (s::real set)"
hoelzl@33714
   372
  unfolding convex_def Ball_def forall_vec1 unfolding vec1_dest_vec1_simps image_iff by auto
hoelzl@33714
   373
himmelma@33175
   374
lemma convex_empty[intro]: "convex {}"
himmelma@33175
   375
  unfolding convex_def by simp
himmelma@33175
   376
himmelma@33175
   377
lemma convex_singleton[intro]: "convex {a}"
himmelma@33175
   378
  unfolding convex_def by (auto simp add:scaleR_left_distrib[THEN sym])
himmelma@33175
   379
himmelma@33175
   380
lemma convex_UNIV[intro]: "convex UNIV"
himmelma@33175
   381
  unfolding convex_def by auto
himmelma@33175
   382
himmelma@33175
   383
lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
himmelma@33175
   384
  unfolding convex_def by auto
himmelma@33175
   385
himmelma@33175
   386
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
himmelma@33175
   387
  unfolding convex_def by auto
himmelma@33175
   388
himmelma@33175
   389
lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
himmelma@33175
   390
  unfolding convex_def apply auto
himmelma@33175
   391
  unfolding inner_add inner_scaleR
himmelma@33175
   392
  by (metis real_convex_bound_le)
himmelma@33175
   393
himmelma@33175
   394
lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
himmelma@33175
   395
proof- have *:"{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" by auto
himmelma@33175
   396
  show ?thesis apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
himmelma@33175
   397
himmelma@33175
   398
lemma convex_hyperplane: "convex {x. inner a x = b}"
himmelma@33175
   399
proof-
himmelma@33175
   400
  have *:"{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
himmelma@33175
   401
  show ?thesis unfolding * apply(rule convex_Int)
himmelma@33175
   402
    using convex_halfspace_le convex_halfspace_ge by auto
himmelma@33175
   403
qed
himmelma@33175
   404
himmelma@33175
   405
lemma convex_halfspace_lt: "convex {x. inner a x < b}"
himmelma@33175
   406
  unfolding convex_def
himmelma@33175
   407
  by(auto simp add: real_convex_bound_lt inner_add)
himmelma@33175
   408
himmelma@33175
   409
lemma convex_halfspace_gt: "convex {x. inner a x > b}"
himmelma@33175
   410
   using convex_halfspace_lt[of "-a" "-b"] by auto
himmelma@33175
   411
hoelzl@34291
   412
lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
himmelma@33175
   413
  unfolding convex_def apply auto apply(erule_tac x=i in allE)+
himmelma@33175
   414
  apply(rule add_nonneg_nonneg) by(auto simp add: mult_nonneg_nonneg)
himmelma@33175
   415
himmelma@33175
   416
subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
himmelma@33175
   417
himmelma@33175
   418
lemma convex: "convex s \<longleftrightarrow>
himmelma@33175
   419
  (\<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@33175
   420
           \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
himmelma@33175
   421
  unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
himmelma@33175
   422
  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 *\<^sub>R x i) \<in> s"
himmelma@33175
   423
    "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
himmelma@33175
   424
  show "u *\<^sub>R x + v *\<^sub>R 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@33175
   425
    by (auto simp add: setsum_head_Suc) 
himmelma@33175
   426
next
himmelma@33175
   427
  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 *\<^sub>R x + v *\<^sub>R y \<in> s" 
himmelma@33175
   428
  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 *\<^sub>R x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
himmelma@33175
   429
  case (Suc k) show ?case proof(cases "u (Suc k) = 1")
himmelma@33175
   430
    case True hence "(\<Sum>i = Suc 0..k. u i *\<^sub>R x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
himmelma@33175
   431
      fix i assume i:"i \<in> {Suc 0..k}" "u i *\<^sub>R x i \<noteq> 0"
himmelma@33175
   432
      hence ui:"u i \<noteq> 0" by auto
himmelma@33175
   433
      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@33175
   434
      hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta) 
himmelma@33175
   435
      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@33175
   436
      thus False using Suc(3) unfolding setsum_cl_ivl_Suc and True by simp qed
himmelma@33175
   437
    thus ?thesis unfolding setsum_cl_ivl_Suc using True and Suc(2) by auto
himmelma@33175
   438
  next
himmelma@33175
   439
    have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
himmelma@33175
   440
    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@33175
   441
    have ***:"\<And>i k. (u i / (1 - u (Suc k))) *\<^sub>R x i = (inverse (1 - u (Suc k))) *\<^sub>R (u i *\<^sub>R x i)" unfolding real_divide_def by (auto simp add: algebra_simps)
himmelma@33175
   442
    case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
himmelma@33175
   443
    have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
himmelma@33175
   444
      apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
himmelma@33175
   445
    hence "(1 - u (Suc k)) *\<^sub>R (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) + u (Suc k) *\<^sub>R x (Suc k) \<in> s"
himmelma@33175
   446
      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@33175
   447
    thus ?thesis unfolding setsum_cl_ivl_Suc and *** and scaleR_right.setsum [symmetric] using nn by auto qed qed auto qed
himmelma@33175
   448
himmelma@33175
   449
himmelma@33175
   450
lemma convex_explicit:
himmelma@33175
   451
  fixes s :: "'a::real_vector set"
himmelma@33175
   452
  shows "convex s \<longleftrightarrow>
himmelma@33175
   453
  (\<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 *\<^sub>R x) t \<in> s)"
himmelma@33175
   454
  unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
himmelma@33175
   455
  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 *\<^sub>R x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
himmelma@33175
   456
  show "u *\<^sub>R x + v *\<^sub>R y \<in> s" proof(cases "x=y")
himmelma@33175
   457
    case True show ?thesis unfolding True and scaleR_left_distrib[THEN sym] using as(3,6) by auto next
himmelma@33175
   458
    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@33175
   459
next 
himmelma@33175
   460
  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 *\<^sub>R x + v *\<^sub>R y \<in> s" "finite (t::'a set)"
himmelma@33175
   461
  (*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
himmelma@33175
   462
  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 *\<^sub>R x) \<in> s" apply(induct_tac t rule:finite_induct)
himmelma@33175
   463
    prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
himmelma@33175
   464
    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 *\<^sub>R x) \<in> s"
himmelma@33175
   465
    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@33175
   466
    show "(\<Sum>x\<in>insert x f. u x *\<^sub>R x) \<in> s" proof(cases "u x = 1")
himmelma@33175
   467
      case True hence "setsum (\<lambda>x. u x *\<^sub>R x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
himmelma@33175
   468
        fix y assume y:"y \<in> f" "u y *\<^sub>R y \<noteq> 0"
himmelma@33175
   469
        hence uy:"u y \<noteq> 0" by auto
himmelma@33175
   470
        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@33175
   471
        hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta) 
himmelma@33175
   472
        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@33175
   473
        thus False using as(2,5) unfolding setsum_clauses(2)[OF as(1)] and True by auto qed
himmelma@33175
   474
      thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2,3) unfolding True by auto
himmelma@33175
   475
    next
himmelma@33175
   476
      have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
himmelma@33175
   477
      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@33175
   478
        using setsum_nonneg[of f u] and as(4) by auto
himmelma@33175
   479
      case False hence "inverse (1 - u x) *\<^sub>R (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s" unfolding scaleR_right.setsum and scaleR_scaleR
himmelma@33175
   480
        apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
himmelma@33175
   481
        unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
himmelma@33175
   482
      hence "u x *\<^sub>R x + (1 - u x) *\<^sub>R ((inverse (1 - u x)) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) f) \<in>s" 
himmelma@33175
   483
        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@33175
   484
      thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
himmelma@33175
   485
  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 *\<^sub>R x) \<in> s" by auto
himmelma@33175
   486
qed
himmelma@33175
   487
himmelma@33175
   488
lemma convex_finite: assumes "finite s"
himmelma@33175
   489
  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
himmelma@33175
   490
                      \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
himmelma@33175
   491
  unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
himmelma@33175
   492
  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 *\<^sub>R x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
himmelma@33175
   493
  have *:"s \<inter> t = t" using as(3) by auto
himmelma@33175
   494
  show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
hoelzl@35577
   495
    unfolding if_smult and setsum_cases[OF assms] using as(2-) * by auto
himmelma@33175
   496
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
himmelma@33175
   497
himmelma@33175
   498
subsection {* Cones. *}
himmelma@33175
   499
himmelma@33175
   500
definition
himmelma@33175
   501
  cone :: "'a::real_vector set \<Rightarrow> bool" where
himmelma@33175
   502
  "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
himmelma@33175
   503
himmelma@33175
   504
lemma cone_empty[intro, simp]: "cone {}"
himmelma@33175
   505
  unfolding cone_def by auto
himmelma@33175
   506
himmelma@33175
   507
lemma cone_univ[intro, simp]: "cone UNIV"
himmelma@33175
   508
  unfolding cone_def by auto
himmelma@33175
   509
himmelma@33175
   510
lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
himmelma@33175
   511
  unfolding cone_def by auto
himmelma@33175
   512
himmelma@33175
   513
subsection {* Conic hull. *}
himmelma@33175
   514
himmelma@33175
   515
lemma cone_cone_hull: "cone (cone hull s)"
himmelma@33175
   516
  unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
himmelma@33175
   517
  by (auto simp add: mem_def)
himmelma@33175
   518
himmelma@33175
   519
lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
himmelma@33175
   520
  apply(rule hull_eq[unfolded mem_def])
himmelma@33175
   521
  using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
himmelma@33175
   522
himmelma@33175
   523
subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
himmelma@33175
   524
himmelma@33175
   525
definition
himmelma@33175
   526
  affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
himmelma@33175
   527
  "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
himmelma@33175
   528
himmelma@33175
   529
lemma affine_dependent_explicit:
himmelma@33175
   530
  "affine_dependent p \<longleftrightarrow>
himmelma@33175
   531
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
himmelma@33175
   532
    (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
himmelma@33175
   533
  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
himmelma@33175
   534
  apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
himmelma@33175
   535
proof-
himmelma@33175
   536
  fix x s u assume as:"x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
himmelma@33175
   537
  have "x\<notin>s" using as(1,4) by auto
himmelma@33175
   538
  show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
himmelma@33175
   539
    apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
himmelma@33175
   540
    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
himmelma@33175
   541
next
himmelma@33175
   542
  fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
himmelma@33175
   543
  have "s \<noteq> {v}" using as(3,6) by auto
himmelma@33175
   544
  thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
himmelma@33175
   545
    apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
himmelma@33175
   546
    unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
himmelma@33175
   547
qed
himmelma@33175
   548
himmelma@33175
   549
lemma affine_dependent_explicit_finite:
himmelma@33175
   550
  fixes s :: "'a::real_vector set" assumes "finite s"
himmelma@33175
   551
  shows "affine_dependent s \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
himmelma@33175
   552
  (is "?lhs = ?rhs")
himmelma@33175
   553
proof
himmelma@33175
   554
  have *:"\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::'a))" by auto
himmelma@33175
   555
  assume ?lhs
himmelma@33175
   556
  then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
himmelma@33175
   557
    unfolding affine_dependent_explicit by auto
himmelma@33175
   558
  thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
himmelma@33175
   559
    apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
himmelma@33175
   560
    unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
himmelma@33175
   561
next
himmelma@33175
   562
  assume ?rhs
himmelma@33175
   563
  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
himmelma@33175
   564
  thus ?lhs unfolding affine_dependent_explicit using assms by auto
himmelma@33175
   565
qed
himmelma@33175
   566
himmelma@33175
   567
subsection {* A general lemma. *}
himmelma@33175
   568
himmelma@33175
   569
lemma convex_connected:
himmelma@33175
   570
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   571
  assumes "convex s" shows "connected s"
himmelma@33175
   572
proof-
himmelma@33175
   573
  { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
himmelma@33175
   574
    assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
himmelma@33175
   575
    then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
himmelma@33175
   576
    hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
himmelma@33175
   577
himmelma@33175
   578
    { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
himmelma@33175
   579
      { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
himmelma@33175
   580
          by (simp add: algebra_simps)
himmelma@33175
   581
        assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
himmelma@33175
   582
        hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
himmelma@33175
   583
          unfolding * and scaleR_right_diff_distrib[THEN sym]
himmelma@33175
   584
          unfolding less_divide_eq using n by auto  }
himmelma@33175
   585
      hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
himmelma@33175
   586
        apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
himmelma@33175
   587
        apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
himmelma@33175
   588
himmelma@33175
   589
    have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
himmelma@33175
   590
      apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
himmelma@33175
   591
      using * apply(simp add: dist_norm)
himmelma@33175
   592
      using as(1,2)[unfolded open_dist] apply simp
himmelma@33175
   593
      using as(1,2)[unfolded open_dist] apply simp
himmelma@33175
   594
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
himmelma@33175
   595
      using as(3) by auto
himmelma@33175
   596
    then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
himmelma@33175
   597
    hence False using as(4) 
himmelma@33175
   598
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
himmelma@33175
   599
      using x1(2) x2(2) by auto  }
himmelma@33175
   600
  thus ?thesis unfolding connected_def by auto
himmelma@33175
   601
qed
himmelma@33175
   602
himmelma@33175
   603
subsection {* One rather trivial consequence. *}
himmelma@33175
   604
hoelzl@34964
   605
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
himmelma@33175
   606
  by(simp add: convex_connected convex_UNIV)
himmelma@33175
   607
himmelma@33175
   608
subsection {* Convex functions into the reals. *}
himmelma@33175
   609
himmelma@33175
   610
definition
himmelma@33175
   611
  convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
himmelma@33175
   612
  "convex_on s f \<longleftrightarrow>
himmelma@33175
   613
  (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
himmelma@33175
   614
himmelma@33175
   615
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
himmelma@33175
   616
  unfolding convex_on_def by auto
himmelma@33175
   617
hoelzl@34964
   618
lemma convex_add[intro]:
himmelma@33175
   619
  assumes "convex_on s f" "convex_on s g"
himmelma@33175
   620
  shows "convex_on s (\<lambda>x. f x + g x)"
himmelma@33175
   621
proof-
himmelma@33175
   622
  { fix x y assume "x\<in>s" "y\<in>s" moreover
himmelma@33175
   623
    fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   624
    ultimately have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
himmelma@33175
   625
      using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
himmelma@33175
   626
      using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
himmelma@33175
   627
      apply - apply(rule add_mono) by auto
himmelma@33175
   628
    hence "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps)  }
himmelma@33175
   629
  thus ?thesis unfolding convex_on_def by auto 
himmelma@33175
   630
qed
himmelma@33175
   631
hoelzl@34964
   632
lemma convex_cmul[intro]:
himmelma@33175
   633
  assumes "0 \<le> (c::real)" "convex_on s f"
himmelma@33175
   634
  shows "convex_on s (\<lambda>x. c * f x)"
himmelma@33175
   635
proof-
himmelma@33175
   636
  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@33175
   637
  show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
himmelma@33175
   638
qed
himmelma@33175
   639
himmelma@33175
   640
lemma convex_lower:
himmelma@33175
   641
  assumes "convex_on s f"  "x\<in>s"  "y \<in> s"  "0 \<le> u"  "0 \<le> v"  "u + v = 1"
himmelma@33175
   642
  shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
himmelma@33175
   643
proof-
himmelma@33175
   644
  let ?m = "max (f x) (f y)"
himmelma@33175
   645
  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@33175
   646
    using assms(4,5) by(auto simp add: mult_mono1)
himmelma@33175
   647
  also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
himmelma@33175
   648
  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@33175
   649
    using assms(2-6) by auto 
himmelma@33175
   650
qed
himmelma@33175
   651
himmelma@33175
   652
lemma convex_local_global_minimum:
himmelma@33175
   653
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   654
  assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
himmelma@33175
   655
  shows "\<forall>y\<in>s. f x \<le> f y"
himmelma@33175
   656
proof(rule ccontr)
himmelma@33175
   657
  have "x\<in>s" using assms(1,3) by auto
himmelma@33175
   658
  assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
himmelma@33175
   659
  then obtain y where "y\<in>s" and y:"f x > f y" by auto
himmelma@33175
   660
  hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
himmelma@33175
   661
himmelma@33175
   662
  then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
himmelma@33175
   663
    using real_lbound_gt_zero[of 1 "e / dist x y"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
himmelma@33175
   664
  hence "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
himmelma@33175
   665
    using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]] by auto
himmelma@33175
   666
  moreover
himmelma@33175
   667
  have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
himmelma@33175
   668
  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
himmelma@33175
   669
    using u unfolding pos_less_divide_eq[OF xy] by auto
himmelma@33175
   670
  hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
himmelma@33175
   671
  ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
himmelma@33175
   672
qed
himmelma@33175
   673
hoelzl@34964
   674
lemma convex_distance[intro]:
himmelma@33175
   675
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   676
  shows "convex_on s (\<lambda>x. dist a x)"
himmelma@33175
   677
proof(auto simp add: convex_on_def dist_norm)
himmelma@33175
   678
  fix x y assume "x\<in>s" "y\<in>s"
himmelma@33175
   679
  fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   680
  have "a = u *\<^sub>R a + v *\<^sub>R a" unfolding scaleR_left_distrib[THEN sym] and `u+v=1` by simp
himmelma@33175
   681
  hence *:"a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
himmelma@33175
   682
    by (auto simp add: algebra_simps)
himmelma@33175
   683
  show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
himmelma@33175
   684
    unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
himmelma@33175
   685
    using `0 \<le> u` `0 \<le> v` by auto
himmelma@33175
   686
qed
himmelma@33175
   687
himmelma@33175
   688
subsection {* Arithmetic operations on sets preserve convexity. *}
himmelma@33175
   689
himmelma@33175
   690
lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *\<^sub>R x) ` s)"
himmelma@33175
   691
  unfolding convex_def and image_iff apply auto
himmelma@33175
   692
  apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by (auto simp add: algebra_simps)
himmelma@33175
   693
himmelma@33175
   694
lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
himmelma@33175
   695
  unfolding convex_def and image_iff apply auto
himmelma@33175
   696
  apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by auto
himmelma@33175
   697
himmelma@33175
   698
lemma convex_sums:
himmelma@33175
   699
  assumes "convex s" "convex t"
himmelma@33175
   700
  shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
himmelma@33175
   701
proof(auto simp add: convex_def image_iff scaleR_right_distrib)
himmelma@33175
   702
  fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
himmelma@33175
   703
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   704
  show "\<exists>x y. u *\<^sub>R xa + u *\<^sub>R ya + (v *\<^sub>R xb + v *\<^sub>R yb) = x + y \<and> x \<in> s \<and> y \<in> t"
himmelma@33175
   705
    apply(rule_tac x="u *\<^sub>R xa + v *\<^sub>R xb" in exI) apply(rule_tac x="u *\<^sub>R ya + v *\<^sub>R yb" in exI)
himmelma@33175
   706
    using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
himmelma@33175
   707
    using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
himmelma@33175
   708
    using uv xy by auto
himmelma@33175
   709
qed
himmelma@33175
   710
himmelma@33175
   711
lemma convex_differences: 
himmelma@33175
   712
  assumes "convex s" "convex t"
himmelma@33175
   713
  shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
himmelma@33175
   714
proof-
himmelma@33175
   715
  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@33175
   716
    apply(rule_tac x=xa in exI) apply(rule_tac x="-y" in exI) apply simp
himmelma@33175
   717
    apply(rule_tac x=xa in exI) apply(rule_tac x=xb in exI) by simp
himmelma@33175
   718
  thus ?thesis using convex_sums[OF assms(1)  convex_negations[OF assms(2)]] by auto
himmelma@33175
   719
qed
himmelma@33175
   720
himmelma@33175
   721
lemma convex_translation: assumes "convex s" shows "convex ((\<lambda>x. a + x) ` s)"
himmelma@33175
   722
proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
himmelma@33175
   723
  thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
himmelma@33175
   724
himmelma@33175
   725
lemma convex_affinity: assumes "convex s" shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
himmelma@33175
   726
proof- have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s" by auto
himmelma@33175
   727
  thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
himmelma@33175
   728
himmelma@33175
   729
lemma convex_linear_image:
himmelma@33175
   730
  assumes c:"convex s" and l:"bounded_linear f"
himmelma@33175
   731
  shows "convex(f ` s)"
himmelma@33175
   732
proof(auto simp add: convex_def)
himmelma@33175
   733
  interpret f: bounded_linear f by fact
himmelma@33175
   734
  fix x y assume xy:"x \<in> s" "y \<in> s"
himmelma@33175
   735
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   736
  show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff
himmelma@33175
   737
    apply(rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in bexI)
himmelma@33175
   738
    unfolding f.add f.scaleR
himmelma@33175
   739
    using c[unfolded convex_def] xy uv by auto
himmelma@33175
   740
qed
himmelma@33175
   741
himmelma@33175
   742
subsection {* Balls, being convex, are connected. *}
himmelma@33175
   743
himmelma@33175
   744
lemma convex_ball:
himmelma@33175
   745
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   746
  shows "convex (ball x e)" 
himmelma@33175
   747
proof(auto simp add: convex_def)
himmelma@33175
   748
  fix y z assume yz:"dist x y < e" "dist x z < e"
himmelma@33175
   749
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   750
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
himmelma@33175
   751
    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@33175
   752
  thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using real_convex_bound_lt[OF yz uv] by auto 
himmelma@33175
   753
qed
himmelma@33175
   754
himmelma@33175
   755
lemma convex_cball:
himmelma@33175
   756
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   757
  shows "convex(cball x e)"
himmelma@33175
   758
proof(auto simp add: convex_def Ball_def mem_cball)
himmelma@33175
   759
  fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
himmelma@33175
   760
  fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   761
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
himmelma@33175
   762
    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@33175
   763
  thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using real_convex_bound_le[OF yz uv] by auto 
himmelma@33175
   764
qed
himmelma@33175
   765
himmelma@33175
   766
lemma connected_ball:
himmelma@33175
   767
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   768
  shows "connected (ball x e)"
himmelma@33175
   769
  using convex_connected convex_ball by auto
himmelma@33175
   770
himmelma@33175
   771
lemma connected_cball:
himmelma@33175
   772
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   773
  shows "connected(cball x e)"
himmelma@33175
   774
  using convex_connected convex_cball by auto
himmelma@33175
   775
himmelma@33175
   776
subsection {* Convex hull. *}
himmelma@33175
   777
himmelma@33175
   778
lemma convex_convex_hull: "convex(convex hull s)"
himmelma@33175
   779
  unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
himmelma@33175
   780
  unfolding mem_def by auto
himmelma@33175
   781
haftmann@34064
   782
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
haftmann@34064
   783
  apply (rule hull_eq [unfolded mem_def])
haftmann@34064
   784
  apply (rule convex_Inter [unfolded Ball_def mem_def])
haftmann@34064
   785
  apply (simp add: le_fun_def le_bool_def)
haftmann@34064
   786
  done
himmelma@33175
   787
himmelma@33175
   788
lemma bounded_convex_hull:
himmelma@33175
   789
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   790
  assumes "bounded s" shows "bounded(convex hull s)"
himmelma@33175
   791
proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
himmelma@33175
   792
  show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
himmelma@33175
   793
    unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
himmelma@33175
   794
    unfolding subset_eq mem_cball dist_norm using B by auto qed
himmelma@33175
   795
himmelma@33175
   796
lemma finite_imp_bounded_convex_hull:
himmelma@33175
   797
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
   798
  shows "finite s \<Longrightarrow> bounded(convex hull s)"
himmelma@33175
   799
  using bounded_convex_hull finite_imp_bounded by auto
himmelma@33175
   800
himmelma@33175
   801
subsection {* Stepping theorems for convex hulls of finite sets. *}
himmelma@33175
   802
himmelma@33175
   803
lemma convex_hull_empty[simp]: "convex hull {} = {}"
himmelma@33175
   804
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@33175
   805
himmelma@33175
   806
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
himmelma@33175
   807
  apply(rule hull_unique) unfolding mem_def by auto
himmelma@33175
   808
himmelma@33175
   809
lemma convex_hull_insert:
himmelma@33175
   810
  fixes s :: "'a::real_vector set"
himmelma@33175
   811
  assumes "s \<noteq> {}"
himmelma@33175
   812
  shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
himmelma@33175
   813
                                    b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
himmelma@33175
   814
 apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
himmelma@33175
   815
 fix x assume x:"x = a \<or> x \<in> s"
himmelma@33175
   816
 thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
himmelma@33175
   817
   apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
himmelma@33175
   818
next
himmelma@33175
   819
  fix x assume "x\<in>?hull"
himmelma@33175
   820
  then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
himmelma@33175
   821
  have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
himmelma@33175
   822
    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
himmelma@33175
   823
  thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
himmelma@33175
   824
    apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
himmelma@33175
   825
next
himmelma@33175
   826
  show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
himmelma@33175
   827
    fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
himmelma@33175
   828
    from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
himmelma@33175
   829
    from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
himmelma@33175
   830
    have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
himmelma@33175
   831
    have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
himmelma@33175
   832
    proof(cases "u * v1 + v * v2 = 0")
himmelma@33175
   833
      have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
himmelma@33175
   834
      case True hence **:"u * v1 = 0" "v * v2 = 0" apply- apply(rule_tac [!] ccontr)
himmelma@33175
   835
        using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by auto
himmelma@33175
   836
      hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
himmelma@33175
   837
      thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
himmelma@33175
   838
    next
himmelma@33175
   839
      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
himmelma@33175
   840
      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
himmelma@33175
   841
      also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
himmelma@33175
   842
      case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
himmelma@33175
   843
        apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
himmelma@33175
   844
        using as(1,2) obt1(1,2) obt2(1,2) by auto 
himmelma@33175
   845
      thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
himmelma@33175
   846
        apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
himmelma@33175
   847
        apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
himmelma@33175
   848
        unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
himmelma@33175
   849
        by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
himmelma@33175
   850
    qed note * = this
himmelma@33175
   851
    have u1:"u1 \<le> 1" apply(rule ccontr) unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
himmelma@33175
   852
    have u2:"u2 \<le> 1" apply(rule ccontr) unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
himmelma@33175
   853
    have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
himmelma@33175
   854
      apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
himmelma@33175
   855
    also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
himmelma@33175
   856
    finally 
himmelma@33175
   857
    show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
himmelma@33175
   858
      apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
himmelma@33175
   859
      using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
himmelma@33175
   860
  qed
himmelma@33175
   861
qed
himmelma@33175
   862
himmelma@33175
   863
himmelma@33175
   864
subsection {* Explicit expression for convex hull. *}
himmelma@33175
   865
himmelma@33175
   866
lemma convex_hull_indexed:
himmelma@33175
   867
  fixes s :: "'a::real_vector set"
himmelma@33175
   868
  shows "convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
himmelma@33175
   869
                            (setsum u {1..k} = 1) \<and>
himmelma@33175
   870
                            (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
himmelma@33175
   871
  apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
himmelma@33175
   872
  apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
himmelma@33175
   873
proof-
himmelma@33175
   874
  fix x assume "x\<in>s"
himmelma@33175
   875
  thus "x \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
himmelma@33175
   876
next
himmelma@33175
   877
  fix t assume as:"s \<subseteq> t" "convex t"
himmelma@33175
   878
  show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
himmelma@33175
   879
    fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
himmelma@33175
   880
    show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
himmelma@33175
   881
      using assm(1,2) as(1) by auto qed
himmelma@33175
   882
next
himmelma@33175
   883
  fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and xy:"x\<in>?hull" "y\<in>?hull"
himmelma@33175
   884
  from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" by auto
himmelma@33175
   885
  from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" by auto
himmelma@33175
   886
  have *:"\<And>P (x1::'a) x2 s1 s2 i.(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
himmelma@33175
   887
    "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
himmelma@33175
   888
    prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
himmelma@33175
   889
  have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
himmelma@33175
   890
  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
himmelma@33175
   891
    apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
himmelma@33175
   892
    apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
hoelzl@35577
   893
    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
himmelma@33175
   894
    unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
himmelma@33175
   895
    fix i assume i:"i \<in> {1..k1+k2}"
himmelma@33175
   896
    show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and> (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
himmelma@33175
   897
    proof(cases "i\<in>{1..k1}")
himmelma@33175
   898
      case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
himmelma@33175
   899
    next def j \<equiv> "i - k1"
himmelma@33175
   900
      case False with i have "j \<in> {1..k2}" unfolding j_def by auto
himmelma@33175
   901
      thus ?thesis unfolding j_def[symmetric] using False
himmelma@33175
   902
        using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
himmelma@33175
   903
  qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
himmelma@33175
   904
qed
himmelma@33175
   905
himmelma@33175
   906
lemma convex_hull_finite:
himmelma@33175
   907
  fixes s :: "'a::real_vector set"
himmelma@33175
   908
  assumes "finite s"
himmelma@33175
   909
  shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
himmelma@33175
   910
         setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
himmelma@33175
   911
proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
himmelma@33175
   912
  fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x" 
himmelma@33175
   913
    apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
himmelma@33175
   914
    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
himmelma@33175
   915
next
himmelma@33175
   916
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
himmelma@33175
   917
  fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
himmelma@33175
   918
  fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
himmelma@33175
   919
  { fix x assume "x\<in>s"
himmelma@33175
   920
    hence "0 \<le> u * ux x + v * uy x" using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
himmelma@33175
   921
      by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
himmelma@33175
   922
  moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
himmelma@33175
   923
    unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
himmelma@33175
   924
  moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
himmelma@33175
   925
    unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
himmelma@33175
   926
  ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
himmelma@33175
   927
    apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
himmelma@33175
   928
next
himmelma@33175
   929
  fix t assume t:"s \<subseteq> t" "convex t" 
himmelma@33175
   930
  fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
himmelma@33175
   931
  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
himmelma@33175
   932
    using assms and t(1) by auto
himmelma@33175
   933
qed
himmelma@33175
   934
himmelma@33175
   935
subsection {* Another formulation from Lars Schewe. *}
himmelma@33175
   936
himmelma@33175
   937
lemma setsum_constant_scaleR:
himmelma@33175
   938
  fixes y :: "'a::real_vector"
himmelma@33175
   939
  shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
himmelma@33175
   940
apply (cases "finite A")
himmelma@33175
   941
apply (induct set: finite)
himmelma@33175
   942
apply (simp_all add: algebra_simps)
himmelma@33175
   943
done
himmelma@33175
   944
himmelma@33175
   945
lemma convex_hull_explicit:
himmelma@33175
   946
  fixes p :: "'a::real_vector set"
himmelma@33175
   947
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
himmelma@33175
   948
             (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
himmelma@33175
   949
proof-
himmelma@33175
   950
  { fix x assume "x\<in>?lhs"
himmelma@33175
   951
    then obtain k u y where obt:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
himmelma@33175
   952
      unfolding convex_hull_indexed by auto
himmelma@33175
   953
himmelma@33175
   954
    have fin:"finite {1..k}" by auto
himmelma@33175
   955
    have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
himmelma@33175
   956
    { fix j assume "j\<in>{1..k}"
himmelma@33175
   957
      hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
himmelma@33175
   958
        using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
himmelma@33175
   959
        apply(rule setsum_nonneg) using obt(1) by auto } 
himmelma@33175
   960
    moreover
himmelma@33175
   961
    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
himmelma@33175
   962
      unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
himmelma@33175
   963
    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
himmelma@33175
   964
      using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
himmelma@33175
   965
      unfolding scaleR_left.setsum using obt(3) by auto
himmelma@33175
   966
    ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
himmelma@33175
   967
      apply(rule_tac x="y ` {1..k}" in exI)
himmelma@33175
   968
      apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
himmelma@33175
   969
    hence "x\<in>?rhs" by auto  }
himmelma@33175
   970
  moreover
himmelma@33175
   971
  { fix y assume "y\<in>?rhs"
himmelma@33175
   972
    then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
himmelma@33175
   973
himmelma@33175
   974
    obtain f where f:"inj_on f {1..card s}" "f ` {1..card s} = s" using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
himmelma@33175
   975
    
himmelma@33175
   976
    { fix i::nat assume "i\<in>{1..card s}"
himmelma@33175
   977
      hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
himmelma@33175
   978
      hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
himmelma@33175
   979
    moreover have *:"finite {1..card s}" by auto
himmelma@33175
   980
    { fix y assume "y\<in>s"
himmelma@33175
   981
      then obtain i where "i\<in>{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"] by auto
himmelma@33175
   982
      hence "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}" apply auto using f(1)[unfolded inj_on_def] apply(erule_tac x=x in ballE) by auto
himmelma@33175
   983
      hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
himmelma@33175
   984
      hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
himmelma@33175
   985
            "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
himmelma@33175
   986
        by (auto simp add: setsum_constant_scaleR)   }
himmelma@33175
   987
himmelma@33175
   988
    hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
himmelma@33175
   989
      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
himmelma@33175
   990
      unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
himmelma@33175
   991
      using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
himmelma@33175
   992
    
himmelma@33175
   993
    ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
himmelma@33175
   994
      apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastsimp
himmelma@33175
   995
    hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
himmelma@33175
   996
  ultimately show ?thesis unfolding expand_set_eq by blast
himmelma@33175
   997
qed
himmelma@33175
   998
himmelma@33175
   999
subsection {* A stepping theorem for that expansion. *}
himmelma@33175
  1000
himmelma@33175
  1001
lemma convex_hull_finite_step:
himmelma@33175
  1002
  fixes s :: "'a::real_vector set" assumes "finite s"
himmelma@33175
  1003
  shows "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
himmelma@33175
  1004
     \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?lhs = ?rhs")
himmelma@33175
  1005
proof(rule, case_tac[!] "a\<in>s")
himmelma@33175
  1006
  assume "a\<in>s" hence *:"insert a s = s" by auto
himmelma@33175
  1007
  assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
himmelma@33175
  1008
next
himmelma@33175
  1009
  assume ?lhs then obtain u where u:"\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
himmelma@33175
  1010
  assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
himmelma@33175
  1011
    apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
himmelma@33175
  1012
next
himmelma@33175
  1013
  assume "a\<in>s" hence *:"insert a s = s" by auto
himmelma@33175
  1014
  have fin:"finite (insert a s)" using assms by auto
himmelma@33175
  1015
  assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
himmelma@33175
  1016
  show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
himmelma@33175
  1017
    unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
himmelma@33175
  1018
next
himmelma@33175
  1019
  assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
himmelma@33175
  1020
  moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
himmelma@33175
  1021
    apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
himmelma@33175
  1022
  ultimately show ?lhs apply(rule_tac x="\<lambda>x. if a = x then v else u x" in exI)  unfolding setsum_clauses(2)[OF assms] by auto
himmelma@33175
  1023
qed
himmelma@33175
  1024
himmelma@33175
  1025
subsection {* Hence some special cases. *}
himmelma@33175
  1026
himmelma@33175
  1027
lemma convex_hull_2:
himmelma@33175
  1028
  "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
himmelma@33175
  1029
proof- have *:"\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" by auto have **:"finite {b}" by auto
himmelma@33175
  1030
show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
himmelma@33175
  1031
  apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
himmelma@33175
  1032
  apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
himmelma@33175
  1033
himmelma@33175
  1034
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
himmelma@33175
  1035
  unfolding convex_hull_2 unfolding Collect_def 
himmelma@33175
  1036
proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
himmelma@33175
  1037
  fix x show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
himmelma@33175
  1038
    unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
himmelma@33175
  1039
himmelma@33175
  1040
lemma convex_hull_3:
himmelma@33175
  1041
  "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
himmelma@33175
  1042
proof-
himmelma@33175
  1043
  have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
himmelma@33175
  1044
  have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
himmelma@34289
  1045
         "\<And>x y z ::real^_. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: ring_simps)
himmelma@33175
  1046
  show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
himmelma@33175
  1047
    unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
himmelma@33175
  1048
    apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
himmelma@33175
  1049
    apply(rule_tac x="1 - v - w" in exI) apply simp apply(rule_tac x=v in exI) apply simp apply(rule_tac x="\<lambda>x. w" in exI) by simp qed
himmelma@33175
  1050
himmelma@33175
  1051
lemma convex_hull_3_alt:
himmelma@33175
  1052
  "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
himmelma@33175
  1053
proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
himmelma@33175
  1054
  show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply (simp add: algebra_simps)
himmelma@33175
  1055
    apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
himmelma@33175
  1056
himmelma@33175
  1057
subsection {* Relations among closure notions and corresponding hulls. *}
himmelma@33175
  1058
himmelma@33175
  1059
text {* TODO: Generalize linear algebra concepts defined in @{text
himmelma@33175
  1060
Euclidean_Space.thy} so that we can generalize these lemmas. *}
himmelma@33175
  1061
himmelma@33175
  1062
lemma subspace_imp_affine:
himmelma@33175
  1063
  fixes s :: "(real ^ _) set" shows "subspace s \<Longrightarrow> affine s"
himmelma@33175
  1064
  unfolding subspace_def affine_def smult_conv_scaleR by auto
himmelma@33175
  1065
himmelma@33175
  1066
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
himmelma@33175
  1067
  unfolding affine_def convex_def by auto
himmelma@33175
  1068
himmelma@33175
  1069
lemma subspace_imp_convex:
himmelma@33175
  1070
  fixes s :: "(real ^ _) set" shows "subspace s \<Longrightarrow> convex s"
himmelma@33175
  1071
  using subspace_imp_affine affine_imp_convex by auto
himmelma@33175
  1072
himmelma@33175
  1073
lemma affine_hull_subset_span:
himmelma@33175
  1074
  fixes s :: "(real ^ _) set" shows "(affine hull s) \<subseteq> (span s)"
himmelma@33175
  1075
  unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
himmelma@33175
  1076
  using subspace_imp_affine  by auto
himmelma@33175
  1077
himmelma@33175
  1078
lemma convex_hull_subset_span:
himmelma@33175
  1079
  fixes s :: "(real ^ _) set" shows "(convex hull s) \<subseteq> (span s)"
himmelma@33175
  1080
  unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
himmelma@33175
  1081
  using subspace_imp_convex by auto
himmelma@33175
  1082
himmelma@33175
  1083
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
himmelma@33175
  1084
  unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
himmelma@33175
  1085
  using affine_imp_convex by auto
himmelma@33175
  1086
himmelma@33175
  1087
lemma affine_dependent_imp_dependent:
himmelma@33175
  1088
  fixes s :: "(real ^ _) set" shows "affine_dependent s \<Longrightarrow> dependent s"
himmelma@33175
  1089
  unfolding affine_dependent_def dependent_def 
himmelma@33175
  1090
  using affine_hull_subset_span by auto
himmelma@33175
  1091
himmelma@33175
  1092
lemma dependent_imp_affine_dependent:
himmelma@33175
  1093
  fixes s :: "(real ^ _) set"
himmelma@33175
  1094
  assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
himmelma@33175
  1095
  shows "affine_dependent (insert a s)"
himmelma@33175
  1096
proof-
himmelma@33175
  1097
  from assms(1)[unfolded dependent_explicit smult_conv_scaleR] obtain S u v 
himmelma@33175
  1098
    where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
himmelma@33175
  1099
  def t \<equiv> "(\<lambda>x. x + a) ` S"
himmelma@33175
  1100
himmelma@33175
  1101
  have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
himmelma@33175
  1102
  have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
himmelma@33175
  1103
  have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
himmelma@33175
  1104
himmelma@33175
  1105
  hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
himmelma@33175
  1106
  moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
himmelma@33175
  1107
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
himmelma@33175
  1108
  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
himmelma@33175
  1109
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
himmelma@33175
  1110
  moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
himmelma@33175
  1111
    apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
himmelma@33175
  1112
  moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
himmelma@33175
  1113
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
himmelma@33175
  1114
  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
himmelma@33175
  1115
    unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
himmelma@33175
  1116
    using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
himmelma@33175
  1117
  hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
himmelma@33175
  1118
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *  vector_smult_lneg) 
himmelma@33175
  1119
  ultimately show ?thesis unfolding affine_dependent_explicit
himmelma@33175
  1120
    apply(rule_tac x="insert a t" in exI) by auto 
himmelma@33175
  1121
qed
himmelma@33175
  1122
himmelma@33175
  1123
lemma convex_cone:
himmelma@33175
  1124
  "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)" (is "?lhs = ?rhs")
himmelma@33175
  1125
proof-
himmelma@33175
  1126
  { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
himmelma@33175
  1127
    hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
himmelma@33175
  1128
    hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
himmelma@33175
  1129
      apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
himmelma@33175
  1130
      apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
paulson@33270
  1131
  thus ?thesis unfolding convex_def cone_def by auto
himmelma@33175
  1132
qed
himmelma@33175
  1133
hoelzl@34291
  1134
lemma affine_dependent_biggerset: fixes s::"(real^'n) set"
himmelma@33175
  1135
  assumes "finite s" "card s \<ge> CARD('n) + 2"
himmelma@33175
  1136
  shows "affine_dependent s"
himmelma@33175
  1137
proof-
himmelma@33175
  1138
  have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
himmelma@33175
  1139
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
himmelma@33175
  1140
  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
himmelma@33175
  1141
    apply(rule card_image) unfolding inj_on_def by auto
himmelma@33175
  1142
  also have "\<dots> > CARD('n)" using assms(2)
himmelma@33175
  1143
    unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
himmelma@33175
  1144
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
himmelma@33175
  1145
    apply(rule dependent_imp_affine_dependent)
himmelma@33175
  1146
    apply(rule dependent_biggerset) by auto qed
himmelma@33175
  1147
himmelma@33175
  1148
lemma affine_dependent_biggerset_general:
hoelzl@34291
  1149
  assumes "finite (s::(real^'n) set)" "card s \<ge> dim s + 2"
himmelma@33175
  1150
  shows "affine_dependent s"
himmelma@33175
  1151
proof-
himmelma@33175
  1152
  from assms(2) have "s \<noteq> {}" by auto
himmelma@33175
  1153
  then obtain a where "a\<in>s" by auto
himmelma@33175
  1154
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
himmelma@33175
  1155
  have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
himmelma@33175
  1156
    apply(rule card_image) unfolding inj_on_def by auto
himmelma@33175
  1157
  have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
himmelma@33175
  1158
    apply(rule subset_le_dim) unfolding subset_eq
himmelma@33175
  1159
    using `a\<in>s` by (auto simp add:span_superset span_sub)
himmelma@33175
  1160
  also have "\<dots> < dim s + 1" by auto
himmelma@33175
  1161
  also have "\<dots> \<le> card (s - {a})" using assms
himmelma@33175
  1162
    using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
himmelma@33175
  1163
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
himmelma@33175
  1164
    apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
himmelma@33175
  1165
himmelma@33175
  1166
subsection {* Caratheodory's theorem. *}
himmelma@33175
  1167
hoelzl@34291
  1168
lemma convex_hull_caratheodory: fixes p::"(real^'n) set"
himmelma@33175
  1169
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and>
himmelma@33175
  1170
  (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
himmelma@33175
  1171
  unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
himmelma@33175
  1172
proof(rule,rule)
himmelma@33175
  1173
  fix y let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
  1174
  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
himmelma@33175
  1175
  then obtain N where "?P N" by auto
himmelma@33175
  1176
  hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
himmelma@33175
  1177
  then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
himmelma@33175
  1178
  then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
himmelma@33175
  1179
himmelma@33175
  1180
  have "card s \<le> CARD('n) + 1" proof(rule ccontr, simp only: not_le)
himmelma@33175
  1181
    assume "CARD('n) + 1 < card s"
himmelma@33175
  1182
    hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
himmelma@33175
  1183
    then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
himmelma@33175
  1184
      using affine_dependent_explicit_finite[OF obt(1)] by auto
himmelma@33175
  1185
    def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
himmelma@33175
  1186
    have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
himmelma@33175
  1187
      assume as:"\<forall>x\<in>s. 0 \<le> w x"
himmelma@33175
  1188
      hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
himmelma@33175
  1189
      hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
himmelma@33175
  1190
        using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
himmelma@33175
  1191
      thus False using wv(1) by auto
himmelma@33175
  1192
    qed hence "i\<noteq>{}" unfolding i_def by auto
himmelma@33175
  1193
himmelma@33175
  1194
    hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
himmelma@33175
  1195
      using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
himmelma@33175
  1196
    have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
himmelma@33175
  1197
      fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
himmelma@33175
  1198
      show"0 \<le> u v + t * w v" proof(cases "w v < 0")
himmelma@33175
  1199
        case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
himmelma@33175
  1200
          using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
himmelma@33175
  1201
        case True hence "t \<le> u v / (- w v)" using `v\<in>s`
himmelma@33175
  1202
          unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
himmelma@33175
  1203
        thus ?thesis unfolding real_0_le_add_iff
himmelma@33175
  1204
          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
himmelma@33175
  1205
      qed qed
himmelma@33175
  1206
himmelma@33175
  1207
    obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
himmelma@33175
  1208
      using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
himmelma@33175
  1209
    hence a:"a\<in>s" "u a + t * w a = 0" by auto
himmelma@33175
  1210
    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@33175
  1211
    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
himmelma@33175
  1212
      unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
himmelma@33175
  1213
    moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y" 
himmelma@33175
  1214
      unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
himmelma@33175
  1215
      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]]
himmelma@33175
  1216
      by (simp add: vector_smult_lneg)
himmelma@33175
  1217
    ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
himmelma@33175
  1218
      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: * scaleR_left_distrib)
himmelma@33175
  1219
    thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
himmelma@33175
  1220
  thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1
himmelma@33175
  1221
    \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" using obt by auto
himmelma@33175
  1222
qed auto
himmelma@33175
  1223
himmelma@33175
  1224
lemma caratheodory:
hoelzl@34291
  1225
 "convex hull p = {x::real^'n. \<exists>s. finite s \<and> s \<subseteq> p \<and>
himmelma@33175
  1226
      card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s}"
himmelma@33175
  1227
  unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
himmelma@33175
  1228
  fix x assume "x \<in> convex hull p"
himmelma@33175
  1229
  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1"
himmelma@33175
  1230
     "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"unfolding convex_hull_caratheodory by auto
himmelma@33175
  1231
  thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
himmelma@33175
  1232
    apply(rule_tac x=s in exI) using hull_subset[of s convex]
himmelma@33175
  1233
  using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
himmelma@33175
  1234
next
himmelma@33175
  1235
  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@33175
  1236
  then obtain s where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1" "x \<in> convex hull s" by auto
himmelma@33175
  1237
  thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
himmelma@33175
  1238
qed
himmelma@33175
  1239
himmelma@33175
  1240
subsection {* Openness and compactness are preserved by convex hull operation. *}
himmelma@33175
  1241
hoelzl@34964
  1242
lemma open_convex_hull[intro]:
himmelma@33175
  1243
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1244
  assumes "open s"
himmelma@33175
  1245
  shows "open(convex hull s)"
himmelma@33175
  1246
  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10) 
himmelma@33175
  1247
proof(rule, rule) fix a
himmelma@33175
  1248
  assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
himmelma@33175
  1249
  then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
himmelma@33175
  1250
himmelma@33175
  1251
  from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
himmelma@33175
  1252
    using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
himmelma@33175
  1253
  have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
himmelma@33175
  1254
himmelma@33175
  1255
  show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
himmelma@33175
  1256
    apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
himmelma@33175
  1257
  proof-
himmelma@33175
  1258
    show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
himmelma@33175
  1259
      using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
himmelma@33175
  1260
  next  fix y assume "y \<in> cball a (Min i)"
himmelma@33175
  1261
    hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
himmelma@33175
  1262
    { fix x assume "x\<in>t"
himmelma@33175
  1263
      hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
himmelma@33175
  1264
      hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
himmelma@33175
  1265
      moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
himmelma@33175
  1266
      ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by auto }
himmelma@33175
  1267
    moreover
himmelma@33175
  1268
    have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
himmelma@33175
  1269
    have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
himmelma@33175
  1270
      unfolding setsum_reindex[OF *] o_def using obt(4) by auto
himmelma@33175
  1271
    moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
himmelma@33175
  1272
      unfolding setsum_reindex[OF *] o_def using obt(4,5)
himmelma@33175
  1273
      by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
himmelma@33175
  1274
    ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
himmelma@33175
  1275
      apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
himmelma@33175
  1276
      using obt(1, 3) by auto
himmelma@33175
  1277
  qed
himmelma@33175
  1278
qed
himmelma@33175
  1279
himmelma@33175
  1280
lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
hoelzl@34964
  1281
unfolding open_vector_def forall_1 by auto
himmelma@33175
  1282
himmelma@33175
  1283
lemma tendsto_dest_vec1 [tendsto_intros]:
himmelma@33175
  1284
  "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
himmelma@33175
  1285
  unfolding tendsto_def
himmelma@33175
  1286
  apply clarify
himmelma@33175
  1287
  apply (drule_tac x="dest_vec1 -` S" in spec)
himmelma@33175
  1288
  apply (simp add: open_dest_vec1_vimage)
himmelma@33175
  1289
  done
himmelma@33175
  1290
himmelma@33175
  1291
lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
himmelma@33175
  1292
  unfolding continuous_def by (rule tendsto_dest_vec1)
himmelma@33175
  1293
himmelma@33175
  1294
(* TODO: move *)
himmelma@33175
  1295
lemma compact_real_interval:
himmelma@33175
  1296
  fixes a b :: real shows "compact {a..b}"
himmelma@33175
  1297
proof -
himmelma@33175
  1298
  have "continuous_on {vec1 a .. vec1 b} dest_vec1"
himmelma@33175
  1299
    unfolding continuous_on
himmelma@33175
  1300
    by (simp add: tendsto_dest_vec1 Lim_at_within Lim_ident_at)
himmelma@33175
  1301
  moreover have "compact {vec1 a .. vec1 b}" by (rule compact_interval)
himmelma@33175
  1302
  ultimately have "compact (dest_vec1 ` {vec1 a .. vec1 b})"
himmelma@33175
  1303
    by (rule compact_continuous_image)
himmelma@33175
  1304
  also have "dest_vec1 ` {vec1 a .. vec1 b} = {a..b}"
himmelma@33175
  1305
    by (auto simp add: image_def Bex_def exists_vec1)
himmelma@33175
  1306
  finally show ?thesis .
himmelma@33175
  1307
qed
himmelma@33175
  1308
himmelma@33175
  1309
lemma compact_convex_combinations:
himmelma@33175
  1310
  fixes s t :: "'a::real_normed_vector set"
himmelma@33175
  1311
  assumes "compact s" "compact t"
himmelma@33175
  1312
  shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
himmelma@33175
  1313
proof-
himmelma@33175
  1314
  let ?X = "{0..1} \<times> s \<times> t"
himmelma@33175
  1315
  let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
himmelma@33175
  1316
  have *:"{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
himmelma@33175
  1317
    apply(rule set_ext) unfolding image_iff mem_Collect_eq
himmelma@33175
  1318
    apply rule apply auto
himmelma@33175
  1319
    apply (rule_tac x=u in rev_bexI, simp)
himmelma@33175
  1320
    apply (erule rev_bexI, erule rev_bexI, simp)
himmelma@33175
  1321
    by auto
himmelma@33175
  1322
  have "continuous_on ({0..1} \<times> s \<times> t)
himmelma@33175
  1323
     (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
himmelma@33175
  1324
    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
himmelma@33175
  1325
  thus ?thesis unfolding *
himmelma@33175
  1326
    apply (rule compact_continuous_image)
himmelma@33175
  1327
    apply (intro compact_Times compact_real_interval assms)
himmelma@33175
  1328
    done
himmelma@33175
  1329
qed
himmelma@33175
  1330
hoelzl@34291
  1331
lemma compact_convex_hull: fixes s::"(real^'n) set"
himmelma@33175
  1332
  assumes "compact s"  shows "compact(convex hull s)"
himmelma@33175
  1333
proof(cases "s={}")
himmelma@33175
  1334
  case True thus ?thesis using compact_empty by simp
himmelma@33175
  1335
next
himmelma@33175
  1336
  case False then obtain w where "w\<in>s" by auto
himmelma@33175
  1337
  show ?thesis unfolding caratheodory[of s]
berghofe@34915
  1338
  proof(induct ("CARD('n) + 1"))
himmelma@33175
  1339
    have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
himmelma@33175
  1340
      using compact_empty by (auto simp add: convex_hull_empty)
himmelma@33175
  1341
    case 0 thus ?case unfolding * by simp
himmelma@33175
  1342
  next
himmelma@33175
  1343
    case (Suc n)
himmelma@33175
  1344
    show ?case proof(cases "n=0")
himmelma@33175
  1345
      case True have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
himmelma@33175
  1346
        unfolding expand_set_eq and mem_Collect_eq proof(rule, rule)
himmelma@33175
  1347
        fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1348
        then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
himmelma@33175
  1349
        show "x\<in>s" proof(cases "card t = 0")
himmelma@33175
  1350
          case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by(simp add: convex_hull_empty)
himmelma@33175
  1351
        next
himmelma@33175
  1352
          case False hence "card t = Suc 0" using t(3) `n=0` by auto
himmelma@33175
  1353
          then obtain a where "t = {a}" unfolding card_Suc_eq by auto
himmelma@33175
  1354
          thus ?thesis using t(2,4) by (simp add: convex_hull_singleton)
himmelma@33175
  1355
        qed
himmelma@33175
  1356
      next
himmelma@33175
  1357
        fix x assume "x\<in>s"
himmelma@33175
  1358
        thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1359
          apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
himmelma@33175
  1360
      qed thus ?thesis using assms by simp
himmelma@33175
  1361
    next
himmelma@33175
  1362
      case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
himmelma@33175
  1363
        { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
himmelma@33175
  1364
        0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
himmelma@33175
  1365
        unfolding expand_set_eq and mem_Collect_eq proof(rule,rule)
himmelma@33175
  1366
        fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
himmelma@33175
  1367
          0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
himmelma@33175
  1368
        then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
himmelma@33175
  1369
          "0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n"  "v \<in> convex hull t" by auto
himmelma@33175
  1370
        moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
himmelma@33175
  1371
          apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
himmelma@33175
  1372
          using obt(7) and hull_mono[of t "insert u t"] by auto
himmelma@33175
  1373
        ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1374
          apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
himmelma@33175
  1375
      next
himmelma@33175
  1376
        fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
himmelma@33175
  1377
        then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
himmelma@33175
  1378
        let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
himmelma@33175
  1379
          0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
himmelma@33175
  1380
        show ?P proof(cases "card t = Suc n")
himmelma@33175
  1381
          case False hence "card t \<le> n" using t(3) by auto
himmelma@33175
  1382
          thus ?P apply(rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) using `w\<in>s` and t
himmelma@33175
  1383
            by(auto intro!: exI[where x=t])
himmelma@33175
  1384
        next
himmelma@33175
  1385
          case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
himmelma@33175
  1386
          show ?P proof(cases "u={}")
himmelma@33175
  1387
            case True hence "x=a" using t(4)[unfolded au] by auto
himmelma@33175
  1388
            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@33175
  1389
              using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
himmelma@33175
  1390
          next
himmelma@33175
  1391
            case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
himmelma@33175
  1392
              using t(4)[unfolded au convex_hull_insert[OF False]] by auto
himmelma@33175
  1393
            have *:"1 - vx = ux" using obt(3) by auto
himmelma@33175
  1394
            show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
himmelma@33175
  1395
              using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
himmelma@33175
  1396
              by(auto intro!: exI[where x=u])
himmelma@33175
  1397
          qed
himmelma@33175
  1398
        qed
himmelma@33175
  1399
      qed
himmelma@33175
  1400
      thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
himmelma@33175
  1401
    qed
himmelma@33175
  1402
  qed 
himmelma@33175
  1403
qed
himmelma@33175
  1404
himmelma@33175
  1405
lemma finite_imp_compact_convex_hull:
himmelma@33175
  1406
  fixes s :: "(real ^ _) set"
himmelma@33175
  1407
  shows "finite s \<Longrightarrow> compact(convex hull s)"
himmelma@33175
  1408
  apply(drule finite_imp_compact, drule compact_convex_hull) by assumption
himmelma@33175
  1409
himmelma@33175
  1410
subsection {* Extremal points of a simplex are some vertices. *}
himmelma@33175
  1411
himmelma@33175
  1412
lemma dist_increases_online:
himmelma@33175
  1413
  fixes a b d :: "'a::real_inner"
himmelma@33175
  1414
  assumes "d \<noteq> 0"
himmelma@33175
  1415
  shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
himmelma@33175
  1416
proof(cases "inner a d - inner b d > 0")
himmelma@33175
  1417
  case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
himmelma@33175
  1418
    apply(rule_tac add_pos_pos) using assms by auto
himmelma@33175
  1419
  thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
himmelma@33175
  1420
    by (simp add: algebra_simps inner_commute)
himmelma@33175
  1421
next
himmelma@33175
  1422
  case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
himmelma@33175
  1423
    apply(rule_tac add_pos_nonneg) using assms by auto
himmelma@33175
  1424
  thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
himmelma@33175
  1425
    by (simp add: algebra_simps inner_commute)
himmelma@33175
  1426
qed
himmelma@33175
  1427
himmelma@33175
  1428
lemma norm_increases_online:
himmelma@33175
  1429
  fixes d :: "'a::real_inner"
himmelma@33175
  1430
  shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
himmelma@33175
  1431
  using dist_increases_online[of d a 0] unfolding dist_norm by auto
himmelma@33175
  1432
himmelma@33175
  1433
lemma simplex_furthest_lt:
himmelma@33175
  1434
  fixes s::"'a::real_inner set" assumes "finite s"
himmelma@33175
  1435
  shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
himmelma@33175
  1436
proof(induct_tac rule: finite_induct[of s])
himmelma@33175
  1437
  fix x s assume as:"finite s" "x\<notin>s" "\<forall>x\<in>convex hull s. x \<notin> s \<longrightarrow> (\<exists>y\<in>convex hull s. norm (x - a) < norm (y - a))"
himmelma@33175
  1438
  show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow> (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
himmelma@33175
  1439
  proof(rule,rule,cases "s = {}")
himmelma@33175
  1440
    case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
himmelma@33175
  1441
    obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *\<^sub>R x + v *\<^sub>R b"
himmelma@33175
  1442
      using y(1)[unfolded convex_hull_insert[OF False]] by auto
himmelma@33175
  1443
    show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
himmelma@33175
  1444
    proof(cases "y\<in>convex hull s")
himmelma@33175
  1445
      case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
himmelma@33175
  1446
        using as(3)[THEN bspec[where x=y]] and y(2) by auto
himmelma@33175
  1447
      thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
himmelma@33175
  1448
    next
himmelma@33175
  1449
      case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
himmelma@33175
  1450
        assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
himmelma@33175
  1451
        thus ?thesis using False and obt(4) by auto
himmelma@33175
  1452
      next
himmelma@33175
  1453
        assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
himmelma@33175
  1454
        thus ?thesis using y(2) by auto
himmelma@33175
  1455
      next
himmelma@33175
  1456
        assume "u\<noteq>0" "v\<noteq>0"
himmelma@33175
  1457
        then obtain w where w:"w>0" "w<u" "w<v" using real_lbound_gt_zero[of u v] and obt(1,2) by auto
himmelma@33175
  1458
        have "x\<noteq>b" proof(rule ccontr) 
himmelma@33175
  1459
          assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
himmelma@33175
  1460
            using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
himmelma@33175
  1461
          thus False using obt(4) and False by simp qed
himmelma@33175
  1462
        hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
himmelma@33175
  1463
        show ?thesis using dist_increases_online[OF *, of a y]
himmelma@33175
  1464
        proof(erule_tac disjE)
himmelma@33175
  1465
          assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
himmelma@33175
  1466
          hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
himmelma@33175
  1467
            unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
himmelma@33175
  1468
          moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
himmelma@33175
  1469
            unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
himmelma@33175
  1470
            apply(rule_tac x="u + w" in exI) apply rule defer 
himmelma@33175
  1471
            apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
himmelma@33175
  1472
          ultimately show ?thesis by auto
himmelma@33175
  1473
        next
himmelma@33175
  1474
          assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
himmelma@33175
  1475
          hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
himmelma@33175
  1476
            unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
himmelma@33175
  1477
          moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
himmelma@33175
  1478
            unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
himmelma@33175
  1479
            apply(rule_tac x="u - w" in exI) apply rule defer 
himmelma@33175
  1480
            apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
himmelma@33175
  1481
          ultimately show ?thesis by auto
himmelma@33175
  1482
        qed
himmelma@33175
  1483
      qed auto
himmelma@33175
  1484
    qed
himmelma@33175
  1485
  qed auto
himmelma@33175
  1486
qed (auto simp add: assms)
himmelma@33175
  1487
himmelma@33175
  1488
lemma simplex_furthest_le:
himmelma@33175
  1489
  fixes s :: "(real ^ _) set"
himmelma@33175
  1490
  assumes "finite s" "s \<noteq> {}"
himmelma@33175
  1491
  shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
himmelma@33175
  1492
proof-
himmelma@33175
  1493
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
himmelma@33175
  1494
  then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
himmelma@33175
  1495
    using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
himmelma@33175
  1496
    unfolding dist_commute[of a] unfolding dist_norm by auto
himmelma@33175
  1497
  thus ?thesis proof(cases "x\<in>s")
himmelma@33175
  1498
    case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
himmelma@33175
  1499
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
himmelma@33175
  1500
    thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
himmelma@33175
  1501
  qed auto
himmelma@33175
  1502
qed
himmelma@33175
  1503
himmelma@33175
  1504
lemma simplex_furthest_le_exists:
himmelma@33175
  1505
  fixes s :: "(real ^ _) set"
himmelma@33175
  1506
  shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
himmelma@33175
  1507
  using simplex_furthest_le[of s] by (cases "s={}")auto
himmelma@33175
  1508
himmelma@33175
  1509
lemma simplex_extremal_le:
himmelma@33175
  1510
  fixes s :: "(real ^ _) set"
himmelma@33175
  1511
  assumes "finite s" "s \<noteq> {}"
himmelma@33175
  1512
  shows "\<exists>u\<in>s. \<exists>v\<in>s. \<forall>x\<in>convex hull s. \<forall>y \<in> convex hull s. norm(x - y) \<le> norm(u - v)"
himmelma@33175
  1513
proof-
himmelma@33175
  1514
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
himmelma@33175
  1515
  then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
himmelma@33175
  1516
    "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
himmelma@33175
  1517
    using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
himmelma@33175
  1518
  thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
himmelma@33175
  1519
    assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
himmelma@33175
  1520
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
himmelma@33175
  1521
    thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
himmelma@33175
  1522
  next
himmelma@33175
  1523
    assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
himmelma@33175
  1524
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
himmelma@33175
  1525
    thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
himmelma@33175
  1526
      by (auto simp add: norm_minus_commute)
himmelma@33175
  1527
  qed auto
himmelma@33175
  1528
qed 
himmelma@33175
  1529
himmelma@33175
  1530
lemma simplex_extremal_le_exists:
himmelma@33175
  1531
  fixes s :: "(real ^ _) set"
himmelma@33175
  1532
  shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
himmelma@33175
  1533
  \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
himmelma@33175
  1534
  using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
himmelma@33175
  1535
himmelma@33175
  1536
subsection {* Closest point of a convex set is unique, with a continuous projection. *}
himmelma@33175
  1537
himmelma@33175
  1538
definition
hoelzl@34291
  1539
  closest_point :: "(real ^ 'n) set \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 'n" where
himmelma@33175
  1540
 "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
himmelma@33175
  1541
himmelma@33175
  1542
lemma closest_point_exists:
himmelma@33175
  1543
  assumes "closed s" "s \<noteq> {}"
himmelma@33175
  1544
  shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
himmelma@33175
  1545
  unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
himmelma@33175
  1546
  using distance_attains_inf[OF assms(1,2), of a] by auto
himmelma@33175
  1547
himmelma@33175
  1548
lemma closest_point_in_set:
himmelma@33175
  1549
  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
himmelma@33175
  1550
  by(meson closest_point_exists)
himmelma@33175
  1551
himmelma@33175
  1552
lemma closest_point_le:
himmelma@33175
  1553
  "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
himmelma@33175
  1554
  using closest_point_exists[of s] by auto
himmelma@33175
  1555
himmelma@33175
  1556
lemma closest_point_self:
himmelma@33175
  1557
  assumes "x \<in> s"  shows "closest_point s x = x"
himmelma@33175
  1558
  unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
himmelma@33175
  1559
  using assms by auto
himmelma@33175
  1560
himmelma@33175
  1561
lemma closest_point_refl:
himmelma@33175
  1562
 "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
himmelma@33175
  1563
  using closest_point_in_set[of s x] closest_point_self[of x s] by auto
himmelma@33175
  1564
himmelma@33175
  1565
(* TODO: move *)
himmelma@33175
  1566
lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
himmelma@33175
  1567
  unfolding norm_eq_sqrt_inner by simp
himmelma@33175
  1568
himmelma@33175
  1569
(* TODO: move *)
himmelma@33175
  1570
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
himmelma@33175
  1571
  unfolding norm_eq_sqrt_inner by simp
himmelma@33175
  1572
hoelzl@34291
  1573
lemma closer_points_lemma: fixes y::"real^'n"
himmelma@33175
  1574
  assumes "inner y z > 0"
himmelma@33175
  1575
  shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
himmelma@33175
  1576
proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
himmelma@33175
  1577
  thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
himmelma@33175
  1578
    fix v assume "0<v" "v \<le> inner y z / inner z z"
himmelma@33175
  1579
    thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
himmelma@33175
  1580
      by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
himmelma@33175
  1581
  qed(rule divide_pos_pos, auto) qed
himmelma@33175
  1582
himmelma@33175
  1583
lemma closer_point_lemma:
hoelzl@34291
  1584
  fixes x y z :: "real ^ 'n"
himmelma@33175
  1585
  assumes "inner (y - x) (z - x) > 0"
himmelma@33175
  1586
  shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
himmelma@33175
  1587
proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
himmelma@33175
  1588
    using closer_points_lemma[OF assms] by auto
himmelma@33175
  1589
  show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
himmelma@33175
  1590
    unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
himmelma@33175
  1591
himmelma@33175
  1592
lemma any_closest_point_dot:
himmelma@33175
  1593
  fixes s :: "(real ^ _) set"
himmelma@33175
  1594
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
himmelma@33175
  1595
  shows "inner (a - x) (y - x) \<le> 0"
himmelma@33175
  1596
proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
himmelma@33175
  1597
  then obtain u where u:"u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
himmelma@33175
  1598
  let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
himmelma@33175
  1599
  thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
himmelma@33175
  1600
himmelma@33175
  1601
(* TODO: move *)
himmelma@33175
  1602
lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> inner x x \<le> a\<twosuperior>"
himmelma@33175
  1603
proof -
himmelma@33175
  1604
  have "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> norm x \<le> a"
himmelma@33175
  1605
    using norm_ge_zero [of x] by arith
himmelma@33175
  1606
  also have "\<dots> \<longleftrightarrow> 0 \<le> a \<and> (norm x)\<twosuperior> \<le> a\<twosuperior>"
himmelma@33175
  1607
    by (auto intro: power_mono dest: power2_le_imp_le)
himmelma@33175
  1608
  also have "\<dots> \<longleftrightarrow> 0 \<le> a \<and> inner x x \<le> a\<twosuperior>"
himmelma@33175
  1609
    unfolding power2_norm_eq_inner ..
himmelma@33175
  1610
  finally show ?thesis .
himmelma@33175
  1611
qed
himmelma@33175
  1612
himmelma@33175
  1613
lemma any_closest_point_unique:
himmelma@33175
  1614
  fixes s :: "(real ^ _) set"
himmelma@33175
  1615
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
himmelma@33175
  1616
  "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
himmelma@33175
  1617
  shows "x = y" using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
himmelma@33175
  1618
  unfolding norm_pths(1) and norm_le_square
himmelma@33175
  1619
  by (auto simp add: algebra_simps)
himmelma@33175
  1620
himmelma@33175
  1621
lemma closest_point_unique:
himmelma@33175
  1622
  assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
himmelma@33175
  1623
  shows "x = closest_point s a"
himmelma@33175
  1624
  using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
himmelma@33175
  1625
  using closest_point_exists[OF assms(2)] and assms(3) by auto
himmelma@33175
  1626
himmelma@33175
  1627
lemma closest_point_dot:
himmelma@33175
  1628
  assumes "convex s" "closed s" "x \<in> s"
himmelma@33175
  1629
  shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
himmelma@33175
  1630
  apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
himmelma@33175
  1631
  using closest_point_exists[OF assms(2)] and assms(3) by auto
himmelma@33175
  1632
himmelma@33175
  1633
lemma closest_point_lt:
himmelma@33175
  1634
  assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
himmelma@33175
  1635
  shows "dist a (closest_point s a) < dist a x"
himmelma@33175
  1636
  apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
himmelma@33175
  1637
  apply(rule closest_point_unique[OF assms(1-3), of a])
himmelma@33175
  1638
  using closest_point_le[OF assms(2), of _ a] by fastsimp
himmelma@33175
  1639
himmelma@33175
  1640
lemma closest_point_lipschitz:
himmelma@33175
  1641
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@33175
  1642
  shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
himmelma@33175
  1643
proof-
himmelma@33175
  1644
  have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
himmelma@33175
  1645
       "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
himmelma@33175
  1646
    apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
himmelma@33175
  1647
    using closest_point_exists[OF assms(2-3)] by auto
himmelma@33175
  1648
  thus ?thesis unfolding dist_norm and norm_le
himmelma@33175
  1649
    using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
himmelma@33175
  1650
    by (simp add: inner_add inner_diff inner_commute) qed
himmelma@33175
  1651
himmelma@33175
  1652
lemma continuous_at_closest_point:
himmelma@33175
  1653
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@33175
  1654
  shows "continuous (at x) (closest_point s)"
himmelma@33175
  1655
  unfolding continuous_at_eps_delta 
himmelma@33175
  1656
  using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
himmelma@33175
  1657
himmelma@33175
  1658
lemma continuous_on_closest_point:
himmelma@33175
  1659
  assumes "convex s" "closed s" "s \<noteq> {}"
himmelma@33175
  1660
  shows "continuous_on t (closest_point s)"
himmelma@33175
  1661
  apply(rule continuous_at_imp_continuous_on) using continuous_at_closest_point[OF assms] by auto
himmelma@33175
  1662
himmelma@33175
  1663
subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
himmelma@33175
  1664
himmelma@33175
  1665
lemma supporting_hyperplane_closed_point:
himmelma@33175
  1666
  fixes s :: "(real ^ _) set"
himmelma@33175
  1667
  assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
himmelma@33175
  1668
  shows "\<exists>a b. \<exists>y\<in>s. inner a z < b \<and> (inner a y = b) \<and> (\<forall>x\<in>s. inner a x \<ge> b)"
himmelma@33175
  1669
proof-
himmelma@33175
  1670
  from distance_attains_inf[OF assms(2-3)] obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x" by auto
himmelma@33175
  1671
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
himmelma@33175
  1672
    apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
himmelma@33175
  1673
    show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
himmelma@33175
  1674
      unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
himmelma@33175
  1675
  next
himmelma@33175
  1676
    fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
himmelma@33175
  1677
      using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
himmelma@33175
  1678
    assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
himmelma@33175
  1679
      "v>0" "v\<le>1" "dist (y + v *\<^sub>R (x - y)) z < dist y z" using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
himmelma@33175
  1680
    thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
himmelma@33175
  1681
  qed auto
himmelma@33175
  1682
qed
himmelma@33175
  1683
himmelma@33175
  1684
lemma separating_hyperplane_closed_point:
himmelma@33175
  1685
  fixes s :: "(real ^ _) set"
himmelma@33175
  1686
  assumes "convex s" "closed s" "z \<notin> s"
himmelma@33175
  1687
  shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
himmelma@33175
  1688
proof(cases "s={}")
himmelma@33175
  1689
  case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
himmelma@33175
  1690
    using less_le_trans[OF _ inner_ge_zero[of z]] by auto
himmelma@33175
  1691
next
himmelma@33175
  1692
  case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
himmelma@33175
  1693
    using distance_attains_inf[OF assms(2) False] by auto
himmelma@33175
  1694
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
himmelma@33175
  1695
    apply rule defer apply rule proof-
himmelma@33175
  1696
    fix x assume "x\<in>s"
himmelma@33175
  1697
    have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
himmelma@33175
  1698
      assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
himmelma@33175
  1699
      then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
himmelma@33175
  1700
      thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
himmelma@33175
  1701
        using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
himmelma@33175
  1702
        using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
himmelma@33175
  1703
    moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
himmelma@33175
  1704
    hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
himmelma@33175
  1705
    ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
himmelma@33175
  1706
      unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
himmelma@33175
  1707
  qed(insert `y\<in>s` `z\<notin>s`, auto)
himmelma@33175
  1708
qed
himmelma@33175
  1709
himmelma@33175
  1710
lemma separating_hyperplane_closed_0:
hoelzl@34291
  1711
  assumes "convex (s::(real^'n) set)" "closed s" "0 \<notin> s"
himmelma@33175
  1712
  shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
himmelma@33175
  1713
  proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
hoelzl@34291
  1714
  case True have "norm ((basis a)::real^'n) = 1" 
himmelma@33175
  1715
    using norm_basis and dimindex_ge_1 by auto
himmelma@33175
  1716
  thus ?thesis apply(rule_tac x="basis a" in exI, rule_tac x=1 in exI) using True by auto
himmelma@33175
  1717
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
himmelma@35542
  1718
    apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
himmelma@33175
  1719
himmelma@33175
  1720
subsection {* Now set-to-set for closed/compact sets. *}
himmelma@33175
  1721
himmelma@33175
  1722
lemma separating_hyperplane_closed_compact:
hoelzl@34291
  1723
  assumes "convex (s::(real^'n) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
himmelma@33175
  1724
  shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
himmelma@33175
  1725
proof(cases "s={}")
himmelma@33175
  1726
  case True
himmelma@33175
  1727
  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@33175
  1728
  obtain z::"real^'n" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
himmelma@33175
  1729
  hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
himmelma@33175
  1730
  then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
himmelma@33175
  1731
    using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
himmelma@33175
  1732
  thus ?thesis using True by auto
himmelma@33175
  1733
next
himmelma@33175
  1734
  case False then obtain y where "y\<in>s" by auto
himmelma@33175
  1735
  obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
himmelma@33175
  1736
    using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
himmelma@33175
  1737
    using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
himmelma@33175
  1738
  hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
paulson@33270
  1739
  def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
himmelma@33175
  1740
  show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
himmelma@33175
  1741
    apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
himmelma@33175
  1742
    from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
himmelma@33175
  1743
      apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
paulson@33270
  1744
    hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
himmelma@33175
  1745
    fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
himmelma@33175
  1746
  next
himmelma@33175
  1747
    fix x assume "x\<in>s" 
paulson@33270
  1748
    hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
himmelma@33175
  1749
      using ab[THEN bspec[where x=x]] by auto
himmelma@33175
  1750
    thus "k + b / 2 < inner a x" using `0 < b` by auto
himmelma@33175
  1751
  qed
himmelma@33175
  1752
qed
himmelma@33175
  1753
himmelma@33175
  1754
lemma separating_hyperplane_compact_closed:
himmelma@33175
  1755
  fixes s :: "(real ^ _) set"
himmelma@33175
  1756
  assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
himmelma@33175
  1757
  shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
himmelma@33175
  1758
proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
himmelma@33175
  1759
    using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
himmelma@33175
  1760
  thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
himmelma@33175
  1761
himmelma@33175
  1762
subsection {* General case without assuming closure and getting non-strict separation. *}
himmelma@33175
  1763
himmelma@33175
  1764
lemma separating_hyperplane_set_0:
hoelzl@34291
  1765
  assumes "convex s" "(0::real^'n) \<notin> s"
himmelma@33175
  1766
  shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
himmelma@33175
  1767
proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> inner c x}"
himmelma@33175
  1768
  have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
himmelma@33175
  1769
    apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
himmelma@33175
  1770
    defer apply(rule,rule,erule conjE) proof-
himmelma@33175
  1771
    fix f assume as:"f \<subseteq> ?k ` s" "finite f"
himmelma@33175
  1772
    obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
himmelma@33175
  1773
    then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
himmelma@33175
  1774
      using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
himmelma@33175
  1775
      using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
himmelma@33175
  1776
      using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
himmelma@33175
  1777
    hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)" apply(rule_tac x="inverse(norm a) *\<^sub>R a" in exI)
himmelma@33175
  1778
       using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
himmelma@33175
  1779
       apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
himmelma@33175
  1780
       by(auto simp add: inner_commute elim!: ballE)
himmelma@33175
  1781
    thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
himmelma@33175
  1782
  qed(insert closed_halfspace_ge, auto)
himmelma@33175
  1783
  then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
himmelma@33175
  1784
  thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
himmelma@33175
  1785
himmelma@33175
  1786
lemma separating_hyperplane_sets:
hoelzl@34291
  1787
  assumes "convex s" "convex (t::(real^'n) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
himmelma@33175
  1788
  shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
himmelma@33175
  1789
proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
paulson@33270
  1790
  obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x" 
paulson@33270
  1791
    using assms(3-5) by auto 
paulson@33270
  1792
  hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
paulson@33270
  1793
    by (force simp add: inner_diff)
paulson@33270
  1794
  thus ?thesis
paulson@33270
  1795
    apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
paulson@33270
  1796
    apply auto
paulson@33270
  1797
    apply (rule Sup[THEN isLubD2]) 
paulson@33270
  1798
    prefer 4
paulson@33270
  1799
    apply (rule Sup_least) 
paulson@33270
  1800
     using assms(3-5) apply (auto simp add: setle_def)
paulson@33270
  1801
    apply (metis COMBC_def Collect_def Collect_mem_eq) 
paulson@33270
  1802
    done
paulson@33270
  1803
qed
himmelma@33175
  1804
himmelma@33175
  1805
subsection {* More convexity generalities. *}
himmelma@33175
  1806
himmelma@33175
  1807
lemma convex_closure:
himmelma@33175
  1808
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1809
  assumes "convex s" shows "convex(closure s)"
himmelma@33175
  1810
  unfolding convex_def Ball_def closure_sequential
himmelma@33175
  1811
  apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
himmelma@33175
  1812
  apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
himmelma@33175
  1813
  apply(rule assms[unfolded convex_def, rule_format]) prefer 6
himmelma@33175
  1814
  apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
himmelma@33175
  1815
himmelma@33175
  1816
lemma convex_interior:
himmelma@33175
  1817
  fixes s :: "'a::real_normed_vector set"
himmelma@33175
  1818
  assumes "convex s" shows "convex(interior s)"
himmelma@33175
  1819
  unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
himmelma@33175
  1820
  fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
himmelma@33175
  1821
  fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
himmelma@33175
  1822
  show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
himmelma@33175
  1823
    apply rule unfolding subset_eq defer apply rule proof-
himmelma@33175
  1824
    fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
himmelma@33175
  1825
    hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
himmelma@33175
  1826
      apply(rule_tac assms[unfolded convex_alt, rule_format])
himmelma@33175
  1827
      using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
himmelma@33175
  1828
    thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
himmelma@33175
  1829
hoelzl@34964
  1830
lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
himmelma@33175
  1831
  using hull_subset[of s convex] convex_hull_empty by auto
himmelma@33175
  1832
himmelma@33175
  1833
subsection {* Moving and scaling convex hulls. *}
himmelma@33175
  1834
himmelma@33175
  1835
lemma convex_hull_translation_lemma:
himmelma@33175
  1836
  "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
himmelma@33175
  1837
  apply(rule hull_minimal, rule image_mono, rule hull_subset) unfolding mem_def
himmelma@33175
  1838
  using convex_translation[OF convex_convex_hull, of a s] by assumption
himmelma@33175
  1839
himmelma@33175
  1840
lemma convex_hull_bilemma: fixes neg
himmelma@33175
  1841
  assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
himmelma@33175
  1842
  shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
himmelma@33175
  1843
  \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
himmelma@33175
  1844
  using assms by(metis subset_antisym) 
himmelma@33175
  1845
himmelma@33175
  1846
lemma convex_hull_translation:
himmelma@33175
  1847
  "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
himmelma@33175
  1848
  apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
himmelma@33175
  1849
himmelma@33175
  1850
lemma convex_hull_scaling_lemma:
himmelma@33175
  1851
 "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
himmelma@33175
  1852
  apply(rule hull_minimal, rule image_mono, rule hull_subset)
himmelma@33175
  1853
  unfolding mem_def by(rule convex_scaling, rule convex_convex_hull)
himmelma@33175
  1854
himmelma@33175
  1855
lemma convex_hull_scaling:
himmelma@33175
  1856
  "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
himmelma@33175
  1857
  apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
himmelma@33175
  1858
  unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv convex_hull_eq_empty)
himmelma@33175
  1859
himmelma@33175
  1860
lemma convex_hull_affinity:
himmelma@33175
  1861
  "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
himmelma@33175
  1862
  unfolding image_image[THEN sym] convex_hull_scaling convex_hull_translation  ..
himmelma@33175
  1863
himmelma@33175
  1864
subsection {* Convex set as intersection of halfspaces. *}
himmelma@33175
  1865
himmelma@33175
  1866
lemma convex_halfspace_intersection:
himmelma@33175
  1867
  fixes s :: "(real ^ _) set"
himmelma@33175
  1868
  assumes "closed s" "convex s"
himmelma@33175
  1869
  shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
himmelma@33175
  1870
  apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
himmelma@33175
  1871
  fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
himmelma@33175
  1872
  hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
himmelma@33175
  1873
  thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
himmelma@33175
  1874
    apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
himmelma@33175
  1875
qed auto
himmelma@33175
  1876
himmelma@33175
  1877
subsection {* Radon's theorem (from Lars Schewe). *}
himmelma@33175
  1878
himmelma@33175
  1879
lemma radon_ex_lemma:
himmelma@33175
  1880
  assumes "finite c" "affine_dependent c"
himmelma@33175
  1881
  shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
himmelma@33175
  1882
proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
himmelma@33175
  1883
  thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
himmelma@33175
  1884
    and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
himmelma@33175
  1885
himmelma@33175
  1886
lemma radon_s_lemma:
himmelma@33175
  1887
  assumes "finite s" "setsum f s = (0::real)"
himmelma@33175
  1888
  shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
himmelma@33175
  1889
proof- have *:"\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" by auto
himmelma@33175
  1890
  show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
himmelma@33175
  1891
    using assms(2) by assumption qed
himmelma@33175
  1892
himmelma@33175
  1893
lemma radon_v_lemma:
himmelma@34289
  1894
  assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::real^_)"
himmelma@33175
  1895
  shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
himmelma@33175
  1896
proof-
himmelma@33175
  1897
  have *:"\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" using assms(3) by auto 
himmelma@33175
  1898
  show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
himmelma@33175
  1899
    using assms(2) by assumption qed
himmelma@33175
  1900
himmelma@33175
  1901
lemma radon_partition:
himmelma@33175
  1902
  assumes "finite c" "affine_dependent c"
himmelma@33175
  1903
  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
himmelma@33175
  1904
  obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
himmelma@33175
  1905
  have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
himmelma@33175
  1906
  def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
himmelma@33175
  1907
  have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
himmelma@33175
  1908
    case False hence "u v < 0" by auto
himmelma@33175
  1909
    thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
himmelma@33175
  1910
      case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
himmelma@33175
  1911
    next
himmelma@33175
  1912
      case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
himmelma@33175
  1913
      thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
himmelma@33175
  1914
  qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
himmelma@33175
  1915
himmelma@33175
  1916
  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@33175
  1917
  moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
himmelma@33175
  1918
    "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
himmelma@33175
  1919
    using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
himmelma@33175
  1920
  hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
himmelma@33175
  1921
   "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)" 
himmelma@33175
  1922
    unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
himmelma@33175
  1923
  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
himmelma@33175
  1924
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
himmelma@33175
  1925
himmelma@33175
  1926
  ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
himmelma@33175
  1927
    apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
himmelma@33175
  1928
    using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
himmelma@33175
  1929
    by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
himmelma@33175
  1930
  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
himmelma@33175
  1931
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
himmelma@33175
  1932
  hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
himmelma@33175
  1933
    apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
himmelma@33175
  1934
    using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
himmelma@33175
  1935
    by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
himmelma@33175
  1936
  ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
himmelma@33175
  1937
qed
himmelma@33175
  1938
himmelma@33175
  1939
lemma radon: assumes "affine_dependent c"
himmelma@33175
  1940
  obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
himmelma@33175
  1941
proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
himmelma@33175
  1942
  hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
himmelma@33175
  1943
  from radon_partition[OF *] guess m .. then guess p ..
himmelma@33175
  1944
  thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
himmelma@33175
  1945
himmelma@33175
  1946
subsection {* Helly's theorem. *}
himmelma@33175
  1947
hoelzl@34291
  1948
lemma helly_induct: fixes f::"(real^'n) set set"
hoelzl@33715
  1949
  assumes "card f = n" "n \<ge> CARD('n) + 1"
himmelma@33175
  1950
  "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
himmelma@33175
  1951
  shows "\<Inter> f \<noteq> {}"
hoelzl@33715
  1952
using assms proof(induct n arbitrary: f)
himmelma@33175
  1953
case (Suc n)
hoelzl@33715
  1954
have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
hoelzl@33715
  1955
show "\<Inter> f \<noteq> {}" apply(cases "n = CARD('n)") apply(rule Suc(5)[rule_format])
hoelzl@33715
  1956
  unfolding `card f = Suc n` proof-
himmelma@33175
  1957
  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
hoelzl@33715
  1958
    apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
hoelzl@33715
  1959
    defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
himmelma@33175
  1960
  then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
himmelma@33175
  1961
  show ?thesis proof(cases "inj_on X f")
himmelma@33175
  1962
    case False then obtain s t where st:"s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t" unfolding inj_on_def by auto
himmelma@33175
  1963
    hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
himmelma@33175
  1964
    show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
himmelma@33175
  1965
      apply(rule, rule X[rule_format]) using X st by auto
himmelma@33175
  1966
  next case True then obtain m p where mp:"m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
himmelma@33175
  1967
      using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
hoelzl@33715
  1968
      unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
himmelma@33175
  1969
    have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
himmelma@33175
  1970
    then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
himmelma@33175
  1971
    hence "f \<union> (g \<union> h) = f" by auto
himmelma@33175
  1972
    hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
himmelma@33175
  1973
      unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
himmelma@33175
  1974
    have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
himmelma@33175
  1975
    have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
hoelzl@33715
  1976
      apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  unfolding mem_def unfolding subset_eq
himmelma@33175
  1977
      apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
himmelma@33175
  1978
      fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
himmelma@33175
  1979
      thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
himmelma@33175
  1980
      fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
himmelma@33175
  1981
      thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
himmelma@33175
  1982
    qed(auto)
himmelma@33175
  1983
    thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
himmelma@33175
  1984
qed(insert dimindex_ge_1, auto) qed(auto)
himmelma@33175
  1985
hoelzl@34291
  1986
lemma helly: fixes f::"(real^'n) set set"
hoelzl@33715
  1987
  assumes "card f \<ge> CARD('n) + 1" "\<forall>s\<in>f. convex s"
himmelma@33175
  1988
          "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
himmelma@33175
  1989
  shows "\<Inter> f \<noteq>{}"
hoelzl@33715
  1990
  apply(rule helly_induct) using assms by auto
himmelma@33175
  1991
himmelma@33175
  1992
subsection {* Convex hull is "preserved" by a linear function. *}
himmelma@33175
  1993
himmelma@33175
  1994
lemma convex_hull_linear_image:
himmelma@33175
  1995
  assumes "bounded_linear f"
himmelma@33175
  1996
  shows "f ` (convex hull s) = convex hull (f ` s)"
himmelma@33175
  1997
  apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
himmelma@33175
  1998
  apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
himmelma@33175
  1999
  apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
himmelma@33175
  2000
proof-
himmelma@33175
  2001
  interpret f: bounded_linear f by fact
himmelma@33175
  2002
  show "convex {x. f x \<in> convex hull f ` s}" 
himmelma@33175
  2003
  unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
himmelma@33175
  2004
  interpret f: bounded_linear f by fact
himmelma@33175
  2005
  show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
himmelma@33175
  2006
    unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
himmelma@33175
  2007
qed auto
himmelma@33175
  2008
himmelma@33175
  2009
lemma in_convex_hull_linear_image:
himmelma@33175
  2010
  assumes "bounded_linear f" "x \<in> convex hull s"
himmelma@33175
  2011
  shows "(f x) \<in> convex hull (f ` s)"
himmelma@33175
  2012
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
himmelma@33175
  2013
himmelma@33175
  2014
subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
himmelma@33175
  2015
himmelma@33175
  2016
lemma compact_frontier_line_lemma:
himmelma@33175
  2017
  fixes s :: "(real ^ _) set"
himmelma@33175
  2018
  assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
himmelma@33175
  2019
  obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
himmelma@33175
  2020
proof-
himmelma@33175
  2021
  obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
himmelma@33175
  2022
  let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
himmelma@33175
  2023
  have A:"?A = (\<lambda>u. dest_vec1 u *\<^sub>R x) ` {0 .. vec1 (b / norm x)}"
himmelma@33175
  2024
    unfolding image_image[of "\<lambda>u. u *\<^sub>R x" "\<lambda>x. dest_vec1 x", THEN sym]
himmelma@33175
  2025
    unfolding dest_vec1_inverval vec1_dest_vec1 by auto
himmelma@33175
  2026
  have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
himmelma@33175
  2027
    apply(rule, rule continuous_vmul)
himmelma@33175
  2028
    apply (rule continuous_dest_vec1)
himmelma@33175
  2029
    apply(rule continuous_at_id) by(rule compact_interval)
himmelma@33175
  2030
  moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
himmelma@33175
  2031
    unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
himmelma@33175
  2032
  ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
himmelma@33175
  2033
    "y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
himmelma@33175
  2034
himmelma@33175
  2035
  have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
himmelma@33175
  2036
  { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
himmelma@33175
  2037
    hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
himmelma@33175
  2038
      using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
himmelma@33175
  2039
    hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
himmelma@33175
  2040
      apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
himmelma@33175
  2041
      using as(1) `u\<ge>0` by(auto simp add:field_simps) 
himmelma@33175
  2042
    hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
himmelma@33175
  2043
  } note u_max = this
himmelma@33175
  2044
himmelma@33175
  2045
  have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[THEN sym]
himmelma@33175
  2046
    prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
himmelma@33175
  2047
    fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
himmelma@33175
  2048
    hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
himmelma@33175
  2049
    thus False using u_max[OF _ as] by auto
himmelma@33175
  2050
  qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
himmelma@33175
  2051
  thus ?thesis apply(rule_tac that[of u]) apply(rule obt(1), assumption)
himmelma@33175
  2052
    apply(rule,rule,rule ccontr) apply(rule u_max) by auto qed
himmelma@33175
  2053
himmelma@33175
  2054
lemma starlike_compact_projective:
hoelzl@34291
  2055
  assumes "compact s" "cball (0::real^'n) 1 \<subseteq> s "
himmelma@33175
  2056
  "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
hoelzl@34291
  2057
  shows "s homeomorphic (cball (0::real^'n) 1)"
himmelma@33175
  2058
proof-
himmelma@33175
  2059
  have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
himmelma@33175
  2060
  def pi \<equiv> "\<lambda>x::real^'n. inverse (norm x) *\<^sub>R x"
himmelma@33175
  2061
  have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
himmelma@33175
  2062
    using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
himmelma@33175
  2063
  have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
himmelma@33175
  2064
himmelma@33175
  2065
  have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
himmelma@33175
  2066
    apply rule unfolding pi_def
himmelma@33175
  2067
    apply (rule continuous_mul)
himmelma@33175
  2068
    apply (rule continuous_at_inv[unfolded o_def])
himmelma@33175
  2069
    apply (rule continuous_at_norm)
himmelma@33175
  2070
    apply simp
himmelma@33175
  2071
    apply (rule continuous_at_id)
himmelma@33175
  2072
    done
himmelma@33175
  2073
  def sphere \<equiv> "{x::real^'n. norm x = 1}"
himmelma@33175
  2074
  have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x" unfolding pi_def sphere_def by auto
himmelma@33175
  2075
himmelma@33175
  2076
  have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
himmelma@33175
  2077
  have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
himmelma@33175
  2078
    fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
himmelma@33175
  2079
    hence "x\<noteq>0" using `0\<notin>frontier s` by auto
himmelma@33175
  2080
    obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
himmelma@33175
  2081
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
himmelma@33175
  2082
    have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
himmelma@33175
  2083
      assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
himmelma@33175
  2084
      assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
himmelma@33175
  2085
        using v and x and fs unfolding inverse_less_1_iff by auto qed
himmelma@33175
  2086
    show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
himmelma@33175
  2087
      assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
himmelma@33175
  2088
        using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
himmelma@33175
  2089
himmelma@33175
  2090
  have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
himmelma@33175
  2091
    apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
himmelma@33175
  2092
    apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_ext,rule) 
himmelma@33175
  2093
    unfolding inj_on_def prefer 3 apply(rule,rule,rule)
himmelma@33175
  2094
  proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
himmelma@33175
  2095
    thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
himmelma@33175
  2096
  next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
himmelma@33175
  2097
    then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
himmelma@33175
  2098
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
himmelma@33175
  2099
    thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
himmelma@33175
  2100
  next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
himmelma@33175
  2101
    hence xys:"x\<in>s" "y\<in>s" using fs by auto
himmelma@33175
  2102
    from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
himmelma@33175
  2103
    from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
himmelma@33175
  2104
    from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto 
himmelma@33175
  2105
    have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
himmelma@33175
  2106
      unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
himmelma@33175
  2107
    hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
himmelma@33175
  2108
      using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
himmelma@33175
  2109
      using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
himmelma@33175
  2110
      using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
himmelma@33175
  2111
    thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
himmelma@33175
  2112
  qed(insert `0 \<notin> frontier s`, auto)
himmelma@33175
  2113
  then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
himmelma@33175
  2114
    "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
himmelma@33175
  2115
  
himmelma@33175
  2116
  have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
himmelma@33175
  2117
    apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
himmelma@33175
  2118
himmelma@33175
  2119
  { fix x assume as:"x \<in> cball (0::real^'n) 1"
himmelma@33175
  2120
    have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
himmelma@33175
  2121
      case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
himmelma@33175
  2122
      thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
himmelma@33175
  2123
        apply(rule_tac fs[unfolded subset_eq, rule_format])
himmelma@33175
  2124
        unfolding surf(5)[THEN sym] by auto
himmelma@33175
  2125
    next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
himmelma@33175
  2126
        unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
himmelma@33175
  2127
himmelma@33175
  2128
  { fix x assume "x\<in>s"
himmelma@33175
  2129
    hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
himmelma@33175
  2130
      case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
himmelma@33175
  2131
    next let ?a = "inverse (norm (surf (pi x)))"
himmelma@33175
  2132
      case False hence invn:"inverse (norm x) \<noteq> 0" by auto
himmelma@33175
  2133
      from False have pix:"pi x\<in>sphere" using pi(1) by auto
himmelma@33175
  2134
      hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
himmelma@33175
  2135
      hence **:"norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
himmelma@33175
  2136
      hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
himmelma@33175
  2137
        apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
himmelma@33175
  2138
      have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
himmelma@33175
  2139
      hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
himmelma@33175
  2140
        unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
himmelma@33175
  2141
      moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))" 
himmelma@33175
  2142
        unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
himmelma@33175
  2143
      moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
himmelma@33175
  2144
      hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
himmelma@33175
  2145
        using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
himmelma@33175
  2146
        using False `x\<in>s` by(auto simp add:field_simps)
himmelma@33175
  2147
      ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
himmelma@33175
  2148
        apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
himmelma@33175
  2149
        unfolding pi(2)[OF `?a > 0`] by auto
himmelma@33175
  2150
    qed } note hom2 = this
himmelma@33175
  2151
himmelma@33175
  2152
  show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
himmelma@33175
  2153
    apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
himmelma@33175
  2154
    prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
himmelma@33175
  2155
    fix x::"real^'n" assume as:"x \<in> cball 0 1"
himmelma@33175
  2156
    thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
himmelma@33175
  2157
      case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
himmelma@33175
  2158
        using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
himmelma@33175
  2159
    next guess a using UNIV_witness[where 'a = 'n] ..
himmelma@33175
  2160
      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@33175
  2161
      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)
himmelma@33175
  2162
        unfolding Ball_def mem_cball dist_norm by (auto simp add: norm_basis[unfolded One_nat_def])
himmelma@33175
  2163
      case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
himmelma@33175
  2164
        apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
himmelma@33175
  2165
        unfolding norm_0 scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
himmelma@33175
  2166
        fix e and x::"real^'n" assume as:"norm x < e / B" "0 < norm x" "0<e"
himmelma@33175
  2167
        hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
himmelma@33175
  2168
        hence "norm (surf (pi x)) \<le> B" using B fs by auto
himmelma@33175
  2169
        hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
himmelma@33175
  2170
        also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
himmelma@33175
  2171
        also have "\<dots> = e" using `B>0` by auto
himmelma@33175