author immler Mon Jan 07 14:06:54 2019 +0100 (4 months ago) changeset 69619 3f7d8e05e0f2 parent 69618 2be1baf40351 child 69620 19d8a59481db
split off Convex.thy: material that does not require Topology_Euclidean_Space
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Analysis/Convex.thy	Mon Jan 07 14:06:54 2019 +0100
1.3 @@ -0,0 +1,4195 @@
1.4 +(* Title:      HOL/Analysis/Convex_Euclidean_Space.thy
1.5 +   Author:     L C Paulson, University of Cambridge
1.6 +   Author:     Robert Himmelmann, TU Muenchen
1.7 +   Author:     Bogdan Grechuk, University of Edinburgh
1.8 +   Author:     Armin Heller, TU Muenchen
1.9 +   Author:     Johannes Hoelzl, TU Muenchen
1.10 +*)
1.11 +
1.12 +section \<open>Convex Sets and Functions\<close>
1.13 +
1.14 +theory Convex
1.15 +imports
1.16 +  Linear_Algebra
1.17 +  "HOL-Library.Set_Algebras"
1.18 +begin
1.19 +
1.20 +lemma substdbasis_expansion_unique:
1.21 +  assumes d: "d \<subseteq> Basis"
1.22 +  shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
1.23 +    (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
1.24 +proof -
1.25 +  have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
1.26 +    by auto
1.27 +  have **: "finite d"
1.28 +    by (auto intro: finite_subset[OF assms])
1.29 +  have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
1.30 +    using d
1.31 +    by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
1.32 +  show ?thesis
1.33 +    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
1.34 +qed
1.35 +
1.36 +lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
1.37 +  by (rule independent_mono[OF independent_Basis])
1.38 +
1.39 +lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
1.40 +  by (rule ccontr) auto
1.41 +
1.42 +lemma subset_translation_eq [simp]:
1.43 +    fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
1.44 +  by auto
1.45 +
1.46 +lemma translate_inj_on:
1.47 +  fixes A :: "'a::ab_group_add set"
1.48 +  shows "inj_on (\<lambda>x. a + x) A"
1.49 +  unfolding inj_on_def by auto
1.50 +
1.51 +lemma translation_assoc:
1.52 +  fixes a b :: "'a::ab_group_add"
1.53 +  shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
1.54 +  by auto
1.55 +
1.56 +lemma translation_invert:
1.57 +  fixes a :: "'a::ab_group_add"
1.58 +  assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
1.59 +  shows "A = B"
1.60 +proof -
1.61 +  have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
1.62 +    using assms by auto
1.63 +  then show ?thesis
1.64 +    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
1.65 +qed
1.66 +
1.67 +lemma translation_galois:
1.68 +  fixes a :: "'a::ab_group_add"
1.69 +  shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
1.70 +  using translation_assoc[of "-a" a S]
1.71 +  apply auto
1.72 +  using translation_assoc[of a "-a" T]
1.73 +  apply auto
1.74 +  done
1.75 +
1.76 +lemma translation_inverse_subset:
1.77 +  assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
1.78 +  shows "V \<le> ((\<lambda>x. a + x) ` S)"
1.79 +proof -
1.80 +  {
1.81 +    fix x
1.82 +    assume "x \<in> V"
1.83 +    then have "x-a \<in> S" using assms by auto
1.84 +    then have "x \<in> {a + v |v. v \<in> S}"
1.85 +      apply auto
1.86 +      apply (rule exI[of _ "x-a"], simp)
1.87 +      done
1.88 +    then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
1.89 +  }
1.90 +  then show ?thesis by auto
1.91 +qed
1.92 +
1.93 +subsection \<open>Convexity\<close>
1.94 +
1.95 +definition%important convex :: "'a::real_vector set \<Rightarrow> bool"
1.96 +  where "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)"
1.97 +
1.98 +lemma convexI:
1.99 +  assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
1.100 +  shows "convex s"
1.101 +  using assms unfolding convex_def by fast
1.102 +
1.103 +lemma convexD:
1.104 +  assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
1.105 +  shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
1.106 +  using assms unfolding convex_def by fast
1.107 +
1.108 +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)"
1.109 +  (is "_ \<longleftrightarrow> ?alt")
1.110 +proof
1.111 +  show "convex s" if alt: ?alt
1.112 +  proof -
1.113 +    {
1.114 +      fix x y and u v :: real
1.115 +      assume mem: "x \<in> s" "y \<in> s"
1.116 +      assume "0 \<le> u" "0 \<le> v"
1.117 +      moreover
1.118 +      assume "u + v = 1"
1.119 +      then have "u = 1 - v" by auto
1.120 +      ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
1.121 +        using alt [rule_format, OF mem] by auto
1.122 +    }
1.123 +    then show ?thesis
1.124 +      unfolding convex_def by auto
1.125 +  qed
1.126 +  show ?alt if "convex s"
1.127 +    using that by (auto simp: convex_def)
1.128 +qed
1.129 +
1.130 +lemma convexD_alt:
1.131 +  assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
1.132 +  shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
1.133 +  using assms unfolding convex_alt by auto
1.134 +
1.135 +lemma mem_convex_alt:
1.136 +  assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
1.137 +  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
1.138 +  apply (rule convexD)
1.139 +  using assms
1.141 +  done
1.142 +
1.143 +lemma convex_empty[intro,simp]: "convex {}"
1.144 +  unfolding convex_def by simp
1.145 +
1.146 +lemma convex_singleton[intro,simp]: "convex {a}"
1.147 +  unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
1.148 +
1.149 +lemma convex_UNIV[intro,simp]: "convex UNIV"
1.150 +  unfolding convex_def by auto
1.151 +
1.152 +lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
1.153 +  unfolding convex_def by auto
1.154 +
1.155 +lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
1.156 +  unfolding convex_def by auto
1.157 +
1.158 +lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
1.159 +  unfolding convex_def by auto
1.160 +
1.161 +lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
1.162 +  unfolding convex_def by auto
1.163 +
1.164 +lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
1.165 +  unfolding convex_def
1.166 +  by (auto simp: inner_add intro!: convex_bound_le)
1.167 +
1.168 +lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
1.169 +proof -
1.170 +  have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
1.171 +    by auto
1.172 +  show ?thesis
1.173 +    unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
1.174 +qed
1.175 +
1.176 +lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
1.177 +proof -
1.178 +  have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
1.179 +    by auto
1.180 +  show ?thesis
1.181 +    unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
1.182 +qed
1.183 +
1.184 +lemma convex_hyperplane: "convex {x. inner a x = b}"
1.185 +proof -
1.186 +  have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
1.187 +    by auto
1.188 +  show ?thesis using convex_halfspace_le convex_halfspace_ge
1.189 +    by (auto intro!: convex_Int simp: *)
1.190 +qed
1.191 +
1.192 +lemma convex_halfspace_lt: "convex {x. inner a x < b}"
1.193 +  unfolding convex_def
1.194 +  by (auto simp: convex_bound_lt inner_add)
1.195 +
1.196 +lemma convex_halfspace_gt: "convex {x. inner a x > b}"
1.197 +  using convex_halfspace_lt[of "-a" "-b"] by auto
1.198 +
1.199 +lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
1.200 +  using convex_halfspace_ge[of b "1::complex"] by simp
1.201 +
1.202 +lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
1.203 +  using convex_halfspace_le[of "1::complex" b] by simp
1.204 +
1.205 +lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
1.206 +  using convex_halfspace_ge[of b \<i>] by simp
1.207 +
1.208 +lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
1.209 +  using convex_halfspace_le[of \<i> b] by simp
1.210 +
1.211 +lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
1.212 +  using convex_halfspace_gt[of b "1::complex"] by simp
1.213 +
1.214 +lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
1.215 +  using convex_halfspace_lt[of "1::complex" b] by simp
1.216 +
1.217 +lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
1.218 +  using convex_halfspace_gt[of b \<i>] by simp
1.219 +
1.220 +lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
1.221 +  using convex_halfspace_lt[of \<i> b] by simp
1.222 +
1.223 +lemma convex_real_interval [iff]:
1.224 +  fixes a b :: "real"
1.225 +  shows "convex {a..}" and "convex {..b}"
1.226 +    and "convex {a<..}" and "convex {..<b}"
1.227 +    and "convex {a..b}" and "convex {a<..b}"
1.228 +    and "convex {a..<b}" and "convex {a<..<b}"
1.229 +proof -
1.230 +  have "{a..} = {x. a \<le> inner 1 x}"
1.231 +    by auto
1.232 +  then show 1: "convex {a..}"
1.233 +    by (simp only: convex_halfspace_ge)
1.234 +  have "{..b} = {x. inner 1 x \<le> b}"
1.235 +    by auto
1.236 +  then show 2: "convex {..b}"
1.237 +    by (simp only: convex_halfspace_le)
1.238 +  have "{a<..} = {x. a < inner 1 x}"
1.239 +    by auto
1.240 +  then show 3: "convex {a<..}"
1.241 +    by (simp only: convex_halfspace_gt)
1.242 +  have "{..<b} = {x. inner 1 x < b}"
1.243 +    by auto
1.244 +  then show 4: "convex {..<b}"
1.245 +    by (simp only: convex_halfspace_lt)
1.246 +  have "{a..b} = {a..} \<inter> {..b}"
1.247 +    by auto
1.248 +  then show "convex {a..b}"
1.249 +    by (simp only: convex_Int 1 2)
1.250 +  have "{a<..b} = {a<..} \<inter> {..b}"
1.251 +    by auto
1.252 +  then show "convex {a<..b}"
1.253 +    by (simp only: convex_Int 3 2)
1.254 +  have "{a..<b} = {a..} \<inter> {..<b}"
1.255 +    by auto
1.256 +  then show "convex {a..<b}"
1.257 +    by (simp only: convex_Int 1 4)
1.258 +  have "{a<..<b} = {a<..} \<inter> {..<b}"
1.259 +    by auto
1.260 +  then show "convex {a<..<b}"
1.261 +    by (simp only: convex_Int 3 4)
1.262 +qed
1.263 +
1.264 +lemma convex_Reals: "convex \<real>"
1.265 +  by (simp add: convex_def scaleR_conv_of_real)
1.266 +
1.267 +
1.268 +subsection%unimportant \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
1.269 +
1.270 +lemma convex_sum:
1.271 +  fixes C :: "'a::real_vector set"
1.272 +  assumes "finite s"
1.273 +    and "convex C"
1.274 +    and "(\<Sum> i \<in> s. a i) = 1"
1.275 +  assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
1.276 +    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
1.277 +  shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
1.278 +  using assms(1,3,4,5)
1.279 +proof (induct arbitrary: a set: finite)
1.280 +  case empty
1.281 +  then show ?case by simp
1.282 +next
1.283 +  case (insert i s) note IH = this(3)
1.284 +  have "a i + sum a s = 1"
1.285 +    and "0 \<le> a i"
1.286 +    and "\<forall>j\<in>s. 0 \<le> a j"
1.287 +    and "y i \<in> C"
1.288 +    and "\<forall>j\<in>s. y j \<in> C"
1.289 +    using insert.hyps(1,2) insert.prems by simp_all
1.290 +  then have "0 \<le> sum a s"
1.291 +    by (simp add: sum_nonneg)
1.292 +  have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
1.293 +  proof (cases "sum a s = 0")
1.294 +    case True
1.295 +    with \<open>a i + sum a s = 1\<close> have "a i = 1"
1.296 +      by simp
1.297 +    from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
1.298 +      by simp
1.299 +    show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
1.300 +      by simp
1.301 +  next
1.302 +    case False
1.303 +    with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
1.304 +      by simp
1.305 +    then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
1.306 +      using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
1.307 +      by (simp add: IH sum_divide_distrib [symmetric])
1.308 +    from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
1.309 +      and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
1.310 +    have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
1.311 +      by (rule convexD)
1.312 +    then show ?thesis
1.313 +      by (simp add: scaleR_sum_right False)
1.314 +  qed
1.315 +  then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
1.316 +    by simp
1.317 +qed
1.318 +
1.319 +lemma convex:
1.320 +  "convex s \<longleftrightarrow> (\<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> (sum u {1..k} = 1)
1.321 +      \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
1.322 +proof safe
1.323 +  fix k :: nat
1.324 +  fix u :: "nat \<Rightarrow> real"
1.325 +  fix x
1.326 +  assume "convex s"
1.327 +    "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
1.328 +    "sum u {1..k} = 1"
1.329 +  with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
1.330 +    by auto
1.331 +next
1.332 +  assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1
1.333 +    \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
1.334 +  {
1.335 +    fix \<mu> :: real
1.336 +    fix x y :: 'a
1.337 +    assume xy: "x \<in> s" "y \<in> s"
1.338 +    assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
1.339 +    let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
1.340 +    let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
1.341 +    have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
1.342 +      by auto
1.343 +    then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
1.344 +      by simp
1.345 +    then have "sum ?u {1 .. 2} = 1"
1.346 +      using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
1.347 +      by auto
1.348 +    with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
1.349 +      using mu xy by auto
1.350 +    have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
1.351 +      using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
1.352 +    from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
1.353 +    have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
1.354 +      by auto
1.355 +    then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
1.356 +      using s by (auto simp: add.commute)
1.357 +  }
1.358 +  then show "convex s"
1.359 +    unfolding convex_alt by auto
1.360 +qed
1.361 +
1.362 +
1.363 +lemma convex_explicit:
1.364 +  fixes s :: "'a::real_vector set"
1.365 +  shows "convex s \<longleftrightarrow>
1.366 +    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
1.367 +proof safe
1.368 +  fix t
1.369 +  fix u :: "'a \<Rightarrow> real"
1.370 +  assume "convex s"
1.371 +    and "finite t"
1.372 +    and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
1.373 +  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
1.374 +    using convex_sum[of t s u "\<lambda> x. x"] by auto
1.375 +next
1.376 +  assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
1.377 +    sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
1.378 +  show "convex s"
1.379 +    unfolding convex_alt
1.380 +  proof safe
1.381 +    fix x y
1.382 +    fix \<mu> :: real
1.383 +    assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
1.384 +    show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
1.385 +    proof (cases "x = y")
1.386 +      case False
1.387 +      then show ?thesis
1.388 +        using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
1.389 +        by auto
1.390 +    next
1.391 +      case True
1.392 +      then show ?thesis
1.393 +        using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
1.394 +        by (auto simp: field_simps real_vector.scale_left_diff_distrib)
1.395 +    qed
1.396 +  qed
1.397 +qed
1.398 +
1.399 +lemma convex_finite:
1.400 +  assumes "finite s"
1.401 +  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
1.402 +  unfolding convex_explicit
1.403 +  apply safe
1.404 +  subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
1.405 +  subgoal for t u
1.406 +  proof -
1.407 +    have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
1.408 +      by simp
1.409 +    assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
1.410 +    assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
1.411 +    assume "t \<subseteq> s"
1.412 +    then have "s \<inter> t = t" by auto
1.413 +    with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
1.414 +      by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
1.415 +  qed
1.416 +  done
1.417 +
1.418 +
1.419 +subsection \<open>Functions that are convex on a set\<close>
1.420 +
1.421 +definition%important convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
1.422 +  where "convex_on s f \<longleftrightarrow>
1.423 +    (\<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)"
1.424 +
1.425 +lemma convex_onI [intro?]:
1.426 +  assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
1.427 +    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
1.428 +  shows "convex_on A f"
1.429 +  unfolding convex_on_def
1.430 +proof clarify
1.431 +  fix x y
1.432 +  fix u v :: real
1.433 +  assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
1.434 +  from A(5) have [simp]: "v = 1 - u"
1.435 +    by (simp add: algebra_simps)
1.436 +  from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
1.437 +    using assms[of u y x]
1.438 +    by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
1.439 +qed
1.440 +
1.441 +lemma convex_on_linorderI [intro?]:
1.442 +  fixes A :: "('a::{linorder,real_vector}) set"
1.443 +  assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
1.444 +    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
1.445 +  shows "convex_on A f"
1.446 +proof
1.447 +  fix x y
1.448 +  fix t :: real
1.449 +  assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
1.450 +  with assms [of t x y] assms [of "1 - t" y x]
1.451 +  show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
1.452 +    by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
1.453 +qed
1.454 +
1.455 +lemma convex_onD:
1.456 +  assumes "convex_on A f"
1.457 +  shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
1.458 +    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
1.459 +  using assms by (auto simp: convex_on_def)
1.460 +
1.461 +lemma convex_onD_Icc:
1.462 +  assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
1.463 +  shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
1.464 +    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
1.465 +  using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
1.466 +
1.467 +lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
1.468 +  unfolding convex_on_def by auto
1.469 +
1.471 +  assumes "convex_on s f"
1.472 +    and "convex_on s g"
1.473 +  shows "convex_on s (\<lambda>x. f x + g x)"
1.474 +proof -
1.475 +  {
1.476 +    fix x y
1.477 +    assume "x \<in> s" "y \<in> s"
1.478 +    moreover
1.479 +    fix u v :: real
1.480 +    assume "0 \<le> u" "0 \<le> v" "u + v = 1"
1.481 +    ultimately
1.482 +    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)"
1.483 +      using assms unfolding convex_on_def by (auto simp: add_mono)
1.484 +    then have "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)"
1.485 +      by (simp add: field_simps)
1.486 +  }
1.487 +  then show ?thesis
1.488 +    unfolding convex_on_def by auto
1.489 +qed
1.490 +
1.491 +lemma convex_on_cmul [intro]:
1.492 +  fixes c :: real
1.493 +  assumes "0 \<le> c"
1.494 +    and "convex_on s f"
1.495 +  shows "convex_on s (\<lambda>x. c * f x)"
1.496 +proof -
1.497 +  have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
1.498 +    for u c fx v fy :: real
1.499 +    by (simp add: field_simps)
1.500 +  show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
1.501 +    unfolding convex_on_def and * by auto
1.502 +qed
1.503 +
1.504 +lemma convex_lower:
1.505 +  assumes "convex_on s f"
1.506 +    and "x \<in> s"
1.507 +    and "y \<in> s"
1.508 +    and "0 \<le> u"
1.509 +    and "0 \<le> v"
1.510 +    and "u + v = 1"
1.511 +  shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
1.512 +proof -
1.513 +  let ?m = "max (f x) (f y)"
1.514 +  have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
1.515 +    using assms(4,5) by (auto simp: mult_left_mono add_mono)
1.516 +  also have "\<dots> = max (f x) (f y)"
1.517 +    using assms(6) by (simp add: distrib_right [symmetric])
1.518 +  finally show ?thesis
1.519 +    using assms unfolding convex_on_def by fastforce
1.520 +qed
1.521 +
1.522 +lemma convex_on_dist [intro]:
1.523 +  fixes s :: "'a::real_normed_vector set"
1.524 +  shows "convex_on s (\<lambda>x. dist a x)"
1.525 +proof (auto simp: convex_on_def dist_norm)
1.526 +  fix x y
1.527 +  assume "x \<in> s" "y \<in> s"
1.528 +  fix u v :: real
1.529 +  assume "0 \<le> u"
1.530 +  assume "0 \<le> v"
1.531 +  assume "u + v = 1"
1.532 +  have "a = u *\<^sub>R a + v *\<^sub>R a"
1.533 +    unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
1.534 +  then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
1.535 +    by (auto simp: algebra_simps)
1.536 +  show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
1.537 +    unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
1.538 +    using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
1.539 +qed
1.540 +
1.541 +
1.542 +subsection%unimportant \<open>Arithmetic operations on sets preserve convexity\<close>
1.543 +
1.544 +lemma convex_linear_image:
1.545 +  assumes "linear f"
1.546 +    and "convex s"
1.547 +  shows "convex (f ` s)"
1.548 +proof -
1.549 +  interpret f: linear f by fact
1.550 +  from \<open>convex s\<close> show "convex (f ` s)"
1.552 +qed
1.553 +
1.554 +lemma convex_linear_vimage:
1.555 +  assumes "linear f"
1.556 +    and "convex s"
1.557 +  shows "convex (f -` s)"
1.558 +proof -
1.559 +  interpret f: linear f by fact
1.560 +  from \<open>convex s\<close> show "convex (f -` s)"
1.562 +qed
1.563 +
1.564 +lemma convex_scaling:
1.565 +  assumes "convex s"
1.566 +  shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
1.567 +proof -
1.568 +  have "linear (\<lambda>x. c *\<^sub>R x)"
1.570 +  then show ?thesis
1.571 +    using \<open>convex s\<close> by (rule convex_linear_image)
1.572 +qed
1.573 +
1.574 +lemma convex_scaled:
1.575 +  assumes "convex S"
1.576 +  shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
1.577 +proof -
1.578 +  have "linear (\<lambda>x. x *\<^sub>R c)"
1.580 +  then show ?thesis
1.581 +    using \<open>convex S\<close> by (rule convex_linear_image)
1.582 +qed
1.583 +
1.584 +lemma convex_negations:
1.585 +  assumes "convex S"
1.586 +  shows "convex ((\<lambda>x. - x) ` S)"
1.587 +proof -
1.588 +  have "linear (\<lambda>x. - x)"
1.589 +    by (simp add: linearI)
1.590 +  then show ?thesis
1.591 +    using \<open>convex S\<close> by (rule convex_linear_image)
1.592 +qed
1.593 +
1.594 +lemma convex_sums:
1.595 +  assumes "convex S"
1.596 +    and "convex T"
1.597 +  shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
1.598 +proof -
1.599 +  have "linear (\<lambda>(x, y). x + y)"
1.600 +    by (auto intro: linearI simp: scaleR_add_right)
1.601 +  with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
1.602 +    by (intro convex_linear_image convex_Times)
1.603 +  also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
1.604 +    by auto
1.605 +  finally show ?thesis .
1.606 +qed
1.607 +
1.608 +lemma convex_differences:
1.609 +  assumes "convex S" "convex T"
1.610 +  shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
1.611 +proof -
1.612 +  have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}"
1.614 +  then show ?thesis
1.615 +    using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
1.616 +qed
1.617 +
1.618 +lemma convex_translation:
1.619 +  assumes "convex S"
1.620 +  shows "convex ((\<lambda>x. a + x) ` S)"
1.621 +proof -
1.622 +  have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (\<lambda>x. a + x) ` S"
1.623 +    by auto
1.624 +  then show ?thesis
1.625 +    using convex_sums[OF convex_singleton[of a] assms] by auto
1.626 +qed
1.627 +
1.628 +lemma convex_affinity:
1.629 +  assumes "convex S"
1.630 +  shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
1.631 +proof -
1.632 +  have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S"
1.633 +    by auto
1.634 +  then show ?thesis
1.635 +    using convex_translation[OF convex_scaling[OF assms], of a c] by auto
1.636 +qed
1.637 +
1.638 +lemma pos_is_convex: "convex {0 :: real <..}"
1.639 +  unfolding convex_alt
1.640 +proof safe
1.641 +  fix y x \<mu> :: real
1.642 +  assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
1.643 +  {
1.644 +    assume "\<mu> = 0"
1.645 +    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
1.646 +      by simp
1.647 +    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
1.648 +      using * by simp
1.649 +  }
1.650 +  moreover
1.651 +  {
1.652 +    assume "\<mu> = 1"
1.653 +    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
1.654 +      using * by simp
1.655 +  }
1.656 +  moreover
1.657 +  {
1.658 +    assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
1.659 +    then have "\<mu> > 0" "(1 - \<mu>) > 0"
1.660 +      using * by auto
1.661 +    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
1.662 +      using * by (auto simp: add_pos_pos)
1.663 +  }
1.664 +  ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
1.665 +    by fastforce
1.666 +qed
1.667 +
1.668 +lemma convex_on_sum:
1.669 +  fixes a :: "'a \<Rightarrow> real"
1.670 +    and y :: "'a \<Rightarrow> 'b::real_vector"
1.671 +    and f :: "'b \<Rightarrow> real"
1.672 +  assumes "finite s" "s \<noteq> {}"
1.673 +    and "convex_on C f"
1.674 +    and "convex C"
1.675 +    and "(\<Sum> i \<in> s. a i) = 1"
1.676 +    and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
1.677 +    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
1.678 +  shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
1.679 +  using assms
1.680 +proof (induct s arbitrary: a rule: finite_ne_induct)
1.681 +  case (singleton i)
1.682 +  then have ai: "a i = 1"
1.683 +    by auto
1.684 +  then show ?case
1.685 +    by auto
1.686 +next
1.687 +  case (insert i s)
1.688 +  then have "convex_on C f"
1.689 +    by simp
1.690 +  from this[unfolded convex_on_def, rule_format]
1.691 +  have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
1.692 +      f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
1.693 +    by simp
1.694 +  show ?case
1.695 +  proof (cases "a i = 1")
1.696 +    case True
1.697 +    then have "(\<Sum> j \<in> s. a j) = 0"
1.698 +      using insert by auto
1.699 +    then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
1.700 +      using insert by (fastforce simp: sum_nonneg_eq_0_iff)
1.701 +    then show ?thesis
1.702 +      using insert by auto
1.703 +  next
1.704 +    case False
1.705 +    from insert have yai: "y i \<in> C" "a i \<ge> 0"
1.706 +      by auto
1.707 +    have fis: "finite (insert i s)"
1.708 +      using insert by auto
1.709 +    then have ai1: "a i \<le> 1"
1.710 +      using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
1.711 +    then have "a i < 1"
1.712 +      using False by auto
1.713 +    then have i0: "1 - a i > 0"
1.714 +      by auto
1.715 +    let ?a = "\<lambda>j. a j / (1 - a i)"
1.716 +    have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
1.717 +      using i0 insert that by fastforce
1.718 +    have "(\<Sum> j \<in> insert i s. a j) = 1"
1.719 +      using insert by auto
1.720 +    then have "(\<Sum> j \<in> s. a j) = 1 - a i"
1.721 +      using sum.insert insert by fastforce
1.722 +    then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
1.723 +      using i0 by auto
1.724 +    then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
1.725 +      unfolding sum_divide_distrib by simp
1.726 +    have "convex C" using insert by auto
1.727 +    then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
1.728 +      using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
1.729 +    have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
1.730 +      using a_nonneg a1 insert by blast
1.731 +    have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
1.732 +      using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
1.733 +      by (auto simp only: add.commute)
1.734 +    also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
1.735 +      using i0 by auto
1.736 +    also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
1.737 +      using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
1.738 +      by (auto simp: algebra_simps)
1.739 +    also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
1.740 +      by (auto simp: divide_inverse)
1.741 +    also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
1.742 +      using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
1.743 +      by (auto simp: add.commute)
1.744 +    also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
1.745 +      using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
1.746 +            OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
1.747 +      by simp
1.748 +    also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
1.749 +      unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
1.750 +      using i0 by auto
1.751 +    also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
1.752 +      using i0 by auto
1.753 +    also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
1.754 +      using insert by auto
1.755 +    finally show ?thesis
1.756 +      by simp
1.757 +  qed
1.758 +qed
1.759 +
1.760 +lemma convex_on_alt:
1.761 +  fixes C :: "'a::real_vector set"
1.762 +  assumes "convex C"
1.763 +  shows "convex_on C f \<longleftrightarrow>
1.764 +    (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
1.765 +      f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
1.766 +proof safe
1.767 +  fix x y
1.768 +  fix \<mu> :: real
1.769 +  assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
1.770 +  from this[unfolded convex_on_def, rule_format]
1.771 +  have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
1.772 +    by auto
1.773 +  from this [of "\<mu>" "1 - \<mu>", simplified] *
1.774 +  show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
1.775 +    by auto
1.776 +next
1.777 +  assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
1.778 +    f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
1.779 +  {
1.780 +    fix x y
1.781 +    fix u v :: real
1.782 +    assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
1.783 +    then have[simp]: "1 - u = v" by auto
1.784 +    from *[rule_format, of x y u]
1.785 +    have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
1.786 +      using ** by auto
1.787 +  }
1.788 +  then show "convex_on C f"
1.789 +    unfolding convex_on_def by auto
1.790 +qed
1.791 +
1.792 +lemma convex_on_diff:
1.793 +  fixes f :: "real \<Rightarrow> real"
1.794 +  assumes f: "convex_on I f"
1.795 +    and I: "x \<in> I" "y \<in> I"
1.796 +    and t: "x < t" "t < y"
1.797 +  shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
1.798 +    and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
1.799 +proof -
1.800 +  define a where "a \<equiv> (t - y) / (x - y)"
1.801 +  with t have "0 \<le> a" "0 \<le> 1 - a"
1.802 +    by (auto simp: field_simps)
1.803 +  with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
1.804 +    by (auto simp: convex_on_def)
1.805 +  have "a * x + (1 - a) * y = a * (x - y) + y"
1.806 +    by (simp add: field_simps)
1.807 +  also have "\<dots> = t"
1.808 +    unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
1.809 +  finally have "f t \<le> a * f x + (1 - a) * f y"
1.810 +    using cvx by simp
1.811 +  also have "\<dots> = a * (f x - f y) + f y"
1.812 +    by (simp add: field_simps)
1.813 +  finally have "f t - f y \<le> a * (f x - f y)"
1.814 +    by simp
1.815 +  with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
1.816 +    by (simp add: le_divide_eq divide_le_eq field_simps a_def)
1.817 +  with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
1.818 +    by (simp add: le_divide_eq divide_le_eq field_simps)
1.819 +qed
1.820 +
1.821 +lemma pos_convex_function:
1.822 +  fixes f :: "real \<Rightarrow> real"
1.823 +  assumes "convex C"
1.824 +    and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
1.825 +  shows "convex_on C f"
1.826 +  unfolding convex_on_alt[OF assms(1)]
1.827 +  using assms
1.828 +proof safe
1.829 +  fix x y \<mu> :: real
1.830 +  let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
1.831 +  assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
1.832 +  then have "1 - \<mu> \<ge> 0" by auto
1.833 +  then have xpos: "?x \<in> C"
1.834 +    using * unfolding convex_alt by fastforce
1.835 +  have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
1.836 +      \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
1.837 +    using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
1.838 +        mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
1.839 +    by auto
1.840 +  then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
1.841 +    by (auto simp: field_simps)
1.842 +  then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
1.843 +    using convex_on_alt by auto
1.844 +qed
1.845 +
1.846 +lemma atMostAtLeast_subset_convex:
1.847 +  fixes C :: "real set"
1.848 +  assumes "convex C"
1.849 +    and "x \<in> C" "y \<in> C" "x < y"
1.850 +  shows "{x .. y} \<subseteq> C"
1.851 +proof safe
1.852 +  fix z assume z: "z \<in> {x .. y}"
1.853 +  have less: "z \<in> C" if *: "x < z" "z < y"
1.854 +  proof -
1.855 +    let ?\<mu> = "(y - z) / (y - x)"
1.856 +    have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
1.857 +      using assms * by (auto simp: field_simps)
1.858 +    then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
1.859 +      using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
1.860 +      by (simp add: algebra_simps)
1.861 +    have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
1.862 +      by (auto simp: field_simps)
1.863 +    also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
1.864 +      using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
1.865 +    also have "\<dots> = z"
1.866 +      using assms by (auto simp: field_simps)
1.867 +    finally show ?thesis
1.868 +      using comb by auto
1.869 +  qed
1.870 +  show "z \<in> C"
1.871 +    using z less assms by (auto simp: le_less)
1.872 +qed
1.873 +
1.874 +lemma f''_imp_f':
1.875 +  fixes f :: "real \<Rightarrow> real"
1.876 +  assumes "convex C"
1.877 +    and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
1.878 +    and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
1.879 +    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
1.880 +    and x: "x \<in> C"
1.881 +    and y: "y \<in> C"
1.882 +  shows "f' x * (y - x) \<le> f y - f x"
1.883 +  using assms
1.884 +proof -
1.885 +  have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
1.886 +    if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
1.887 +  proof -
1.888 +    from * have ge: "y - x > 0" "y - x \<ge> 0"
1.889 +      by auto
1.890 +    from * have le: "x - y < 0" "x - y \<le> 0"
1.891 +      by auto
1.892 +    then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
1.893 +      using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
1.894 +          THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
1.895 +      by auto
1.896 +    then have "z1 \<in> C"
1.897 +      using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
1.898 +      by fastforce
1.899 +    from z1 have z1': "f x - f y = (x - y) * f' z1"
1.900 +      by (simp add: field_simps)
1.901 +    obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
1.902 +      using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
1.903 +          THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
1.904 +      by auto
1.905 +    obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
1.906 +      using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
1.907 +          THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
1.908 +      by auto
1.909 +    have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
1.910 +      using * z1' by auto
1.911 +    also have "\<dots> = (y - z1) * f'' z3"
1.912 +      using z3 by auto
1.913 +    finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
1.914 +      by simp
1.915 +    have A': "y - z1 \<ge> 0"
1.916 +      using z1 by auto
1.917 +    have "z3 \<in> C"
1.918 +      using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
1.919 +      by fastforce
1.920 +    then have B': "f'' z3 \<ge> 0"
1.921 +      using assms by auto
1.922 +    from A' B' have "(y - z1) * f'' z3 \<ge> 0"
1.923 +      by auto
1.924 +    from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
1.925 +      by auto
1.926 +    from mult_right_mono_neg[OF this le(2)]
1.927 +    have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
1.928 +      by (simp add: algebra_simps)
1.929 +    then have "f' y * (x - y) - (f x - f y) \<le> 0"
1.930 +      using le by auto
1.931 +    then have res: "f' y * (x - y) \<le> f x - f y"
1.932 +      by auto
1.933 +    have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
1.934 +      using * z1 by auto
1.935 +    also have "\<dots> = (z1 - x) * f'' z2"
1.936 +      using z2 by auto
1.937 +    finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
1.938 +      by simp
1.939 +    have A: "z1 - x \<ge> 0"
1.940 +      using z1 by auto
1.941 +    have "z2 \<in> C"
1.942 +      using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
1.943 +      by fastforce
1.944 +    then have B: "f'' z2 \<ge> 0"
1.945 +      using assms by auto
1.946 +    from A B have "(z1 - x) * f'' z2 \<ge> 0"
1.947 +      by auto
1.948 +    with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
1.949 +      by auto
1.950 +    from mult_right_mono[OF this ge(2)]
1.951 +    have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
1.952 +      by (simp add: algebra_simps)
1.953 +    then have "f y - f x - f' x * (y - x) \<ge> 0"
1.954 +      using ge by auto
1.955 +    then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
1.956 +      using res by auto
1.957 +  qed
1.958 +  show ?thesis
1.959 +  proof (cases "x = y")
1.960 +    case True
1.961 +    with x y show ?thesis by auto
1.962 +  next
1.963 +    case False
1.964 +    with less_imp x y show ?thesis
1.965 +      by (auto simp: neq_iff)
1.966 +  qed
1.967 +qed
1.968 +
1.969 +lemma f''_ge0_imp_convex:
1.970 +  fixes f :: "real \<Rightarrow> real"
1.971 +  assumes conv: "convex C"
1.972 +    and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
1.973 +    and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
1.974 +    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
1.975 +  shows "convex_on C f"
1.976 +  using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
1.977 +  by fastforce
1.978 +
1.979 +lemma minus_log_convex:
1.980 +  fixes b :: real
1.981 +  assumes "b > 1"
1.982 +  shows "convex_on {0 <..} (\<lambda> x. - log b x)"
1.983 +proof -
1.984 +  have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
1.985 +    using DERIV_log by auto
1.986 +  then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
1.987 +    by (auto simp: DERIV_minus)
1.988 +  have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
1.989 +    using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
1.990 +  from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
1.991 +  have "\<And>z::real. z > 0 \<Longrightarrow>
1.992 +    DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
1.993 +    by auto
1.994 +  then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
1.995 +    DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
1.996 +    unfolding inverse_eq_divide by (auto simp: mult.assoc)
1.997 +  have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
1.998 +    using \<open>b > 1\<close> by (auto intro!: less_imp_le)
1.999 +  from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
1.1000 +  show ?thesis
1.1001 +    by auto
1.1002 +qed
1.1003 +
1.1004 +
1.1005 +subsection%unimportant \<open>Convexity of real functions\<close>
1.1006 +
1.1007 +lemma convex_on_realI:
1.1008 +  assumes "connected A"
1.1009 +    and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
1.1010 +    and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
1.1011 +  shows "convex_on A f"
1.1012 +proof (rule convex_on_linorderI)
1.1013 +  fix t x y :: real
1.1014 +  assume t: "t > 0" "t < 1"
1.1015 +  assume xy: "x \<in> A" "y \<in> A" "x < y"
1.1016 +  define z where "z = (1 - t) * x + t * y"
1.1017 +  with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
1.1018 +    using connected_contains_Icc by blast
1.1019 +
1.1020 +  from xy t have xz: "z > x"
1.1021 +    by (simp add: z_def algebra_simps)
1.1022 +  have "y - z = (1 - t) * (y - x)"
1.1023 +    by (simp add: z_def algebra_simps)
1.1024 +  also from xy t have "\<dots> > 0"
1.1025 +    by (intro mult_pos_pos) simp_all
1.1026 +  finally have yz: "z < y"
1.1027 +    by simp
1.1028 +
1.1029 +  from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
1.1030 +    by (intro MVT2) (auto intro!: assms(2))
1.1031 +  then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
1.1032 +    by auto
1.1033 +  from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
1.1034 +    by (intro MVT2) (auto intro!: assms(2))
1.1035 +  then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
1.1036 +    by auto
1.1037 +
1.1038 +  from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
1.1039 +  also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
1.1040 +    by auto
1.1041 +  with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
1.1042 +    by (intro assms(3)) auto
1.1043 +  also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
1.1044 +  finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
1.1045 +    using xz yz by (simp add: field_simps)
1.1046 +  also have "z - x = t * (y - x)"
1.1047 +    by (simp add: z_def algebra_simps)
1.1048 +  also have "y - z = (1 - t) * (y - x)"
1.1049 +    by (simp add: z_def algebra_simps)
1.1050 +  finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
1.1051 +    using xy by simp
1.1052 +  then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
1.1053 +    by (simp add: z_def algebra_simps)
1.1054 +qed
1.1055 +
1.1056 +lemma convex_on_inverse:
1.1057 +  assumes "A \<subseteq> {0<..}"
1.1058 +  shows "convex_on A (inverse :: real \<Rightarrow> real)"
1.1059 +proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
1.1060 +  fix u v :: real
1.1061 +  assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
1.1062 +  with assms show "-inverse (u^2) \<le> -inverse (v^2)"
1.1063 +    by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
1.1064 +qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
1.1065 +
1.1066 +lemma convex_onD_Icc':
1.1067 +  assumes "convex_on {x..y} f" "c \<in> {x..y}"
1.1068 +  defines "d \<equiv> y - x"
1.1069 +  shows "f c \<le> (f y - f x) / d * (c - x) + f x"
1.1070 +proof (cases x y rule: linorder_cases)
1.1071 +  case less
1.1072 +  then have d: "d > 0"
1.1073 +    by (simp add: d_def)
1.1074 +  from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
1.1075 +    by (simp_all add: d_def divide_simps)
1.1076 +  have "f c = f (x + (c - x) * 1)"
1.1077 +    by simp
1.1078 +  also from less have "1 = ((y - x) / d)"
1.1079 +    by (simp add: d_def)
1.1080 +  also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
1.1081 +    by (simp add: field_simps)
1.1082 +  also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
1.1083 +    using assms less by (intro convex_onD_Icc) simp_all
1.1084 +  also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
1.1085 +    by (simp add: field_simps)
1.1086 +  finally show ?thesis .
1.1087 +qed (insert assms(2), simp_all)
1.1088 +
1.1089 +lemma convex_onD_Icc'':
1.1090 +  assumes "convex_on {x..y} f" "c \<in> {x..y}"
1.1091 +  defines "d \<equiv> y - x"
1.1092 +  shows "f c \<le> (f x - f y) / d * (y - c) + f y"
1.1093 +proof (cases x y rule: linorder_cases)
1.1094 +  case less
1.1095 +  then have d: "d > 0"
1.1096 +    by (simp add: d_def)
1.1097 +  from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
1.1098 +    by (simp_all add: d_def divide_simps)
1.1099 +  have "f c = f (y - (y - c) * 1)"
1.1100 +    by simp
1.1101 +  also from less have "1 = ((y - x) / d)"
1.1102 +    by (simp add: d_def)
1.1103 +  also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
1.1104 +    by (simp add: field_simps)
1.1105 +  also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
1.1106 +    using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
1.1107 +  also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
1.1108 +    by (simp add: field_simps)
1.1109 +  finally show ?thesis .
1.1110 +qed (insert assms(2), simp_all)
1.1111 +
1.1112 +lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
1.1113 +  by (metis convex_translation translation_galois)
1.1114 +
1.1115 +lemma convex_linear_image_eq [simp]:
1.1116 +    fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
1.1117 +    shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
1.1118 +    by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
1.1119 +
1.1120 +lemma fst_linear: "linear fst"
1.1121 +  unfolding linear_iff by (simp add: algebra_simps)
1.1122 +
1.1123 +lemma snd_linear: "linear snd"
1.1124 +  unfolding linear_iff by (simp add: algebra_simps)
1.1125 +
1.1126 +lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
1.1127 +  unfolding linear_iff by (simp add: algebra_simps)
1.1128 +
1.1129 +lemma vector_choose_size:
1.1130 +  assumes "0 \<le> c"
1.1131 +  obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
1.1132 +proof -
1.1133 +  obtain a::'a where "a \<noteq> 0"
1.1134 +    using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
1.1135 +  then show ?thesis
1.1136 +    by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
1.1137 +qed
1.1138 +
1.1139 +lemma vector_choose_dist:
1.1140 +  assumes "0 \<le> c"
1.1141 +  obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
1.1142 +by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
1.1143 +
1.1144 +lemma sum_delta_notmem:
1.1145 +  assumes "x \<notin> s"
1.1146 +  shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
1.1147 +    and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
1.1148 +    and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
1.1149 +    and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
1.1150 +  apply (rule_tac [!] sum.cong)
1.1151 +  using assms
1.1152 +  apply auto
1.1153 +  done
1.1154 +
1.1155 +lemma sum_delta'':
1.1156 +  fixes s::"'a::real_vector set"
1.1157 +  assumes "finite s"
1.1158 +  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)"
1.1159 +proof -
1.1160 +  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)"
1.1161 +    by auto
1.1162 +  show ?thesis
1.1163 +    unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
1.1164 +qed
1.1165 +
1.1166 +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)"
1.1167 +  by (fact if_distrib)
1.1168 +
1.1169 +lemma dist_triangle_eq:
1.1170 +  fixes x y z :: "'a::real_inner"
1.1171 +  shows "dist x z = dist x y + dist y z \<longleftrightarrow>
1.1172 +    norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
1.1173 +proof -
1.1174 +  have *: "x - y + (y - z) = x - z" by auto
1.1175 +  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
1.1176 +    by (auto simp:norm_minus_commute)
1.1177 +qed
1.1178 +
1.1179 +
1.1180 +subsection \<open>Affine set and affine hull\<close>
1.1181 +
1.1182 +definition%important affine :: "'a::real_vector set \<Rightarrow> bool"
1.1183 +  where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
1.1184 +
1.1185 +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)"
1.1186 +  unfolding affine_def by (metis eq_diff_eq')
1.1187 +
1.1188 +lemma affine_empty [iff]: "affine {}"
1.1189 +  unfolding affine_def by auto
1.1190 +
1.1191 +lemma affine_sing [iff]: "affine {x}"
1.1192 +  unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
1.1193 +
1.1194 +lemma affine_UNIV [iff]: "affine UNIV"
1.1195 +  unfolding affine_def by auto
1.1196 +
1.1197 +lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
1.1198 +  unfolding affine_def by auto
1.1199 +
1.1200 +lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
1.1201 +  unfolding affine_def by auto
1.1202 +
1.1203 +lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
1.1204 +  apply (clarsimp simp add: affine_def)
1.1205 +  apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
1.1206 +  apply (auto simp: algebra_simps)
1.1207 +  done
1.1208 +
1.1209 +lemma affine_affine_hull [simp]: "affine(affine hull s)"
1.1210 +  unfolding hull_def
1.1211 +  using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
1.1212 +
1.1213 +lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
1.1214 +  by (metis affine_affine_hull hull_same)
1.1215 +
1.1216 +lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
1.1217 +  by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
1.1218 +
1.1219 +
1.1220 +subsubsection%unimportant \<open>Some explicit formulations\<close>
1.1221 +
1.1222 +text "Formalized by Lars Schewe."
1.1223 +
1.1224 +lemma affine:
1.1225 +  fixes V::"'a::real_vector set"
1.1226 +  shows "affine V \<longleftrightarrow>
1.1227 +         (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
1.1228 +proof -
1.1229 +  have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
1.1230 +    and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
1.1231 +  proof (cases "x = y")
1.1232 +    case True
1.1233 +    then show ?thesis
1.1234 +      using that by (metis scaleR_add_left scaleR_one)
1.1235 +  next
1.1236 +    case False
1.1237 +    then show ?thesis
1.1238 +      using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
1.1239 +  qed
1.1240 +  moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
1.1241 +                if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
1.1242 +                  and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
1.1243 +  proof -
1.1244 +    define n where "n = card S"
1.1245 +    consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
1.1246 +    then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
1.1247 +    proof cases
1.1248 +      assume "card S = 1"
1.1249 +      then obtain a where "S={a}"
1.1250 +        by (auto simp: card_Suc_eq)
1.1251 +      then show ?thesis
1.1252 +        using that by simp
1.1253 +    next
1.1254 +      assume "card S = 2"
1.1255 +      then obtain a b where "S = {a, b}"
1.1256 +        by (metis Suc_1 card_1_singletonE card_Suc_eq)
1.1257 +      then show ?thesis
1.1258 +        using *[of a b] that
1.1259 +        by (auto simp: sum_clauses(2))
1.1260 +    next
1.1261 +      assume "card S > 2"
1.1262 +      then show ?thesis using that n_def
1.1263 +      proof (induct n arbitrary: u S)
1.1264 +        case 0
1.1265 +        then show ?case by auto
1.1266 +      next
1.1267 +        case (Suc n u S)
1.1268 +        have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
1.1269 +          using that unfolding card_eq_sum by auto
1.1270 +        with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
1.1271 +        have c: "card (S - {x}) = card S - 1"
1.1272 +          by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
1.1273 +        have "sum u (S - {x}) = 1 - u x"
1.1274 +          by (simp add: Suc.prems sum_diff1_ring \<open>x \<in> S\<close>)
1.1275 +        with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
1.1276 +          by auto
1.1277 +        have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
1.1278 +        proof (cases "card (S - {x}) > 2")
1.1279 +          case True
1.1280 +          then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
1.1281 +            using Suc.prems c by force+
1.1282 +          show ?thesis
1.1283 +          proof (rule Suc.hyps)
1.1284 +            show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
1.1285 +              by (auto simp: eq1 sum_distrib_left[symmetric])
1.1286 +          qed (use S Suc.prems True in auto)
1.1287 +        next
1.1288 +          case False
1.1289 +          then have "card (S - {x}) = Suc (Suc 0)"
1.1290 +            using Suc.prems c by auto
1.1291 +          then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
1.1292 +            unfolding card_Suc_eq by auto
1.1293 +          then show ?thesis
1.1294 +            using eq1 \<open>S \<subseteq> V\<close>
1.1295 +            by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
1.1296 +        qed
1.1297 +        have "u x + (1 - u x) = 1 \<Longrightarrow>
1.1298 +          u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
1.1299 +          by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
1.1300 +        moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
1.1301 +          by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
1.1302 +        ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
1.1303 +          by (simp add: x)
1.1304 +      qed
1.1305 +    qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
1.1306 +  qed
1.1307 +  ultimately show ?thesis
1.1308 +    unfolding affine_def by meson
1.1309 +qed
1.1310 +
1.1311 +
1.1312 +lemma affine_hull_explicit:
1.1313 +  "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
1.1314 +  (is "_ = ?rhs")
1.1315 +proof (rule hull_unique)
1.1316 +  show "p \<subseteq> ?rhs"
1.1317 +  proof (intro subsetI CollectI exI conjI)
1.1318 +    show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
1.1319 +      by auto
1.1320 +  qed auto
1.1321 +  show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
1.1322 +    using that unfolding affine by blast
1.1323 +  show "affine ?rhs"
1.1324 +    unfolding affine_def
1.1325 +  proof clarify
1.1326 +    fix u v :: real and sx ux sy uy
1.1327 +    assume uv: "u + v = 1"
1.1328 +      and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
1.1329 +      and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
1.1330 +    have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
1.1331 +      by auto
1.1332 +    show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
1.1333 +        sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
1.1334 +    proof (intro exI conjI)
1.1335 +      show "finite (sx \<union> sy)"
1.1336 +        using x y by auto
1.1337 +      show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
1.1338 +        using x y uv
1.1339 +        by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
1.1340 +      have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
1.1341 +          = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
1.1342 +        using x y
1.1343 +        unfolding scaleR_left_distrib scaleR_zero_left if_smult
1.1344 +        by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric]  **)
1.1345 +      also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
1.1346 +        unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
1.1347 +      finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
1.1348 +                  = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
1.1349 +    qed (use x y in auto)
1.1350 +  qed
1.1351 +qed
1.1352 +
1.1353 +lemma affine_hull_finite:
1.1354 +  assumes "finite S"
1.1355 +  shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
1.1356 +proof -
1.1357 +  have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
1.1358 +    if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
1.1359 +  proof -
1.1360 +    have "S \<inter> F = F"
1.1361 +      using that by auto
1.1362 +    show ?thesis
1.1363 +    proof (intro exI conjI)
1.1364 +      show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
1.1365 +        by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
1.1366 +      show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
1.1367 +        by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
1.1368 +    qed
1.1369 +  qed
1.1370 +  show ?thesis
1.1371 +    unfolding affine_hull_explicit using assms
1.1372 +    by (fastforce dest: *)
1.1373 +qed
1.1374 +
1.1375 +
1.1376 +subsubsection%unimportant \<open>Stepping theorems and hence small special cases\<close>
1.1377 +
1.1378 +lemma affine_hull_empty[simp]: "affine hull {} = {}"
1.1379 +  by simp
1.1380 +
1.1381 +lemma affine_hull_finite_step:
1.1382 +  fixes y :: "'a::real_vector"
1.1383 +  shows "finite S \<Longrightarrow>
1.1384 +      (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
1.1385 +      (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
1.1386 +proof -
1.1387 +  assume fin: "finite S"
1.1388 +  show "?lhs = ?rhs"
1.1389 +  proof
1.1390 +    assume ?lhs
1.1391 +    then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
1.1392 +      by auto
1.1393 +    show ?rhs
1.1394 +    proof (cases "a \<in> S")
1.1395 +      case True
1.1396 +      then show ?thesis
1.1397 +        using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
1.1398 +    next
1.1399 +      case False
1.1400 +      show ?thesis
1.1401 +        by (rule exI [where x="u a"]) (use u fin False in auto)
1.1402 +    qed
1.1403 +  next
1.1404 +    assume ?rhs
1.1405 +    then obtain v u where vu: "sum u S = w - v"  "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
1.1406 +      by auto
1.1407 +    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)"
1.1408 +      by auto
1.1409 +    show ?lhs
1.1410 +    proof (cases "a \<in> S")
1.1411 +      case True
1.1412 +      show ?thesis
1.1413 +        by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
1.1414 +           (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
1.1415 +    next
1.1416 +      case False
1.1417 +      then show ?thesis
1.1418 +        apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
1.1419 +        apply (simp add: vu sum_clauses(2)[OF fin] *)
1.1420 +        by (simp add: sum_delta_notmem(3) vu)
1.1421 +    qed
1.1422 +  qed
1.1423 +qed
1.1424 +
1.1425 +lemma affine_hull_2:
1.1426 +  fixes a b :: "'a::real_vector"
1.1427 +  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
1.1428 +  (is "?lhs = ?rhs")
1.1429 +proof -
1.1430 +  have *:
1.1431 +    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
1.1432 +    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
1.1433 +  have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
1.1434 +    using affine_hull_finite[of "{a,b}"] by auto
1.1435 +  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
1.1436 +    by (simp add: affine_hull_finite_step[of "{b}" a])
1.1437 +  also have "\<dots> = ?rhs" unfolding * by auto
1.1438 +  finally show ?thesis by auto
1.1439 +qed
1.1440 +
1.1441 +lemma affine_hull_3:
1.1442 +  fixes a b c :: "'a::real_vector"
1.1443 +  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}"
1.1444 +proof -
1.1445 +  have *:
1.1446 +    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
1.1447 +    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
1.1448 +  show ?thesis
1.1449 +    apply (simp add: affine_hull_finite affine_hull_finite_step)
1.1450 +    unfolding *
1.1451 +    apply safe
1.1453 +    apply (rule_tac x=u in exI, force)
1.1454 +    done
1.1455 +qed
1.1456 +
1.1457 +lemma mem_affine:
1.1458 +  assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
1.1459 +  shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
1.1460 +  using assms affine_def[of S] by auto
1.1461 +
1.1462 +lemma mem_affine_3:
1.1463 +  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
1.1464 +  shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
1.1465 +proof -
1.1466 +  have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
1.1467 +    using affine_hull_3[of x y z] assms by auto
1.1468 +  moreover
1.1469 +  have "affine hull {x, y, z} \<subseteq> affine hull S"
1.1470 +    using hull_mono[of "{x, y, z}" "S"] assms by auto
1.1471 +  moreover
1.1472 +  have "affine hull S = S"
1.1473 +    using assms affine_hull_eq[of S] by auto
1.1474 +  ultimately show ?thesis by auto
1.1475 +qed
1.1476 +
1.1477 +lemma mem_affine_3_minus:
1.1478 +  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
1.1479 +  shows "x + v *\<^sub>R (y-z) \<in> S"
1.1480 +  using mem_affine_3[of S x y z 1 v "-v"] assms
1.1481 +  by (simp add: algebra_simps)
1.1482 +
1.1483 +corollary mem_affine_3_minus2:
1.1484 +    "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
1.1485 +  by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
1.1486 +
1.1487 +
1.1488 +subsubsection%unimportant \<open>Some relations between affine hull and subspaces\<close>
1.1489 +
1.1490 +lemma affine_hull_insert_subset_span:
1.1491 +  "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
1.1492 +proof -
1.1493 +  have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
1.1494 +    if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
1.1495 +    for x F u
1.1496 +  proof -
1.1497 +    have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
1.1498 +      using that by auto
1.1499 +    show ?thesis
1.1500 +    proof (intro exI conjI)
1.1501 +      show "finite ((\<lambda>x. x - a) ` (F - {a}))"
1.1502 +        by (simp add: that(1))
1.1503 +      show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
1.1504 +        by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
1.1505 +            sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
1.1506 +    qed (use \<open>F \<subseteq> insert a S\<close> in auto)
1.1507 +  qed
1.1508 +  then show ?thesis
1.1509 +    unfolding affine_hull_explicit span_explicit by blast
1.1510 +qed
1.1511 +
1.1512 +lemma affine_hull_insert_span:
1.1513 +  assumes "a \<notin> S"
1.1514 +  shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x.  x \<in> S}}"
1.1515 +proof -
1.1516 +  have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
1.1517 +    if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
1.1518 +  proof -
1.1519 +    from that
1.1520 +    obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
1.1521 +      unfolding span_explicit by auto
1.1522 +    define F where "F = (\<lambda>x. x + a) ` T"
1.1523 +    have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
1.1524 +      unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
1.1525 +    have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
1.1526 +      using F assms by auto
1.1527 +    show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
1.1528 +      apply (rule_tac x = "insert a F" in exI)
1.1529 +      apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
1.1530 +      using assms F
1.1531 +      apply (auto simp:  sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
1.1532 +      done
1.1533 +  qed
1.1534 +  show ?thesis
1.1535 +    by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
1.1536 +qed
1.1537 +
1.1538 +lemma affine_hull_span:
1.1539 +  assumes "a \<in> S"
1.1540 +  shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
1.1541 +  using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
1.1542 +
1.1543 +
1.1544 +subsubsection%unimportant \<open>Parallel affine sets\<close>
1.1545 +
1.1546 +definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
1.1547 +  where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
1.1548 +
1.1549 +lemma affine_parallel_expl_aux:
1.1550 +  fixes S T :: "'a::real_vector set"
1.1551 +  assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
1.1552 +  shows "T = (\<lambda>x. a + x) ` S"
1.1553 +proof -
1.1554 +  have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
1.1555 +    using that
1.1557 +  moreover have "T \<ge> (\<lambda>x. a + x) ` S"
1.1558 +    using assms by auto
1.1559 +  ultimately show ?thesis by auto
1.1560 +qed
1.1561 +
1.1562 +lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
1.1563 +  unfolding affine_parallel_def
1.1564 +  using affine_parallel_expl_aux[of S _ T] by auto
1.1565 +
1.1566 +lemma affine_parallel_reflex: "affine_parallel S S"
1.1567 +  unfolding affine_parallel_def
1.1568 +  using image_add_0 by blast
1.1569 +
1.1570 +lemma affine_parallel_commut:
1.1571 +  assumes "affine_parallel A B"
1.1572 +  shows "affine_parallel B A"
1.1573 +proof -
1.1574 +  from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
1.1575 +    unfolding affine_parallel_def by auto
1.1576 +  have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
1.1577 +  from B show ?thesis
1.1578 +    using translation_galois [of B a A]
1.1579 +    unfolding affine_parallel_def by auto
1.1580 +qed
1.1581 +
1.1582 +lemma affine_parallel_assoc:
1.1583 +  assumes "affine_parallel A B"
1.1584 +    and "affine_parallel B C"
1.1585 +  shows "affine_parallel A C"
1.1586 +proof -
1.1587 +  from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
1.1588 +    unfolding affine_parallel_def by auto
1.1589 +  moreover
1.1590 +  from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
1.1591 +    unfolding affine_parallel_def by auto
1.1592 +  ultimately show ?thesis
1.1593 +    using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
1.1594 +qed
1.1595 +
1.1596 +lemma affine_translation_aux:
1.1597 +  fixes a :: "'a::real_vector"
1.1598 +  assumes "affine ((\<lambda>x. a + x) ` S)"
1.1599 +  shows "affine S"
1.1600 +proof -
1.1601 +  {
1.1602 +    fix x y u v
1.1603 +    assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
1.1604 +    then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
1.1605 +      by auto
1.1606 +    then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
1.1607 +      using xy assms unfolding affine_def by auto
1.1608 +    have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
1.1609 +      by (simp add: algebra_simps)
1.1610 +    also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
1.1611 +      using \<open>u + v = 1\<close> by auto
1.1612 +    ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
1.1613 +      using h1 by auto
1.1614 +    then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
1.1615 +  }
1.1616 +  then show ?thesis unfolding affine_def by auto
1.1617 +qed
1.1618 +
1.1619 +lemma affine_translation:
1.1620 +  fixes a :: "'a::real_vector"
1.1621 +  shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
1.1622 +proof -
1.1623 +  have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
1.1624 +    using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
1.1625 +    using translation_assoc[of "-a" a S] by auto
1.1626 +  then show ?thesis using affine_translation_aux by auto
1.1627 +qed
1.1628 +
1.1629 +lemma parallel_is_affine:
1.1630 +  fixes S T :: "'a::real_vector set"
1.1631 +  assumes "affine S" "affine_parallel S T"
1.1632 +  shows "affine T"
1.1633 +proof -
1.1634 +  from assms obtain a where "T = (\<lambda>x. a + x) ` S"
1.1635 +    unfolding affine_parallel_def by auto
1.1636 +  then show ?thesis
1.1637 +    using affine_translation assms by auto
1.1638 +qed
1.1639 +
1.1640 +lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
1.1641 +  unfolding subspace_def affine_def by auto
1.1642 +
1.1643 +
1.1644 +subsubsection%unimportant \<open>Subspace parallel to an affine set\<close>
1.1645 +
1.1646 +lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
1.1647 +proof -
1.1648 +  have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
1.1649 +    using subspace_imp_affine[of S] subspace_0 by auto
1.1650 +  {
1.1651 +    assume assm: "affine S \<and> 0 \<in> S"
1.1652 +    {
1.1653 +      fix c :: real
1.1654 +      fix x
1.1655 +      assume x: "x \<in> S"
1.1656 +      have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
1.1657 +      moreover
1.1658 +      have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
1.1659 +        using affine_alt[of S] assm x by auto
1.1660 +      ultimately have "c *\<^sub>R x \<in> S" by auto
1.1661 +    }
1.1662 +    then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
1.1663 +
1.1664 +    {
1.1665 +      fix x y
1.1666 +      assume xy: "x \<in> S" "y \<in> S"
1.1667 +      define u where "u = (1 :: real)/2"
1.1668 +      have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
1.1669 +        by auto
1.1670 +      moreover
1.1671 +      have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
1.1672 +        by (simp add: algebra_simps)
1.1673 +      moreover
1.1674 +      have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
1.1675 +        using affine_alt[of S] assm xy by auto
1.1676 +      ultimately
1.1677 +      have "(1/2) *\<^sub>R (x+y) \<in> S"
1.1678 +        using u_def by auto
1.1679 +      moreover
1.1680 +      have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
1.1681 +        by auto
1.1682 +      ultimately
1.1683 +      have "x + y \<in> S"
1.1684 +        using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
1.1685 +    }
1.1686 +    then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
1.1687 +      by auto
1.1688 +    then have "subspace S"
1.1689 +      using h1 assm unfolding subspace_def by auto
1.1690 +  }
1.1691 +  then show ?thesis using h0 by metis
1.1692 +qed
1.1693 +
1.1694 +lemma affine_diffs_subspace:
1.1695 +  assumes "affine S" "a \<in> S"
1.1696 +  shows "subspace ((\<lambda>x. (-a)+x) ` S)"
1.1697 +proof -
1.1698 +  have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
1.1699 +  have "affine ((\<lambda>x. (-a)+x) ` S)"
1.1700 +    using  affine_translation assms by auto
1.1701 +  moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
1.1702 +    using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
1.1703 +  ultimately show ?thesis using subspace_affine by auto
1.1704 +qed
1.1705 +
1.1706 +lemma parallel_subspace_explicit:
1.1707 +  assumes "affine S"
1.1708 +    and "a \<in> S"
1.1709 +  assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
1.1710 +  shows "subspace L \<and> affine_parallel S L"
1.1711 +proof -
1.1712 +  from assms have "L = plus (- a) ` S" by auto
1.1713 +  then have par: "affine_parallel S L"
1.1714 +    unfolding affine_parallel_def ..
1.1715 +  then have "affine L" using assms parallel_is_affine by auto
1.1716 +  moreover have "0 \<in> L"
1.1717 +    using assms by auto
1.1718 +  ultimately show ?thesis
1.1719 +    using subspace_affine par by auto
1.1720 +qed
1.1721 +
1.1722 +lemma parallel_subspace_aux:
1.1723 +  assumes "subspace A"
1.1724 +    and "subspace B"
1.1725 +    and "affine_parallel A B"
1.1726 +  shows "A \<supseteq> B"
1.1727 +proof -
1.1728 +  from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
1.1729 +    using affine_parallel_expl[of A B] by auto
1.1730 +  then have "-a \<in> A"
1.1731 +    using assms subspace_0[of B] by auto
1.1732 +  then have "a \<in> A"
1.1733 +    using assms subspace_neg[of A "-a"] by auto
1.1734 +  then show ?thesis
1.1735 +    using assms a unfolding subspace_def by auto
1.1736 +qed
1.1737 +
1.1738 +lemma parallel_subspace:
1.1739 +  assumes "subspace A"
1.1740 +    and "subspace B"
1.1741 +    and "affine_parallel A B"
1.1742 +  shows "A = B"
1.1743 +proof
1.1744 +  show "A \<supseteq> B"
1.1745 +    using assms parallel_subspace_aux by auto
1.1746 +  show "A \<subseteq> B"
1.1747 +    using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
1.1748 +qed
1.1749 +
1.1750 +lemma affine_parallel_subspace:
1.1751 +  assumes "affine S" "S \<noteq> {}"
1.1752 +  shows "\<exists>!L. subspace L \<and> affine_parallel S L"
1.1753 +proof -
1.1754 +  have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
1.1755 +    using assms parallel_subspace_explicit by auto
1.1756 +  {
1.1757 +    fix L1 L2
1.1758 +    assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
1.1759 +    then have "affine_parallel L1 L2"
1.1760 +      using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
1.1761 +    then have "L1 = L2"
1.1762 +      using ass parallel_subspace by auto
1.1763 +  }
1.1764 +  then show ?thesis using ex by auto
1.1765 +qed
1.1766 +
1.1767 +
1.1768 +subsection \<open>Cones\<close>
1.1769 +
1.1770 +definition%important cone :: "'a::real_vector set \<Rightarrow> bool"
1.1771 +  where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
1.1772 +
1.1773 +lemma cone_empty[intro, simp]: "cone {}"
1.1774 +  unfolding cone_def by auto
1.1775 +
1.1776 +lemma cone_univ[intro, simp]: "cone UNIV"
1.1777 +  unfolding cone_def by auto
1.1778 +
1.1779 +lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
1.1780 +  unfolding cone_def by auto
1.1781 +
1.1782 +lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
1.1783 +  by (simp add: cone_def subspace_scale)
1.1784 +
1.1785 +
1.1786 +subsubsection \<open>Conic hull\<close>
1.1787 +
1.1788 +lemma cone_cone_hull: "cone (cone hull s)"
1.1789 +  unfolding hull_def by auto
1.1790 +
1.1791 +lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
1.1792 +  apply (rule hull_eq)
1.1793 +  using cone_Inter
1.1794 +  unfolding subset_eq
1.1795 +  apply auto
1.1796 +  done
1.1797 +
1.1798 +lemma mem_cone:
1.1799 +  assumes "cone S" "x \<in> S" "c \<ge> 0"
1.1800 +  shows "c *\<^sub>R x \<in> S"
1.1801 +  using assms cone_def[of S] by auto
1.1802 +
1.1803 +lemma cone_contains_0:
1.1804 +  assumes "cone S"
1.1805 +  shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
1.1806 +proof -
1.1807 +  {
1.1808 +    assume "S \<noteq> {}"
1.1809 +    then obtain a where "a \<in> S" by auto
1.1810 +    then have "0 \<in> S"
1.1811 +      using assms mem_cone[of S a 0] by auto
1.1812 +  }
1.1813 +  then show ?thesis by auto
1.1814 +qed
1.1815 +
1.1816 +lemma cone_0: "cone {0}"
1.1817 +  unfolding cone_def by auto
1.1818 +
1.1819 +lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
1.1820 +  unfolding cone_def by blast
1.1821 +
1.1822 +lemma cone_iff:
1.1823 +  assumes "S \<noteq> {}"
1.1824 +  shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
1.1825 +proof -
1.1826 +  {
1.1827 +    assume "cone S"
1.1828 +    {
1.1829 +      fix c :: real
1.1830 +      assume "c > 0"
1.1831 +      {
1.1832 +        fix x
1.1833 +        assume "x \<in> S"
1.1834 +        then have "x \<in> ((*\<^sub>R) c) ` S"
1.1835 +          unfolding image_def
1.1836 +          using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
1.1837 +            exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
1.1838 +          by auto
1.1839 +      }
1.1840 +      moreover
1.1841 +      {
1.1842 +        fix x
1.1843 +        assume "x \<in> ((*\<^sub>R) c) ` S"
1.1844 +        then have "x \<in> S"
1.1845 +          using \<open>cone S\<close> \<open>c > 0\<close>
1.1846 +          unfolding cone_def image_def \<open>c > 0\<close> by auto
1.1847 +      }
1.1848 +      ultimately have "((*\<^sub>R) c) ` S = S" by auto
1.1849 +    }
1.1850 +    then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
1.1851 +      using \<open>cone S\<close> cone_contains_0[of S] assms by auto
1.1852 +  }
1.1853 +  moreover
1.1854 +  {
1.1855 +    assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
1.1856 +    {
1.1857 +      fix x
1.1858 +      assume "x \<in> S"
1.1859 +      fix c1 :: real
1.1860 +      assume "c1 \<ge> 0"
1.1861 +      then have "c1 = 0 \<or> c1 > 0" by auto
1.1862 +      then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
1.1863 +    }
1.1864 +    then have "cone S" unfolding cone_def by auto
1.1865 +  }
1.1866 +  ultimately show ?thesis by blast
1.1867 +qed
1.1868 +
1.1869 +lemma cone_hull_empty: "cone hull {} = {}"
1.1870 +  by (metis cone_empty cone_hull_eq)
1.1871 +
1.1872 +lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
1.1873 +  by (metis bot_least cone_hull_empty hull_subset xtrans(5))
1.1874 +
1.1875 +lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
1.1876 +  using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
1.1877 +  by auto
1.1878 +
1.1879 +lemma mem_cone_hull:
1.1880 +  assumes "x \<in> S" "c \<ge> 0"
1.1881 +  shows "c *\<^sub>R x \<in> cone hull S"
1.1882 +  by (metis assms cone_cone_hull hull_inc mem_cone)
1.1883 +
1.1884 +proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
1.1885 +  (is "?lhs = ?rhs")
1.1886 +proof -
1.1887 +  {
1.1888 +    fix x
1.1889 +    assume "x \<in> ?rhs"
1.1890 +    then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
1.1891 +      by auto
1.1892 +    fix c :: real
1.1893 +    assume c: "c \<ge> 0"
1.1894 +    then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
1.1895 +      using x by (simp add: algebra_simps)
1.1896 +    moreover
1.1897 +    have "c * cx \<ge> 0" using c x by auto
1.1898 +    ultimately
1.1899 +    have "c *\<^sub>R x \<in> ?rhs" using x by auto
1.1900 +  }
1.1901 +  then have "cone ?rhs"
1.1902 +    unfolding cone_def by auto
1.1903 +  then have "?rhs \<in> Collect cone"
1.1904 +    unfolding mem_Collect_eq by auto
1.1905 +  {
1.1906 +    fix x
1.1907 +    assume "x \<in> S"
1.1908 +    then have "1 *\<^sub>R x \<in> ?rhs"
1.1909 +      apply auto
1.1910 +      apply (rule_tac x = 1 in exI, auto)
1.1911 +      done
1.1912 +    then have "x \<in> ?rhs" by auto
1.1913 +  }
1.1914 +  then have "S \<subseteq> ?rhs" by auto
1.1915 +  then have "?lhs \<subseteq> ?rhs"
1.1916 +    using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
1.1917 +  moreover
1.1918 +  {
1.1919 +    fix x
1.1920 +    assume "x \<in> ?rhs"
1.1921 +    then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
1.1922 +      by auto
1.1923 +    then have "xx \<in> cone hull S"
1.1924 +      using hull_subset[of S] by auto
1.1925 +    then have "x \<in> ?lhs"
1.1926 +      using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
1.1927 +  }
1.1928 +  ultimately show ?thesis by auto
1.1929 +qed
1.1930 +
1.1931 +
1.1932 +subsection \<open>Affine dependence and consequential theorems\<close>
1.1933 +
1.1934 +text "Formalized by Lars Schewe."
1.1935 +
1.1936 +definition%important affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
1.1937 +  where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
1.1938 +
1.1939 +lemma affine_dependent_subset:
1.1940 +   "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
1.1941 +apply (simp add: affine_dependent_def Bex_def)
1.1942 +apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
1.1943 +done
1.1944 +
1.1945 +lemma affine_independent_subset:
1.1946 +  shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
1.1947 +by (metis affine_dependent_subset)
1.1948 +
1.1949 +lemma affine_independent_Diff:
1.1950 +   "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
1.1951 +by (meson Diff_subset affine_dependent_subset)
1.1952 +
1.1953 +proposition affine_dependent_explicit:
1.1954 +  "affine_dependent p \<longleftrightarrow>
1.1955 +    (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
1.1956 +proof -
1.1957 +  have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
1.1958 +    if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
1.1959 +  proof (intro exI conjI)
1.1960 +    have "x \<notin> S"
1.1961 +      using that by auto
1.1962 +    then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
1.1963 +      using that by (simp add: sum_delta_notmem)
1.1964 +    show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
1.1965 +      using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
1.1966 +  qed (use that in auto)
1.1967 +  moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
1.1968 +    if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
1.1969 +  proof (intro bexI exI conjI)
1.1970 +    have "S \<noteq> {v}"
1.1971 +      using that by auto
1.1972 +    then show "S - {v} \<noteq> {}"
1.1973 +      using that by auto
1.1974 +    show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
1.1975 +      unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
1.1976 +    show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
1.1977 +      unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
1.1978 +                scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
1.1979 +      using that by auto
1.1980 +    show "S - {v} \<subseteq> p - {v}"
1.1981 +      using that by auto
1.1982 +  qed (use that in auto)
1.1983 +  ultimately show ?thesis
1.1984 +    unfolding affine_dependent_def affine_hull_explicit by auto
1.1985 +qed
1.1986 +
1.1987 +lemma affine_dependent_explicit_finite:
1.1988 +  fixes S :: "'a::real_vector set"
1.1989 +  assumes "finite S"
1.1990 +  shows "affine_dependent S \<longleftrightarrow>
1.1991 +    (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
1.1992 +  (is "?lhs = ?rhs")
1.1993 +proof
1.1994 +  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)"
1.1995 +    by auto
1.1996 +  assume ?lhs
1.1997 +  then obtain t u v where
1.1998 +    "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
1.1999 +    unfolding affine_dependent_explicit by auto
1.2000 +  then show ?rhs
1.2001 +    apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
1.2002 +    apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
1.2003 +    done
1.2004 +next
1.2005 +  assume ?rhs
1.2006 +  then obtain u v where "sum u S = 0"  "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
1.2007 +    by auto
1.2008 +  then show ?lhs unfolding affine_dependent_explicit
1.2009 +    using assms by auto
1.2010 +qed
1.2011 +
1.2012 +
1.2013 +subsection%unimportant \<open>Connectedness of convex sets\<close>
1.2014 +
1.2015 +lemma connectedD:
1.2016 +  "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
1.2017 +  by (rule Topological_Spaces.topological_space_class.connectedD)
1.2018 +
1.2019 +lemma convex_connected:
1.2020 +  fixes S :: "'a::real_normed_vector set"
1.2021 +  assumes "convex S"
1.2022 +  shows "connected S"
1.2023 +proof (rule connectedI)
1.2024 +  fix A B
1.2025 +  assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
1.2026 +  moreover
1.2027 +  assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
1.2028 +  then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
1.2029 +  define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
1.2030 +  then have "continuous_on {0 .. 1} f"
1.2031 +    by (auto intro!: continuous_intros)
1.2032 +  then have "connected (f ` {0 .. 1})"
1.2033 +    by (auto intro!: connected_continuous_image)
1.2034 +  note connectedD[OF this, of A B]
1.2035 +  moreover have "a \<in> A \<inter> f ` {0 .. 1}"
1.2036 +    using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
1.2037 +  moreover have "b \<in> B \<inter> f ` {0 .. 1}"
1.2038 +    using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
1.2039 +  moreover have "f ` {0 .. 1} \<subseteq> S"
1.2040 +    using \<open>convex S\<close> a b unfolding convex_def f_def by auto
1.2041 +  ultimately show False by auto
1.2042 +qed
1.2043 +
1.2044 +corollary connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
1.2045 +  by (simp add: convex_connected)
1.2046 +
1.2047 +lemma convex_prod:
1.2048 +  assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
1.2049 +  shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
1.2050 +  using assms unfolding convex_def
1.2051 +  by (auto simp: inner_add_left)
1.2052 +
1.2053 +lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
1.2054 +  by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
1.2055 +
1.2056 +subsection \<open>Convex hull\<close>
1.2057 +
1.2058 +lemma convex_convex_hull [iff]: "convex (convex hull s)"
1.2059 +  unfolding hull_def
1.2060 +  using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
1.2061 +  by auto
1.2062 +
1.2063 +lemma convex_hull_subset:
1.2064 +    "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
1.2065 +  by (simp add: convex_convex_hull subset_hull)
1.2066 +
1.2067 +lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
1.2068 +  by (metis convex_convex_hull hull_same)
1.2069 +
1.2070 +subsubsection%unimportant \<open>Convex hull is "preserved" by a linear function\<close>
1.2071 +
1.2072 +lemma convex_hull_linear_image:
1.2073 +  assumes f: "linear f"
1.2074 +  shows "f ` (convex hull s) = convex hull (f ` s)"
1.2075 +proof
1.2076 +  show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
1.2077 +    by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
1.2078 +  show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
1.2079 +  proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
1.2080 +    show "s \<subseteq> f -` (convex hull (f ` s))"
1.2081 +      by (fast intro: hull_inc)
1.2082 +    show "convex (f -` (convex hull (f ` s)))"
1.2083 +      by (intro convex_linear_vimage [OF f] convex_convex_hull)
1.2084 +  qed
1.2085 +qed
1.2086 +
1.2087 +lemma in_convex_hull_linear_image:
1.2088 +  assumes "linear f"
1.2089 +    and "x \<in> convex hull s"
1.2090 +  shows "f x \<in> convex hull (f ` s)"
1.2091 +  using convex_hull_linear_image[OF assms(1)] assms(2) by auto
1.2092 +
1.2093 +lemma convex_hull_Times:
1.2094 +  "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
1.2095 +proof
1.2096 +  show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
1.2097 +    by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
1.2098 +  have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
1.2099 +  proof (rule hull_induct [OF x], rule hull_induct [OF y])
1.2100 +    fix x y assume "x \<in> s" and "y \<in> t"
1.2101 +    then show "(x, y) \<in> convex hull (s \<times> t)"
1.2102 +      by (simp add: hull_inc)
1.2103 +  next
1.2104 +    fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
1.2105 +    have "convex ?S"
1.2106 +      by (intro convex_linear_vimage convex_translation convex_convex_hull,
1.2108 +    also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
1.2109 +      by (auto simp: image_def Bex_def)
1.2110 +    finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
1.2111 +  next
1.2112 +    show "convex {x. (x, y) \<in> convex hull s \<times> t}"
1.2113 +    proof -
1.2114 +      fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
1.2115 +      have "convex ?S"
1.2116 +      by (intro convex_linear_vimage convex_translation convex_convex_hull,
1.2118 +      also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
1.2119 +        by (auto simp: image_def Bex_def)
1.2120 +      finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
1.2121 +    qed
1.2122 +  qed
1.2123 +  then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
1.2124 +    unfolding subset_eq split_paired_Ball_Sigma by blast
1.2125 +qed
1.2126 +
1.2127 +
1.2128 +subsubsection%unimportant \<open>Stepping theorems for convex hulls of finite sets\<close>
1.2129 +
1.2130 +lemma convex_hull_empty[simp]: "convex hull {} = {}"
1.2131 +  by (rule hull_unique) auto
1.2132 +
1.2133 +lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
1.2134 +  by (rule hull_unique) auto
1.2135 +
1.2136 +lemma convex_hull_insert:
1.2137 +  fixes S :: "'a::real_vector set"
1.2138 +  assumes "S \<noteq> {}"
1.2139 +  shows "convex hull (insert a S) =
1.2140 +         {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
1.2141 +  (is "_ = ?hull")
1.2142 +proof (intro equalityI hull_minimal subsetI)
1.2143 +  fix x
1.2144 +  assume "x \<in> insert a S"
1.2145 +  then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
1.2146 +  unfolding insert_iff
1.2147 +  proof
1.2148 +    assume "x = a"
1.2149 +    then show ?thesis
1.2150 +      by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
1.2151 +  next
1.2152 +    assume "x \<in> S"
1.2153 +    with hull_subset[of S convex] show ?thesis
1.2154 +      by force
1.2155 +  qed
1.2156 +  then show "x \<in> ?hull"
1.2157 +    by simp
1.2158 +next
1.2159 +  fix x
1.2160 +  assume "x \<in> ?hull"
1.2161 +  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"
1.2162 +    by auto
1.2163 +  have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
1.2164 +    using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
1.2165 +    by auto
1.2166 +  then show "x \<in> convex hull insert a S"
1.2167 +    unfolding obt(5) using obt(1-3)
1.2168 +    by (rule convexD [OF convex_convex_hull])
1.2169 +next
1.2170 +  show "convex ?hull"
1.2171 +  proof (rule convexI)
1.2172 +    fix x y u v
1.2173 +    assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
1.2174 +    from x obtain u1 v1 b1 where
1.2175 +      obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
1.2176 +      by auto
1.2177 +    from y obtain u2 v2 b2 where
1.2178 +      obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
1.2179 +      by auto
1.2180 +    have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
1.2181 +      by (auto simp: algebra_simps)
1.2182 +    have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
1.2183 +      (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
1.2184 +    proof (cases "u * v1 + v * v2 = 0")
1.2185 +      case True
1.2186 +      have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
1.2187 +        by (auto simp: algebra_simps)
1.2188 +      have eq0: "u * v1 = 0" "v * v2 = 0"
1.2189 +        using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
1.2190 +        by arith+
1.2191 +      then have "u * u1 + v * u2 = 1"
1.2192 +        using as(3) obt1(3) obt2(3) by auto
1.2193 +      then show ?thesis
1.2194 +        using "*" eq0 as obt1(4) xeq yeq by auto
1.2195 +    next
1.2196 +      case False
1.2197 +      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
1.2198 +        using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
1.2199 +      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
1.2200 +        using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
1.2201 +      also have "\<dots> = u * v1 + v * v2"
1.2202 +        by simp
1.2203 +      finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
1.2204 +      let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
1.2205 +      have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
1.2206 +        using as(1,2) obt1(1,2) obt2(1,2) by auto
1.2207 +      show ?thesis
1.2208 +      proof
1.2209 +        show "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)"
1.2210 +          unfolding xeq yeq * **
1.2211 +          using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
1.2212 +        show "?b \<in> convex hull S"
1.2213 +          using False zeroes obt1(4) obt2(4)
1.2214 +          by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib  add_divide_distrib[symmetric]  zero_le_divide_iff)
1.2215 +      qed
1.2216 +    qed
1.2217 +    then obtain b where b: "b \<in> convex hull S"
1.2218 +       "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)" ..
1.2219 +
1.2220 +    have u1: "u1 \<le> 1"
1.2221 +      unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
1.2222 +    have u2: "u2 \<le> 1"
1.2223 +      unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
1.2224 +    have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
1.2226 +      show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
1.2227 +        by (simp_all add: as mult_right_mono)
1.2228 +    qed
1.2229 +    also have "\<dots> \<le> 1"
1.2230 +      unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
1.2231 +    finally have le1: "u1 * u + u2 * v \<le> 1" .
1.2232 +    show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
1.2233 +    proof (intro CollectI exI conjI)
1.2234 +      show "0 \<le> u * u1 + v * u2"
1.2235 +        by (simp add: as(1) as(2) obt1(1) obt2(1))
1.2236 +      show "0 \<le> 1 - u * u1 - v * u2"
1.2238 +    qed (use b in \<open>auto simp: algebra_simps\<close>)
1.2239 +  qed
1.2240 +qed
1.2241 +
1.2242 +lemma convex_hull_insert_alt:
1.2243 +   "convex hull (insert a S) =
1.2244 +     (if S = {} then {a}
1.2245 +      else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
1.2246 +  apply (auto simp: convex_hull_insert)
1.2247 +  using diff_eq_eq apply fastforce
1.2249 +
1.2250 +subsubsection%unimportant \<open>Explicit expression for convex hull\<close>
1.2251 +
1.2252 +proposition convex_hull_indexed:
1.2253 +  fixes S :: "'a::real_vector set"
1.2254 +  shows "convex hull S =
1.2255 +    {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
1.2256 +                (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
1.2257 +    (is "?xyz = ?hull")
1.2258 +proof (rule hull_unique [OF _ convexI])
1.2259 +  show "S \<subseteq> ?hull"
1.2260 +    by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
1.2261 +next
1.2262 +  fix T
1.2263 +  assume "S \<subseteq> T" "convex T"
1.2264 +  then show "?hull \<subseteq> T"
1.2265 +    by (blast intro: convex_sum)
1.2266 +next
1.2267 +  fix x y u v
1.2268 +  assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
1.2269 +  assume xy: "x \<in> ?hull" "y \<in> ?hull"
1.2270 +  from xy obtain k1 u1 x1 where
1.2271 +    x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S"
1.2272 +                      "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
1.2273 +    by auto
1.2274 +  from xy obtain k2 u2 x2 where
1.2275 +    y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S"
1.2276 +                     "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
1.2277 +    by auto
1.2278 +  have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
1.2279 +          "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
1.2280 +    by auto
1.2281 +  have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
1.2282 +    unfolding inj_on_def by auto
1.2283 +  let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
1.2284 +  let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
1.2285 +  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
1.2286 +  proof (intro CollectI exI conjI ballI)
1.2287 +    show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
1.2288 +      using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
1.2289 +    show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1"  "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
1.2290 +      unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
1.2291 +        sum.reindex[OF inj] Collect_mem_eq o_def
1.2292 +      unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
1.2293 +      by (simp_all add: sum_distrib_left[symmetric]  x(2,3) y(2,3) uv(3))
1.2294 +  qed
1.2295 +qed
1.2296 +
1.2297 +lemma convex_hull_finite:
1.2298 +  fixes S :: "'a::real_vector set"
1.2299 +  assumes "finite S"
1.2300 +  shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
1.2301 +  (is "?HULL = _")
1.2302 +proof (rule hull_unique [OF _ convexI]; clarify)
1.2303 +  fix x
1.2304 +  assume "x \<in> S"
1.2305 +  then show "\<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = x"
1.2306 +    by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
1.2307 +next
1.2308 +  fix u v :: real
1.2309 +  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
1.2310 +  fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
1.2311 +  fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
1.2312 +  have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
1.2313 +    by (simp add: that uv ux(1) uy(1))
1.2314 +  moreover
1.2315 +  have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
1.2316 +    unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
1.2317 +    using uv(3) by auto
1.2318 +  moreover
1.2319 +  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)"
1.2320 +    unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
1.2321 +    by auto
1.2322 +  ultimately
1.2323 +  show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
1.2324 +             (\<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)"
1.2325 +    by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
1.2326 +qed (use assms in \<open>auto simp: convex_explicit\<close>)
1.2327 +
1.2328 +
1.2329 +subsubsection%unimportant \<open>Another formulation\<close>
1.2330 +
1.2331 +text "Formalized by Lars Schewe."
1.2332 +
1.2333 +lemma convex_hull_explicit:
1.2334 +  fixes p :: "'a::real_vector set"
1.2335 +  shows "convex hull p =
1.2336 +    {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
1.2337 +  (is "?lhs = ?rhs")
1.2338 +proof -
1.2339 +  {
1.2340 +    fix x
1.2341 +    assume "x\<in>?lhs"
1.2342 +    then obtain k u y where
1.2343 +        obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
1.2344 +      unfolding convex_hull_indexed by auto
1.2345 +
1.2346 +    have fin: "finite {1..k}" by auto
1.2347 +    have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
1.2348 +    {
1.2349 +      fix j
1.2350 +      assume "j\<in>{1..k}"
1.2351 +      then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
1.2352 +        using obt(1)[THEN bspec[where x=j]] and obt(2)
1.2353 +        apply simp
1.2354 +        apply (rule sum_nonneg)
1.2355 +        using obt(1)
1.2356 +        apply auto
1.2357 +        done
1.2358 +    }
1.2359 +    moreover
1.2360 +    have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
1.2361 +      unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
1.2362 +    moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
1.2363 +      using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
1.2364 +      unfolding scaleR_left.sum using obt(3) by auto
1.2365 +    ultimately
1.2366 +    have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
1.2367 +      apply (rule_tac x="y ` {1..k}" in exI)
1.2368 +      apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
1.2369 +      done
1.2370 +    then have "x\<in>?rhs" by auto
1.2371 +  }
1.2372 +  moreover
1.2373 +  {
1.2374 +    fix y
1.2375 +    assume "y\<in>?rhs"
1.2376 +    then obtain S u where
1.2377 +      obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
1.2378 +      by auto
1.2379 +
1.2380 +    obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
1.2381 +      using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
1.2382 +
1.2383 +    {
1.2384 +      fix i :: nat
1.2385 +      assume "i\<in>{1..card S}"
1.2386 +      then have "f i \<in> S"
1.2387 +        using f(2) by blast
1.2388 +      then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
1.2389 +    }
1.2390 +    moreover have *: "finite {1..card S}" by auto
1.2391 +    {
1.2392 +      fix y
1.2393 +      assume "y\<in>S"
1.2394 +      then obtain i where "i\<in>{1..card S}" "f i = y"
1.2395 +        using f using image_iff[of y f "{1..card S}"]
1.2396 +        by auto
1.2397 +      then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
1.2398 +        apply auto
1.2399 +        using f(1)[unfolded inj_on_def]
1.2400 +        by (metis One_nat_def atLeastAtMost_iff)
1.2401 +      then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
1.2402 +      then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
1.2403 +          "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
1.2404 +        by (auto simp: sum_constant_scaleR)
1.2405 +    }
1.2406 +    then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
1.2407 +      unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
1.2408 +        and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
1.2409 +      unfolding f
1.2410 +      using sum.cong [of S 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"]
1.2411 +      using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
1.2412 +      unfolding obt(4,5)
1.2413 +      by auto
1.2414 +    ultimately
1.2415 +    have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
1.2416 +        (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
1.2417 +      apply (rule_tac x="card S" in exI)
1.2418 +      apply (rule_tac x="u \<circ> f" in exI)
1.2419 +      apply (rule_tac x=f in exI, fastforce)
1.2420 +      done
1.2421 +    then have "y \<in> ?lhs"
1.2422 +      unfolding convex_hull_indexed by auto
1.2423 +  }
1.2424 +  ultimately show ?thesis
1.2425 +    unfolding set_eq_iff by blast
1.2426 +qed
1.2427 +
1.2428 +
1.2429 +subsubsection%unimportant \<open>A stepping theorem for that expansion\<close>
1.2430 +
1.2431 +lemma convex_hull_finite_step:
1.2432 +  fixes S :: "'a::real_vector set"
1.2433 +  assumes "finite S"
1.2434 +  shows
1.2435 +    "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
1.2436 +      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
1.2437 +  (is "?lhs = ?rhs")
1.2438 +proof (rule, case_tac[!] "a\<in>S")
1.2439 +  assume "a \<in> S"
1.2440 +  then have *: "insert a S = S" by auto
1.2441 +  assume ?lhs
1.2442 +  then show ?rhs
1.2443 +    unfolding *  by (rule_tac x=0 in exI, auto)
1.2444 +next
1.2445 +  assume ?lhs
1.2446 +  then obtain u where
1.2447 +      u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
1.2448 +    by auto
1.2449 +  assume "a \<notin> S"
1.2450 +  then show ?rhs
1.2451 +    apply (rule_tac x="u a" in exI)
1.2452 +    using u(1)[THEN bspec[where x=a]]
1.2453 +    apply simp
1.2454 +    apply (rule_tac x=u in exI)
1.2455 +    using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
1.2456 +    apply auto
1.2457 +    done
1.2458 +next
1.2459 +  assume "a \<in> S"
1.2460 +  then have *: "insert a S = S" by auto
1.2461 +  have fin: "finite (insert a S)" using assms by auto
1.2462 +  assume ?rhs
1.2463 +  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
1.2464 +    by auto
1.2465 +  show ?lhs
1.2466 +    apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
1.2467 +    unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
1.2468 +    unfolding sum_clauses(2)[OF assms]
1.2469 +    using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
1.2470 +    apply auto
1.2471 +    done
1.2472 +next
1.2473 +  assume ?rhs
1.2474 +  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
1.2475 +    by auto
1.2476 +  moreover assume "a \<notin> S"
1.2477 +  moreover
1.2478 +  have "(\<Sum>x\<in>S. if a = x then v else u x) = sum 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)"
1.2479 +    using \<open>a \<notin> S\<close>
1.2480 +    by (auto simp: intro!: sum.cong)
1.2481 +  ultimately show ?lhs
1.2482 +    by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
1.2483 +qed
1.2484 +
1.2485 +
1.2486 +subsubsection%unimportant \<open>Hence some special cases\<close>
1.2487 +
1.2488 +lemma convex_hull_2:
1.2489 +  "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}"
1.2490 +proof -
1.2491 +  have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
1.2492 +    by auto
1.2493 +  have **: "finite {b}" by auto
1.2494 +  show ?thesis
1.2495 +    apply (simp add: convex_hull_finite)
1.2496 +    unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
1.2497 +    apply auto
1.2498 +    apply (rule_tac x=v in exI)
1.2499 +    apply (rule_tac x="1 - v" in exI, simp)
1.2500 +    apply (rule_tac x=u in exI, simp)
1.2501 +    apply (rule_tac x="\<lambda>x. v" in exI, simp)
1.2502 +    done
1.2503 +qed
1.2504 +
1.2505 +lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
1.2506 +  unfolding convex_hull_2
1.2507 +proof (rule Collect_cong)
1.2508 +  have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
1.2509 +    by auto
1.2510 +  fix x
1.2511 +  show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
1.2512 +    (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
1.2513 +    unfolding *
1.2514 +    apply auto
1.2515 +    apply (rule_tac[!] x=u in exI)
1.2516 +    apply (auto simp: algebra_simps)
1.2517 +    done
1.2518 +qed
1.2519 +
1.2520 +lemma convex_hull_3:
1.2521 +  "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}"
1.2522 +proof -
1.2523 +  have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
1.2524 +    by auto
1.2525 +  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
1.2526 +    by (auto simp: field_simps)
1.2527 +  show ?thesis
1.2528 +    unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
1.2529 +    unfolding convex_hull_finite_step[OF fin(3)]
1.2530 +    apply (rule Collect_cong, simp)
1.2531 +    apply auto
1.2532 +    apply (rule_tac x=va in exI)
1.2533 +    apply (rule_tac x="u c" in exI, simp)
1.2534 +    apply (rule_tac x="1 - v - w" in exI, simp)
1.2535 +    apply (rule_tac x=v in exI, simp)
1.2536 +    apply (rule_tac x="\<lambda>x. w" in exI, simp)
1.2537 +    done
1.2538 +qed
1.2539 +
1.2540 +lemma convex_hull_3_alt:
1.2541 +  "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}"
1.2542 +proof -
1.2543 +  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
1.2544 +    by auto
1.2545 +  show ?thesis
1.2546 +    unfolding convex_hull_3
1.2547 +    apply (auto simp: *)
1.2548 +    apply (rule_tac x=v in exI)
1.2549 +    apply (rule_tac x=w in exI)
1.2550 +    apply (simp add: algebra_simps)
1.2551 +    apply (rule_tac x=u in exI)
1.2552 +    apply (rule_tac x=v in exI)
1.2553 +    apply (simp add: algebra_simps)
1.2554 +    done
1.2555 +qed
1.2556 +
1.2557 +
1.2558 +subsection%unimportant \<open>Relations among closure notions and corresponding hulls\<close>
1.2559 +
1.2560 +lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
1.2561 +  unfolding affine_def convex_def by auto
1.2562 +
1.2563 +lemma convex_affine_hull [simp]: "convex (affine hull S)"
1.2564 +  by (simp add: affine_imp_convex)
1.2565 +
1.2566 +lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
1.2567 +  using subspace_imp_affine affine_imp_convex by auto
1.2568 +
1.2569 +lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
1.2570 +  by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
1.2571 +
1.2572 +lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
1.2573 +  by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
1.2574 +
1.2575 +lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
1.2576 +  by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
1.2577 +
1.2578 +lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
1.2579 +  unfolding affine_dependent_def dependent_def
1.2580 +  using affine_hull_subset_span by auto
1.2581 +
1.2582 +lemma dependent_imp_affine_dependent:
1.2583 +  assumes "dependent {x - a| x . x \<in> s}"
1.2584 +    and "a \<notin> s"
1.2585 +  shows "affine_dependent (insert a s)"
1.2586 +proof -
1.2587 +  from assms(1)[unfolded dependent_explicit] obtain S u v
1.2588 +    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"
1.2589 +    by auto
1.2590 +  define t where "t = (\<lambda>x. x + a) ` S"
1.2591 +
1.2592 +  have inj: "inj_on (\<lambda>x. x + a) S"
1.2593 +    unfolding inj_on_def by auto
1.2594 +  have "0 \<notin> S"
1.2595 +    using obt(2) assms(2) unfolding subset_eq by auto
1.2596 +  have fin: "finite t" and "t \<subseteq> s"
1.2597 +    unfolding t_def using obt(1,2) by auto
1.2598 +  then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
1.2599 +    by auto
1.2600 +  moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
1.2601 +    apply (rule sum.cong)
1.2602 +    using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
1.2603 +    apply auto
1.2604 +    done
1.2605 +  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
1.2606 +    unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
1.2607 +  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"
1.2608 +    using obt(3,4) \<open>0\<notin>S\<close>
1.2609 +    by (rule_tac x="v + a" in bexI) (auto simp: t_def)
1.2610 +  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)"
1.2611 +    using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
1.2612 +  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
1.2613 +    unfolding scaleR_left.sum
1.2614 +    unfolding t_def and sum.reindex[OF inj] and o_def
1.2615 +    using obt(5)
1.2616 +    by (auto simp: sum.distrib scaleR_right_distrib)
1.2617 +  then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
1.2618 +    unfolding sum_clauses(2)[OF fin]
1.2619 +    using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
1.2620 +    by (auto simp: *)
1.2621 +  ultimately show ?thesis
1.2622 +    unfolding affine_dependent_explicit
1.2623 +    apply (rule_tac x="insert a t" in exI, auto)
1.2624 +    done
1.2625 +qed
1.2626 +
1.2627 +lemma convex_cone:
1.2628 +  "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)"
1.2629 +  (is "?lhs = ?rhs")
1.2630 +proof -
1.2631 +  {
1.2632 +    fix x y
1.2633 +    assume "x\<in>s" "y\<in>s" and ?lhs
1.2634 +    then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
1.2635 +      unfolding cone_def by auto
1.2636 +    then have "x + y \<in> s"
1.2637 +      using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
1.2638 +      apply (erule_tac x="2*\<^sub>R x" in ballE)
1.2639 +      apply (erule_tac x="2*\<^sub>R y" in ballE)
1.2640 +      apply (erule_tac x="1/2" in allE, simp)
1.2641 +      apply (erule_tac x="1/2" in allE, auto)
1.2642 +      done
1.2643 +  }
1.2644 +  then show ?thesis
1.2645 +    unfolding convex_def cone_def by blast
1.2646 +qed
1.2647 +
1.2648 +lemma affine_dependent_biggerset:
1.2649 +  fixes s :: "'a::euclidean_space set"
1.2650 +  assumes "finite s" "card s \<ge> DIM('a) + 2"
1.2651 +  shows "affine_dependent s"
1.2652 +proof -
1.2653 +  have "s \<noteq> {}" using assms by auto
1.2654 +  then obtain a where "a\<in>s" by auto
1.2655 +  have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
1.2656 +    by auto
1.2657 +  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
1.2658 +    unfolding * by (simp add: card_image inj_on_def)
1.2659 +  also have "\<dots> > DIM('a)" using assms(2)
1.2660 +    unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
1.2661 +  finally show ?thesis
1.2662 +    apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
1.2663 +    apply (rule dependent_imp_affine_dependent)
1.2664 +    apply (rule dependent_biggerset, auto)
1.2665 +    done
1.2666 +qed
1.2667 +
1.2668 +lemma affine_dependent_biggerset_general:
1.2669 +  assumes "finite (S :: 'a::euclidean_space set)"
1.2670 +    and "card S \<ge> dim S + 2"
1.2671 +  shows "affine_dependent S"
1.2672 +proof -
1.2673 +  from assms(2) have "S \<noteq> {}" by auto
1.2674 +  then obtain a where "a\<in>S" by auto
1.2675 +  have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
1.2676 +    by auto
1.2677 +  have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
1.2678 +    by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
1.2679 +  have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
1.2680 +    using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
1.2681 +  also have "\<dots> < dim S + 1" by auto
1.2682 +  also have "\<dots> \<le> card (S - {a})"
1.2683 +    using assms
1.2684 +    using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
1.2685 +    by auto
1.2686 +  finally show ?thesis
1.2687 +    apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
1.2688 +    apply (rule dependent_imp_affine_dependent)
1.2689 +    apply (rule dependent_biggerset_general)
1.2690 +    unfolding **
1.2691 +    apply auto
1.2692 +    done
1.2693 +qed
1.2694 +
1.2695 +
1.2696 +subsection%unimportant \<open>Some Properties of Affine Dependent Sets\<close>
1.2697 +
1.2698 +lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
1.2699 +  by (simp add: affine_dependent_def)
1.2700 +
1.2701 +lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
1.2702 +  by (simp add: affine_dependent_def)
1.2703 +
1.2704 +lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
1.2705 +  by (simp add: affine_dependent_def insert_Diff_if hull_same)
1.2706 +
1.2707 +lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
1.2708 +proof -
1.2709 +  have "affine ((\<lambda>x. a + x) ` (affine hull S))"
1.2710 +    using affine_translation affine_affine_hull by blast
1.2711 +  moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
1.2712 +    using hull_subset[of S] by auto
1.2713 +  ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
1.2714 +    by (metis hull_minimal)
1.2715 +  have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
1.2716 +    using affine_translation affine_affine_hull by blast
1.2717 +  moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
1.2718 +    using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
1.2719 +  moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
1.2720 +    using translation_assoc[of "-a" a] by auto
1.2721 +  ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
1.2722 +    by (metis hull_minimal)
1.2723 +  then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
1.2724 +    by auto
1.2725 +  then show ?thesis using h1 by auto
1.2726 +qed
1.2727 +
1.2728 +lemma affine_dependent_translation:
1.2729 +  assumes "affine_dependent S"
1.2730 +  shows "affine_dependent ((\<lambda>x. a + x) ` S)"
1.2731 +proof -
1.2732 +  obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
1.2733 +    using assms affine_dependent_def by auto
1.2734 +  have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
1.2735 +    by auto
1.2736 +  then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
1.2737 +    using affine_hull_translation[of a "S - {x}"] x by auto
1.2738 +  moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
1.2739 +    using x by auto
1.2740 +  ultimately show ?thesis
1.2741 +    unfolding affine_dependent_def by auto
1.2742 +qed
1.2743 +
1.2744 +lemma affine_dependent_translation_eq:
1.2745 +  "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
1.2746 +proof -
1.2747 +  {
1.2748 +    assume "affine_dependent ((\<lambda>x. a + x) ` S)"
1.2749 +    then have "affine_dependent S"
1.2750 +      using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
1.2751 +      by auto
1.2752 +  }
1.2753 +  then show ?thesis
1.2754 +    using affine_dependent_translation by auto
1.2755 +qed
1.2756 +
1.2757 +lemma affine_hull_0_dependent:
1.2758 +  assumes "0 \<in> affine hull S"
1.2759 +  shows "dependent S"
1.2760 +proof -
1.2761 +  obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
1.2762 +    using assms affine_hull_explicit[of S] by auto
1.2763 +  then have "\<exists>v\<in>s. u v \<noteq> 0"
1.2764 +    using sum_not_0[of "u" "s"] by auto
1.2765 +  then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
1.2766 +    using s_u by auto
1.2767 +  then show ?thesis
1.2768 +    unfolding dependent_explicit[of S] by auto
1.2769 +qed
1.2770 +
1.2771 +lemma affine_dependent_imp_dependent2:
1.2772 +  assumes "affine_dependent (insert 0 S)"
1.2773 +  shows "dependent S"
1.2774 +proof -
1.2775 +  obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
1.2776 +    using affine_dependent_def[of "(insert 0 S)"] assms by blast
1.2777 +  then have "x \<in> span (insert 0 S - {x})"
1.2778 +    using affine_hull_subset_span by auto
1.2779 +  moreover have "span (insert 0 S - {x}) = span (S - {x})"
1.2780 +    using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
1.2781 +  ultimately have "x \<in> span (S - {x})" by auto
1.2782 +  then have "x \<noteq> 0 \<Longrightarrow> dependent S"
1.2783 +    using x dependent_def by auto
1.2784 +  moreover
1.2785 +  {
1.2786 +    assume "x = 0"
1.2787 +    then have "0 \<in> affine hull S"
1.2788 +      using x hull_mono[of "S - {0}" S] by auto
1.2789 +    then have "dependent S"
1.2790 +      using affine_hull_0_dependent by auto
1.2791 +  }
1.2792 +  ultimately show ?thesis by auto
1.2793 +qed
1.2794 +
1.2795 +lemma affine_dependent_iff_dependent:
1.2796 +  assumes "a \<notin> S"
1.2797 +  shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
1.2798 +proof -
1.2799 +  have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
1.2800 +  then show ?thesis
1.2801 +    using affine_dependent_translation_eq[of "(insert a S)" "-a"]
1.2802 +      affine_dependent_imp_dependent2 assms
1.2803 +      dependent_imp_affine_dependent[of a S]
1.2804 +    by (auto simp del: uminus_add_conv_diff)
1.2805 +qed
1.2806 +
1.2807 +lemma affine_dependent_iff_dependent2:
1.2808 +  assumes "a \<in> S"
1.2809 +  shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
1.2810 +proof -
1.2811 +  have "insert a (S - {a}) = S"
1.2812 +    using assms by auto
1.2813 +  then show ?thesis
1.2814 +    using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
1.2815 +qed
1.2816 +
1.2817 +lemma affine_hull_insert_span_gen:
1.2818 +  "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
1.2819 +proof -
1.2820 +  have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
1.2821 +    by auto
1.2822 +  {
1.2823 +    assume "a \<notin> s"
1.2824 +    then have ?thesis
1.2825 +      using affine_hull_insert_span[of a s] h1 by auto
1.2826 +  }
1.2827 +  moreover
1.2828 +  {
1.2829 +    assume a1: "a \<in> s"
1.2830 +    have "\<exists>x. x \<in> s \<and> -a+x=0"
1.2831 +      apply (rule exI[of _ a])
1.2832 +      using a1
1.2833 +      apply auto
1.2834 +      done
1.2835 +    then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
1.2836 +      by auto
1.2837 +    then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
1.2838 +      using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
1.2839 +    moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
1.2840 +      by auto
1.2841 +    moreover have "insert a (s - {a}) = insert a s"
1.2842 +      by auto
1.2843 +    ultimately have ?thesis
1.2844 +      using affine_hull_insert_span[of "a" "s-{a}"] by auto
1.2845 +  }
1.2846 +  ultimately show ?thesis by auto
1.2847 +qed
1.2848 +
1.2849 +lemma affine_hull_span2:
1.2850 +  assumes "a \<in> s"
1.2851 +  shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
1.2852 +  using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
1.2853 +  by auto
1.2854 +
1.2855 +lemma affine_hull_span_gen:
1.2856 +  assumes "a \<in> affine hull s"
1.2857 +  shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
1.2858 +proof -
1.2859 +  have "affine hull (insert a s) = affine hull s"
1.2860 +    using hull_redundant[of a affine s] assms by auto
1.2861 +  then show ?thesis
1.2862 +    using affine_hull_insert_span_gen[of a "s"] by auto
1.2863 +qed
1.2864 +
1.2865 +lemma affine_hull_span_0:
1.2866 +  assumes "0 \<in> affine hull S"
1.2867 +  shows "affine hull S = span S"
1.2868 +  using affine_hull_span_gen[of "0" S] assms by auto
1.2869 +
1.2870 +lemma extend_to_affine_basis_nonempty:
1.2871 +  fixes S V :: "'n::euclidean_space set"
1.2872 +  assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
1.2873 +  shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
1.2874 +proof -
1.2875 +  obtain a where a: "a \<in> S"
1.2876 +    using assms by auto
1.2877 +  then have h0: "independent  ((\<lambda>x. -a + x) ` (S-{a}))"
1.2878 +    using affine_dependent_iff_dependent2 assms by auto
1.2879 +  obtain B where B:
1.2880 +    "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
1.2881 +    using assms
1.2882 +    by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
1.2883 +  define T where "T = (\<lambda>x. a+x) ` insert 0 B"
1.2884 +  then have "T = insert a ((\<lambda>x. a+x) ` B)"
1.2885 +    by auto
1.2886 +  then have "affine hull T = (\<lambda>x. a+x) ` span B"
1.2887 +    using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
1.2888 +    by auto
1.2889 +  then have "V \<subseteq> affine hull T"
1.2890 +    using B assms translation_inverse_subset[of a V "span B"]
1.2891 +    by auto
1.2892 +  moreover have "T \<subseteq> V"
1.2893 +    using T_def B a assms by auto
1.2894 +  ultimately have "affine hull T = affine hull V"
1.2895 +    by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
1.2896 +  moreover have "S \<subseteq> T"
1.2897 +    using T_def B translation_inverse_subset[of a "S-{a}" B]
1.2898 +    by auto
1.2899 +  moreover have "\<not> affine_dependent T"
1.2900 +    using T_def affine_dependent_translation_eq[of "insert 0 B"]
1.2901 +      affine_dependent_imp_dependent2 B
1.2902 +    by auto
1.2903 +  ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
1.2904 +qed
1.2905 +
1.2906 +lemma affine_basis_exists:
1.2907 +  fixes V :: "'n::euclidean_space set"
1.2908 +  shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
1.2909 +proof (cases "V = {}")
1.2910 +  case True
1.2911 +  then show ?thesis
1.2912 +    using affine_independent_0 by auto
1.2913 +next
1.2914 +  case False
1.2915 +  then obtain x where "x \<in> V" by auto
1.2916 +  then show ?thesis
1.2917 +    using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
1.2918 +    by auto
1.2919 +qed
1.2920 +
1.2921 +proposition extend_to_affine_basis:
1.2922 +  fixes S V :: "'n::euclidean_space set"
1.2923 +  assumes "\<not> affine_dependent S" "S \<subseteq> V"
1.2924 +  obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
1.2925 +proof (cases "S = {}")
1.2926 +  case True then show ?thesis
1.2927 +    using affine_basis_exists by (metis empty_subsetI that)
1.2928 +next
1.2929 +  case False
1.2930 +  then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
1.2931 +qed
1.2932 +
1.2933 +subsection \<open>Affine Dimension of a Set\<close>
1.2934 +
1.2935 +definition%important aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
1.2936 +  where "aff_dim V =
1.2937 +  (SOME d :: int.
1.2938 +    \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
1.2939 +
1.2940 +lemma aff_dim_basis_exists:
1.2941 +  fixes V :: "('n::euclidean_space) set"
1.2942 +  shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
1.2943 +proof -
1.2944 +  obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
1.2945 +    using affine_basis_exists[of V] by auto
1.2946 +  then show ?thesis
1.2947 +    unfolding aff_dim_def
1.2948 +      some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
1.2949 +    apply auto
1.2950 +    apply (rule exI[of _ "int (card B) - (1 :: int)"])
1.2951 +    apply (rule exI[of _ "B"], auto)
1.2952 +    done
1.2953 +qed
1.2954 +
1.2955 +lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
1.2956 +proof -
1.2957 +  have "S = {} \<Longrightarrow> affine hull S = {}"
1.2958 +    using affine_hull_empty by auto
1.2959 +  moreover have "affine hull S = {} \<Longrightarrow> S = {}"
1.2960 +    unfolding hull_def by auto
1.2961 +  ultimately show ?thesis by blast
1.2962 +qed
1.2963 +
1.2964 +lemma aff_dim_parallel_subspace_aux:
1.2965 +  fixes B :: "'n::euclidean_space set"
1.2966 +  assumes "\<not> affine_dependent B" "a \<in> B"
1.2967 +  shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
1.2968 +proof -
1.2969 +  have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
1.2970 +    using affine_dependent_iff_dependent2 assms by auto
1.2971 +  then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
1.2972 +    "finite ((\<lambda>x. -a + x) ` (B - {a}))"
1.2973 +    using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
1.2974 +  show ?thesis
1.2975 +  proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
1.2976 +    case True
1.2977 +    have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
1.2978 +      using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
1.2979 +    then have "B = {a}" using True by auto
1.2980 +    then show ?thesis using assms fin by auto
1.2981 +  next
1.2982 +    case False
1.2983 +    then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
1.2984 +      using fin by auto
1.2985 +    moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
1.2986 +      by (rule card_image) (use translate_inj_on in blast)
1.2987 +    ultimately have "card (B-{a}) > 0" by auto
1.2988 +    then have *: "finite (B - {a})"
1.2989 +      using card_gt_0_iff[of "(B - {a})"] by auto
1.2990 +    then have "card (B - {a}) = card B - 1"
1.2991 +      using card_Diff_singleton assms by auto
1.2992 +    with * show ?thesis using fin h1 by auto
1.2993 +  qed
1.2994 +qed
1.2995 +
1.2996 +lemma aff_dim_parallel_subspace:
1.2997 +  fixes V L :: "'n::euclidean_space set"
1.2998 +  assumes "V \<noteq> {}"
1.2999 +    and "subspace L"
1.3000 +    and "affine_parallel (affine hull V) L"
1.3001 +  shows "aff_dim V = int (dim L)"
1.3002 +proof -
1.3003 +  obtain B where
1.3004 +    B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
1.3005 +    using aff_dim_basis_exists by auto
1.3006 +  then have "B \<noteq> {}"
1.3007 +    using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
1.3008 +    by auto
1.3009 +  then obtain a where a: "a \<in> B" by auto
1.3010 +  define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
1.3011 +  moreover have "affine_parallel (affine hull B) Lb"
1.3012 +    using Lb_def B assms affine_hull_span2[of a B] a
1.3013 +      affine_parallel_commut[of "Lb" "(affine hull B)"]
1.3014 +    unfolding affine_parallel_def
1.3015 +    by auto
1.3016 +  moreover have "subspace Lb"
1.3017 +    using Lb_def subspace_span by auto
1.3018 +  moreover have "affine hull B \<noteq> {}"
1.3019 +    using assms B affine_hull_nonempty[of V] by auto
1.3020 +  ultimately have "L = Lb"
1.3021 +    using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
1.3022 +    by auto
1.3023 +  then have "dim L = dim Lb"
1.3024 +    by auto
1.3025 +  moreover have "card B - 1 = dim Lb" and "finite B"
1.3026 +    using Lb_def aff_dim_parallel_subspace_aux a B by auto
1.3027 +  ultimately show ?thesis
1.3028 +    using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
1.3029 +qed
1.3030 +
1.3031 +lemma aff_independent_finite:
1.3032 +  fixes B :: "'n::euclidean_space set"
1.3033 +  assumes "\<not> affine_dependent B"
1.3034 +  shows "finite B"
1.3035 +proof -
1.3036 +  {
1.3037 +    assume "B \<noteq> {}"
1.3038 +    then obtain a where "a \<in> B" by auto
1.3039 +    then have ?thesis
1.3040 +      using aff_dim_parallel_subspace_aux assms by auto
1.3041 +  }
1.3042 +  then show ?thesis by auto
1.3043 +qed
1.3044 +
1.3045 +lemmas independent_finite = independent_imp_finite
1.3046 +
1.3047 +lemma span_substd_basis:
1.3048 +  assumes d: "d \<subseteq> Basis"
1.3049 +  shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
1.3050 +  (is "_ = ?B")
1.3051 +proof -
1.3052 +  have "d \<subseteq> ?B"
1.3053 +    using d by (auto simp: inner_Basis)
1.3054 +  moreover have s: "subspace ?B"
1.3055 +    using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
1.3056 +  ultimately have "span d \<subseteq> ?B"
1.3057 +    using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
1.3058 +  moreover have *: "card d \<le> dim (span d)"
1.3059 +    using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
1.3060 +      span_superset[of d]
1.3061 +    by auto
1.3062 +  moreover from * have "dim ?B \<le> dim (span d)"
1.3063 +    using dim_substandard[OF assms] by auto
1.3064 +  ultimately show ?thesis
1.3065 +    using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
1.3066 +qed
1.3067 +
1.3068 +lemma basis_to_substdbasis_subspace_isomorphism:
1.3069 +  fixes B :: "'a::euclidean_space set"
1.3070 +  assumes "independent B"
1.3071 +  shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
1.3072 +    f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
1.3073 +proof -
1.3074 +  have B: "card B = dim B"
1.3075 +    using dim_unique[of B B "card B"] assms span_superset[of B] by auto
1.3076 +  have "dim B \<le> card (Basis :: 'a set)"
1.3077 +    using dim_subset_UNIV[of B] by simp
1.3078 +  from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
1.3079 +    by auto
1.3080 +  let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
1.3081 +  have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
1.3082 +  proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
1.3083 +    show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
1.3084 +      using d inner_not_same_Basis by blast
1.3085 +  qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
1.3086 +  with t \<open>card B = dim B\<close> d show ?thesis by auto
1.3087 +qed
1.3088 +
1.3089 +lemma aff_dim_empty:
1.3090 +  fixes S :: "'n::euclidean_space set"
1.3091 +  shows "S = {} \<longleftrightarrow> aff_dim S = -1"
1.3092 +proof -
1.3093 +  obtain B where *: "affine hull B = affine hull S"
1.3094 +    and "\<not> affine_dependent B"
1.3095 +    and "int (card B) = aff_dim S + 1"
1.3096 +    using aff_dim_basis_exists by auto
1.3097 +  moreover
1.3098 +  from * have "S = {} \<longleftrightarrow> B = {}"
1.3099 +    using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
1.3100 +  ultimately show ?thesis
1.3101 +    using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
1.3102 +qed
1.3103 +
1.3104 +lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
1.3105 +  by (simp add: aff_dim_empty [symmetric])
1.3106 +
1.3107 +lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
1.3108 +  unfolding aff_dim_def using hull_hull[of _ S] by auto
1.3109 +
1.3110 +lemma aff_dim_affine_hull2:
1.3111 +  assumes "affine hull S = affine hull T"
1.3112 +  shows "aff_dim S = aff_dim T"
1.3113 +  unfolding aff_dim_def using assms by auto
1.3114 +
1.3115 +lemma aff_dim_unique:
1.3116 +  fixes B V :: "'n::euclidean_space set"
1.3117 +  assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
1.3118 +  shows "of_nat (card B) = aff_dim V + 1"
1.3119 +proof (cases "B = {}")
1.3120 +  case True
1.3121 +  then have "V = {}"
1.3122 +    using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
1.3123 +    by auto
1.3124 +  then have "aff_dim V = (-1::int)"
1.3125 +    using aff_dim_empty by auto
1.3126 +  then show ?thesis
1.3127 +    using \<open>B = {}\<close> by auto
1.3128 +next
1.3129 +  case False
1.3130 +  then obtain a where a: "a \<in> B" by auto
1.3131 +  define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
1.3132 +  have "affine_parallel (affine hull B) Lb"
1.3133 +    using Lb_def affine_hull_span2[of a B] a
1.3134 +      affine_parallel_commut[of "Lb" "(affine hull B)"]
1.3135 +    unfolding affine_parallel_def by auto
1.3136 +  moreover have "subspace Lb"
1.3137 +    using Lb_def subspace_span by auto
1.3138 +  ultimately have "aff_dim B = int(dim Lb)"
1.3139 +    using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
1.3140 +  moreover have "(card B) - 1 = dim Lb" "finite B"
1.3141 +    using Lb_def aff_dim_parallel_subspace_aux a assms by auto
1.3142 +  ultimately have "of_nat (card B) = aff_dim B + 1"
1.3143 +    using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
1.3144 +  then show ?thesis
1.3145 +    using aff_dim_affine_hull2 assms by auto
1.3146 +qed
1.3147 +
1.3148 +lemma aff_dim_affine_independent:
1.3149 +  fixes B :: "'n::euclidean_space set"
1.3150 +  assumes "\<not> affine_dependent B"
1.3151 +  shows "of_nat (card B) = aff_dim B + 1"
1.3152 +  using aff_dim_unique[of B B] assms by auto
1.3153 +
1.3154 +lemma affine_independent_iff_card:
1.3155 +    fixes s :: "'a::euclidean_space set"
1.3156 +    shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
1.3157 +  apply (rule iffI)
1.3158 +  apply (simp add: aff_dim_affine_independent aff_independent_finite)
1.3159 +  by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
1.3160 +
1.3161 +lemma aff_dim_sing [simp]:
1.3162 +  fixes a :: "'n::euclidean_space"
1.3163 +  shows "aff_dim {a} = 0"
1.3164 +  using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
1.3165 +
1.3166 +lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
1.3167 +proof (clarsimp)
1.3168 +  assume "a \<noteq> b"
1.3169 +  then have "aff_dim{a,b} = card{a,b} - 1"
1.3170 +    using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
1.3171 +  also have "\<dots> = 1"
1.3172 +    using \<open>a \<noteq> b\<close> by simp
1.3173 +  finally show "aff_dim {a, b} = 1" .
1.3174 +qed
1.3175 +
1.3176 +lemma aff_dim_inner_basis_exists:
1.3177 +  fixes V :: "('n::euclidean_space) set"
1.3178 +  shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
1.3179 +    \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
1.3180 +proof -
1.3181 +  obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
1.3182 +    using affine_basis_exists[of V] by auto
1.3183 +  then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
1.3184 +  with B show ?thesis by auto
1.3185 +qed
1.3186 +
1.3187 +lemma aff_dim_le_card:
1.3188 +  fixes V :: "'n::euclidean_space set"
1.3189 +  assumes "finite V"
1.3190 +  shows "aff_dim V \<le> of_nat (card V) - 1"
1.3191 +proof -
1.3192 +  obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
1.3193 +    using aff_dim_inner_basis_exists[of V] by auto
1.3194 +  then have "card B \<le> card V"
1.3195 +    using assms card_mono by auto
1.3196 +  with B show ?thesis by auto
1.3197 +qed
1.3198 +
1.3199 +lemma aff_dim_parallel_eq:
1.3200 +  fixes S T :: "'n::euclidean_space set"
1.3201 +  assumes "affine_parallel (affine hull S) (affine hull T)"
1.3202 +  shows "aff_dim S = aff_dim T"
1.3203 +proof -
1.3204 +  {
1.3205 +    assume "T \<noteq> {}" "S \<noteq> {}"
1.3206 +    then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
1.3207 +      using affine_parallel_subspace[of "affine hull T"]
1.3208 +        affine_affine_hull[of T] affine_hull_nonempty
1.3209 +      by auto
1.3210 +    then have "aff_dim T = int (dim L)"
1.3211 +      using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
1.3212 +    moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
1.3213 +       using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
1.3214 +    moreover from * have "aff_dim S = int (dim L)"
1.3215 +      using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
1.3216 +    ultimately have ?thesis by auto
1.3217 +  }
1.3218 +  moreover
1.3219 +  {
1.3220 +    assume "S = {}"
1.3221 +    then have "S = {}" and "T = {}"
1.3222 +      using assms affine_hull_nonempty
1.3223 +      unfolding affine_parallel_def
1.3224 +      by auto
1.3225 +    then have ?thesis using aff_dim_empty by auto
1.3226 +  }
1.3227 +  moreover
1.3228 +  {
1.3229 +    assume "T = {}"
1.3230 +    then have "S = {}" and "T = {}"
1.3231 +      using assms affine_hull_nonempty
1.3232 +      unfolding affine_parallel_def
1.3233 +      by auto
1.3234 +    then have ?thesis
1.3235 +      using aff_dim_empty by auto
1.3236 +  }
1.3237 +  ultimately show ?thesis by blast
1.3238 +qed
1.3239 +
1.3240 +lemma aff_dim_translation_eq:
1.3241 +  fixes a :: "'n::euclidean_space"
1.3242 +  shows "aff_dim ((\<lambda>x. a + x) ` S) = aff_dim S"
1.3243 +proof -
1.3244 +  have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
1.3245 +    unfolding affine_parallel_def
1.3246 +    apply (rule exI[of _ "a"])
1.3247 +    using affine_hull_translation[of a S]
1.3248 +    apply auto
1.3249 +    done
1.3250 +  then show ?thesis
1.3251 +    using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
1.3252 +qed
1.3253 +
1.3254 +lemma aff_dim_affine:
1.3255 +  fixes S L :: "'n::euclidean_space set"
1.3256 +  assumes "S \<noteq> {}"
1.3257 +    and "affine S"
1.3258 +    and "subspace L"
1.3259 +    and "affine_parallel S L"
1.3260 +  shows "aff_dim S = int (dim L)"
1.3261 +proof -
1.3262 +  have *: "affine hull S = S"
1.3263 +    using assms affine_hull_eq[of S] by auto
1.3264 +  then have "affine_parallel (affine hull S) L"
1.3265 +    using assms by (simp add: *)
1.3266 +  then show ?thesis
1.3267 +    using assms aff_dim_parallel_subspace[of S L] by blast
1.3268 +qed
1.3269 +
1.3270 +lemma dim_affine_hull:
1.3271 +  fixes S :: "'n::euclidean_space set"
1.3272 +  shows "dim (affine hull S) = dim S"
1.3273 +proof -
1.3274 +  have "dim (affine hull S) \<ge> dim S"
1.3275 +    using dim_subset by auto
1.3276 +  moreover have "dim (span S) \<ge> dim (affine hull S)"
1.3277 +    using dim_subset affine_hull_subset_span by blast
1.3278 +  moreover have "dim (span S) = dim S"
1.3279 +    using dim_span by auto
1.3280 +  ultimately show ?thesis by auto
1.3281 +qed
1.3282 +
1.3283 +lemma aff_dim_subspace:
1.3284 +  fixes S :: "'n::euclidean_space set"
1.3285 +  assumes "subspace S"
1.3286 +  shows "aff_dim S = int (dim S)"
1.3287 +proof (cases "S={}")
1.3288 +  case True with assms show ?thesis
1.3289 +    by (simp add: subspace_affine)
1.3290 +next
1.3291 +  case False
1.3292 +  with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
1.3293 +  show ?thesis by auto
1.3294 +qed
1.3295 +
1.3296 +lemma aff_dim_zero:
1.3297 +  fixes S :: "'n::euclidean_space set"
1.3298 +  assumes "0 \<in> affine hull S"
1.3299 +  shows "aff_dim S = int (dim S)"
1.3300 +proof -
1.3301 +  have "subspace (affine hull S)"
1.3302 +    using subspace_affine[of "affine hull S"] affine_affine_hull assms
1.3303 +    by auto
1.3304 +  then have "aff_dim (affine hull S) = int (dim (affine hull S))"
1.3305 +    using assms aff_dim_subspace[of "affine hull S"] by auto
1.3306 +  then show ?thesis
1.3307 +    using aff_dim_affine_hull[of S] dim_affine_hull[of S]
1.3308 +    by auto
1.3309 +qed
1.3310 +
1.3311 +lemma aff_dim_eq_dim:
1.3312 +  fixes S :: "'n::euclidean_space set"
1.3313 +  assumes "a \<in> affine hull S"
1.3314 +  shows "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
1.3315 +proof -
1.3316 +  have "0 \<in> affine hull ((\<lambda>x. -a+x) ` S)"
1.3317 +    unfolding affine_hull_translation
1.3319 +  with aff_dim_zero show ?thesis
1.3320 +    by (metis aff_dim_translation_eq)
1.3321 +qed
1.3322 +
1.3323 +lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
1.3324 +  using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
1.3325 +    dim_UNIV[where 'a="'n::euclidean_space"]
1.3326 +  by auto
1.3327 +
1.3328 +lemma aff_dim_geq:
1.3329 +  fixes V :: "'n::euclidean_space set"
1.3330 +  shows "aff_dim V \<ge> -1"
1.3331 +proof -
1.3332 +  obtain B where "affine hull B = affine hull V"
1.3333 +    and "\<not> affine_dependent B"
1.3334 +    and "int (card B) = aff_dim V + 1"
1.3335 +    using aff_dim_basis_exists by auto
1.3336 +  then show ?thesis by auto
1.3337 +qed
1.3338 +
1.3339 +lemma aff_dim_negative_iff [simp]:
1.3340 +  fixes S :: "'n::euclidean_space set"
1.3341 +  shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
1.3342 +by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
1.3343 +
1.3344 +lemma aff_lowdim_subset_hyperplane:
1.3345 +  fixes S :: "'a::euclidean_space set"
1.3346 +  assumes "aff_dim S < DIM('a)"
1.3347 +  obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
1.3348 +proof (cases "S={}")
1.3349 +  case True
1.3350 +  moreover
1.3351 +  have "(SOME b. b \<in> Basis) \<noteq> 0"
1.3352 +    by (metis norm_some_Basis norm_zero zero_neq_one)
1.3353 +  ultimately show ?thesis
1.3354 +    using that by blast
1.3355 +next
1.3356 +  case False
1.3357 +  then obtain c S' where "c \<notin> S'" "S = insert c S'"
1.3358 +    by (meson equals0I mk_disjoint_insert)
1.3359 +  have "dim ((+) (-c) ` S) < DIM('a)"
1.3360 +    by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
1.3361 +  then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
1.3362 +    using lowdim_subset_hyperplane by blast
1.3363 +  moreover
1.3364 +  have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
1.3365 +  proof -
1.3366 +    have "w-c \<in> span ((+) (- c) ` S)"
1.3367 +      by (simp add: span_base \<open>w \<in> S\<close>)
1.3368 +    with that have "w-c \<in> {x. a \<bullet> x = 0}"
1.3369 +      by blast
1.3370 +    then show ?thesis
1.3371 +      by (auto simp: algebra_simps)
1.3372 +  qed
1.3373 +  ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
1.3374 +    by blast
1.3375 +  then show ?thesis
1.3376 +    by (rule that[OF \<open>a \<noteq> 0\<close>])
1.3377 +qed
1.3378 +
1.3379 +lemma affine_independent_card_dim_diffs:
1.3380 +  fixes S :: "'a :: euclidean_space set"
1.3381 +  assumes "\<not> affine_dependent S" "a \<in> S"
1.3382 +    shows "card S = dim {x - a|x. x \<in> S} + 1"
1.3383 +proof -
1.3384 +  have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
1.3385 +  have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
1.3386 +  proof (cases "x = a")
1.3387 +    case True then show ?thesis by (simp add: span_clauses)
1.3388 +  next
1.3389 +    case False then show ?thesis
1.3390 +      using assms by (blast intro: span_base that)
1.3391 +  qed
1.3392 +  have "\<not> affine_dependent (insert a S)"
1.3393 +    by (simp add: assms insert_absorb)
1.3394 +  then have 3: "independent {b - a |b. b \<in> S - {a}}"
1.3395 +      using dependent_imp_affine_dependent by fastforce
1.3396 +  have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
1.3397 +    by blast
1.3398 +  then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
1.3399 +    by simp
1.3400 +  also have "\<dots> = card (S - {a})"
1.3401 +    by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
1.3402 +  also have "\<dots> = card S - 1"
1.3403 +    by (simp add: aff_independent_finite assms)
1.3404 +  finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
1.3405 +  have "finite S"
1.3406 +    by (meson assms aff_independent_finite)
1.3407 +  with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
1.3408 +  moreover have "dim {x - a |x. x \<in> S} = card S - 1"
1.3409 +    using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
1.3410 +  ultimately show ?thesis
1.3411 +    by auto
1.3412 +qed
1.3413 +
1.3414 +lemma independent_card_le_aff_dim:
1.3415 +  fixes B :: "'n::euclidean_space set"
1.3416 +  assumes "B \<subseteq> V"
1.3417 +  assumes "\<not> affine_dependent B"
1.3418 +  shows "int (card B) \<le> aff_dim V + 1"
1.3419 +proof -
1.3420 +  obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
1.3421 +    by (metis assms extend_to_affine_basis[of B V])
1.3422 +  then have "of_nat (card T) = aff_dim V + 1"
1.3423 +    using aff_dim_unique by auto
1.3424 +  then show ?thesis
1.3425 +    using T card_mono[of T B] aff_independent_finite[of T] by auto
1.3426 +qed
1.3427 +
1.3428 +lemma aff_dim_subset:
1.3429 +  fixes S T :: "'n::euclidean_space set"
1.3430 +  assumes "S \<subseteq> T"
1.3431 +  shows "aff_dim S \<le> aff_dim T"
1.3432 +proof -
1.3433 +  obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
1.3434 +    "of_nat (card B) = aff_dim S + 1"
1.3435 +    using aff_dim_inner_basis_exists[of S] by auto
1.3436 +  then have "int (card B) \<le> aff_dim T + 1"
1.3437 +    using assms independent_card_le_aff_dim[of B T] by auto
1.3438 +  with B show ?thesis by auto
1.3439 +qed
1.3440 +
1.3441 +lemma aff_dim_le_DIM:
1.3442 +  fixes S :: "'n::euclidean_space set"
1.3443 +  shows "aff_dim S \<le> int (DIM('n))"
1.3444 +proof -
1.3445 +  have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
1.3446 +    using aff_dim_UNIV by auto
1.3447 +  then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
1.3448 +    using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
1.3449 +qed
1.3450 +
1.3451 +lemma affine_dim_equal:
1.3452 +  fixes S :: "'n::euclidean_space set"
1.3453 +  assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
1.3454 +  shows "S = T"
1.3455 +proof -
1.3456 +  obtain a where "a \<in> S" using assms by auto
1.3457 +  then have "a \<in> T" using assms by auto
1.3458 +  define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
1.3459 +  then have ls: "subspace LS" "affine_parallel S LS"
1.3460 +    using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
1.3461 +  then have h1: "int(dim LS) = aff_dim S"
1.3462 +    using assms aff_dim_affine[of S LS] by auto
1.3463 +  have "T \<noteq> {}" using assms by auto
1.3464 +  define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
1.3465 +  then have lt: "subspace LT \<and> affine_parallel T LT"
1.3466 +    using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
1.3467 +  then have "int(dim LT) = aff_dim T"
1.3468 +    using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
1.3469 +  then have "dim LS = dim LT"
1.3470 +    using h1 assms by auto
1.3471 +  moreover have "LS \<le> LT"
1.3472 +    using LS_def LT_def assms by auto
1.3473 +  ultimately have "LS = LT"
1.3474 +    using subspace_dim_equal[of LS LT] ls lt by auto
1.3475 +  moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
1.3476 +    using LS_def by auto
1.3477 +  moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
1.3478 +    using LT_def by auto
1.3479 +  ultimately show ?thesis by auto
1.3480 +qed
1.3481 +
1.3482 +lemma aff_dim_eq_0:
1.3483 +  fixes S :: "'a::euclidean_space set"
1.3484 +  shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
1.3485 +proof (cases "S = {}")
1.3486 +  case True
1.3487 +  then show ?thesis
1.3488 +    by auto
1.3489 +next
1.3490 +  case False
1.3491 +  then obtain a where "a \<in> S" by auto
1.3492 +  show ?thesis
1.3493 +  proof safe
1.3494 +    assume 0: "aff_dim S = 0"
1.3495 +    have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
1.3496 +      by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
1.3497 +    then show "\<exists>a. S = {a}"
1.3498 +      using \<open>a \<in> S\<close> by blast
1.3499 +  qed auto
1.3500 +qed
1.3501 +
1.3502 +lemma affine_hull_UNIV:
1.3503 +  fixes S :: "'n::euclidean_space set"
1.3504 +  assumes "aff_dim S = int(DIM('n))"
1.3505 +  shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
1.3506 +proof -
1.3507 +  have "S \<noteq> {}"
1.3508 +    using assms aff_dim_empty[of S] by auto
1.3509 +  have h0: "S \<subseteq> affine hull S"
1.3510 +    using hull_subset[of S _] by auto
1.3511 +  have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
1.3512 +    using aff_dim_UNIV assms by auto
1.3513 +  then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
1.3514 +    using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
1.3515 +  have h3: "aff_dim S \<le> aff_dim (affine hull S)"
1.3516 +    using h0 aff_dim_subset[of S "affine hull S"] assms by auto
1.3517 +  then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
1.3518 +    using h0 h1 h2 by auto
1.3519 +  then show ?thesis
1.3520 +    using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
1.3521 +      affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
1.3522 +    by auto
1.3523 +qed
1.3524 +
1.3525 +lemma disjoint_affine_hull:
1.3526 +  fixes s :: "'n::euclidean_space set"
1.3527 +  assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
1.3528 +    shows "(affine hull t) \<inter> (affine hull u) = {}"
1.3529 +proof -
1.3530 +  have "finite s" using assms by (simp add: aff_independent_finite)
1.3531 +  then have "finite t" "finite u" using assms finite_subset by blast+
1.3532 +  { fix y
1.3533 +    assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
1.3534 +    then obtain a b
1.3535 +           where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
1.3536 +             and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
1.3537 +      by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
1.3538 +    define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
1.3539 +    have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
1.3540 +    have "sum c s = 0"
1.3542 +    moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
1.3543 +      by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one)
1.3544 +    moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
1.3545 +      by (simp add: c_def if_smult sum_negf
1.3547 +    ultimately have False
1.3548 +      using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
1.3549 +  }
1.3550 +  then show ?thesis by blast
1.3551 +qed
1.3552 +
1.3553 +lemma aff_dim_convex_hull:
1.3554 +  fixes S :: "'n::euclidean_space set"
1.3555 +  shows "aff_dim (convex hull S) = aff_dim S"
1.3556 +  using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
1.3557 +    hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
1.3558 +    aff_dim_subset[of "convex hull S" "affine hull S"]
1.3559 +  by auto
1.3560 +
1.3561 +subsection \<open>Caratheodory's theorem\<close>
1.3562 +
1.3563 +lemma convex_hull_caratheodory_aff_dim:
1.3564 +  fixes p :: "('a::euclidean_space) set"
1.3565 +  shows "convex hull p =
1.3566 +    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
1.3567 +      (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
1.3568 +  unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
1.3569 +proof (intro allI iffI)
1.3570 +  fix y
1.3571 +  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>
1.3572 +    sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
1.3573 +  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
1.3574 +  then obtain N where "?P N" by auto
1.3575 +  then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
1.3576 +    apply (rule_tac ex_least_nat_le, auto)
1.3577 +    done
1.3578 +  then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
1.3579 +    by blast
1.3580 +  then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
1.3581 +    "sum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
1.3582 +
1.3583 +  have "card s \<le> aff_dim p + 1"
1.3584 +  proof (rule ccontr, simp only: not_le)
1.3585 +    assume "aff_dim p + 1 < card s"
1.3586 +    then have "affine_dependent s"
1.3587 +      using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
1.3588 +      by blast
1.3589 +    then obtain w v where wv: "sum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
1.3590 +      using affine_dependent_explicit_finite[OF obt(1)] by auto
1.3591 +    define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
1.3592 +    define t where "t = Min i"
1.3593 +    have "\<exists>x\<in>s. w x < 0"
1.3594 +    proof (rule ccontr, simp add: not_less)
1.3595 +      assume as:"\<forall>x\<in>s. 0 \<le> w x"
1.3596 +      then have "sum w (s - {v}) \<ge> 0"
1.3597 +        apply (rule_tac sum_nonneg, auto)
1.3598 +        done
1.3599 +      then have "sum w s > 0"
1.3600 +        unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
1.3601 +        using as[THEN bspec[where x=v]]  \<open>v\<in>s\<close>  \<open>w v \<noteq> 0\<close> by auto
1.3602 +      then show False using wv(1) by auto
1.3603 +    qed
1.3604 +    then have "i \<noteq> {}" unfolding i_def by auto
1.3605 +    then have "t \<ge> 0"
1.3606 +      using Min_ge_iff[of i 0 ] and obt(1)
1.3607 +      unfolding t_def i_def
1.3608 +      using obt(4)[unfolded le_less]
1.3609 +      by (auto simp: divide_le_0_iff)
1.3610 +    have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
1.3611 +    proof
1.3612 +      fix v
1.3613 +      assume "v \<in> s"
1.3614 +      then have v: "0 \<le> u v"
1.3615 +        using obt(4)[THEN bspec[where x=v]] by auto
1.3616 +      show "0 \<le> u v + t * w v"
1.3617 +      proof (cases "w v < 0")
1.3618 +        case False
1.3619 +        thus ?thesis using v \<open>t\<ge>0\<close> by auto
1.3620 +      next
1.3621 +        case True
1.3622 +        then have "t \<le> u v / (- w v)"
1.3623 +          using \<open>v\<in>s\<close> unfolding t_def i_def
1.3624 +          apply (rule_tac Min_le)
1.3625 +          using obt(1) apply auto
1.3626 +          done
1.3627 +        then show ?thesis
1.3629 +          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
1.3630 +          by auto
1.3631 +      qed
1.3632 +    qed
1.3633 +    obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
1.3634 +      using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
1.3635 +    then have a: "a \<in> s" "u a + t * w a = 0" by auto
1.3636 +    have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)"
1.3637 +      unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
1.3638 +    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
1.3639 +      unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
1.3640 +    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"
1.3641 +      unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
1.3642 +      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
1.3643 +    ultimately have "?P (n - 1)"
1.3644 +      apply (rule_tac x="(s - {a})" in exI)
1.3645 +      apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
1.3646 +      using obt(1-3) and t and a
1.3647 +      apply (auto simp: * scaleR_left_distrib)
1.3648 +      done
1.3649 +    then show False
1.3650 +      using smallest[THEN spec[where x="n - 1"]] by auto
1.3651 +  qed
1.3652 +  then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
1.3653 +      (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
1.3654 +    using obt by auto
1.3655 +qed auto
1.3656 +
1.3657 +lemma caratheodory_aff_dim:
1.3658 +  fixes p :: "('a::euclidean_space) set"
1.3659 +  shows "convex hull p = {x. \<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> x \<in> convex hull s}"
1.3660 +        (is "?lhs = ?rhs")
1.3661 +proof
1.3662 +  show "?lhs \<subseteq> ?rhs"
1.3663 +    apply (subst convex_hull_caratheodory_aff_dim, clarify)
1.3664 +    apply (rule_tac x=s in exI)
1.3665 +    apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
1.3666 +    done
1.3667 +next
1.3668 +  show "?rhs \<subseteq> ?lhs"
1.3669 +    using hull_mono by blast
1.3670 +qed
1.3671 +
1.3672 +lemma convex_hull_caratheodory:
1.3673 +  fixes p :: "('a::euclidean_space) set"
1.3674 +  shows "convex hull p =
1.3675 +            {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
1.3676 +              (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
1.3677 +        (is "?lhs = ?rhs")
1.3678 +proof (intro set_eqI iffI)
1.3679 +  fix x
1.3680 +  assume "x \<in> ?lhs" then show "x \<in> ?rhs"
1.3681 +    apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
1.3682 +    apply (erule ex_forward)+
1.3683 +    using aff_dim_le_DIM [of p]
1.3684 +    apply simp
1.3685 +    done
1.3686 +next
1.3687 +  fix x
1.3688 +  assume "x \<in> ?rhs" then show "x \<in> ?lhs"
1.3689 +    by (auto simp: convex_hull_explicit)
1.3690 +qed
1.3691 +
1.3692 +theorem caratheodory:
1.3693 +  "convex hull p =
1.3694 +    {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
1.3695 +      card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
1.3696 +proof safe
1.3697 +  fix x
1.3698 +  assume "x \<in> convex hull p"
1.3699 +  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
1.3700 +    "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
1.3701 +    unfolding convex_hull_caratheodory by auto
1.3702 +  then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
1.3703 +    apply (rule_tac x=s in exI)
1.3704 +    using hull_subset[of s convex]
1.3705 +    using convex_convex_hull[simplified convex_explicit, of s,
1.3706 +      THEN spec[where x=s], THEN spec[where x=u]]
1.3707 +    apply auto
1.3708 +    done
1.3709 +next
1.3710 +  fix x s
1.3711 +  assume  "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
1.3712 +  then show "x \<in> convex hull p"
1.3713 +    using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
1.3714 +qed
1.3715 +
1.3716 +subsection%unimportant\<open>Some Properties of subset of standard basis\<close>
1.3717 +
1.3718 +lemma affine_hull_substd_basis:
1.3719 +  assumes "d \<subseteq> Basis"
1.3720 +  shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
1.3721 +  (is "affine hull (insert 0 ?A) = ?B")
1.3722 +proof -
1.3723 +  have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
1.3724 +    by auto
1.3725 +  show ?thesis
1.3726 +    unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
1.3727 +qed
1.3728 +
1.3729 +lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
1.3730 +  by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
1.3731 +
1.3732 +
1.3733 +subsection%unimportant \<open>Moving and scaling convex hulls\<close>
1.3734 +
1.3735 +lemma convex_hull_set_plus:
1.3736 +  "convex hull (S + T) = convex hull S + convex hull T"
1.3737 +  unfolding set_plus_image
1.3738 +  apply (subst convex_hull_linear_image [symmetric])
1.3739 +  apply (simp add: linear_iff scaleR_right_distrib)
1.3740 +  apply (simp add: convex_hull_Times)
1.3741 +  done
1.3742 +
1.3743 +lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
1.3744 +  unfolding set_plus_def by auto
1.3745 +
1.3746 +lemma convex_hull_translation:
1.3747 +  "convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
1.3748 +  unfolding translation_eq_singleton_plus
1.3749 +  by (simp only: convex_hull_set_plus convex_hull_singleton)
1.3750 +
1.3751 +lemma convex_hull_scaling:
1.3752 +  "convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
1.3753 +  using linear_scaleR by (rule convex_hull_linear_image [symmetric])
1.3754 +
1.3755 +lemma convex_hull_affinity:
1.3756 +  "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
1.3757 +  by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
1.3758 +
1.3759 +
1.3760 +subsection%unimportant \<open>Convexity of cone hulls\<close>
1.3761 +
1.3762 +lemma convex_cone_hull:
1.3763 +  assumes "convex S"
1.3764 +  shows "convex (cone hull S)"
1.3765 +proof (rule convexI)
1.3766 +  fix x y
1.3767 +  assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
1.3768 +  then have "S \<noteq> {}"
1.3769 +    using cone_hull_empty_iff[of S] by auto
1.3770 +  fix u v :: real
1.3771 +  assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
1.3772 +  then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
1.3773 +    using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
1.3774 +  from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
1.3775 +    using cone_hull_expl[of S] by auto
1.3776 +  from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
1.3777 +    using cone_hull_expl[of S] by auto
1.3778 +  {
1.3779 +    assume "cx + cy \<le> 0"
1.3780 +    then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
1.3781 +      using x y by auto
1.3782 +    then have "u *\<^sub>R x + v *\<^sub>R y = 0"
1.3783 +      by auto
1.3784 +    then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
1.3785 +      using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
1.3786 +  }
1.3787 +  moreover
1.3788 +  {
1.3789 +    assume "cx + cy > 0"
1.3790 +    then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
1.3791 +      using assms mem_convex_alt[of S xx yy cx cy] x y by auto
1.3792 +    then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
1.3793 +      using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close>
1.3794 +      by (auto simp: scaleR_right_distrib)
1.3795 +    then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
1.3796 +      using x y by auto
1.3797 +  }
1.3798 +  moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
1.3799 +  ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
1.3800 +qed
1.3801 +
1.3802 +lemma cone_convex_hull:
1.3803 +  assumes "cone S"
1.3804 +  shows "cone (convex hull S)"
1.3805 +proof (cases "S = {}")
1.3806 +  case True
1.3807 +  then show ?thesis by auto
1.3808 +next
1.3809 +  case False
1.3810 +  then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
1.3811 +    using cone_iff[of S] assms by auto
1.3812 +  {
1.3813 +    fix c :: real
1.3814 +    assume "c > 0"
1.3815 +    then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)"
1.3816 +      using convex_hull_scaling[of _ S] by auto
1.3817 +    also have "\<dots> = convex hull S"
1.3818 +      using * \<open>c > 0\<close> by auto
1.3819 +    finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S"
1.3820 +      by auto
1.3821 +  }
1.3822 +  then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)"
1.3823 +    using * hull_subset[of S convex] by auto
1.3824 +  then show ?thesis
1.3825 +    using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
1.3826 +qed
1.3827 +
1.3829 +
1.3830 +text "Formalized by Lars Schewe."
1.3831 +
1.3833 +  assumes "finite c" "affine_dependent c"
1.3834 +  shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
1.3835 +proof -
1.3836 +  from assms(2)[unfolded affine_dependent_explicit]
1.3837 +  obtain s u where
1.3838 +      "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
1.3839 +    by blast
1.3840 +  then show ?thesis
1.3841 +    apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
1.3842 +    unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric]
1.3843 +    apply (auto simp: Int_absorb1)
1.3844 +    done
1.3845 +qed
1.3846 +
1.3848 +  assumes "finite s"
1.3849 +    and "sum f s = (0::real)"
1.3850 +  shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}"
1.3851 +proof -
1.3852 +  have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
1.3853 +    by auto
1.3854 +  show ?thesis
1.3855 +    unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
1.3856 +      and sum.distrib[symmetric] and *
1.3857 +    using assms(2)
1.3858 +    by assumption
1.3859 +qed
1.3860 +
1.3862 +  assumes "finite s"
1.3863 +    and "sum f s = 0"
1.3864 +    and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
1.3865 +  shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}"
1.3866 +proof -
1.3867 +  have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
1.3868 +    using assms(3) by auto
1.3869 +  show ?thesis
1.3870 +    unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
1.3871 +      and sum.distrib[symmetric] and *
1.3872 +    using assms(2)
1.3873 +    apply assumption
1.3874 +    done
1.3875 +qed
1.3876 +
1.3878 +  assumes "finite c" "affine_dependent c"
1.3879 +  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
1.3880 +proof -
1.3881 +  obtain u v where uv: "sum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
1.3882 +    using Radon_ex_lemma[OF assms] by auto
1.3883 +  have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
1.3884 +    using assms(1) by auto
1.3885 +  define z  where "z = inverse (sum u {x\<in>c. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
1.3886 +  have "sum u {x \<in> c. 0 < u x} \<noteq> 0"
1.3887 +  proof (cases "u v \<ge> 0")
1.3888 +    case False
1.3889 +    then have "u v < 0" by auto
1.3890 +    then show ?thesis
1.3891 +    proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
1.3892 +      case True
1.3893 +      then show ?thesis
1.3894 +        using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
1.3895 +    next
1.3896 +      case False
1.3897 +      then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c"
1.3898 +        apply (rule_tac sum_mono, auto)
1.3899 +        done
1.3900 +      then show ?thesis
1.3901 +        unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
1.3902 +    qed
1.3903 +  qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
1.3904 +
1.3905 +  then have *: "sum u {x\<in>c. u x > 0} > 0"
1.3906 +    unfolding less_le
1.3907 +    apply (rule_tac conjI)
1.3908 +    apply (rule_tac sum_nonneg, auto)
1.3909 +    done
1.3910 +  moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c"
1.3911 +    "(\<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)"
1.3912 +    using assms(1)
1.3913 +    apply (rule_tac[!] sum.mono_neutral_left, auto)
1.3914 +    done
1.3915 +  then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}"
1.3916 +    "(\<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)"
1.3918 +    using uv(1,4)
1.3919 +    by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
1.3920 +  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x"
1.3921 +    apply rule
1.3922 +    apply (rule mult_nonneg_nonneg)
1.3923 +    using *
1.3924 +    apply auto
1.3925 +    done
1.3926 +  ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
1.3927 +    unfolding convex_hull_explicit mem_Collect_eq
1.3928 +    apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
1.3929 +    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI)
1.3930 +    using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
1.3931 +    apply (auto simp: sum_negf sum_distrib_left[symmetric])
1.3932 +    done
1.3933 +  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
1.3934 +    apply rule
1.3935 +    apply (rule mult_nonneg_nonneg)
1.3936 +    using *
1.3937 +    apply auto
1.3938 +    done
1.3939 +  then have "z \<in> convex hull {v \<in> c. u v > 0}"
1.3940 +    unfolding convex_hull_explicit mem_Collect_eq
1.3941 +    apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
1.3942 +    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI)
1.3943 +    using assms(1)
1.3944 +    unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
1.3945 +    using *
1.3946 +    apply (auto simp: sum_negf sum_distrib_left[symmetric])
1.3947 +    done
1.3948 +  ultimately show ?thesis
1.3949 +    apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
1.3950 +    apply (rule_tac x="{v\<in>c. u v > 0}" in exI, auto)
1.3951 +    done
1.3952 +qed
1.3953 +
1.3955 +  assumes "affine_dependent c"
1.3956 +  obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
1.3957 +proof -
1.3958 +  from assms[unfolded affine_dependent_explicit]
1.3959 +  obtain s u where
1.3960 +      "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
1.3961 +    by blast
1.3962 +  then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
1.3963 +    unfolding affine_dependent_explicit by auto
1.3965 +  obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
1.3966 +    by blast
1.3967 +  then show ?thesis
1.3968 +    apply (rule_tac that[of p m])
1.3969 +    using s
1.3970 +    apply auto
1.3971 +    done
1.3972 +qed
1.3973 +
1.3974 +
1.3975 +subsection \<open>Helly's theorem\<close>
1.3976 +
1.3977 +lemma Helly_induct:
1.3978 +  fixes f :: "'a::euclidean_space set set"
1.3979 +  assumes "card f = n"
1.3980 +    and "n \<ge> DIM('a) + 1"
1.3981 +    and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
1.3982 +  shows "\<Inter>f \<noteq> {}"
1.3983 +  using assms
1.3984 +proof (induction n arbitrary: f)
1.3985 +  case 0
1.3986 +  then show ?case by auto
1.3987 +next
1.3988 +  case (Suc n)
1.3989 +  have "finite f"
1.3990 +    using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
1.3991 +  show "\<Inter>f \<noteq> {}"
1.3992 +  proof (cases "n = DIM('a)")
1.3993 +    case True
1.3994 +    then show ?thesis
1.3995 +      by (simp add: Suc.prems(1) Suc.prems(4))
1.3996 +  next
1.3997 +    case False
1.3998 +    have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
1.3999 +    proof (rule Suc.IH[rule_format])
1.4000 +      show "card (f - {s}) = n"
1.4001 +        by (simp add: Suc.prems(1) \<open>finite f\<close> that)
1.4002 +      show "DIM('a) + 1 \<le> n"
1.4003 +        using False Suc.prems(2) by linarith
1.4004 +      show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
1.4005 +        by (simp add: Suc.prems(4) subset_Diff_insert)
1.4006 +    qed (use Suc in auto)
1.4007 +    then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
1.4008 +      by blast
1.4009 +    then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
1.4010 +      by metis
1.4011 +    show ?thesis
1.4012 +    proof (cases "inj_on X f")
1.4013 +      case False
1.4014 +      then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
1.4015 +        unfolding inj_on_def by auto
1.4016 +      then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
1.4017 +      show ?thesis
1.4018 +        by (metis "*" X disjoint_iff_not_equal st)
1.4019 +    next
1.4020 +      case True
1.4021 +      then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
1.4022 +        using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
1.4023 +        unfolding card_image[OF True] and \<open>card f = Suc n\<close>
1.4024 +        using Suc(3) \<open>finite f\<close> and False
1.4025 +        by auto
1.4026 +      have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
1.4027 +        using mp(2) by auto
1.4028 +      then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
1.4029 +        unfolding subset_image_iff by auto
1.4030 +      then have "f \<union> (g \<union> h) = f" by auto
1.4031 +      then have f: "f = g \<union> h"
1.4032 +        using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
1.4033 +        unfolding mp(2)[unfolded image_Un[symmetric] gh]
1.4034 +        by auto
1.4035 +      have *: "g \<inter> h = {}"
1.4036 +        using mp(1)
1.4037 +        unfolding gh
1.4038 +        using inj_on_image_Int[OF True gh(3,4)]
1.4039 +        by auto
1.4040 +      have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
1.4041 +        by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
1.4042 +      then show ?thesis
1.4043 +        unfolding f using mp(3)[unfolded gh] by blast
1.4044 +    qed
1.4045 +  qed
1.4046 +qed
1.4047 +
1.4048 +theorem Helly:
1.4049 +  fixes f :: "'a::euclidean_space set set"
1.4050 +  assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
1.4051 +    and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
1.4052 +  shows "\<Inter>f \<noteq> {}"
1.4053 +  apply (rule Helly_induct)
1.4054 +  using assms
1.4055 +  apply auto
1.4056 +  done
1.4057 +
1.4058 +subsection \<open>Epigraphs of convex functions\<close>
1.4059 +
1.4060 +definition%important "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
1.4061 +
1.4062 +lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
1.4063 +  unfolding epigraph_def by auto
1.4064 +
1.4065 +lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
1.4066 +proof safe
1.4067 +  assume L: "convex (epigraph S f)"
1.4068 +  then show "convex_on S f"
1.4069 +    by (auto simp: convex_def convex_on_def epigraph_def)
1.4070 +  show "convex S"
1.4071 +    using L
1.4072 +    apply (clarsimp simp: convex_def convex_on_def epigraph_def)
1.4073 +    apply (erule_tac x=x in allE)
1.4074 +    apply (erule_tac x="f x" in allE, safe)
1.4075 +    apply (erule_tac x=y in allE)
1.4076 +    apply (erule_tac x="f y" in allE)
1.4077 +    apply (auto simp: )
1.4078 +    done
1.4079 +next
1.4080 +  assume "convex_on S f" "convex S"
1.4081 +  then show "convex (epigraph S f)"
1.4082 +    unfolding convex_def convex_on_def epigraph_def
1.4083 +    apply safe
1.4084 +     apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
1.4085 +      apply (auto intro!:mult_left_mono add_mono)
1.4086 +    done
1.4087 +qed
1.4088 +
1.4089 +lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
1.4090 +  unfolding convex_epigraph by auto
1.4091 +
1.4092 +lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
1.4093 +  by (simp add: convex_epigraph)
1.4094 +
1.4095 +
1.4096 +subsubsection%unimportant \<open>Use this to derive general bound property of convex function\<close>
1.4097 +
1.4098 +lemma convex_on:
1.4099 +  assumes "convex S"
1.4100 +  shows "convex_on S f \<longleftrightarrow>
1.4101 +    (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
1.4102 +      f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
1.4103 +
1.4104 +  unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
1.4105 +  unfolding fst_sum snd_sum fst_scaleR snd_scaleR
1.4106 +  apply safe
1.4107 +    apply (drule_tac x=k in spec)
1.4108 +    apply (drule_tac x=u in spec)
1.4109 +    apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
1.4110 +    apply simp
1.4111 +  using assms[unfolded convex] apply simp
1.4112 +  apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
1.4113 +   apply (rule sum_mono)
1.4114 +   apply (erule_tac x=i in allE)
1.4115 +  unfolding real_scaleR_def
1.4116 +   apply (rule mult_left_mono)
1.4117 +  using assms[unfolded convex] apply auto
1.4118 +  done
1.4119 +
1.4120 +subsection%unimportant \<open>A bound within a convex hull\<close>
1.4121 +
1.4122 +lemma convex_on_convex_hull_bound:
1.4123 +  assumes "convex_on (convex hull s) f"
1.4124 +    and "\<forall>x\<in>s. f x \<le> b"
1.4125 +  shows "\<forall>x\<in> convex hull s. f x \<le> b"
1.4126 +proof
1.4127 +  fix x
1.4128 +  assume "x \<in> convex hull s"
1.4129 +  then obtain k u v where
1.4130 +    obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
1.4131 +    unfolding convex_hull_indexed mem_Collect_eq by auto
1.4132 +  have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
1.4133 +    using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
1.4134 +    unfolding sum_distrib_right[symmetric] obt(2) mult_1
1.4135 +    apply (drule_tac meta_mp)
1.4136 +    apply (rule mult_left_mono)
1.4137 +    using assms(2) obt(1)
1.4138 +    apply auto
1.4139 +    done
1.4140 +  then show "f x \<le> b"
1.4141 +    using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
1.4142 +    unfolding obt(2-3)
1.4143 +    using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
1.4144 +    by auto
1.4145 +qed
1.4146 +
1.4147 +lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
1.4148 +  by (simp add: inner_sum_left sum.If_cases inner_Basis)
1.4149 +
1.4150 +lemma convex_set_plus:
1.4151 +  assumes "convex S" and "convex T" shows "convex (S + T)"
1.4152 +proof -
1.4153 +  have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
1.4154 +    using assms by (rule convex_sums)
1.4155 +  moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T"
1.4156 +    unfolding set_plus_def by auto
1.4157 +  finally show "convex (S + T)" .
1.4158 +qed
1.4159 +
1.4160 +lemma convex_set_sum:
1.4161 +  assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
1.4162 +  shows "convex (\<Sum>i\<in>A. B i)"
1.4163 +proof (cases "finite A")
1.4164 +  case True then show ?thesis using assms
1.4165 +    by induct (auto simp: convex_set_plus)
1.4166 +qed auto
1.4167 +
1.4168 +lemma finite_set_sum:
1.4169 +  assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
1.4170 +  using assms by (induct set: finite, simp, simp add: finite_set_plus)
1.4171 +
1.4172 +lemma box_eq_set_sum_Basis:
1.4173 +  shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
1.4174 +  apply (subst set_sum_alt [OF finite_Basis], safe)
1.4175 +  apply (fast intro: euclidean_representation [symmetric])
1.4176 +  apply (subst inner_sum_left)
1.4177 +apply (rename_tac f)
1.4178 +  apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
1.4179 +  apply (drule (1) bspec)
1.4180 +  apply clarsimp
1.4181 +  apply (frule sum.remove [OF finite_Basis])
1.4182 +  apply (erule trans, simp)
1.4183 +  apply (rule sum.neutral, clarsimp)
1.4184 +  apply (frule_tac x=i in bspec, assumption)
1.4185 +  apply (drule_tac x=x in bspec, assumption, clarsimp)
1.4186 +  apply (cut_tac u=x and v=i in inner_Basis, assumption+)
1.4187 +  apply (rule ccontr, simp)
1.4188 +  done
1.4189 +
1.4190 +lemma convex_hull_set_sum:
1.4191 +  "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
1.4192 +proof (cases "finite A")
1.4193 +  assume "finite A" then show ?thesis
1.4194 +    by (induct set: finite, simp, simp add: convex_hull_set_plus)
1.4195 +qed simp
1.4196 +
1.4197 +
1.4198 +end
1.4199 \ No newline at end of file
```
```     2.1 --- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Jan 07 13:33:29 2019 +0100
2.2 +++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Jan 07 14:06:54 2019 +0100
2.3 @@ -6,1143 +6,15 @@
2.4     Author:     Johannes Hoelzl, TU Muenchen
2.5  *)
2.6
2.7 -section \<open>Convex Sets and Functions\<close>
2.8 +section \<open>Convex Sets and Functions on (Normed) Euclidean Spaces\<close>
2.9
2.10  theory Convex_Euclidean_Space
2.11  imports
2.12 +  Convex
2.13    Topology_Euclidean_Space
2.14 -  "HOL-Library.Set_Algebras"
2.15  begin
2.16
2.17 -lemma swap_continuous: (*move to Topological_Spaces?*)
2.18 -  assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
2.19 -    shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
2.20 -proof -
2.21 -  have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
2.22 -    by auto
2.23 -  then show ?thesis
2.24 -    apply (rule ssubst)
2.25 -    apply (rule continuous_on_compose)
2.26 -    apply (simp add: split_def)
2.27 -    apply (rule continuous_intros | simp add: assms)+
2.28 -    done
2.29 -qed
2.30 -
2.31 -lemma substdbasis_expansion_unique:
2.32 -  assumes d: "d \<subseteq> Basis"
2.33 -  shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
2.34 -    (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
2.35 -proof -
2.36 -  have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
2.37 -    by auto
2.38 -  have **: "finite d"
2.39 -    by (auto intro: finite_subset[OF assms])
2.40 -  have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
2.41 -    using d
2.42 -    by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
2.43 -  show ?thesis
2.44 -    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
2.45 -qed
2.46 -
2.47 -lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
2.48 -  by (rule independent_mono[OF independent_Basis])
2.49 -
2.50 -lemma dim_cball:
2.51 -  assumes "e > 0"
2.52 -  shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
2.53 -proof -
2.54 -  {
2.55 -    fix x :: "'n::euclidean_space"
2.56 -    define y where "y = (e / norm x) *\<^sub>R x"
2.57 -    then have "y \<in> cball 0 e"
2.58 -      using assms by auto
2.59 -    moreover have *: "x = (norm x / e) *\<^sub>R y"
2.60 -      using y_def assms by simp
2.61 -    moreover from * have "x = (norm x/e) *\<^sub>R y"
2.62 -      by auto
2.63 -    ultimately have "x \<in> span (cball 0 e)"
2.64 -      using span_scale[of y "cball 0 e" "norm x/e"]
2.65 -        span_superset[of "cball 0 e"]
2.66 -      by (simp add: span_base)
2.67 -  }
2.68 -  then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
2.69 -    by auto
2.70 -  then show ?thesis
2.71 -    using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
2.72 -qed
2.73 -
2.74 -lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
2.75 -  by (rule ccontr) auto
2.76 -
2.77 -lemma subset_translation_eq [simp]:
2.78 -    fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
2.79 -  by auto
2.80 -
2.81 -lemma translate_inj_on:
2.82 -  fixes A :: "'a::ab_group_add set"
2.83 -  shows "inj_on (\<lambda>x. a + x) A"
2.84 -  unfolding inj_on_def by auto
2.85 -
2.86 -lemma translation_assoc:
2.87 -  fixes a b :: "'a::ab_group_add"
2.88 -  shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
2.89 -  by auto
2.90 -
2.91 -lemma translation_invert:
2.92 -  fixes a :: "'a::ab_group_add"
2.93 -  assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
2.94 -  shows "A = B"
2.95 -proof -
2.96 -  have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
2.97 -    using assms by auto
2.98 -  then show ?thesis
2.99 -    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
2.100 -qed
2.101 -
2.102 -lemma translation_galois:
2.103 -  fixes a :: "'a::ab_group_add"
2.104 -  shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
2.105 -  using translation_assoc[of "-a" a S]
2.106 -  apply auto
2.107 -  using translation_assoc[of a "-a" T]
2.108 -  apply auto
2.109 -  done
2.110 -
2.111 -lemma translation_inverse_subset:
2.112 -  assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
2.113 -  shows "V \<le> ((\<lambda>x. a + x) ` S)"
2.114 -proof -
2.115 -  {
2.116 -    fix x
2.117 -    assume "x \<in> V"
2.118 -    then have "x-a \<in> S" using assms by auto
2.119 -    then have "x \<in> {a + v |v. v \<in> S}"
2.120 -      apply auto
2.121 -      apply (rule exI[of _ "x-a"], simp)
2.122 -      done
2.123 -    then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
2.124 -  }
2.125 -  then show ?thesis by auto
2.126 -qed
2.127 -
2.128 -subsection \<open>Convexity\<close>
2.129 -
2.130 -definition%important convex :: "'a::real_vector set \<Rightarrow> bool"
2.131 -  where "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)"
2.132 -
2.133 -lemma convexI:
2.134 -  assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
2.135 -  shows "convex s"
2.136 -  using assms unfolding convex_def by fast
2.137 -
2.138 -lemma convexD:
2.139 -  assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
2.140 -  shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
2.141 -  using assms unfolding convex_def by fast
2.142 -
2.143 -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)"
2.144 -  (is "_ \<longleftrightarrow> ?alt")
2.145 -proof
2.146 -  show "convex s" if alt: ?alt
2.147 -  proof -
2.148 -    {
2.149 -      fix x y and u v :: real
2.150 -      assume mem: "x \<in> s" "y \<in> s"
2.151 -      assume "0 \<le> u" "0 \<le> v"
2.152 -      moreover
2.153 -      assume "u + v = 1"
2.154 -      then have "u = 1 - v" by auto
2.155 -      ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
2.156 -        using alt [rule_format, OF mem] by auto
2.157 -    }
2.158 -    then show ?thesis
2.159 -      unfolding convex_def by auto
2.160 -  qed
2.161 -  show ?alt if "convex s"
2.162 -    using that by (auto simp: convex_def)
2.163 -qed
2.164 -
2.165 -lemma convexD_alt:
2.166 -  assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
2.167 -  shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
2.168 -  using assms unfolding convex_alt by auto
2.169 -
2.170 -lemma mem_convex_alt:
2.171 -  assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
2.172 -  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
2.173 -  apply (rule convexD)
2.174 -  using assms
2.176 -  done
2.177 -
2.178 -lemma convex_empty[intro,simp]: "convex {}"
2.179 -  unfolding convex_def by simp
2.180 -
2.181 -lemma convex_singleton[intro,simp]: "convex {a}"
2.182 -  unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
2.183 -
2.184 -lemma convex_UNIV[intro,simp]: "convex UNIV"
2.185 -  unfolding convex_def by auto
2.186 -
2.187 -lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
2.188 -  unfolding convex_def by auto
2.189 -
2.190 -lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
2.191 -  unfolding convex_def by auto
2.192 -
2.193 -lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
2.194 -  unfolding convex_def by auto
2.195 -
2.196 -lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
2.197 -  unfolding convex_def by auto
2.198 -
2.199 -lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
2.200 -  unfolding convex_def
2.201 -  by (auto simp: inner_add intro!: convex_bound_le)
2.202 -
2.203 -lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
2.204 -proof -
2.205 -  have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
2.206 -    by auto
2.207 -  show ?thesis
2.208 -    unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
2.209 -qed
2.210 -
2.211 -lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
2.212 -proof -
2.213 -  have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
2.214 -    by auto
2.215 -  show ?thesis
2.216 -    unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
2.217 -qed
2.218 -
2.219 -lemma convex_hyperplane: "convex {x. inner a x = b}"
2.220 -proof -
2.221 -  have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
2.222 -    by auto
2.223 -  show ?thesis using convex_halfspace_le convex_halfspace_ge
2.224 -    by (auto intro!: convex_Int simp: *)
2.225 -qed
2.226 -
2.227 -lemma convex_halfspace_lt: "convex {x. inner a x < b}"
2.228 -  unfolding convex_def
2.229 -  by (auto simp: convex_bound_lt inner_add)
2.230 -
2.231 -lemma convex_halfspace_gt: "convex {x. inner a x > b}"
2.232 -  using convex_halfspace_lt[of "-a" "-b"] by auto
2.233 -
2.234 -lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
2.235 -  using convex_halfspace_ge[of b "1::complex"] by simp
2.236 -
2.237 -lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
2.238 -  using convex_halfspace_le[of "1::complex" b] by simp
2.239 -
2.240 -lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
2.241 -  using convex_halfspace_ge[of b \<i>] by simp
2.242 -
2.243 -lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
2.244 -  using convex_halfspace_le[of \<i> b] by simp
2.245 -
2.246 -lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
2.247 -  using convex_halfspace_gt[of b "1::complex"] by simp
2.248 -
2.249 -lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
2.250 -  using convex_halfspace_lt[of "1::complex" b] by simp
2.251 -
2.252 -lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
2.253 -  using convex_halfspace_gt[of b \<i>] by simp
2.254 -
2.255 -lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
2.256 -  using convex_halfspace_lt[of \<i> b] by simp
2.257 -
2.258 -lemma convex_real_interval [iff]:
2.259 -  fixes a b :: "real"
2.260 -  shows "convex {a..}" and "convex {..b}"
2.261 -    and "convex {a<..}" and "convex {..<b}"
2.262 -    and "convex {a..b}" and "convex {a<..b}"
2.263 -    and "convex {a..<b}" and "convex {a<..<b}"
2.264 -proof -
2.265 -  have "{a..} = {x. a \<le> inner 1 x}"
2.266 -    by auto
2.267 -  then show 1: "convex {a..}"
2.268 -    by (simp only: convex_halfspace_ge)
2.269 -  have "{..b} = {x. inner 1 x \<le> b}"
2.270 -    by auto
2.271 -  then show 2: "convex {..b}"
2.272 -    by (simp only: convex_halfspace_le)
2.273 -  have "{a<..} = {x. a < inner 1 x}"
2.274 -    by auto
2.275 -  then show 3: "convex {a<..}"
2.276 -    by (simp only: convex_halfspace_gt)
2.277 -  have "{..<b} = {x. inner 1 x < b}"
2.278 -    by auto
2.279 -  then show 4: "convex {..<b}"
2.280 -    by (simp only: convex_halfspace_lt)
2.281 -  have "{a..b} = {a..} \<inter> {..b}"
2.282 -    by auto
2.283 -  then show "convex {a..b}"
2.284 -    by (simp only: convex_Int 1 2)
2.285 -  have "{a<..b} = {a<..} \<inter> {..b}"
2.286 -    by auto
2.287 -  then show "convex {a<..b}"
2.288 -    by (simp only: convex_Int 3 2)
2.289 -  have "{a..<b} = {a..} \<inter> {..<b}"
2.290 -    by auto
2.291 -  then show "convex {a..<b}"
2.292 -    by (simp only: convex_Int 1 4)
2.293 -  have "{a<..<b} = {a<..} \<inter> {..<b}"
2.294 -    by auto
2.295 -  then show "convex {a<..<b}"
2.296 -    by (simp only: convex_Int 3 4)
2.297 -qed
2.298 -
2.299 -lemma convex_Reals: "convex \<real>"
2.300 -  by (simp add: convex_def scaleR_conv_of_real)
2.301 -
2.302 -
2.303 -subsection%unimportant \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
2.304 -
2.305 -lemma convex_sum:
2.306 -  fixes C :: "'a::real_vector set"
2.307 -  assumes "finite s"
2.308 -    and "convex C"
2.309 -    and "(\<Sum> i \<in> s. a i) = 1"
2.310 -  assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
2.311 -    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
2.312 -  shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
2.313 -  using assms(1,3,4,5)
2.314 -proof (induct arbitrary: a set: finite)
2.315 -  case empty
2.316 -  then show ?case by simp
2.317 -next
2.318 -  case (insert i s) note IH = this(3)
2.319 -  have "a i + sum a s = 1"
2.320 -    and "0 \<le> a i"
2.321 -    and "\<forall>j\<in>s. 0 \<le> a j"
2.322 -    and "y i \<in> C"
2.323 -    and "\<forall>j\<in>s. y j \<in> C"
2.324 -    using insert.hyps(1,2) insert.prems by simp_all
2.325 -  then have "0 \<le> sum a s"
2.326 -    by (simp add: sum_nonneg)
2.327 -  have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
2.328 -  proof (cases "sum a s = 0")
2.329 -    case True
2.330 -    with \<open>a i + sum a s = 1\<close> have "a i = 1"
2.331 -      by simp
2.332 -    from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
2.333 -      by simp
2.334 -    show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
2.335 -      by simp
2.336 -  next
2.337 -    case False
2.338 -    with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
2.339 -      by simp
2.340 -    then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
2.341 -      using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
2.342 -      by (simp add: IH sum_divide_distrib [symmetric])
2.343 -    from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
2.344 -      and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
2.345 -    have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
2.346 -      by (rule convexD)
2.347 -    then show ?thesis
2.348 -      by (simp add: scaleR_sum_right False)
2.349 -  qed
2.350 -  then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
2.351 -    by simp
2.352 -qed
2.353 -
2.354 -lemma convex:
2.355 -  "convex s \<longleftrightarrow> (\<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> (sum u {1..k} = 1)
2.356 -      \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
2.357 -proof safe
2.358 -  fix k :: nat
2.359 -  fix u :: "nat \<Rightarrow> real"
2.360 -  fix x
2.361 -  assume "convex s"
2.362 -    "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
2.363 -    "sum u {1..k} = 1"
2.364 -  with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
2.365 -    by auto
2.366 -next
2.367 -  assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1
2.368 -    \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
2.369 -  {
2.370 -    fix \<mu> :: real
2.371 -    fix x y :: 'a
2.372 -    assume xy: "x \<in> s" "y \<in> s"
2.373 -    assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
2.374 -    let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
2.375 -    let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
2.376 -    have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
2.377 -      by auto
2.378 -    then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
2.379 -      by simp
2.380 -    then have "sum ?u {1 .. 2} = 1"
2.381 -      using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
2.382 -      by auto
2.383 -    with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
2.384 -      using mu xy by auto
2.385 -    have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
2.386 -      using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
2.387 -    from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
2.388 -    have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
2.389 -      by auto
2.390 -    then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
2.391 -      using s by (auto simp: add.commute)
2.392 -  }
2.393 -  then show "convex s"
2.394 -    unfolding convex_alt by auto
2.395 -qed
2.396 -
2.397 -
2.398 -lemma convex_explicit:
2.399 -  fixes s :: "'a::real_vector set"
2.400 -  shows "convex s \<longleftrightarrow>
2.401 -    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
2.402 -proof safe
2.403 -  fix t
2.404 -  fix u :: "'a \<Rightarrow> real"
2.405 -  assume "convex s"
2.406 -    and "finite t"
2.407 -    and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
2.408 -  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
2.409 -    using convex_sum[of t s u "\<lambda> x. x"] by auto
2.410 -next
2.411 -  assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
2.412 -    sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
2.413 -  show "convex s"
2.414 -    unfolding convex_alt
2.415 -  proof safe
2.416 -    fix x y
2.417 -    fix \<mu> :: real
2.418 -    assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
2.419 -    show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
2.420 -    proof (cases "x = y")
2.421 -      case False
2.422 -      then show ?thesis
2.423 -        using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
2.424 -        by auto
2.425 -    next
2.426 -      case True
2.427 -      then show ?thesis
2.428 -        using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
2.429 -        by (auto simp: field_simps real_vector.scale_left_diff_distrib)
2.430 -    qed
2.431 -  qed
2.432 -qed
2.433 -
2.434 -lemma convex_finite:
2.435 -  assumes "finite s"
2.436 -  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
2.437 -  unfolding convex_explicit
2.438 -  apply safe
2.439 -  subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
2.440 -  subgoal for t u
2.441 -  proof -
2.442 -    have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
2.443 -      by simp
2.444 -    assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
2.445 -    assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
2.446 -    assume "t \<subseteq> s"
2.447 -    then have "s \<inter> t = t" by auto
2.448 -    with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
2.449 -      by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
2.450 -  qed
2.451 -  done
2.452 -
2.453 -
2.454 -subsection \<open>Functions that are convex on a set\<close>
2.455 -
2.456 -definition%important convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
2.457 -  where "convex_on s f \<longleftrightarrow>
2.458 -    (\<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)"
2.459 -
2.460 -lemma convex_onI [intro?]:
2.461 -  assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
2.462 -    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
2.463 -  shows "convex_on A f"
2.464 -  unfolding convex_on_def
2.465 -proof clarify
2.466 -  fix x y
2.467 -  fix u v :: real
2.468 -  assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
2.469 -  from A(5) have [simp]: "v = 1 - u"
2.470 -    by (simp add: algebra_simps)
2.471 -  from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
2.472 -    using assms[of u y x]
2.473 -    by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
2.474 -qed
2.475 -
2.476 -lemma convex_on_linorderI [intro?]:
2.477 -  fixes A :: "('a::{linorder,real_vector}) set"
2.478 -  assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
2.479 -    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
2.480 -  shows "convex_on A f"
2.481 -proof
2.482 -  fix x y
2.483 -  fix t :: real
2.484 -  assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
2.485 -  with assms [of t x y] assms [of "1 - t" y x]
2.486 -  show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
2.487 -    by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
2.488 -qed
2.489 -
2.490 -lemma convex_onD:
2.491 -  assumes "convex_on A f"
2.492 -  shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
2.493 -    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
2.494 -  using assms by (auto simp: convex_on_def)
2.495 -
2.496 -lemma convex_onD_Icc:
2.497 -  assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
2.498 -  shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
2.499 -    f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
2.500 -  using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
2.501 -
2.502 -lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
2.503 -  unfolding convex_on_def by auto
2.504 -
2.506 -  assumes "convex_on s f"
2.507 -    and "convex_on s g"
2.508 -  shows "convex_on s (\<lambda>x. f x + g x)"
2.509 -proof -
2.510 -  {
2.511 -    fix x y
2.512 -    assume "x \<in> s" "y \<in> s"
2.513 -    moreover
2.514 -    fix u v :: real
2.515 -    assume "0 \<le> u" "0 \<le> v" "u + v = 1"
2.516 -    ultimately
2.517 -    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)"
2.518 -      using assms unfolding convex_on_def by (auto simp: add_mono)
2.519 -    then have "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)"
2.520 -      by (simp add: field_simps)
2.521 -  }
2.522 -  then show ?thesis
2.523 -    unfolding convex_on_def by auto
2.524 -qed
2.525 -
2.526 -lemma convex_on_cmul [intro]:
2.527 -  fixes c :: real
2.528 -  assumes "0 \<le> c"
2.529 -    and "convex_on s f"
2.530 -  shows "convex_on s (\<lambda>x. c * f x)"
2.531 -proof -
2.532 -  have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
2.533 -    for u c fx v fy :: real
2.534 -    by (simp add: field_simps)
2.535 -  show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
2.536 -    unfolding convex_on_def and * by auto
2.537 -qed
2.538 -
2.539 -lemma convex_lower:
2.540 -  assumes "convex_on s f"
2.541 -    and "x \<in> s"
2.542 -    and "y \<in> s"
2.543 -    and "0 \<le> u"
2.544 -    and "0 \<le> v"
2.545 -    and "u + v = 1"
2.546 -  shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
2.547 -proof -
2.548 -  let ?m = "max (f x) (f y)"
2.549 -  have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
2.550 -    using assms(4,5) by (auto simp: mult_left_mono add_mono)
2.551 -  also have "\<dots> = max (f x) (f y)"
2.552 -    using assms(6) by (simp add: distrib_right [symmetric])
2.553 -  finally show ?thesis
2.554 -    using assms unfolding convex_on_def by fastforce
2.555 -qed
2.556 -
2.557 -lemma convex_on_dist [intro]:
2.558 -  fixes s :: "'a::real_normed_vector set"
2.559 -  shows "convex_on s (\<lambda>x. dist a x)"
2.560 -proof (auto simp: convex_on_def dist_norm)
2.561 -  fix x y
2.562 -  assume "x \<in> s" "y \<in> s"
2.563 -  fix u v :: real
2.564 -  assume "0 \<le> u"
2.565 -  assume "0 \<le> v"
2.566 -  assume "u + v = 1"
2.567 -  have "a = u *\<^sub>R a + v *\<^sub>R a"
2.568 -    unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
2.569 -  then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
2.570 -    by (auto simp: algebra_simps)
2.571 -  show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
2.572 -    unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
2.573 -    using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
2.574 -qed
2.575 -
2.576 -
2.577 -subsection%unimportant \<open>Arithmetic operations on sets preserve convexity\<close>
2.578 -
2.579 -lemma convex_linear_image:
2.580 -  assumes "linear f"
2.581 -    and "convex s"
2.582 -  shows "convex (f ` s)"
2.583 -proof -
2.584 -  interpret f: linear f by fact
2.585 -  from \<open>convex s\<close> show "convex (f ` s)"
2.587 -qed
2.588 -
2.589 -lemma convex_linear_vimage:
2.590 -  assumes "linear f"
2.591 -    and "convex s"
2.592 -  shows "convex (f -` s)"
2.593 -proof -
2.594 -  interpret f: linear f by fact
2.595 -  from \<open>convex s\<close> show "convex (f -` s)"
2.597 -qed
2.598 -
2.599 -lemma convex_scaling:
2.600 -  assumes "convex s"
2.601 -  shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
2.602 -proof -
2.603 -  have "linear (\<lambda>x. c *\<^sub>R x)"
2.605 -  then show ?thesis
2.606 -    using \<open>convex s\<close> by (rule convex_linear_image)
2.607 -qed
2.608 -
2.609 -lemma convex_scaled:
2.610 -  assumes "convex S"
2.611 -  shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
2.612 -proof -
2.613 -  have "linear (\<lambda>x. x *\<^sub>R c)"
2.615 -  then show ?thesis
2.616 -    using \<open>convex S\<close> by (rule convex_linear_image)
2.617 -qed
2.618 -
2.619 -lemma convex_negations:
2.620 -  assumes "convex S"
2.621 -  shows "convex ((\<lambda>x. - x) ` S)"
2.622 -proof -
2.623 -  have "linear (\<lambda>x. - x)"
2.624 -    by (simp add: linearI)
2.625 -  then show ?thesis
2.626 -    using \<open>convex S\<close> by (rule convex_linear_image)
2.627 -qed
2.628 -
2.629 -lemma convex_sums:
2.630 -  assumes "convex S"
2.631 -    and "convex T"
2.632 -  shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
2.633 -proof -
2.634 -  have "linear (\<lambda>(x, y). x + y)"
2.635 -    by (auto intro: linearI simp: scaleR_add_right)
2.636 -  with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
2.637 -    by (intro convex_linear_image convex_Times)
2.638 -  also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
2.639 -    by auto
2.640 -  finally show ?thesis .
2.641 -qed
2.642 -
2.643 -lemma convex_differences:
2.644 -  assumes "convex S" "convex T"
2.645 -  shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
2.646 -proof -
2.647 -  have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}"
2.649 -  then show ?thesis
2.650 -    using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
2.651 -qed
2.652 -
2.653 -lemma convex_translation:
2.654 -  assumes "convex S"
2.655 -  shows "convex ((\<lambda>x. a + x) ` S)"
2.656 -proof -
2.657 -  have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (\<lambda>x. a + x) ` S"
2.658 -    by auto
2.659 -  then show ?thesis
2.660 -    using convex_sums[OF convex_singleton[of a] assms] by auto
2.661 -qed
2.662 -
2.663 -lemma convex_affinity:
2.664 -  assumes "convex S"
2.665 -  shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
2.666 -proof -
2.667 -  have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S"
2.668 -    by auto
2.669 -  then show ?thesis
2.670 -    using convex_translation[OF convex_scaling[OF assms], of a c] by auto
2.671 -qed
2.672 -
2.673 -lemma pos_is_convex: "convex {0 :: real <..}"
2.674 -  unfolding convex_alt
2.675 -proof safe
2.676 -  fix y x \<mu> :: real
2.677 -  assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
2.678 -  {
2.679 -    assume "\<mu> = 0"
2.680 -    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
2.681 -      by simp
2.682 -    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
2.683 -      using * by simp
2.684 -  }
2.685 -  moreover
2.686 -  {
2.687 -    assume "\<mu> = 1"
2.688 -    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
2.689 -      using * by simp
2.690 -  }
2.691 -  moreover
2.692 -  {
2.693 -    assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
2.694 -    then have "\<mu> > 0" "(1 - \<mu>) > 0"
2.695 -      using * by auto
2.696 -    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
2.697 -      using * by (auto simp: add_pos_pos)
2.698 -  }
2.699 -  ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
2.700 -    by fastforce
2.701 -qed
2.702 -
2.703 -lemma convex_on_sum:
2.704 -  fixes a :: "'a \<Rightarrow> real"
2.705 -    and y :: "'a \<Rightarrow> 'b::real_vector"
2.706 -    and f :: "'b \<Rightarrow> real"
2.707 -  assumes "finite s" "s \<noteq> {}"
2.708 -    and "convex_on C f"
2.709 -    and "convex C"
2.710 -    and "(\<Sum> i \<in> s. a i) = 1"
2.711 -    and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
2.712 -    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
2.713 -  shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
2.714 -  using assms
2.715 -proof (induct s arbitrary: a rule: finite_ne_induct)
2.716 -  case (singleton i)
2.717 -  then have ai: "a i = 1"
2.718 -    by auto
2.719 -  then show ?case
2.720 -    by auto
2.721 -next
2.722 -  case (insert i s)
2.723 -  then have "convex_on C f"
2.724 -    by simp
2.725 -  from this[unfolded convex_on_def, rule_format]
2.726 -  have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
2.727 -      f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
2.728 -    by simp
2.729 -  show ?case
2.730 -  proof (cases "a i = 1")
2.731 -    case True
2.732 -    then have "(\<Sum> j \<in> s. a j) = 0"
2.733 -      using insert by auto
2.734 -    then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
2.735 -      using insert by (fastforce simp: sum_nonneg_eq_0_iff)
2.736 -    then show ?thesis
2.737 -      using insert by auto
2.738 -  next
2.739 -    case False
2.740 -    from insert have yai: "y i \<in> C" "a i \<ge> 0"
2.741 -      by auto
2.742 -    have fis: "finite (insert i s)"
2.743 -      using insert by auto
2.744 -    then have ai1: "a i \<le> 1"
2.745 -      using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
2.746 -    then have "a i < 1"
2.747 -      using False by auto
2.748 -    then have i0: "1 - a i > 0"
2.749 -      by auto
2.750 -    let ?a = "\<lambda>j. a j / (1 - a i)"
2.751 -    have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
2.752 -      using i0 insert that by fastforce
2.753 -    have "(\<Sum> j \<in> insert i s. a j) = 1"
2.754 -      using insert by auto
2.755 -    then have "(\<Sum> j \<in> s. a j) = 1 - a i"
2.756 -      using sum.insert insert by fastforce
2.757 -    then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
2.758 -      using i0 by auto
2.759 -    then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
2.760 -      unfolding sum_divide_distrib by simp
2.761 -    have "convex C" using insert by auto
2.762 -    then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
2.763 -      using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
2.764 -    have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
2.765 -      using a_nonneg a1 insert by blast
2.766 -    have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
2.767 -      using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
2.768 -      by (auto simp only: add.commute)
2.769 -    also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
2.770 -      using i0 by auto
2.771 -    also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
2.772 -      using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
2.773 -      by (auto simp: algebra_simps)
2.774 -    also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
2.775 -      by (auto simp: divide_inverse)
2.776 -    also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
2.777 -      using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
2.778 -      by (auto simp: add.commute)
2.779 -    also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
2.780 -      using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
2.781 -            OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
2.782 -      by simp
2.783 -    also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
2.784 -      unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
2.785 -      using i0 by auto
2.786 -    also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
2.787 -      using i0 by auto
2.788 -    also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
2.789 -      using insert by auto
2.790 -    finally show ?thesis
2.791 -      by simp
2.792 -  qed
2.793 -qed
2.794 -
2.795 -lemma convex_on_alt:
2.796 -  fixes C :: "'a::real_vector set"
2.797 -  assumes "convex C"
2.798 -  shows "convex_on C f \<longleftrightarrow>
2.799 -    (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
2.800 -      f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
2.801 -proof safe
2.802 -  fix x y
2.803 -  fix \<mu> :: real
2.804 -  assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
2.805 -  from this[unfolded convex_on_def, rule_format]
2.806 -  have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
2.807 -    by auto
2.808 -  from this [of "\<mu>" "1 - \<mu>", simplified] *
2.809 -  show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
2.810 -    by auto
2.811 -next
2.812 -  assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
2.813 -    f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
2.814 -  {
2.815 -    fix x y
2.816 -    fix u v :: real
2.817 -    assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
2.818 -    then have[simp]: "1 - u = v" by auto
2.819 -    from *[rule_format, of x y u]
2.820 -    have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
2.821 -      using ** by auto
2.822 -  }
2.823 -  then show "convex_on C f"
2.824 -    unfolding convex_on_def by auto
2.825 -qed
2.826 -
2.827 -lemma convex_on_diff:
2.828 -  fixes f :: "real \<Rightarrow> real"
2.829 -  assumes f: "convex_on I f"
2.830 -    and I: "x \<in> I" "y \<in> I"
2.831 -    and t: "x < t" "t < y"
2.832 -  shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
2.833 -    and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
2.834 -proof -
2.835 -  define a where "a \<equiv> (t - y) / (x - y)"
2.836 -  with t have "0 \<le> a" "0 \<le> 1 - a"
2.837 -    by (auto simp: field_simps)
2.838 -  with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
2.839 -    by (auto simp: convex_on_def)
2.840 -  have "a * x + (1 - a) * y = a * (x - y) + y"
2.841 -    by (simp add: field_simps)
2.842 -  also have "\<dots> = t"
2.843 -    unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
2.844 -  finally have "f t \<le> a * f x + (1 - a) * f y"
2.845 -    using cvx by simp
2.846 -  also have "\<dots> = a * (f x - f y) + f y"
2.847 -    by (simp add: field_simps)
2.848 -  finally have "f t - f y \<le> a * (f x - f y)"
2.849 -    by simp
2.850 -  with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
2.851 -    by (simp add: le_divide_eq divide_le_eq field_simps a_def)
2.852 -  with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
2.853 -    by (simp add: le_divide_eq divide_le_eq field_simps)
2.854 -qed
2.855 -
2.856 -lemma pos_convex_function:
2.857 -  fixes f :: "real \<Rightarrow> real"
2.858 -  assumes "convex C"
2.859 -    and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
2.860 -  shows "convex_on C f"
2.861 -  unfolding convex_on_alt[OF assms(1)]
2.862 -  using assms
2.863 -proof safe
2.864 -  fix x y \<mu> :: real
2.865 -  let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
2.866 -  assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
2.867 -  then have "1 - \<mu> \<ge> 0" by auto
2.868 -  then have xpos: "?x \<in> C"
2.869 -    using * unfolding convex_alt by fastforce
2.870 -  have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
2.871 -      \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
2.872 -    using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
2.873 -        mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
2.874 -    by auto
2.875 -  then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
2.876 -    by (auto simp: field_simps)
2.877 -  then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
2.878 -    using convex_on_alt by auto
2.879 -qed
2.880 -
2.881 -lemma atMostAtLeast_subset_convex:
2.882 -  fixes C :: "real set"
2.883 -  assumes "convex C"
2.884 -    and "x \<in> C" "y \<in> C" "x < y"
2.885 -  shows "{x .. y} \<subseteq> C"
2.886 -proof safe
2.887 -  fix z assume z: "z \<in> {x .. y}"
2.888 -  have less: "z \<in> C" if *: "x < z" "z < y"
2.889 -  proof -
2.890 -    let ?\<mu> = "(y - z) / (y - x)"
2.891 -    have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
2.892 -      using assms * by (auto simp: field_simps)
2.893 -    then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
2.894 -      using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
2.895 -      by (simp add: algebra_simps)
2.896 -    have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
2.897 -      by (auto simp: field_simps)
2.898 -    also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
2.899 -      using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
2.900 -    also have "\<dots> = z"
2.901 -      using assms by (auto simp: field_simps)
2.902 -    finally show ?thesis
2.903 -      using comb by auto
2.904 -  qed
2.905 -  show "z \<in> C"
2.906 -    using z less assms by (auto simp: le_less)
2.907 -qed
2.908 -
2.909 -lemma f''_imp_f':
2.910 -  fixes f :: "real \<Rightarrow> real"
2.911 -  assumes "convex C"
2.912 -    and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
2.913 -    and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
2.914 -    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
2.915 -    and x: "x \<in> C"
2.916 -    and y: "y \<in> C"
2.917 -  shows "f' x * (y - x) \<le> f y - f x"
2.918 -  using assms
2.919 -proof -
2.920 -  have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
2.921 -    if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
2.922 -  proof -
2.923 -    from * have ge: "y - x > 0" "y - x \<ge> 0"
2.924 -      by auto
2.925 -    from * have le: "x - y < 0" "x - y \<le> 0"
2.926 -      by auto
2.927 -    then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
2.928 -      using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
2.929 -          THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
2.930 -      by auto
2.931 -    then have "z1 \<in> C"
2.932 -      using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
2.933 -      by fastforce
2.934 -    from z1 have z1': "f x - f y = (x - y) * f' z1"
2.935 -      by (simp add: field_simps)
2.936 -    obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
2.937 -      using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
2.938 -          THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
2.939 -      by auto
2.940 -    obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
2.941 -      using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
2.942 -          THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
2.943 -      by auto
2.944 -    have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
2.945 -      using * z1' by auto
2.946 -    also have "\<dots> = (y - z1) * f'' z3"
2.947 -      using z3 by auto
2.948 -    finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
2.949 -      by simp
2.950 -    have A': "y - z1 \<ge> 0"
2.951 -      using z1 by auto
2.952 -    have "z3 \<in> C"
2.953 -      using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
2.954 -      by fastforce
2.955 -    then have B': "f'' z3 \<ge> 0"
2.956 -      using assms by auto
2.957 -    from A' B' have "(y - z1) * f'' z3 \<ge> 0"
2.958 -      by auto
2.959 -    from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
2.960 -      by auto
2.961 -    from mult_right_mono_neg[OF this le(2)]
2.962 -    have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
2.963 -      by (simp add: algebra_simps)
2.964 -    then have "f' y * (x - y) - (f x - f y) \<le> 0"
2.965 -      using le by auto
2.966 -    then have res: "f' y * (x - y) \<le> f x - f y"
2.967 -      by auto
2.968 -    have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
2.969 -      using * z1 by auto
2.970 -    also have "\<dots> = (z1 - x) * f'' z2"
2.971 -      using z2 by auto
2.972 -    finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
2.973 -      by simp
2.974 -    have A: "z1 - x \<ge> 0"
2.975 -      using z1 by auto
2.976 -    have "z2 \<in> C"
2.977 -      using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
2.978 -      by fastforce
2.979 -    then have B: "f'' z2 \<ge> 0"
2.980 -      using assms by auto
2.981 -    from A B have "(z1 - x) * f'' z2 \<ge> 0"
2.982 -      by auto
2.983 -    with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
2.984 -      by auto
2.985 -    from mult_right_mono[OF this ge(2)]
2.986 -    have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
2.987 -      by (simp add: algebra_simps)
2.988 -    then have "f y - f x - f' x * (y - x) \<ge> 0"
2.989 -      using ge by auto
2.990 -    then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
2.991 -      using res by auto
2.992 -  qed
2.993 -  show ?thesis
2.994 -  proof (cases "x = y")
2.995 -    case True
2.996 -    with x y show ?thesis by auto
2.997 -  next
2.998 -    case False
2.999 -    with less_imp x y show ?thesis
2.1000 -      by (auto simp: neq_iff)
2.1001 -  qed
2.1002 -qed
2.1003 -
2.1004 -lemma f''_ge0_imp_convex:
2.1005 -  fixes f :: "real \<Rightarrow> real"
2.1006 -  assumes conv: "convex C"
2.1007 -    and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
2.1008 -    and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
2.1009 -    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
2.1010 -  shows "convex_on C f"
2.1011 -  using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
2.1012 -  by fastforce
2.1013 -
2.1014 -lemma minus_log_convex:
2.1015 -  fixes b :: real
2.1016 -  assumes "b > 1"
2.1017 -  shows "convex_on {0 <..} (\<lambda> x. - log b x)"
2.1018 -proof -
2.1019 -  have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
2.1020 -    using DERIV_log by auto
2.1021 -  then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
2.1022 -    by (auto simp: DERIV_minus)
2.1023 -  have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
2.1024 -    using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
2.1025 -  from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
2.1026 -  have "\<And>z::real. z > 0 \<Longrightarrow>
2.1027 -    DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
2.1028 -    by auto
2.1029 -  then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
2.1030 -    DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
2.1031 -    unfolding inverse_eq_divide by (auto simp: mult.assoc)
2.1032 -  have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
2.1033 -    using \<open>b > 1\<close> by (auto intro!: less_imp_le)
2.1034 -  from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
2.1035 -  show ?thesis
2.1036 -    by auto
2.1037 -qed
2.1038 -
2.1039 -
2.1040 -subsection%unimportant \<open>Convexity of real functions\<close>
2.1041 -
2.1042 -lemma convex_on_realI:
2.1043 -  assumes "connected A"
2.1044 -    and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
2.1045 -    and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
2.1046 -  shows "convex_on A f"
2.1047 -proof (rule convex_on_linorderI)
2.1048 -  fix t x y :: real
2.1049 -  assume t: "t > 0" "t < 1"
2.1050 -  assume xy: "x \<in> A" "y \<in> A" "x < y"
2.1051 -  define z where "z = (1 - t) * x + t * y"
2.1052 -  with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
2.1053 -    using connected_contains_Icc by blast
2.1054 -
2.1055 -  from xy t have xz: "z > x"
2.1056 -    by (simp add: z_def algebra_simps)
2.1057 -  have "y - z = (1 - t) * (y - x)"
2.1058 -    by (simp add: z_def algebra_simps)
2.1059 -  also from xy t have "\<dots> > 0"
2.1060 -    by (intro mult_pos_pos) simp_all
2.1061 -  finally have yz: "z < y"
2.1062 -    by simp
2.1063 -
2.1064 -  from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
2.1065 -    by (intro MVT2) (auto intro!: assms(2))
2.1066 -  then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
2.1067 -    by auto
2.1068 -  from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
2.1069 -    by (intro MVT2) (auto intro!: assms(2))
2.1070 -  then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
2.1071 -    by auto
2.1072 -
2.1073 -  from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
2.1074 -  also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
2.1075 -    by auto
2.1076 -  with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
2.1077 -    by (intro assms(3)) auto
2.1078 -  also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
2.1079 -  finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
2.1080 -    using xz yz by (simp add: field_simps)
2.1081 -  also have "z - x = t * (y - x)"
2.1082 -    by (simp add: z_def algebra_simps)
2.1083 -  also have "y - z = (1 - t) * (y - x)"
2.1084 -    by (simp add: z_def algebra_simps)
2.1085 -  finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
2.1086 -    using xy by simp
2.1087 -  then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
2.1088 -    by (simp add: z_def algebra_simps)
2.1089 -qed
2.1090 -
2.1091 -lemma convex_on_inverse:
2.1092 -  assumes "A \<subseteq> {0<..}"
2.1093 -  shows "convex_on A (inverse :: real \<Rightarrow> real)"
2.1094 -proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
2.1095 -  fix u v :: real
2.1096 -  assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
2.1097 -  with assms show "-inverse (u^2) \<le> -inverse (v^2)"
2.1098 -    by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
2.1099 -qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
2.1100 -
2.1101 -lemma convex_onD_Icc':
2.1102 -  assumes "convex_on {x..y} f" "c \<in> {x..y}"
2.1103 -  defines "d \<equiv> y - x"
2.1104 -  shows "f c \<le> (f y - f x) / d * (c - x) + f x"
2.1105 -proof (cases x y rule: linorder_cases)
2.1106 -  case less
2.1107 -  then have d: "d > 0"
2.1108 -    by (simp add: d_def)
2.1109 -  from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
2.1110 -    by (simp_all add: d_def divide_simps)
2.1111 -  have "f c = f (x + (c - x) * 1)"
2.1112 -    by simp
2.1113 -  also from less have "1 = ((y - x) / d)"
2.1114 -    by (simp add: d_def)
2.1115 -  also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
2.1116 -    by (simp add: field_simps)
2.1117 -  also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
2.1118 -    using assms less by (intro convex_onD_Icc) simp_all
2.1119 -  also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
2.1120 -    by (simp add: field_simps)
2.1121 -  finally show ?thesis .
2.1122 -qed (insert assms(2), simp_all)
2.1123 -
2.1124 -lemma convex_onD_Icc'':
2.1125 -  assumes "convex_on {x..y} f" "c \<in> {x..y}"
2.1126 -  defines "d \<equiv> y - x"
2.1127 -  shows "f c \<le> (f x - f y) / d * (y - c) + f y"
2.1128 -proof (cases x y rule: linorder_cases)
2.1129 -  case less
2.1130 -  then have d: "d > 0"
2.1131 -    by (simp add: d_def)
2.1132 -  from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
2.1133 -    by (simp_all add: d_def divide_simps)
2.1134 -  have "f c = f (y - (y - c) * 1)"
2.1135 -    by simp
2.1136 -  also from less have "1 = ((y - x) / d)"
2.1137 -    by (simp add: d_def)
2.1138 -  also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
2.1139 -    by (simp add: field_simps)
2.1140 -  also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
2.1141 -    using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
2.1142 -  also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
2.1143 -    by (simp add: field_simps)
2.1144 -  finally show ?thesis .
2.1145 -qed (insert assms(2), simp_all)
2.1146 +subsection%unimportant \<open>Topological Properties of Convex Sets and Functions\<close>
2.1147
2.1148  lemma convex_supp_sum:
2.1149    assumes "convex S" and 1: "supp_sum u I = 1"
2.1150 @@ -1160,14 +32,6 @@
2.1151      done
2.1152  qed
2.1153
2.1154 -lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
2.1155 -  by (metis convex_translation translation_galois)
2.1156 -
2.1157 -lemma convex_linear_image_eq [simp]:
2.1158 -    fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
2.1159 -    shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
2.1160 -    by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
2.1161 -
2.1162  lemma closure_bounded_linear_image_subset:
2.1163    assumes f: "bounded_linear f"
2.1164    shows "f ` closure S \<subseteq> closure (f ` S)"
2.1165 @@ -1238,822 +102,11 @@
2.1166      by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
2.1167  qed
2.1168
2.1169 -lemma fst_linear: "linear fst"
2.1170 -  unfolding linear_iff by (simp add: algebra_simps)
2.1171 -
2.1172 -lemma snd_linear: "linear snd"
2.1173 -  unfolding linear_iff by (simp add: algebra_simps)
2.1174 -
2.1175 -lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
2.1176 -  unfolding linear_iff by (simp add: algebra_simps)
2.1177 -
2.1178 -lemma vector_choose_size:
2.1179 -  assumes "0 \<le> c"
2.1180 -  obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
2.1181 -proof -
2.1182 -  obtain a::'a where "a \<noteq> 0"
2.1183 -    using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
2.1184 -  then show ?thesis
2.1185 -    by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
2.1186 -qed
2.1187 -
2.1188 -lemma vector_choose_dist:
2.1189 -  assumes "0 \<le> c"
2.1190 -  obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
2.1191 -by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
2.1192 -
2.1193  lemma sphere_eq_empty [simp]:
2.1194    fixes a :: "'a::{real_normed_vector, perfect_space}"
2.1195    shows "sphere a r = {} \<longleftrightarrow> r < 0"
2.1196  by (auto simp: sphere_def dist_norm) (metis dist_norm le_less_linear vector_choose_dist)
2.1197
2.1198 -lemma sum_delta_notmem:
2.1199 -  assumes "x \<notin> s"
2.1200 -  shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
2.1201 -    and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
2.1202 -    and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
2.1203 -    and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
2.1204 -  apply (rule_tac [!] sum.cong)
2.1205 -  using assms
2.1206 -  apply auto
2.1207 -  done
2.1208 -
2.1209 -lemma sum_delta'':
2.1210 -  fixes s::"'a::real_vector set"
2.1211 -  assumes "finite s"
2.1212 -  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)"
2.1213 -proof -
2.1214 -  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)"
2.1215 -    by auto
2.1216 -  show ?thesis
2.1217 -    unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
2.1218 -qed
2.1219 -
2.1220 -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)"
2.1221 -  by (fact if_distrib)
2.1222 -
2.1223 -lemma dist_triangle_eq:
2.1224 -  fixes x y z :: "'a::real_inner"
2.1225 -  shows "dist x z = dist x y + dist y z \<longleftrightarrow>
2.1226 -    norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
2.1227 -proof -
2.1228 -  have *: "x - y + (y - z) = x - z" by auto
2.1229 -  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
2.1230 -    by (auto simp:norm_minus_commute)
2.1231 -qed
2.1232 -
2.1233 -
2.1234 -subsection \<open>Affine set and affine hull\<close>
2.1235 -
2.1236 -definition%important affine :: "'a::real_vector set \<Rightarrow> bool"
2.1237 -  where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
2.1238 -
2.1239 -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)"
2.1240 -  unfolding affine_def by (metis eq_diff_eq')
2.1241 -
2.1242 -lemma affine_empty [iff]: "affine {}"
2.1243 -  unfolding affine_def by auto
2.1244 -
2.1245 -lemma affine_sing [iff]: "affine {x}"
2.1246 -  unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
2.1247 -
2.1248 -lemma affine_UNIV [iff]: "affine UNIV"
2.1249 -  unfolding affine_def by auto
2.1250 -
2.1251 -lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
2.1252 -  unfolding affine_def by auto
2.1253 -
2.1254 -lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
2.1255 -  unfolding affine_def by auto
2.1256 -
2.1257 -lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
2.1258 -  apply (clarsimp simp add: affine_def)
2.1259 -  apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
2.1260 -  apply (auto simp: algebra_simps)
2.1261 -  done
2.1262 -
2.1263 -lemma affine_affine_hull [simp]: "affine(affine hull s)"
2.1264 -  unfolding hull_def
2.1265 -  using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
2.1266 -
2.1267 -lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
2.1268 -  by (metis affine_affine_hull hull_same)
2.1269 -
2.1270 -lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
2.1271 -  by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
2.1272 -
2.1273 -
2.1274 -subsubsection%unimportant \<open>Some explicit formulations\<close>
2.1275 -
2.1276 -text "Formalized by Lars Schewe."
2.1277 -
2.1278 -lemma affine:
2.1279 -  fixes V::"'a::real_vector set"
2.1280 -  shows "affine V \<longleftrightarrow>
2.1281 -         (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
2.1282 -proof -
2.1283 -  have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
2.1284 -    and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
2.1285 -  proof (cases "x = y")
2.1286 -    case True
2.1287 -    then show ?thesis
2.1288 -      using that by (metis scaleR_add_left scaleR_one)
2.1289 -  next
2.1290 -    case False
2.1291 -    then show ?thesis
2.1292 -      using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
2.1293 -  qed
2.1294 -  moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
2.1295 -                if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
2.1296 -                  and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
2.1297 -  proof -
2.1298 -    define n where "n = card S"
2.1299 -    consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
2.1300 -    then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
2.1301 -    proof cases
2.1302 -      assume "card S = 1"
2.1303 -      then obtain a where "S={a}"
2.1304 -        by (auto simp: card_Suc_eq)
2.1305 -      then show ?thesis
2.1306 -        using that by simp
2.1307 -    next
2.1308 -      assume "card S = 2"
2.1309 -      then obtain a b where "S = {a, b}"
2.1310 -        by (metis Suc_1 card_1_singletonE card_Suc_eq)
2.1311 -      then show ?thesis
2.1312 -        using *[of a b] that
2.1313 -        by (auto simp: sum_clauses(2))
2.1314 -    next
2.1315 -      assume "card S > 2"
2.1316 -      then show ?thesis using that n_def
2.1317 -      proof (induct n arbitrary: u S)
2.1318 -        case 0
2.1319 -        then show ?case by auto
2.1320 -      next
2.1321 -        case (Suc n u S)
2.1322 -        have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
2.1323 -          using that unfolding card_eq_sum by auto
2.1324 -        with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
2.1325 -        have c: "card (S - {x}) = card S - 1"
2.1326 -          by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
2.1327 -        have "sum u (S - {x}) = 1 - u x"
2.1328 -          by (simp add: Suc.prems sum_diff1_ring \<open>x \<in> S\<close>)
2.1329 -        with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
2.1330 -          by auto
2.1331 -        have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
2.1332 -        proof (cases "card (S - {x}) > 2")
2.1333 -          case True
2.1334 -          then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
2.1335 -            using Suc.prems c by force+
2.1336 -          show ?thesis
2.1337 -          proof (rule Suc.hyps)
2.1338 -            show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
2.1339 -              by (auto simp: eq1 sum_distrib_left[symmetric])
2.1340 -          qed (use S Suc.prems True in auto)
2.1341 -        next
2.1342 -          case False
2.1343 -          then have "card (S - {x}) = Suc (Suc 0)"
2.1344 -            using Suc.prems c by auto
2.1345 -          then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
2.1346 -            unfolding card_Suc_eq by auto
2.1347 -          then show ?thesis
2.1348 -            using eq1 \<open>S \<subseteq> V\<close>
2.1349 -            by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
2.1350 -        qed
2.1351 -        have "u x + (1 - u x) = 1 \<Longrightarrow>
2.1352 -          u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
2.1353 -          by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
2.1354 -        moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
2.1355 -          by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
2.1356 -        ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
2.1357 -          by (simp add: x)
2.1358 -      qed
2.1359 -    qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
2.1360 -  qed
2.1361 -  ultimately show ?thesis
2.1362 -    unfolding affine_def by meson
2.1363 -qed
2.1364 -
2.1365 -
2.1366 -lemma affine_hull_explicit:
2.1367 -  "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
2.1368 -  (is "_ = ?rhs")
2.1369 -proof (rule hull_unique)
2.1370 -  show "p \<subseteq> ?rhs"
2.1371 -  proof (intro subsetI CollectI exI conjI)
2.1372 -    show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
2.1373 -      by auto
2.1374 -  qed auto
2.1375 -  show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
2.1376 -    using that unfolding affine by blast
2.1377 -  show "affine ?rhs"
2.1378 -    unfolding affine_def
2.1379 -  proof clarify
2.1380 -    fix u v :: real and sx ux sy uy
2.1381 -    assume uv: "u + v = 1"
2.1382 -      and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
2.1383 -      and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
2.1384 -    have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
2.1385 -      by auto
2.1386 -    show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
2.1387 -        sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
2.1388 -    proof (intro exI conjI)
2.1389 -      show "finite (sx \<union> sy)"
2.1390 -        using x y by auto
2.1391 -      show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
2.1392 -        using x y uv
2.1393 -        by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
2.1394 -      have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
2.1395 -          = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
2.1396 -        using x y
2.1397 -        unfolding scaleR_left_distrib scaleR_zero_left if_smult
2.1398 -        by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric]  **)
2.1399 -      also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
2.1400 -        unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
2.1401 -      finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
2.1402 -                  = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
2.1403 -    qed (use x y in auto)
2.1404 -  qed
2.1405 -qed
2.1406 -
2.1407 -lemma affine_hull_finite:
2.1408 -  assumes "finite S"
2.1409 -  shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
2.1410 -proof -
2.1411 -  have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
2.1412 -    if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
2.1413 -  proof -
2.1414 -    have "S \<inter> F = F"
2.1415 -      using that by auto
2.1416 -    show ?thesis
2.1417 -    proof (intro exI conjI)
2.1418 -      show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
2.1419 -        by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
2.1420 -      show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
2.1421 -        by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
2.1422 -    qed
2.1423 -  qed
2.1424 -  show ?thesis
2.1425 -    unfolding affine_hull_explicit using assms
2.1426 -    by (fastforce dest: *)
2.1427 -qed
2.1428 -
2.1429 -
2.1430 -subsubsection%unimportant \<open>Stepping theorems and hence small special cases\<close>
2.1431 -
2.1432 -lemma affine_hull_empty[simp]: "affine hull {} = {}"
2.1433 -  by simp
2.1434 -
2.1435 -lemma affine_hull_finite_step:
2.1436 -  fixes y :: "'a::real_vector"
2.1437 -  shows "finite S \<Longrightarrow>
2.1438 -      (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
2.1439 -      (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
2.1440 -proof -
2.1441 -  assume fin: "finite S"
2.1442 -  show "?lhs = ?rhs"
2.1443 -  proof
2.1444 -    assume ?lhs
2.1445 -    then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
2.1446 -      by auto
2.1447 -    show ?rhs
2.1448 -    proof (cases "a \<in> S")
2.1449 -      case True
2.1450 -      then show ?thesis
2.1451 -        using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
2.1452 -    next
2.1453 -      case False
2.1454 -      show ?thesis
2.1455 -        by (rule exI [where x="u a"]) (use u fin False in auto)
2.1456 -    qed
2.1457 -  next
2.1458 -    assume ?rhs
2.1459 -    then obtain v u where vu: "sum u S = w - v"  "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
2.1460 -      by auto
2.1461 -    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)"
2.1462 -      by auto
2.1463 -    show ?lhs
2.1464 -    proof (cases "a \<in> S")
2.1465 -      case True
2.1466 -      show ?thesis
2.1467 -        by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
2.1468 -           (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
2.1469 -    next
2.1470 -      case False
2.1471 -      then show ?thesis
2.1472 -        apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
2.1473 -        apply (simp add: vu sum_clauses(2)[OF fin] *)
2.1474 -        by (simp add: sum_delta_notmem(3) vu)
2.1475 -    qed
2.1476 -  qed
2.1477 -qed
2.1478 -
2.1479 -lemma affine_hull_2:
2.1480 -  fixes a b :: "'a::real_vector"
2.1481 -  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
2.1482 -  (is "?lhs = ?rhs")
2.1483 -proof -
2.1484 -  have *:
2.1485 -    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
2.1486 -    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
2.1487 -  have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
2.1488 -    using affine_hull_finite[of "{a,b}"] by auto
2.1489 -  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
2.1490 -    by (simp add: affine_hull_finite_step[of "{b}" a])
2.1491 -  also have "\<dots> = ?rhs" unfolding * by auto
2.1492 -  finally show ?thesis by auto
2.1493 -qed
2.1494 -
2.1495 -lemma affine_hull_3:
2.1496 -  fixes a b c :: "'a::real_vector"
2.1497 -  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}"
2.1498 -proof -
2.1499 -  have *:
2.1500 -    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
2.1501 -    "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
2.1502 -  show ?thesis
2.1503 -    apply (simp add: affine_hull_finite affine_hull_finite_step)
2.1504 -    unfolding *
2.1505 -    apply safe
2.1507 -    apply (rule_tac x=u in exI, force)
2.1508 -    done
2.1509 -qed
2.1510 -
2.1511 -lemma mem_affine:
2.1512 -  assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
2.1513 -  shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
2.1514 -  using assms affine_def[of S] by auto
2.1515 -
2.1516 -lemma mem_affine_3:
2.1517 -  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
2.1518 -  shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
2.1519 -proof -
2.1520 -  have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
2.1521 -    using affine_hull_3[of x y z] assms by auto
2.1522 -  moreover
2.1523 -  have "affine hull {x, y, z} \<subseteq> affine hull S"
2.1524 -    using hull_mono[of "{x, y, z}" "S"] assms by auto
2.1525 -  moreover
2.1526 -  have "affine hull S = S"
2.1527 -    using assms affine_hull_eq[of S] by auto
2.1528 -  ultimately show ?thesis by auto
2.1529 -qed
2.1530 -
2.1531 -lemma mem_affine_3_minus:
2.1532 -  assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
2.1533 -  shows "x + v *\<^sub>R (y-z) \<in> S"
2.1534 -  using mem_affine_3[of S x y z 1 v "-v"] assms
2.1535 -  by (simp add: algebra_simps)
2.1536 -
2.1537 -corollary mem_affine_3_minus2:
2.1538 -    "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
2.1539 -  by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
2.1540 -
2.1541 -
2.1542 -subsubsection%unimportant \<open>Some relations between affine hull and subspaces\<close>
2.1543 -
2.1544 -lemma affine_hull_insert_subset_span:
2.1545 -  "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
2.1546 -proof -
2.1547 -  have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
2.1548 -    if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
2.1549 -    for x F u
2.1550 -  proof -
2.1551 -    have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
2.1552 -      using that by auto
2.1553 -    show ?thesis
2.1554 -    proof (intro exI conjI)
2.1555 -      show "finite ((\<lambda>x. x - a) ` (F - {a}))"
2.1556 -        by (simp add: that(1))
2.1557 -      show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
2.1558 -        by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
2.1559 -            sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
2.1560 -    qed (use \<open>F \<subseteq> insert a S\<close> in auto)
2.1561 -  qed
2.1562 -  then show ?thesis
2.1563 -    unfolding affine_hull_explicit span_explicit by blast
2.1564 -qed
2.1565 -
2.1566 -lemma affine_hull_insert_span:
2.1567 -  assumes "a \<notin> S"
2.1568 -  shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x.  x \<in> S}}"
2.1569 -proof -
2.1570 -  have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
2.1571 -    if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
2.1572 -  proof -
2.1573 -    from that
2.1574 -    obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
2.1575 -      unfolding span_explicit by auto
2.1576 -    define F where "F = (\<lambda>x. x + a) ` T"
2.1577 -    have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
2.1578 -      unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
2.1579 -    have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
2.1580 -      using F assms by auto
2.1581 -    show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
2.1582 -      apply (rule_tac x = "insert a F" in exI)
2.1583 -      apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
2.1584 -      using assms F
2.1585 -      apply (auto simp:  sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
2.1586 -      done
2.1587 -  qed
2.1588 -  show ?thesis
2.1589 -    by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
2.1590 -qed
2.1591 -
2.1592 -lemma affine_hull_span:
2.1593 -  assumes "a \<in> S"
2.1594 -  shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
2.1595 -  using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
2.1596 -
2.1597 -
2.1598 -subsubsection%unimportant \<open>Parallel affine sets\<close>
2.1599 -
2.1600 -definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
2.1601 -  where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
2.1602 -
2.1603 -lemma affine_parallel_expl_aux:
2.1604 -  fixes S T :: "'a::real_vector set"
2.1605 -  assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
2.1606 -  shows "T = (\<lambda>x. a + x) ` S"
2.1607 -proof -
2.1608 -  have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
2.1609 -    using that
2.1611 -  moreover have "T \<ge> (\<lambda>x. a + x) ` S"
2.1612 -    using assms by auto
2.1613 -  ultimately show ?thesis by auto
2.1614 -qed
2.1615 -
2.1616 -lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
2.1617 -  unfolding affine_parallel_def
2.1618 -  using affine_parallel_expl_aux[of S _ T] by auto
2.1619 -
2.1620 -lemma affine_parallel_reflex: "affine_parallel S S"
2.1621 -  unfolding affine_parallel_def
2.1622 -  using image_add_0 by blast
2.1623 -
2.1624 -lemma affine_parallel_commut:
2.1625 -  assumes "affine_parallel A B"
2.1626 -  shows "affine_parallel B A"
2.1627 -proof -
2.1628 -  from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
2.1629 -    unfolding affine_parallel_def by auto
2.1630 -  have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
2.1631 -  from B show ?thesis
2.1632 -    using translation_galois [of B a A]
2.1633 -    unfolding affine_parallel_def by auto
2.1634 -qed
2.1635 -
2.1636 -lemma affine_parallel_assoc:
2.1637 -  assumes "affine_parallel A B"
2.1638 -    and "affine_parallel B C"
2.1639 -  shows "affine_parallel A C"
2.1640 -proof -
2.1641 -  from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
2.1642 -    unfolding affine_parallel_def by auto
2.1643 -  moreover
2.1644 -  from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
2.1645 -    unfolding affine_parallel_def by auto
2.1646 -  ultimately show ?thesis
2.1647 -    using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
2.1648 -qed
2.1649 -
2.1650 -lemma affine_translation_aux:
2.1651 -  fixes a :: "'a::real_vector"
2.1652 -  assumes "affine ((\<lambda>x. a + x) ` S)"
2.1653 -  shows "affine S"
2.1654 -proof -
2.1655 -  {
2.1656 -    fix x y u v
2.1657 -    assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
2.1658 -    then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
2.1659 -      by auto
2.1660 -    then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
2.1661 -      using xy assms unfolding affine_def by auto
2.1662 -    have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
2.1663 -      by (simp add: algebra_simps)
2.1664 -    also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
2.1665 -      using \<open>u + v = 1\<close> by auto
2.1666 -    ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
2.1667 -      using h1 by auto
2.1668 -    then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
2.1669 -  }
2.1670 -  then show ?thesis unfolding affine_def by auto
2.1671 -qed
2.1672 -
2.1673 -lemma affine_translation:
2.1674 -  fixes a :: "'a::real_vector"
2.1675 -  shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
2.1676 -proof -
2.1677 -  have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
2.1678 -    using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
2.1679 -    using translation_assoc[of "-a" a S] by auto
2.1680 -  then show ?thesis using affine_translation_aux by auto
2.1681 -qed
2.1682 -
2.1683 -lemma parallel_is_affine:
2.1684 -  fixes S T :: "'a::real_vector set"
2.1685 -  assumes "affine S" "affine_parallel S T"
2.1686 -  shows "affine T"
2.1687 -proof -
2.1688 -  from assms obtain a where "T = (\<lambda>x. a + x) ` S"
2.1689 -    unfolding affine_parallel_def by auto
2.1690 -  then show ?thesis
2.1691 -    using affine_translation assms by auto
2.1692 -qed
2.1693 -
2.1694 -lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
2.1695 -  unfolding subspace_def affine_def by auto
2.1696 -
2.1697 -
2.1698 -subsubsection%unimportant \<open>Subspace parallel to an affine set\<close>
2.1699 -
2.1700 -lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
2.1701 -proof -
2.1702 -  have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
2.1703 -    using subspace_imp_affine[of S] subspace_0 by auto
2.1704 -  {
2.1705 -    assume assm: "affine S \<and> 0 \<in> S"
2.1706 -    {
2.1707 -      fix c :: real
2.1708 -      fix x
2.1709 -      assume x: "x \<in> S"
2.1710 -      have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
2.1711 -      moreover
2.1712 -      have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
2.1713 -        using affine_alt[of S] assm x by auto
2.1714 -      ultimately have "c *\<^sub>R x \<in> S" by auto
2.1715 -    }
2.1716 -    then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
2.1717 -
2.1718 -    {
2.1719 -      fix x y
2.1720 -      assume xy: "x \<in> S" "y \<in> S"
2.1721 -      define u where "u = (1 :: real)/2"
2.1722 -      have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
2.1723 -        by auto
2.1724 -      moreover
2.1725 -      have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
2.1726 -        by (simp add: algebra_simps)
2.1727 -      moreover
2.1728 -      have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
2.1729 -        using affine_alt[of S] assm xy by auto
2.1730 -      ultimately
2.1731 -      have "(1/2) *\<^sub>R (x+y) \<in> S"
2.1732 -        using u_def by auto
2.1733 -      moreover
2.1734 -      have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
2.1735 -        by auto
2.1736 -      ultimately
2.1737 -      have "x + y \<in> S"
2.1738 -        using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
2.1739 -    }
2.1740 -    then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
2.1741 -      by auto
2.1742 -    then have "subspace S"
2.1743 -      using h1 assm unfolding subspace_def by auto
2.1744 -  }
2.1745 -  then show ?thesis using h0 by metis
2.1746 -qed
2.1747 -
2.1748 -lemma affine_diffs_subspace:
2.1749 -  assumes "affine S" "a \<in> S"
2.1750 -  shows "subspace ((\<lambda>x. (-a)+x) ` S)"
2.1751 -proof -
2.1752 -  have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
2.1753 -  have "affine ((\<lambda>x. (-a)+x) ` S)"
2.1754 -    using  affine_translation assms by auto
2.1755 -  moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
2.1756 -    using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
2.1757 -  ultimately show ?thesis using subspace_affine by auto
2.1758 -qed
2.1759 -
2.1760 -lemma parallel_subspace_explicit:
2.1761 -  assumes "affine S"
2.1762 -    and "a \<in> S"
2.1763 -  assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
2.1764 -  shows "subspace L \<and> affine_parallel S L"
2.1765 -proof -
2.1766 -  from assms have "L = plus (- a) ` S" by auto
2.1767 -  then have par: "affine_parallel S L"
2.1768 -    unfolding affine_parallel_def ..
2.1769 -  then have "affine L" using assms parallel_is_affine by auto
2.1770 -  moreover have "0 \<in> L"
2.1771 -    using assms by auto
2.1772 -  ultimately show ?thesis
2.1773 -    using subspace_affine par by auto
2.1774 -qed
2.1775 -
2.1776 -lemma parallel_subspace_aux:
2.1777 -  assumes "subspace A"
2.1778 -    and "subspace B"
2.1779 -    and "affine_parallel A B"
2.1780 -  shows "A \<supseteq> B"
2.1781 -proof -
2.1782 -  from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
2.1783 -    using affine_parallel_expl[of A B] by auto
2.1784 -  then have "-a \<in> A"
2.1785 -    using assms subspace_0[of B] by auto
2.1786 -  then have "a \<in> A"
2.1787 -    using assms subspace_neg[of A "-a"] by auto
2.1788 -  then show ?thesis
2.1789 -    using assms a unfolding subspace_def by auto
2.1790 -qed
2.1791 -
2.1792 -lemma parallel_subspace:
2.1793 -  assumes "subspace A"
2.1794 -    and "subspace B"
2.1795 -    and "affine_parallel A B"
2.1796 -  shows "A = B"
2.1797 -proof
2.1798 -  show "A \<supseteq> B"
2.1799 -    using assms parallel_subspace_aux by auto
2.1800 -  show "A \<subseteq> B"
2.1801 -    using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
2.1802 -qed
2.1803 -
2.1804 -lemma affine_parallel_subspace:
2.1805 -  assumes "affine S" "S \<noteq> {}"
2.1806 -  shows "\<exists>!L. subspace L \<and> affine_parallel S L"
2.1807 -proof -
2.1808 -  have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
2.1809 -    using assms parallel_subspace_explicit by auto
2.1810 -  {
2.1811 -    fix L1 L2
2.1812 -    assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
2.1813 -    then have "affine_parallel L1 L2"
2.1814 -      using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
2.1815 -    then have "L1 = L2"
2.1816 -      using ass parallel_subspace by auto
2.1817 -  }
2.1818 -  then show ?thesis using ex by auto
2.1819 -qed
2.1820 -
2.1821 -
2.1822 -subsection \<open>Cones\<close>
2.1823 -
2.1824 -definition%important cone :: "'a::real_vector set \<Rightarrow> bool"
2.1825 -  where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
2.1826 -
2.1827 -lemma cone_empty[intro, simp]: "cone {}"
2.1828 -  unfolding cone_def by auto
2.1829 -
2.1830 -lemma cone_univ[intro, simp]: "cone UNIV"
2.1831 -  unfolding cone_def by auto
2.1832 -
2.1833 -lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
2.1834 -  unfolding cone_def by auto
2.1835 -
2.1836 -lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
2.1837 -  by (simp add: cone_def subspace_scale)
2.1838 -
2.1839 -
2.1840 -subsubsection \<open>Conic hull\<close>
2.1841 -
2.1842 -lemma cone_cone_hull: "cone (cone hull s)"
2.1843 -  unfolding hull_def by auto
2.1844 -
2.1845 -lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
2.1846 -  apply (rule hull_eq)
2.1847 -  using cone_Inter
2.1848 -  unfolding subset_eq
2.1849 -  apply auto
2.1850 -  done
2.1851 -
2.1852 -lemma mem_cone:
2.1853 -  assumes "cone S" "x \<in> S" "c \<ge> 0"
2.1854 -  shows "c *\<^sub>R x \<in> S"
2.1855 -  using assms cone_def[of S] by auto
2.1856 -
2.1857 -lemma cone_contains_0:
2.1858 -  assumes "cone S"
2.1859 -  shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
2.1860 -proof -
2.1861 -  {
2.1862 -    assume "S \<noteq> {}"
2.1863 -    then obtain a where "a \<in> S" by auto
2.1864 -    then have "0 \<in> S"
2.1865 -      using assms mem_cone[of S a 0] by auto
2.1866 -  }
2.1867 -  then show ?thesis by auto
2.1868 -qed
2.1869 -
2.1870 -lemma cone_0: "cone {0}"
2.1871 -  unfolding cone_def by auto
2.1872 -
2.1873 -lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
2.1874 -  unfolding cone_def by blast
2.1875 -
2.1876 -lemma cone_iff:
2.1877 -  assumes "S \<noteq> {}"
2.1878 -  shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
2.1879 -proof -
2.1880 -  {
2.1881 -    assume "cone S"
2.1882 -    {
2.1883 -      fix c :: real
2.1884 -      assume "c > 0"
2.1885 -      {
2.1886 -        fix x
2.1887 -        assume "x \<in> S"
2.1888 -        then have "x \<in> ((*\<^sub>R) c) ` S"
2.1889 -          unfolding image_def
2.1890 -          using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
2.1891 -            exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
2.1892 -          by auto
2.1893 -      }
2.1894 -      moreover
2.1895 -      {
2.1896 -        fix x
2.1897 -        assume "x \<in> ((*\<^sub>R) c) ` S"
2.1898 -        then have "x \<in> S"
2.1899 -          using \<open>cone S\<close> \<open>c > 0\<close>
2.1900 -          unfolding cone_def image_def \<open>c > 0\<close> by auto
2.1901 -      }
2.1902 -      ultimately have "((*\<^sub>R) c) ` S = S" by auto
2.1903 -    }
2.1904 -    then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
2.1905 -      using \<open>cone S\<close> cone_contains_0[of S] assms by auto
2.1906 -  }
2.1907 -  moreover
2.1908 -  {
2.1909 -    assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
2.1910 -    {
2.1911 -      fix x
2.1912 -      assume "x \<in> S"
2.1913 -      fix c1 :: real
2.1914 -      assume "c1 \<ge> 0"
2.1915 -      then have "c1 = 0 \<or> c1 > 0" by auto
2.1916 -      then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
2.1917 -    }
2.1918 -    then have "cone S" unfolding cone_def by auto
2.1919 -  }
2.1920 -  ultimately show ?thesis by blast
2.1921 -qed
2.1922 -
2.1923 -lemma cone_hull_empty: "cone hull {} = {}"
2.1924 -  by (metis cone_empty cone_hull_eq)
2.1925 -
2.1926 -lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
2.1927 -  by (metis bot_least cone_hull_empty hull_subset xtrans(5))
2.1928 -
2.1929 -lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
2.1930 -  using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
2.1931 -  by auto
2.1932 -
2.1933 -lemma mem_cone_hull:
2.1934 -  assumes "x \<in> S" "c \<ge> 0"
2.1935 -  shows "c *\<^sub>R x \<in> cone hull S"
2.1936 -  by (metis assms cone_cone_hull hull_inc mem_cone)
2.1937 -
2.1938 -proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
2.1939 -  (is "?lhs = ?rhs")
2.1940 -proof -
2.1941 -  {
2.1942 -    fix x
2.1943 -    assume "x \<in> ?rhs"
2.1944 -    then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
2.1945 -      by auto
2.1946 -    fix c :: real
2.1947 -    assume c: "c \<ge> 0"
2.1948 -    then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
2.1949 -      using x by (simp add: algebra_simps)
2.1950 -    moreover
2.1951 -    have "c * cx \<ge> 0" using c x by auto
2.1952 -    ultimately
2.1953 -    have "c *\<^sub>R x \<in> ?rhs" using x by auto
2.1954 -  }
2.1955 -  then have "cone ?rhs"
2.1956 -    unfolding cone_def by auto
2.1957 -  then have "?rhs \<in> Collect cone"
2.1958 -    unfolding mem_Collect_eq by auto
2.1959 -  {
2.1960 -    fix x
2.1961 -    assume "x \<in> S"
2.1962 -    then have "1 *\<^sub>R x \<in> ?rhs"
2.1963 -      apply auto
2.1964 -      apply (rule_tac x = 1 in exI, auto)
2.1965 -      done
2.1966 -    then have "x \<in> ?rhs" by auto
2.1967 -  }
2.1968 -  then have "S \<subseteq> ?rhs" by auto
2.1969 -  then have "?lhs \<subseteq> ?rhs"
2.1970 -    using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
2.1971 -  moreover
2.1972 -  {
2.1973 -    fix x
2.1974 -    assume "x \<in> ?rhs"
2.1975 -    then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
2.1976 -      by auto
2.1977 -    then have "xx \<in> cone hull S"
2.1978 -      using hull_subset[of S] by auto
2.1979 -    then have "x \<in> ?lhs"
2.1980 -      using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
2.1981 -  }
2.1982 -  ultimately show ?thesis by auto
2.1983 -qed
2.1984 -
2.1985  lemma cone_closure:
2.1986    fixes S :: "'a::real_normed_vector set"
2.1987    assumes "cone S"
2.1988 @@ -2071,122 +124,6 @@
2.1989      using False cone_iff[of "closure S"] by auto
2.1990  qed
2.1991
2.1992 -
2.1993 -subsection \<open>Affine dependence and consequential theorems\<close>
2.1994 -
2.1995 -text "Formalized by Lars Schewe."
2.1996 -
2.1997 -definition%important affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
2.1998 -  where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
2.1999 -
2.2000 -lemma affine_dependent_subset:
2.2001 -   "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
2.2002 -apply (simp add: affine_dependent_def Bex_def)
2.2003 -apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
2.2004 -done
2.2005 -
2.2006 -lemma affine_independent_subset:
2.2007 -  shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
2.2008 -by (metis affine_dependent_subset)
2.2009 -
2.2010 -lemma affine_independent_Diff:
2.2011 -   "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
2.2012 -by (meson Diff_subset affine_dependent_subset)
2.2013 -
2.2014 -proposition affine_dependent_explicit:
2.2015 -  "affine_dependent p \<longleftrightarrow>
2.2016 -    (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
2.2017 -proof -
2.2018 -  have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
2.2019 -    if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
2.2020 -  proof (intro exI conjI)
2.2021 -    have "x \<notin> S"
2.2022 -      using that by auto
2.2023 -    then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
2.2024 -      using that by (simp add: sum_delta_notmem)
2.2025 -    show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
2.2026 -      using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
2.2027 -  qed (use that in auto)
2.2028 -  moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
2.2029 -    if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
2.2030 -  proof (intro bexI exI conjI)
2.2031 -    have "S \<noteq> {v}"
2.2032 -      using that by auto
2.2033 -    then show "S - {v} \<noteq> {}"
2.2034 -      using that by auto
2.2035 -    show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
2.2036 -      unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
2.2037 -    show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
2.2038 -      unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
2.2039 -                scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
2.2040 -      using that by auto
2.2041 -    show "S - {v} \<subseteq> p - {v}"
2.2042 -      using that by auto
2.2043 -  qed (use that in auto)
2.2044 -  ultimately show ?thesis
2.2045 -    unfolding affine_dependent_def affine_hull_explicit by auto
2.2046 -qed
2.2047 -
2.2048 -lemma affine_dependent_explicit_finite:
2.2049 -  fixes S :: "'a::real_vector set"
2.2050 -  assumes "finite S"
2.2051 -  shows "affine_dependent S \<longleftrightarrow>
2.2052 -    (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
2.2053 -  (is "?lhs = ?rhs")
2.2054 -proof
2.2055 -  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)"
2.2056 -    by auto
2.2057 -  assume ?lhs
2.2058 -  then obtain t u v where
2.2059 -    "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
2.2060 -    unfolding affine_dependent_explicit by auto
2.2061 -  then show ?rhs
2.2062 -    apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
2.2063 -    apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
2.2064 -    done
2.2065 -next
2.2066 -  assume ?rhs
2.2067 -  then obtain u v where "sum u S = 0"  "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
2.2068 -    by auto
2.2069 -  then show ?lhs unfolding affine_dependent_explicit
2.2070 -    using assms by auto
2.2071 -qed
2.2072 -
2.2073 -
2.2074 -subsection%unimportant \<open>Connectedness of convex sets\<close>
2.2075 -
2.2076 -lemma connectedD:
2.2077 -  "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
2.2078 -  by (rule Topological_Spaces.topological_space_class.connectedD)
2.2079 -
2.2080 -lemma convex_connected:
2.2081 -  fixes S :: "'a::real_normed_vector set"
2.2082 -  assumes "convex S"
2.2083 -  shows "connected S"
2.2084 -proof (rule connectedI)
2.2085 -  fix A B
2.2086 -  assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
2.2087 -  moreover
2.2088 -  assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
2.2089 -  then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
2.2090 -  define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
2.2091 -  then have "continuous_on {0 .. 1} f"
2.2092 -    by (auto intro!: continuous_intros)
2.2093 -  then have "connected (f ` {0 .. 1})"
2.2094 -    by (auto intro!: connected_continuous_image)
2.2095 -  note connectedD[OF this, of A B]
2.2096 -  moreover have "a \<in> A \<inter> f ` {0 .. 1}"
2.2097 -    using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
2.2098 -  moreover have "b \<in> B \<inter> f ` {0 .. 1}"
2.2099 -    using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
2.2100 -  moreover have "f ` {0 .. 1} \<subseteq> S"
2.2101 -    using \<open>convex S\<close> a b unfolding convex_def f_def by auto
2.2102 -  ultimately show False by auto
2.2103 -qed
2.2104 -
2.2105 -corollary connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
2.2106 -  by (simp add: convex_connected)
2.2107 -
2.2108  corollary component_complement_connected:
2.2109    fixes S :: "'a::real_normed_vector set"
2.2110    assumes "connected S" "C \<in> components (-S)"
2.2111 @@ -2216,15 +153,6 @@
2.2112
2.2113  text \<open>Balls, being convex, are connected.\<close>
2.2114
2.2115 -lemma convex_prod:
2.2116 -  assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
2.2117 -  shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
2.2118 -  using assms unfolding convex_def
2.2119 -  by (auto simp: inner_add_left)
2.2120 -
2.2121 -lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
2.2122 -  by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
2.2123 -
2.2124  lemma convex_local_global_minimum:
2.2125    fixes s :: "'a::real_normed_vector set"
2.2126    assumes "e > 0"
2.2127 @@ -2310,20 +238,6 @@
2.2128    using convex_connected convex_cball by auto
2.2129
2.2130
2.2131 -subsection \<open>Convex hull\<close>
2.2132 -
2.2133 -lemma convex_convex_hull [iff]: "convex (convex hull s)"
2.2134 -  unfolding hull_def
2.2135 -  using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
2.2136 -  by auto
2.2137 -
2.2138 -lemma convex_hull_subset:
2.2139 -    "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
2.2140 -  by (simp add: convex_convex_hull subset_hull)
2.2141 -
2.2142 -lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
2.2143 -  by (metis convex_convex_hull hull_same)
2.2144 -
2.2145  lemma bounded_convex_hull:
2.2146    fixes s :: "'a::real_normed_vector set"
2.2147    assumes "bounded s"
2.2148 @@ -2345,1499 +259,6 @@
2.2149    using bounded_convex_hull finite_imp_bounded
2.2150    by auto
2.2151
2.2152 -
2.2153 -subsubsection%unimportant \<open>Convex hull is "preserved" by a linear function\<close>
2.2154 -
2.2155 -lemma convex_hull_linear_image:
2.2156 -  assumes f: "linear f"
2.2157 -  shows "f ` (convex hull s) = convex hull (f ` s)"
2.2158 -proof
2.2159 -  show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
2.2160 -    by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
2.2161 -  show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
2.2162 -  proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
2.2163 -    show "s \<subseteq> f -` (convex hull (f ` s))"
2.2164 -      by (fast intro: hull_inc)
2.2165 -    show "convex (f -` (convex hull (f ` s)))"
2.2166 -      by (intro convex_linear_vimage [OF f] convex_convex_hull)
2.2167 -  qed
2.2168 -qed
2.2169 -
2.2170 -lemma in_convex_hull_linear_image:
2.2171 -  assumes "linear f"
2.2172 -    and "x \<in> convex hull s"
2.2173 -  shows "f x \<in> convex hull (f ` s)"
2.2174 -  using convex_hull_linear_image[OF assms(1)] assms(2) by auto
2.2175 -
2.2176 -lemma convex_hull_Times:
2.2177 -  "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
2.2178 -proof
2.2179 -  show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
2.2180 -    by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
2.2181 -  have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
2.2182 -  proof (rule hull_induct [OF x], rule hull_induct [OF y])
2.2183 -    fix x y assume "x \<in> s" and "y \<in> t"
2.2184 -    then show "(x, y) \<in> convex hull (s \<times> t)"
2.2185 -      by (simp add: hull_inc)
2.2186 -  next
2.2187 -    fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
2.2188 -    have "convex ?S"
2.2189 -      by (intro convex_linear_vimage convex_translation convex_convex_hull,
2.2191 -    also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
2.2192 -      by (auto simp: image_def Bex_def)
2.2193 -    finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
2.2194 -  next
2.2195 -    show "convex {x. (x, y) \<in> convex hull s \<times> t}"
2.2196 -    proof -
2.2197 -      fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
2.2198 -      have "convex ?S"
2.2199 -      by (intro convex_linear_vimage convex_translation convex_convex_hull,
2.2201 -      also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
2.2202 -        by (auto simp: image_def Bex_def)
2.2203 -      finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
2.2204 -    qed
2.2205 -  qed
2.2206 -  then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
2.2207 -    unfolding subset_eq split_paired_Ball_Sigma by blast
2.2208 -qed
2.2209 -
2.2210 -
2.2211 -subsubsection%unimportant \<open>Stepping theorems for convex hulls of finite sets\<close>
2.2212 -
2.2213 -lemma convex_hull_empty[simp]: "convex hull {} = {}"
2.2214 -  by (rule hull_unique) auto
2.2215 -
2.2216 -lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
2.2217 -  by (rule hull_unique) auto
2.2218 -
2.2219 -lemma convex_hull_insert:
2.2220 -  fixes S :: "'a::real_vector set"
2.2221 -  assumes "S \<noteq> {}"
2.2222 -  shows "convex hull (insert a S) =
2.2223 -         {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
2.2224 -  (is "_ = ?hull")
2.2225 -proof (intro equalityI hull_minimal subsetI)
2.2226 -  fix x
2.2227 -  assume "x \<in> insert a S"
2.2228 -  then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
2.2229 -  unfolding insert_iff
2.2230 -  proof
2.2231 -    assume "x = a"
2.2232 -    then show ?thesis
2.2233 -      by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
2.2234 -  next
2.2235 -    assume "x \<in> S"
2.2236 -    with hull_subset[of S convex] show ?thesis
2.2237 -      by force
2.2238 -  qed
2.2239 -  then show "x \<in> ?hull"
2.2240 -    by simp
2.2241 -next
2.2242 -  fix x
2.2243 -  assume "x \<in> ?hull"
2.2244 -  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"
2.2245 -    by auto
2.2246 -  have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
2.2247 -    using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
2.2248 -    by auto
2.2249 -  then show "x \<in> convex hull insert a S"
2.2250 -    unfolding obt(5) using obt(1-3)
2.2251 -    by (rule convexD [OF convex_convex_hull])
2.2252 -next
2.2253 -  show "convex ?hull"
2.2254 -  proof (rule convexI)
2.2255 -    fix x y u v
2.2256 -    assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
2.2257 -    from x obtain u1 v1 b1 where
2.2258 -      obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
2.2259 -      by auto
2.2260 -    from y obtain u2 v2 b2 where
2.2261 -      obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
2.2262 -      by auto
2.2263 -    have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
2.2264 -      by (auto simp: algebra_simps)
2.2265 -    have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
2.2266 -      (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
2.2267 -    proof (cases "u * v1 + v * v2 = 0")
2.2268 -      case True
2.2269 -      have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
2.2270 -        by (auto simp: algebra_simps)
2.2271 -      have eq0: "u * v1 = 0" "v * v2 = 0"
2.2272 -        using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
2.2273 -        by arith+
2.2274 -      then have "u * u1 + v * u2 = 1"
2.2275 -        using as(3) obt1(3) obt2(3) by auto
2.2276 -      then show ?thesis
2.2277 -        using "*" eq0 as obt1(4) xeq yeq by auto
2.2278 -    next
2.2279 -      case False
2.2280 -      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
2.2281 -        using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
2.2282 -      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
2.2283 -        using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
2.2284 -      also have "\<dots> = u * v1 + v * v2"
2.2285 -        by simp
2.2286 -      finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
2.2287 -      let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
2.2288 -      have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
2.2289 -        using as(1,2) obt1(1,2) obt2(1,2) by auto
2.2290 -      show ?thesis
2.2291 -      proof
2.2292 -        show "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)"
2.2293 -          unfolding xeq yeq * **
2.2294 -          using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
2.2295 -        show "?b \<in> convex hull S"
2.2296 -          using False zeroes obt1(4) obt2(4)
2.2297 -          by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib  add_divide_distrib[symmetric]  zero_le_divide_iff)
2.2298 -      qed
2.2299 -    qed
2.2300 -    then obtain b where b: "b \<in> convex hull S"
2.2301 -       "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)" ..
2.2302 -
2.2303 -    have u1: "u1 \<le> 1"
2.2304 -      unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
2.2305 -    have u2: "u2 \<le> 1"
2.2306 -      unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
2.2307 -    have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
2.2309 -      show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
2.2310 -        by (simp_all add: as mult_right_mono)
2.2311 -    qed
2.2312 -    also have "\<dots> \<le> 1"
2.2313 -      unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
2.2314 -    finally have le1: "u1 * u + u2 * v \<le> 1" .
2.2315 -    show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
2.2316 -    proof (intro CollectI exI conjI)
2.2317 -      show "0 \<le> u * u1 + v * u2"
2.2318 -        by (simp add: as(1) as(2) obt1(1) obt2(1))
2.2319 -      show "0 \<le> 1 - u * u1 - v * u2"
2.2321 -    qed (use b in \<open>auto simp: algebra_simps\<close>)
2.2322 -  qed
2.2323 -qed
2.2324 -
2.2325 -lemma convex_hull_insert_alt:
2.2326 -   "convex hull (insert a S) =
2.2327 -     (if S = {} then {a}
2.2328 -      else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
2.2329 -  apply (auto simp: convex_hull_insert)
2.2330 -  using diff_eq_eq apply fastforce
2.2332 -
2.2333 -subsubsection%unimportant \<open>Explicit expression for convex hull\<close>
2.2334 -
2.2335 -proposition convex_hull_indexed:
2.2336 -  fixes S :: "'a::real_vector set"
2.2337 -  shows "convex hull S =
2.2338 -    {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
2.2339 -                (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
2.2340 -    (is "?xyz = ?hull")
2.2341 -proof (rule hull_unique [OF _ convexI])
2.2342 -  show "S \<subseteq> ?hull"
2.2343 -    by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
2.2344 -next
2.2345 -  fix T
2.2346 -  assume "S \<subseteq> T" "convex T"
2.2347 -  then show "?hull \<subseteq> T"
2.2348 -    by (blast intro: convex_sum)
2.2349 -next
2.2350 -  fix x y u v
2.2351 -  assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
2.2352 -  assume xy: "x \<in> ?hull" "y \<in> ?hull"
2.2353 -  from xy obtain k1 u1 x1 where
2.2354 -    x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S"
2.2355 -                      "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
2.2356 -    by auto
2.2357 -  from xy obtain k2 u2 x2 where
2.2358 -    y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S"
2.2359 -                     "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
2.2360 -    by auto
2.2361 -  have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
2.2362 -          "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
2.2363 -    by auto
2.2364 -  have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
2.2365 -    unfolding inj_on_def by auto
2.2366 -  let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
2.2367 -  let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
2.2368 -  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
2.2369 -  proof (intro CollectI exI conjI ballI)
2.2370 -    show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
2.2371 -      using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
2.2372 -    show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1"  "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
2.2373 -      unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
2.2374 -        sum.reindex[OF inj] Collect_mem_eq o_def
2.2375 -      unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
2.2376 -      by (simp_all add: sum_distrib_left[symmetric]  x(2,3) y(2,3) uv(3))
2.2377 -  qed
2.2378 -qed
2.2379 -
2.2380 -lemma convex_hull_finite:
2.2381 -  fixes S :: "'a::real_vector set"
2.2382 -  assumes "finite S"
2.2383 -  shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
2.2384 -  (is "?HULL = _")
2.2385 -proof (rule hull_unique [OF _ convexI]; clarify)
2.2386 -  fix x
2.2387 -  assume "x \<in> S"
2.2388 -  then show "\<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = x"
2.2389 -    by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
2.2390 -next
2.2391 -  fix u v :: real
2.2392 -  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
2.2393 -  fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
2.2394 -  fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
2.2395 -  have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
2.2396 -    by (simp add: that uv ux(1) uy(1))
2.2397 -  moreover
2.2398 -  have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
2.2399 -    unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
2.2400 -    using uv(3) by auto
2.2401 -  moreover
2.2402 -  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)"
2.2403 -    unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
2.2404 -    by auto
2.2405 -  ultimately
2.2406 -  show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
2.2407 -             (\<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)"
2.2408 -    by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
2.2409 -qed (use assms in \<open>auto simp: convex_explicit\<close>)
2.2410 -
2.2411 -
2.2412 -subsubsection%unimportant \<open>Another formulation\<close>
2.2413 -
2.2414 -text "Formalized by Lars Schewe."
2.2415 -
2.2416 -lemma convex_hull_explicit:
2.2417 -  fixes p :: "'a::real_vector set"
2.2418 -  shows "convex hull p =
2.2419 -    {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
2.2420 -  (is "?lhs = ?rhs")
2.2421 -proof -
2.2422 -  {
2.2423 -    fix x
2.2424 -    assume "x\<in>?lhs"
2.2425 -    then obtain k u y where
2.2426 -        obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
2.2427 -      unfolding convex_hull_indexed by auto
2.2428 -
2.2429 -    have fin: "finite {1..k}" by auto
2.2430 -    have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
2.2431 -    {
2.2432 -      fix j
2.2433 -      assume "j\<in>{1..k}"
2.2434 -      then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
2.2435 -        using obt(1)[THEN bspec[where x=j]] and obt(2)
2.2436 -        apply simp
2.2437 -        apply (rule sum_nonneg)
2.2438 -        using obt(1)
2.2439 -        apply auto
2.2440 -        done
2.2441 -    }
2.2442 -    moreover
2.2443 -    have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
2.2444 -      unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
2.2445 -    moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
2.2446 -      using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
2.2447 -      unfolding scaleR_left.sum using obt(3) by auto
2.2448 -    ultimately
2.2449 -    have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
2.2450 -      apply (rule_tac x="y ` {1..k}" in exI)
2.2451 -      apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
2.2452 -      done
2.2453 -    then have "x\<in>?rhs" by auto
2.2454 -  }
2.2455 -  moreover
2.2456 -  {
2.2457 -    fix y
2.2458 -    assume "y\<in>?rhs"
2.2459 -    then obtain S u where
2.2460 -      obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
2.2461 -      by auto
2.2462 -
2.2463 -    obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
2.2464 -      using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
2.2465 -
2.2466 -    {
2.2467 -      fix i :: nat
2.2468 -      assume "i\<in>{1..card S}"
2.2469 -      then have "f i \<in> S"
2.2470 -        using f(2) by blast
2.2471 -      then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
2.2472 -    }
2.2473 -    moreover have *: "finite {1..card S}" by auto
2.2474 -    {
2.2475 -      fix y
2.2476 -      assume "y\<in>S"
2.2477 -      then obtain i where "i\<in>{1..card S}" "f i = y"
2.2478 -        using f using image_iff[of y f "{1..card S}"]
2.2479 -        by auto
2.2480 -      then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
2.2481 -        apply auto
2.2482 -        using f(1)[unfolded inj_on_def]
2.2483 -        by (metis One_nat_def atLeastAtMost_iff)
2.2484 -      then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
2.2485 -      then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
2.2486 -          "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
2.2487 -        by (auto simp: sum_constant_scaleR)
2.2488 -    }
2.2489 -    then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
2.2490 -      unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
2.2491 -        and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
2.2492 -      unfolding f
2.2493 -      using sum.cong [of S 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"]
2.2494 -      using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
2.2495 -      unfolding obt(4,5)
2.2496 -      by auto
2.2497 -    ultimately
2.2498 -    have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
2.2499 -        (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
2.2500 -      apply (rule_tac x="card S" in exI)
2.2501 -      apply (rule_tac x="u \<circ> f" in exI)
2.2502 -      apply (rule_tac x=f in exI, fastforce)
2.2503 -      done
2.2504 -    then have "y \<in> ?lhs"
2.2505 -      unfolding convex_hull_indexed by auto
2.2506 -  }
2.2507 -  ultimately show ?thesis
2.2508 -    unfolding set_eq_iff by blast
2.2509 -qed
2.2510 -
2.2511 -
2.2512 -subsubsection%unimportant \<open>A stepping theorem for that expansion\<close>
2.2513 -
2.2514 -lemma convex_hull_finite_step:
2.2515 -  fixes S :: "'a::real_vector set"
2.2516 -  assumes "finite S"
2.2517 -  shows
2.2518 -    "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
2.2519 -      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
2.2520 -  (is "?lhs = ?rhs")
2.2521 -proof (rule, case_tac[!] "a\<in>S")
2.2522 -  assume "a \<in> S"
2.2523 -  then have *: "insert a S = S" by auto
2.2524 -  assume ?lhs
2.2525 -  then show ?rhs
2.2526 -    unfolding *  by (rule_tac x=0 in exI, auto)
2.2527 -next
2.2528 -  assume ?lhs
2.2529 -  then obtain u where
2.2530 -      u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
2.2531 -    by auto
2.2532 -  assume "a \<notin> S"
2.2533 -  then show ?rhs
2.2534 -    apply (rule_tac x="u a" in exI)
2.2535 -    using u(1)[THEN bspec[where x=a]]
2.2536 -    apply simp
2.2537 -    apply (rule_tac x=u in exI)
2.2538 -    using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
2.2539 -    apply auto
2.2540 -    done
2.2541 -next
2.2542 -  assume "a \<in> S"
2.2543 -  then have *: "insert a S = S" by auto
2.2544 -  have fin: "finite (insert a S)" using assms by auto
2.2545 -  assume ?rhs
2.2546 -  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
2.2547 -    by auto
2.2548 -  show ?lhs
2.2549 -    apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
2.2550 -    unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
2.2551 -    unfolding sum_clauses(2)[OF assms]
2.2552 -    using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
2.2553 -    apply auto
2.2554 -    done
2.2555 -next
2.2556 -  assume ?rhs
2.2557 -  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
2.2558 -    by auto
2.2559 -  moreover assume "a \<notin> S"
2.2560 -  moreover
2.2561 -  have "(\<Sum>x\<in>S. if a = x then v else u x) = sum 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)"
2.2562 -    using \<open>a \<notin> S\<close>
2.2563 -    by (auto simp: intro!: sum.cong)
2.2564 -  ultimately show ?lhs
2.2565 -    by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
2.2566 -qed
2.2567 -
2.2568 -
2.2569 -subsubsection%unimportant \<open>Hence some special cases\<close>
2.2570 -
2.2571 -lemma convex_hull_2:
2.2572 -  "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}"
2.2573 -proof -
2.2574 -  have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
2.2575 -    by auto
2.2576 -  have **: "finite {b}" by auto
2.2577 -  show ?thesis
2.2578 -    apply (simp add: convex_hull_finite)
2.2579 -    unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
2.2580 -    apply auto
2.2581 -    apply (rule_tac x=v in exI)
2.2582 -    apply (rule_tac x="1 - v" in exI, simp)
2.2583 -    apply (rule_tac x=u in exI, simp)
2.2584 -    apply (rule_tac x="\<lambda>x. v" in exI, simp)
2.2585 -    done
2.2586 -qed
2.2587 -
2.2588 -lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
2.2589 -  unfolding convex_hull_2
2.2590 -proof (rule Collect_cong)
2.2591 -  have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
2.2592 -    by auto
2.2593 -  fix x
2.2594 -  show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
2.2595 -    (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
2.2596 -    unfolding *
2.2597 -    apply auto
2.2598 -    apply (rule_tac[!] x=u in exI)
2.2599 -    apply (auto simp: algebra_simps)
2.2600 -    done
2.2601 -qed
2.2602 -
2.2603 -lemma convex_hull_3:
2.2604 -  "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}"
2.2605 -proof -
2.2606 -  have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
2.2607 -    by auto
2.2608 -  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
2.2609 -    by (auto simp: field_simps)
2.2610 -  show ?thesis
2.2611 -    unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
2.2612 -    unfolding convex_hull_finite_step[OF fin(3)]
2.2613 -    apply (rule Collect_cong, simp)
2.2614 -    apply auto
2.2615 -    apply (rule_tac x=va in exI)
2.2616 -    apply (rule_tac x="u c" in exI, simp)
2.2617 -    apply (rule_tac x="1 - v - w" in exI, simp)
2.2618 -    apply (rule_tac x=v in exI, simp)
2.2619 -    apply (rule_tac x="\<lambda>x. w" in exI, simp)
2.2620 -    done
2.2621 -qed
2.2622 -
2.2623 -lemma convex_hull_3_alt:
2.2624 -  "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}"
2.2625 -proof -
2.2626 -  have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
2.2627 -    by auto
2.2628 -  show ?thesis
2.2629 -    unfolding convex_hull_3
2.2630 -    apply (auto simp: *)
2.2631 -    apply (rule_tac x=v in exI)
2.2632 -    apply (rule_tac x=w in exI)
2.2633 -    apply (simp add: algebra_simps)
2.2634 -    apply (rule_tac x=u in exI)
2.2635 -    apply (rule_tac x=v in exI)
2.2636 -    apply (simp add: algebra_simps)
2.2637 -    done
2.2638 -qed
2.2639 -
2.2640 -
2.2641 -subsection%unimportant \<open>Relations among closure notions and corresponding hulls\<close>
2.2642 -
2.2643 -lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
2.2644 -  unfolding affine_def convex_def by auto
2.2645 -
2.2646 -lemma convex_affine_hull [simp]: "convex (affine hull S)"
2.2647 -  by (simp add: affine_imp_convex)
2.2648 -
2.2649 -lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
2.2650 -  using subspace_imp_affine affine_imp_convex by auto
2.2651 -
2.2652 -lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
2.2653 -  by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
2.2654 -
2.2655 -lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
2.2656 -  by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
2.2657 -
2.2658 -lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
2.2659 -  by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
2.2660 -
2.2661 -lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
2.2662 -  unfolding affine_dependent_def dependent_def
2.2663 -  using affine_hull_subset_span by auto
2.2664 -
2.2665 -lemma dependent_imp_affine_dependent:
2.2666 -  assumes "dependent {x - a| x . x \<in> s}"
2.2667 -    and "a \<notin> s"
2.2668 -  shows "affine_dependent (insert a s)"
2.2669 -proof -
2.2670 -  from assms(1)[unfolded dependent_explicit] obtain S u v
2.2671 -    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"
2.2672 -    by auto
2.2673 -  define t where "t = (\<lambda>x. x + a) ` S"
2.2674 -
2.2675 -  have inj: "inj_on (\<lambda>x. x + a) S"
2.2676 -    unfolding inj_on_def by auto
2.2677 -  have "0 \<notin> S"
2.2678 -    using obt(2) assms(2) unfolding subset_eq by auto
2.2679 -  have fin: "finite t" and "t \<subseteq> s"
2.2680 -    unfolding t_def using obt(1,2) by auto
2.2681 -  then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
2.2682 -    by auto
2.2683 -  moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
2.2684 -    apply (rule sum.cong)
2.2685 -    using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
2.2686 -    apply auto
2.2687 -    done
2.2688 -  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
2.2689 -    unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
2.2690 -  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"
2.2691 -    using obt(3,4) \<open>0\<notin>S\<close>
2.2692 -    by (rule_tac x="v + a" in bexI) (auto simp: t_def)
2.2693 -  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)"
2.2694 -    using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
2.2695 -  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
2.2696 -    unfolding scaleR_left.sum
2.2697 -    unfolding t_def and sum.reindex[OF inj] and o_def
2.2698 -    using obt(5)
2.2699 -    by (auto simp: sum.distrib scaleR_right_distrib)
2.2700 -  then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
2.2701 -    unfolding sum_clauses(2)[OF fin]
2.2702 -    using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
2.2703 -    by (auto simp: *)
2.2704 -  ultimately show ?thesis
2.2705 -    unfolding affine_dependent_explicit
2.2706 -    apply (rule_tac x="insert a t" in exI, auto)
2.2707 -    done
2.2708 -qed
2.2709 -
2.2710 -lemma convex_cone:
2.2711 -  "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)"
2.2712 -  (is "?lhs = ?rhs")
2.2713 -proof -
2.2714 -  {
2.2715 -    fix x y
2.2716 -    assume "x\<in>s" "y\<in>s" and ?lhs
2.2717 -    then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
2.2718 -      unfolding cone_def by auto
2.2719 -    then have "x + y \<in> s"
2.2720 -      using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
2.2721 -      apply (erule_tac x="2*\<^sub>R x" in ballE)
2.2722 -      apply (erule_tac x="2*\<^sub>R y" in ballE)
2.2723 -      apply (erule_tac x="1/2" in allE, simp)
2.2724 -      apply (erule_tac x="1/2" in allE, auto)
2.2725 -      done
2.2726 -  }
2.2727 -  then show ?thesis
2.2728 -    unfolding convex_def cone_def by blast
2.2729 -qed
2.2730 -
2.2731 -lemma affine_dependent_biggerset:
2.2732 -  fixes s :: "'a::euclidean_space set"
2.2733 -  assumes "finite s" "card s \<ge> DIM('a) + 2"
2.2734 -  shows "affine_dependent s"
2.2735 -proof -
2.2736 -  have "s \<noteq> {}" using assms by auto
2.2737 -  then obtain a where "a\<in>s" by auto
2.2738 -  have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
2.2739 -    by auto
2.2740 -  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
2.2741 -    unfolding * by (simp add: card_image inj_on_def)
2.2742 -  also have "\<dots> > DIM('a)" using assms(2)
2.2743 -    unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
2.2744 -  finally show ?thesis
2.2745 -    apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
2.2746 -    apply (rule dependent_imp_affine_dependent)
2.2747 -    apply (rule dependent_biggerset, auto)
2.2748 -    done
2.2749 -qed
2.2750 -
2.2751 -lemma affine_dependent_biggerset_general:
2.2752 -  assumes "finite (S :: 'a::euclidean_space set)"
2.2753 -    and "card S \<ge> dim S + 2"
2.2754 -  shows "affine_dependent S"
2.2755 -proof -
2.2756 -  from assms(2) have "S \<noteq> {}" by auto
2.2757 -  then obtain a where "a\<in>S" by auto
2.2758 -  have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
2.2759 -    by auto
2.2760 -  have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
2.2761 -    by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
2.2762 -  have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
2.2763 -    using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
2.2764 -  also have "\<dots> < dim S + 1" by auto
2.2765 -  also have "\<dots> \<le> card (S - {a})"
2.2766 -    using assms
2.2767 -    using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
2.2768 -    by auto
2.2769 -  finally show ?thesis
2.2770 -    apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
2.2771 -    apply (rule dependent_imp_affine_dependent)
2.2772 -    apply (rule dependent_biggerset_general)
2.2773 -    unfolding **
2.2774 -    apply auto
2.2775 -    done
2.2776 -qed
2.2777 -
2.2778 -
2.2779 -subsection%unimportant \<open>Some Properties of Affine Dependent Sets\<close>
2.2780 -
2.2781 -lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
2.2782 -  by (simp add: affine_dependent_def)
2.2783 -
2.2784 -lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
2.2785 -  by (simp add: affine_dependent_def)
2.2786 -
2.2787 -lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
2.2788 -  by (simp add: affine_dependent_def insert_Diff_if hull_same)
2.2789 -
2.2790 -lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
2.2791 -proof -
2.2792 -  have "affine ((\<lambda>x. a + x) ` (affine hull S))"
2.2793 -    using affine_translation affine_affine_hull by blast
2.2794 -  moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
2.2795 -    using hull_subset[of S] by auto
2.2796 -  ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
2.2797 -    by (metis hull_minimal)
2.2798 -  have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
2.2799 -    using affine_translation affine_affine_hull by blast
2.2800 -  moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
2.2801 -    using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
2.2802 -  moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
2.2803 -    using translation_assoc[of "-a" a] by auto
2.2804 -  ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
2.2805 -    by (metis hull_minimal)
2.2806 -  then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
2.2807 -    by auto
2.2808 -  then show ?thesis using h1 by auto
2.2809 -qed
2.2810 -
2.2811 -lemma affine_dependent_translation:
2.2812 -  assumes "affine_dependent S"
2.2813 -  shows "affine_dependent ((\<lambda>x. a + x) ` S)"
2.2814 -proof -
2.2815 -  obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
2.2816 -    using assms affine_dependent_def by auto
2.2817 -  have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
2.2818 -    by auto
2.2819 -  then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
2.2820 -    using affine_hull_translation[of a "S - {x}"] x by auto
2.2821 -  moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
2.2822 -    using x by auto
2.2823 -  ultimately show ?thesis
2.2824 -    unfolding affine_dependent_def by auto
2.2825 -qed
2.2826 -
2.2827 -lemma affine_dependent_translation_eq:
2.2828 -  "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
2.2829 -proof -
2.2830 -  {
2.2831 -    assume "affine_dependent ((\<lambda>x. a + x) ` S)"
2.2832 -    then have "affine_dependent S"
2.2833 -      using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
2.2834 -      by auto
2.2835 -  }
2.2836 -  then show ?thesis
2.2837 -    using affine_dependent_translation by auto
2.2838 -qed
2.2839 -
2.2840 -lemma affine_hull_0_dependent:
2.2841 -  assumes "0 \<in> affine hull S"
2.2842 -  shows "dependent S"
2.2843 -proof -
2.2844 -  obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
2.2845 -    using assms affine_hull_explicit[of S] by auto
2.2846 -  then have "\<exists>v\<in>s. u v \<noteq> 0"
2.2847 -    using sum_not_0[of "u" "s"] by auto
2.2848 -  then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
2.2849 -    using s_u by auto
2.2850 -  then show ?thesis
2.2851 -    unfolding dependent_explicit[of S] by auto
2.2852 -qed
2.2853 -
2.2854 -lemma affine_dependent_imp_dependent2:
2.2855 -  assumes "affine_dependent (insert 0 S)"
2.2856 -  shows "dependent S"
2.2857 -proof -
2.2858 -  obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
2.2859 -    using affine_dependent_def[of "(insert 0 S)"] assms by blast
2.2860 -  then have "x \<in> span (insert 0 S - {x})"
2.2861 -    using affine_hull_subset_span by auto
2.2862 -  moreover have "span (insert 0 S - {x}) = span (S - {x})"
2.2863 -    using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
2.2864 -  ultimately have "x \<in> span (S - {x})" by auto
2.2865 -  then have "x \<noteq> 0 \<Longrightarrow> dependent S"
2.2866 -    using x dependent_def by auto
2.2867 -  moreover
2.2868 -  {
2.2869 -    assume "x = 0"
2.2870 -    then have "0 \<in> affine hull S"
2.2871 -      using x hull_mono[of "S - {0}" S] by auto
2.2872 -    then have "dependent S"
2.2873 -      using affine_hull_0_dependent by auto
2.2874 -  }
2.2875 -  ultimately show ?thesis by auto
2.2876 -qed
2.2877 -
2.2878 -lemma affine_dependent_iff_dependent:
2.2879 -  assumes "a \<notin> S"
2.2880 -  shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
2.2881 -proof -
2.2882 -  have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
2.2883 -  then show ?thesis
2.2884 -    using affine_dependent_translation_eq[of "(insert a S)" "-a"]
2.2885 -      affine_dependent_imp_dependent2 assms
2.2886 -      dependent_imp_affine_dependent[of a S]
2.2887 -    by (auto simp del: uminus_add_conv_diff)
2.2888 -qed
2.2889 -
2.2890 -lemma affine_dependent_iff_dependent2:
2.2891 -  assumes "a \<in> S"
2.2892 -  shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
2.2893 -proof -
2.2894 -  have "insert a (S - {a}) = S"
2.2895 -    using assms by auto
2.2896 -  then show ?thesis
2.2897 -    using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
2.2898 -qed
2.2899 -
2.2900 -lemma affine_hull_insert_span_gen:
2.2901 -  "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
2.2902 -proof -
2.2903 -  have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
2.2904 -    by auto
2.2905 -  {
2.2906 -    assume "a \<notin> s"
2.2907 -    then have ?thesis
2.2908 -      using affine_hull_insert_span[of a s] h1 by auto
2.2909 -  }
2.2910 -  moreover
2.2911 -  {
2.2912 -    assume a1: "a \<in> s"
2.2913 -    have "\<exists>x. x \<in> s \<and> -a+x=0"
2.2914 -      apply (rule exI[of _ a])
2.2915 -      using a1
2.2916 -      apply auto
2.2917 -      done
2.2918 -    then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
2.2919 -      by auto
2.2920 -    then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
2.2921 -      using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
2.2922 -    moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
2.2923 -      by auto
2.2924 -    moreover have "insert a (s - {a}) = insert a s"
2.2925 -      by auto
```