src/HOL/Analysis/Convex_Euclidean_Space.thy
 author immler Tue Jul 10 09:38:35 2018 +0200 (11 months ago) changeset 68607 67bb59e49834 parent 68527 2f4e2aab190a child 69064 5840724b1d71 permissions -rw-r--r--
make theorem, corollary, and proposition %important for HOL-Analysis manual
```     1 (* Title:      HOL/Analysis/Convex_Euclidean_Space.thy
```
```     2    Author:     L C Paulson, University of Cambridge
```
```     3    Author:     Robert Himmelmann, TU Muenchen
```
```     4    Author:     Bogdan Grechuk, University of Edinburgh
```
```     5    Author:     Armin Heller, TU Muenchen
```
```     6    Author:     Johannes Hoelzl, TU Muenchen
```
```     7 *)
```
```     8
```
```     9 section \<open>Convex sets, functions and related things\<close>
```
```    10
```
```    11 theory Convex_Euclidean_Space
```
```    12 imports
```
```    13   Connected
```
```    14   "HOL-Library.Set_Algebras"
```
```    15 begin
```
```    16
```
```    17 lemma swap_continuous: (*move to Topological_Spaces?*)
```
```    18   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
```
```    19     shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
```
```    20 proof -
```
```    21   have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
```
```    22     by auto
```
```    23   then show ?thesis
```
```    24     apply (rule ssubst)
```
```    25     apply (rule continuous_on_compose)
```
```    26     apply (simp add: split_def)
```
```    27     apply (rule continuous_intros | simp add: assms)+
```
```    28     done
```
```    29 qed
```
```    30
```
```    31 lemma substdbasis_expansion_unique:
```
```    32   assumes d: "d \<subseteq> Basis"
```
```    33   shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
```
```    34     (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
```
```    35 proof -
```
```    36   have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
```
```    37     by auto
```
```    38   have **: "finite d"
```
```    39     by (auto intro: finite_subset[OF assms])
```
```    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)"
```
```    41     using d
```
```    42     by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
```
```    43   show ?thesis
```
```    44     unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
```
```    45 qed
```
```    46
```
```    47 lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
```
```    48   by (rule independent_mono[OF independent_Basis])
```
```    49
```
```    50 lemma dim_cball:
```
```    51   assumes "e > 0"
```
```    52   shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
```
```    53 proof -
```
```    54   {
```
```    55     fix x :: "'n::euclidean_space"
```
```    56     define y where "y = (e / norm x) *\<^sub>R x"
```
```    57     then have "y \<in> cball 0 e"
```
```    58       using assms by auto
```
```    59     moreover have *: "x = (norm x / e) *\<^sub>R y"
```
```    60       using y_def assms by simp
```
```    61     moreover from * have "x = (norm x/e) *\<^sub>R y"
```
```    62       by auto
```
```    63     ultimately have "x \<in> span (cball 0 e)"
```
```    64       using span_scale[of y "cball 0 e" "norm x/e"]
```
```    65         span_superset[of "cball 0 e"]
```
```    66       by (simp add: span_base)
```
```    67   }
```
```    68   then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
```
```    69     by auto
```
```    70   then show ?thesis
```
```    71     using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
```
```    72 qed
```
```    73
```
```    74 lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
```
```    75   by (rule ccontr) auto
```
```    76
```
```    77 lemma subset_translation_eq [simp]:
```
```    78     fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
```
```    79   by auto
```
```    80
```
```    81 lemma translate_inj_on:
```
```    82   fixes A :: "'a::ab_group_add set"
```
```    83   shows "inj_on (\<lambda>x. a + x) A"
```
```    84   unfolding inj_on_def by auto
```
```    85
```
```    86 lemma translation_assoc:
```
```    87   fixes a b :: "'a::ab_group_add"
```
```    88   shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
```
```    89   by auto
```
```    90
```
```    91 lemma translation_invert:
```
```    92   fixes a :: "'a::ab_group_add"
```
```    93   assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
```
```    94   shows "A = B"
```
```    95 proof -
```
```    96   have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
```
```    97     using assms by auto
```
```    98   then show ?thesis
```
```    99     using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
```
```   100 qed
```
```   101
```
```   102 lemma translation_galois:
```
```   103   fixes a :: "'a::ab_group_add"
```
```   104   shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
```
```   105   using translation_assoc[of "-a" a S]
```
```   106   apply auto
```
```   107   using translation_assoc[of a "-a" T]
```
```   108   apply auto
```
```   109   done
```
```   110
```
```   111 lemma translation_inverse_subset:
```
```   112   assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
```
```   113   shows "V \<le> ((\<lambda>x. a + x) ` S)"
```
```   114 proof -
```
```   115   {
```
```   116     fix x
```
```   117     assume "x \<in> V"
```
```   118     then have "x-a \<in> S" using assms by auto
```
```   119     then have "x \<in> {a + v |v. v \<in> S}"
```
```   120       apply auto
```
```   121       apply (rule exI[of _ "x-a"], simp)
```
```   122       done
```
```   123     then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
```
```   124   }
```
```   125   then show ?thesis by auto
```
```   126 qed
```
```   127
```
```   128 subsection \<open>Convexity\<close>
```
```   129
```
```   130 definition%important convex :: "'a::real_vector set \<Rightarrow> bool"
```
```   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)"
```
```   132
```
```   133 lemma convexI:
```
```   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"
```
```   135   shows "convex s"
```
```   136   using assms unfolding convex_def by fast
```
```   137
```
```   138 lemma convexD:
```
```   139   assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
```
```   140   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
```
```   141   using assms unfolding convex_def by fast
```
```   142
```
```   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)"
```
```   144   (is "_ \<longleftrightarrow> ?alt")
```
```   145 proof
```
```   146   show "convex s" if alt: ?alt
```
```   147   proof -
```
```   148     {
```
```   149       fix x y and u v :: real
```
```   150       assume mem: "x \<in> s" "y \<in> s"
```
```   151       assume "0 \<le> u" "0 \<le> v"
```
```   152       moreover
```
```   153       assume "u + v = 1"
```
```   154       then have "u = 1 - v" by auto
```
```   155       ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
```
```   156         using alt [rule_format, OF mem] by auto
```
```   157     }
```
```   158     then show ?thesis
```
```   159       unfolding convex_def by auto
```
```   160   qed
```
```   161   show ?alt if "convex s"
```
```   162     using that by (auto simp: convex_def)
```
```   163 qed
```
```   164
```
```   165 lemma convexD_alt:
```
```   166   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
```
```   167   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
```
```   168   using assms unfolding convex_alt by auto
```
```   169
```
```   170 lemma mem_convex_alt:
```
```   171   assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
```
```   172   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
```
```   173   apply (rule convexD)
```
```   174   using assms
```
```   175        apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
```
```   176   done
```
```   177
```
```   178 lemma convex_empty[intro,simp]: "convex {}"
```
```   179   unfolding convex_def by simp
```
```   180
```
```   181 lemma convex_singleton[intro,simp]: "convex {a}"
```
```   182   unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
```
```   183
```
```   184 lemma convex_UNIV[intro,simp]: "convex UNIV"
```
```   185   unfolding convex_def by auto
```
```   186
```
```   187 lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
```
```   188   unfolding convex_def by auto
```
```   189
```
```   190 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
```
```   191   unfolding convex_def by auto
```
```   192
```
```   193 lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
```
```   194   unfolding convex_def by auto
```
```   195
```
```   196 lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
```
```   197   unfolding convex_def by auto
```
```   198
```
```   199 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
```
```   200   unfolding convex_def
```
```   201   by (auto simp: inner_add intro!: convex_bound_le)
```
```   202
```
```   203 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
```
```   204 proof -
```
```   205   have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
```
```   206     by auto
```
```   207   show ?thesis
```
```   208     unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
```
```   209 qed
```
```   210
```
```   211 lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
```
```   212 proof -
```
```   213   have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
```
```   214     by auto
```
```   215   show ?thesis
```
```   216     unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
```
```   217 qed
```
```   218
```
```   219 lemma convex_hyperplane: "convex {x. inner a x = b}"
```
```   220 proof -
```
```   221   have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
```
```   222     by auto
```
```   223   show ?thesis using convex_halfspace_le convex_halfspace_ge
```
```   224     by (auto intro!: convex_Int simp: *)
```
```   225 qed
```
```   226
```
```   227 lemma convex_halfspace_lt: "convex {x. inner a x < b}"
```
```   228   unfolding convex_def
```
```   229   by (auto simp: convex_bound_lt inner_add)
```
```   230
```
```   231 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
```
```   232   using convex_halfspace_lt[of "-a" "-b"] by auto
```
```   233
```
```   234 lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
```
```   235   using convex_halfspace_ge[of b "1::complex"] by simp
```
```   236
```
```   237 lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
```
```   238   using convex_halfspace_le[of "1::complex" b] by simp
```
```   239
```
```   240 lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
```
```   241   using convex_halfspace_ge[of b \<i>] by simp
```
```   242
```
```   243 lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
```
```   244   using convex_halfspace_le[of \<i> b] by simp
```
```   245
```
```   246 lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
```
```   247   using convex_halfspace_gt[of b "1::complex"] by simp
```
```   248
```
```   249 lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
```
```   250   using convex_halfspace_lt[of "1::complex" b] by simp
```
```   251
```
```   252 lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
```
```   253   using convex_halfspace_gt[of b \<i>] by simp
```
```   254
```
```   255 lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
```
```   256   using convex_halfspace_lt[of \<i> b] by simp
```
```   257
```
```   258 lemma convex_real_interval [iff]:
```
```   259   fixes a b :: "real"
```
```   260   shows "convex {a..}" and "convex {..b}"
```
```   261     and "convex {a<..}" and "convex {..<b}"
```
```   262     and "convex {a..b}" and "convex {a<..b}"
```
```   263     and "convex {a..<b}" and "convex {a<..<b}"
```
```   264 proof -
```
```   265   have "{a..} = {x. a \<le> inner 1 x}"
```
```   266     by auto
```
```   267   then show 1: "convex {a..}"
```
```   268     by (simp only: convex_halfspace_ge)
```
```   269   have "{..b} = {x. inner 1 x \<le> b}"
```
```   270     by auto
```
```   271   then show 2: "convex {..b}"
```
```   272     by (simp only: convex_halfspace_le)
```
```   273   have "{a<..} = {x. a < inner 1 x}"
```
```   274     by auto
```
```   275   then show 3: "convex {a<..}"
```
```   276     by (simp only: convex_halfspace_gt)
```
```   277   have "{..<b} = {x. inner 1 x < b}"
```
```   278     by auto
```
```   279   then show 4: "convex {..<b}"
```
```   280     by (simp only: convex_halfspace_lt)
```
```   281   have "{a..b} = {a..} \<inter> {..b}"
```
```   282     by auto
```
```   283   then show "convex {a..b}"
```
```   284     by (simp only: convex_Int 1 2)
```
```   285   have "{a<..b} = {a<..} \<inter> {..b}"
```
```   286     by auto
```
```   287   then show "convex {a<..b}"
```
```   288     by (simp only: convex_Int 3 2)
```
```   289   have "{a..<b} = {a..} \<inter> {..<b}"
```
```   290     by auto
```
```   291   then show "convex {a..<b}"
```
```   292     by (simp only: convex_Int 1 4)
```
```   293   have "{a<..<b} = {a<..} \<inter> {..<b}"
```
```   294     by auto
```
```   295   then show "convex {a<..<b}"
```
```   296     by (simp only: convex_Int 3 4)
```
```   297 qed
```
```   298
```
```   299 lemma convex_Reals: "convex \<real>"
```
```   300   by (simp add: convex_def scaleR_conv_of_real)
```
```   301
```
```   302
```
```   303 subsection%unimportant \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
```
```   304
```
```   305 lemma convex_sum:
```
```   306   fixes C :: "'a::real_vector set"
```
```   307   assumes "finite s"
```
```   308     and "convex C"
```
```   309     and "(\<Sum> i \<in> s. a i) = 1"
```
```   310   assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```   311     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
```
```   312   shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
```
```   313   using assms(1,3,4,5)
```
```   314 proof (induct arbitrary: a set: finite)
```
```   315   case empty
```
```   316   then show ?case by simp
```
```   317 next
```
```   318   case (insert i s) note IH = this(3)
```
```   319   have "a i + sum a s = 1"
```
```   320     and "0 \<le> a i"
```
```   321     and "\<forall>j\<in>s. 0 \<le> a j"
```
```   322     and "y i \<in> C"
```
```   323     and "\<forall>j\<in>s. y j \<in> C"
```
```   324     using insert.hyps(1,2) insert.prems by simp_all
```
```   325   then have "0 \<le> sum a s"
```
```   326     by (simp add: sum_nonneg)
```
```   327   have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
```
```   328   proof (cases "sum a s = 0")
```
```   329     case True
```
```   330     with \<open>a i + sum a s = 1\<close> have "a i = 1"
```
```   331       by simp
```
```   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"
```
```   333       by simp
```
```   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>
```
```   335       by simp
```
```   336   next
```
```   337     case False
```
```   338     with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
```
```   339       by simp
```
```   340     then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
```
```   341       using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
```
```   342       by (simp add: IH sum_divide_distrib [symmetric])
```
```   343     from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
```
```   344       and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
```
```   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"
```
```   346       by (rule convexD)
```
```   347     then show ?thesis
```
```   348       by (simp add: scaleR_sum_right False)
```
```   349   qed
```
```   350   then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
```
```   351     by simp
```
```   352 qed
```
```   353
```
```   354 lemma convex:
```
```   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)
```
```   356       \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
```
```   357 proof safe
```
```   358   fix k :: nat
```
```   359   fix u :: "nat \<Rightarrow> real"
```
```   360   fix x
```
```   361   assume "convex s"
```
```   362     "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
```
```   363     "sum u {1..k} = 1"
```
```   364   with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
```
```   365     by auto
```
```   366 next
```
```   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
```
```   368     \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
```
```   369   {
```
```   370     fix \<mu> :: real
```
```   371     fix x y :: 'a
```
```   372     assume xy: "x \<in> s" "y \<in> s"
```
```   373     assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
```
```   374     let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
```
```   375     let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
```
```   376     have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
```
```   377       by auto
```
```   378     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
```
```   379       by simp
```
```   380     then have "sum ?u {1 .. 2} = 1"
```
```   381       using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
```
```   382       by auto
```
```   383     with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
```
```   384       using mu xy by auto
```
```   385     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
```
```   386       using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
```
```   387     from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
```
```   388     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
```
```   389       by auto
```
```   390     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
```
```   391       using s by (auto simp: add.commute)
```
```   392   }
```
```   393   then show "convex s"
```
```   394     unfolding convex_alt by auto
```
```   395 qed
```
```   396
```
```   397
```
```   398 lemma convex_explicit:
```
```   399   fixes s :: "'a::real_vector set"
```
```   400   shows "convex s \<longleftrightarrow>
```
```   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)"
```
```   402 proof safe
```
```   403   fix t
```
```   404   fix u :: "'a \<Rightarrow> real"
```
```   405   assume "convex s"
```
```   406     and "finite t"
```
```   407     and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
```
```   408   then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
```
```   409     using convex_sum[of t s u "\<lambda> x. x"] by auto
```
```   410 next
```
```   411   assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
```
```   412     sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
```
```   413   show "convex s"
```
```   414     unfolding convex_alt
```
```   415   proof safe
```
```   416     fix x y
```
```   417     fix \<mu> :: real
```
```   418     assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
```
```   419     show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
```
```   420     proof (cases "x = y")
```
```   421       case False
```
```   422       then show ?thesis
```
```   423         using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
```
```   424         by auto
```
```   425     next
```
```   426       case True
```
```   427       then show ?thesis
```
```   428         using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
```
```   429         by (auto simp: field_simps real_vector.scale_left_diff_distrib)
```
```   430     qed
```
```   431   qed
```
```   432 qed
```
```   433
```
```   434 lemma convex_finite:
```
```   435   assumes "finite s"
```
```   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)"
```
```   437   unfolding convex_explicit
```
```   438   apply safe
```
```   439   subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
```
```   440   subgoal for t u
```
```   441   proof -
```
```   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)"
```
```   443       by simp
```
```   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"
```
```   445     assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
```
```   446     assume "t \<subseteq> s"
```
```   447     then have "s \<inter> t = t" by auto
```
```   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"
```
```   449       by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
```
```   450   qed
```
```   451   done
```
```   452
```
```   453
```
```   454 subsection \<open>Functions that are convex on a set\<close>
```
```   455
```
```   456 definition%important convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
```
```   457   where "convex_on s f \<longleftrightarrow>
```
```   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)"
```
```   459
```
```   460 lemma convex_onI [intro?]:
```
```   461   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
```
```   462     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
```
```   463   shows "convex_on A f"
```
```   464   unfolding convex_on_def
```
```   465 proof clarify
```
```   466   fix x y
```
```   467   fix u v :: real
```
```   468   assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
```
```   469   from A(5) have [simp]: "v = 1 - u"
```
```   470     by (simp add: algebra_simps)
```
```   471   from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
```
```   472     using assms[of u y x]
```
```   473     by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
```
```   474 qed
```
```   475
```
```   476 lemma convex_on_linorderI [intro?]:
```
```   477   fixes A :: "('a::{linorder,real_vector}) set"
```
```   478   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
```
```   479     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
```
```   480   shows "convex_on A f"
```
```   481 proof
```
```   482   fix x y
```
```   483   fix t :: real
```
```   484   assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
```
```   485   with assms [of t x y] assms [of "1 - t" y x]
```
```   486   show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
```
```   487     by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
```
```   488 qed
```
```   489
```
```   490 lemma convex_onD:
```
```   491   assumes "convex_on A f"
```
```   492   shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
```
```   493     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
```
```   494   using assms by (auto simp: convex_on_def)
```
```   495
```
```   496 lemma convex_onD_Icc:
```
```   497   assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
```
```   498   shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
```
```   499     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
```
```   500   using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
```
```   501
```
```   502 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
```
```   503   unfolding convex_on_def by auto
```
```   504
```
```   505 lemma convex_on_add [intro]:
```
```   506   assumes "convex_on s f"
```
```   507     and "convex_on s g"
```
```   508   shows "convex_on s (\<lambda>x. f x + g x)"
```
```   509 proof -
```
```   510   {
```
```   511     fix x y
```
```   512     assume "x \<in> s" "y \<in> s"
```
```   513     moreover
```
```   514     fix u v :: real
```
```   515     assume "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```   516     ultimately
```
```   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)"
```
```   518       using assms unfolding convex_on_def by (auto simp: add_mono)
```
```   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)"
```
```   520       by (simp add: field_simps)
```
```   521   }
```
```   522   then show ?thesis
```
```   523     unfolding convex_on_def by auto
```
```   524 qed
```
```   525
```
```   526 lemma convex_on_cmul [intro]:
```
```   527   fixes c :: real
```
```   528   assumes "0 \<le> c"
```
```   529     and "convex_on s f"
```
```   530   shows "convex_on s (\<lambda>x. c * f x)"
```
```   531 proof -
```
```   532   have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
```
```   533     for u c fx v fy :: real
```
```   534     by (simp add: field_simps)
```
```   535   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
```
```   536     unfolding convex_on_def and * by auto
```
```   537 qed
```
```   538
```
```   539 lemma convex_lower:
```
```   540   assumes "convex_on s f"
```
```   541     and "x \<in> s"
```
```   542     and "y \<in> s"
```
```   543     and "0 \<le> u"
```
```   544     and "0 \<le> v"
```
```   545     and "u + v = 1"
```
```   546   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
```
```   547 proof -
```
```   548   let ?m = "max (f x) (f y)"
```
```   549   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
```
```   550     using assms(4,5) by (auto simp: mult_left_mono add_mono)
```
```   551   also have "\<dots> = max (f x) (f y)"
```
```   552     using assms(6) by (simp add: distrib_right [symmetric])
```
```   553   finally show ?thesis
```
```   554     using assms unfolding convex_on_def by fastforce
```
```   555 qed
```
```   556
```
```   557 lemma convex_on_dist [intro]:
```
```   558   fixes s :: "'a::real_normed_vector set"
```
```   559   shows "convex_on s (\<lambda>x. dist a x)"
```
```   560 proof (auto simp: convex_on_def dist_norm)
```
```   561   fix x y
```
```   562   assume "x \<in> s" "y \<in> s"
```
```   563   fix u v :: real
```
```   564   assume "0 \<le> u"
```
```   565   assume "0 \<le> v"
```
```   566   assume "u + v = 1"
```
```   567   have "a = u *\<^sub>R a + v *\<^sub>R a"
```
```   568     unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
```
```   569   then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
```
```   570     by (auto simp: algebra_simps)
```
```   571   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
```
```   572     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
```
```   573     using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
```
```   574 qed
```
```   575
```
```   576
```
```   577 subsection%unimportant \<open>Arithmetic operations on sets preserve convexity\<close>
```
```   578
```
```   579 lemma convex_linear_image:
```
```   580   assumes "linear f"
```
```   581     and "convex s"
```
```   582   shows "convex (f ` s)"
```
```   583 proof -
```
```   584   interpret f: linear f by fact
```
```   585   from \<open>convex s\<close> show "convex (f ` s)"
```
```   586     by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
```
```   587 qed
```
```   588
```
```   589 lemma convex_linear_vimage:
```
```   590   assumes "linear f"
```
```   591     and "convex s"
```
```   592   shows "convex (f -` s)"
```
```   593 proof -
```
```   594   interpret f: linear f by fact
```
```   595   from \<open>convex s\<close> show "convex (f -` s)"
```
```   596     by (simp add: convex_def f.add f.scaleR)
```
```   597 qed
```
```   598
```
```   599 lemma convex_scaling:
```
```   600   assumes "convex s"
```
```   601   shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
```
```   602 proof -
```
```   603   have "linear (\<lambda>x. c *\<^sub>R x)"
```
```   604     by (simp add: linearI scaleR_add_right)
```
```   605   then show ?thesis
```
```   606     using \<open>convex s\<close> by (rule convex_linear_image)
```
```   607 qed
```
```   608
```
```   609 lemma convex_scaled:
```
```   610   assumes "convex S"
```
```   611   shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
```
```   612 proof -
```
```   613   have "linear (\<lambda>x. x *\<^sub>R c)"
```
```   614     by (simp add: linearI scaleR_add_left)
```
```   615   then show ?thesis
```
```   616     using \<open>convex S\<close> by (rule convex_linear_image)
```
```   617 qed
```
```   618
```
```   619 lemma convex_negations:
```
```   620   assumes "convex S"
```
```   621   shows "convex ((\<lambda>x. - x) ` S)"
```
```   622 proof -
```
```   623   have "linear (\<lambda>x. - x)"
```
```   624     by (simp add: linearI)
```
```   625   then show ?thesis
```
```   626     using \<open>convex S\<close> by (rule convex_linear_image)
```
```   627 qed
```
```   628
```
```   629 lemma convex_sums:
```
```   630   assumes "convex S"
```
```   631     and "convex T"
```
```   632   shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
```
```   633 proof -
```
```   634   have "linear (\<lambda>(x, y). x + y)"
```
```   635     by (auto intro: linearI simp: scaleR_add_right)
```
```   636   with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
```
```   637     by (intro convex_linear_image convex_Times)
```
```   638   also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
```
```   639     by auto
```
```   640   finally show ?thesis .
```
```   641 qed
```
```   642
```
```   643 lemma convex_differences:
```
```   644   assumes "convex S" "convex T"
```
```   645   shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
```
```   646 proof -
```
```   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}"
```
```   648     by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
```
```   649   then show ?thesis
```
```   650     using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
```
```   651 qed
```
```   652
```
```   653 lemma convex_translation:
```
```   654   assumes "convex S"
```
```   655   shows "convex ((\<lambda>x. a + x) ` S)"
```
```   656 proof -
```
```   657   have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (\<lambda>x. a + x) ` S"
```
```   658     by auto
```
```   659   then show ?thesis
```
```   660     using convex_sums[OF convex_singleton[of a] assms] by auto
```
```   661 qed
```
```   662
```
```   663 lemma convex_affinity:
```
```   664   assumes "convex S"
```
```   665   shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
```
```   666 proof -
```
```   667   have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` ( *\<^sub>R) c ` S"
```
```   668     by auto
```
```   669   then show ?thesis
```
```   670     using convex_translation[OF convex_scaling[OF assms], of a c] by auto
```
```   671 qed
```
```   672
```
```   673 lemma pos_is_convex: "convex {0 :: real <..}"
```
```   674   unfolding convex_alt
```
```   675 proof safe
```
```   676   fix y x \<mu> :: real
```
```   677   assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
```
```   678   {
```
```   679     assume "\<mu> = 0"
```
```   680     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
```
```   681       by simp
```
```   682     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
```
```   683       using * by simp
```
```   684   }
```
```   685   moreover
```
```   686   {
```
```   687     assume "\<mu> = 1"
```
```   688     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
```
```   689       using * by simp
```
```   690   }
```
```   691   moreover
```
```   692   {
```
```   693     assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
```
```   694     then have "\<mu> > 0" "(1 - \<mu>) > 0"
```
```   695       using * by auto
```
```   696     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
```
```   697       using * by (auto simp: add_pos_pos)
```
```   698   }
```
```   699   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
```
```   700     by fastforce
```
```   701 qed
```
```   702
```
```   703 lemma convex_on_sum:
```
```   704   fixes a :: "'a \<Rightarrow> real"
```
```   705     and y :: "'a \<Rightarrow> 'b::real_vector"
```
```   706     and f :: "'b \<Rightarrow> real"
```
```   707   assumes "finite s" "s \<noteq> {}"
```
```   708     and "convex_on C f"
```
```   709     and "convex C"
```
```   710     and "(\<Sum> i \<in> s. a i) = 1"
```
```   711     and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```   712     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
```
```   713   shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
```
```   714   using assms
```
```   715 proof (induct s arbitrary: a rule: finite_ne_induct)
```
```   716   case (singleton i)
```
```   717   then have ai: "a i = 1"
```
```   718     by auto
```
```   719   then show ?case
```
```   720     by auto
```
```   721 next
```
```   722   case (insert i s)
```
```   723   then have "convex_on C f"
```
```   724     by simp
```
```   725   from this[unfolded convex_on_def, rule_format]
```
```   726   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
```
```   727       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   728     by simp
```
```   729   show ?case
```
```   730   proof (cases "a i = 1")
```
```   731     case True
```
```   732     then have "(\<Sum> j \<in> s. a j) = 0"
```
```   733       using insert by auto
```
```   734     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
```
```   735       using insert by (fastforce simp: sum_nonneg_eq_0_iff)
```
```   736     then show ?thesis
```
```   737       using insert by auto
```
```   738   next
```
```   739     case False
```
```   740     from insert have yai: "y i \<in> C" "a i \<ge> 0"
```
```   741       by auto
```
```   742     have fis: "finite (insert i s)"
```
```   743       using insert by auto
```
```   744     then have ai1: "a i \<le> 1"
```
```   745       using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
```
```   746     then have "a i < 1"
```
```   747       using False by auto
```
```   748     then have i0: "1 - a i > 0"
```
```   749       by auto
```
```   750     let ?a = "\<lambda>j. a j / (1 - a i)"
```
```   751     have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
```
```   752       using i0 insert that by fastforce
```
```   753     have "(\<Sum> j \<in> insert i s. a j) = 1"
```
```   754       using insert by auto
```
```   755     then have "(\<Sum> j \<in> s. a j) = 1 - a i"
```
```   756       using sum.insert insert by fastforce
```
```   757     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
```
```   758       using i0 by auto
```
```   759     then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
```
```   760       unfolding sum_divide_distrib by simp
```
```   761     have "convex C" using insert by auto
```
```   762     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
```
```   763       using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
```
```   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))"
```
```   765       using a_nonneg a1 insert by blast
```
```   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)"
```
```   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
```
```   768       by (auto simp only: add.commute)
```
```   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)"
```
```   770       using i0 by auto
```
```   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)"
```
```   772       using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
```
```   773       by (auto simp: algebra_simps)
```
```   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)"
```
```   775       by (auto simp: divide_inverse)
```
```   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)"
```
```   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]
```
```   778       by (auto simp: add.commute)
```
```   779     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
```
```   780       using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
```
```   781             OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
```
```   782       by simp
```
```   783     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
```
```   784       unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
```
```   785       using i0 by auto
```
```   786     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
```
```   787       using i0 by auto
```
```   788     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
```
```   789       using insert by auto
```
```   790     finally show ?thesis
```
```   791       by simp
```
```   792   qed
```
```   793 qed
```
```   794
```
```   795 lemma convex_on_alt:
```
```   796   fixes C :: "'a::real_vector set"
```
```   797   assumes "convex C"
```
```   798   shows "convex_on C f \<longleftrightarrow>
```
```   799     (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
```
```   800       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
```
```   801 proof safe
```
```   802   fix x y
```
```   803   fix \<mu> :: real
```
```   804   assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
```
```   805   from this[unfolded convex_on_def, rule_format]
```
```   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
```
```   807     by auto
```
```   808   from this [of "\<mu>" "1 - \<mu>", simplified] *
```
```   809   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   810     by auto
```
```   811 next
```
```   812   assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
```
```   813     f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   814   {
```
```   815     fix x y
```
```   816     fix u v :: real
```
```   817     assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
```
```   818     then have[simp]: "1 - u = v" by auto
```
```   819     from *[rule_format, of x y u]
```
```   820     have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
```
```   821       using ** by auto
```
```   822   }
```
```   823   then show "convex_on C f"
```
```   824     unfolding convex_on_def by auto
```
```   825 qed
```
```   826
```
```   827 lemma convex_on_diff:
```
```   828   fixes f :: "real \<Rightarrow> real"
```
```   829   assumes f: "convex_on I f"
```
```   830     and I: "x \<in> I" "y \<in> I"
```
```   831     and t: "x < t" "t < y"
```
```   832   shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
```
```   833     and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
```
```   834 proof -
```
```   835   define a where "a \<equiv> (t - y) / (x - y)"
```
```   836   with t have "0 \<le> a" "0 \<le> 1 - a"
```
```   837     by (auto simp: field_simps)
```
```   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"
```
```   839     by (auto simp: convex_on_def)
```
```   840   have "a * x + (1 - a) * y = a * (x - y) + y"
```
```   841     by (simp add: field_simps)
```
```   842   also have "\<dots> = t"
```
```   843     unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
```
```   844   finally have "f t \<le> a * f x + (1 - a) * f y"
```
```   845     using cvx by simp
```
```   846   also have "\<dots> = a * (f x - f y) + f y"
```
```   847     by (simp add: field_simps)
```
```   848   finally have "f t - f y \<le> a * (f x - f y)"
```
```   849     by simp
```
```   850   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
```
```   851     by (simp add: le_divide_eq divide_le_eq field_simps a_def)
```
```   852   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
```
```   853     by (simp add: le_divide_eq divide_le_eq field_simps)
```
```   854 qed
```
```   855
```
```   856 lemma pos_convex_function:
```
```   857   fixes f :: "real \<Rightarrow> real"
```
```   858   assumes "convex C"
```
```   859     and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
```
```   860   shows "convex_on C f"
```
```   861   unfolding convex_on_alt[OF assms(1)]
```
```   862   using assms
```
```   863 proof safe
```
```   864   fix x y \<mu> :: real
```
```   865   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
```
```   866   assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
```
```   867   then have "1 - \<mu> \<ge> 0" by auto
```
```   868   then have xpos: "?x \<in> C"
```
```   869     using * unfolding convex_alt by fastforce
```
```   870   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
```
```   871       \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
```
```   872     using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
```
```   873         mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
```
```   874     by auto
```
```   875   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
```
```   876     by (auto simp: field_simps)
```
```   877   then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   878     using convex_on_alt by auto
```
```   879 qed
```
```   880
```
```   881 lemma atMostAtLeast_subset_convex:
```
```   882   fixes C :: "real set"
```
```   883   assumes "convex C"
```
```   884     and "x \<in> C" "y \<in> C" "x < y"
```
```   885   shows "{x .. y} \<subseteq> C"
```
```   886 proof safe
```
```   887   fix z assume z: "z \<in> {x .. y}"
```
```   888   have less: "z \<in> C" if *: "x < z" "z < y"
```
```   889   proof -
```
```   890     let ?\<mu> = "(y - z) / (y - x)"
```
```   891     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
```
```   892       using assms * by (auto simp: field_simps)
```
```   893     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
```
```   894       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
```
```   895       by (simp add: algebra_simps)
```
```   896     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
```
```   897       by (auto simp: field_simps)
```
```   898     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
```
```   899       using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
```
```   900     also have "\<dots> = z"
```
```   901       using assms by (auto simp: field_simps)
```
```   902     finally show ?thesis
```
```   903       using comb by auto
```
```   904   qed
```
```   905   show "z \<in> C"
```
```   906     using z less assms by (auto simp: le_less)
```
```   907 qed
```
```   908
```
```   909 lemma f''_imp_f':
```
```   910   fixes f :: "real \<Rightarrow> real"
```
```   911   assumes "convex C"
```
```   912     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
```
```   913     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
```
```   914     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
```
```   915     and x: "x \<in> C"
```
```   916     and y: "y \<in> C"
```
```   917   shows "f' x * (y - x) \<le> f y - f x"
```
```   918   using assms
```
```   919 proof -
```
```   920   have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
```
```   921     if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
```
```   922   proof -
```
```   923     from * have ge: "y - x > 0" "y - x \<ge> 0"
```
```   924       by auto
```
```   925     from * have le: "x - y < 0" "x - y \<le> 0"
```
```   926       by auto
```
```   927     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
```
```   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>],
```
```   929           THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
```
```   930       by auto
```
```   931     then have "z1 \<in> C"
```
```   932       using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
```
```   933       by fastforce
```
```   934     from z1 have z1': "f x - f y = (x - y) * f' z1"
```
```   935       by (simp add: field_simps)
```
```   936     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
```
```   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>],
```
```   938           THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
```
```   939       by auto
```
```   940     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
```
```   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>],
```
```   942           THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
```
```   943       by auto
```
```   944     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
```
```   945       using * z1' by auto
```
```   946     also have "\<dots> = (y - z1) * f'' z3"
```
```   947       using z3 by auto
```
```   948     finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
```
```   949       by simp
```
```   950     have A': "y - z1 \<ge> 0"
```
```   951       using z1 by auto
```
```   952     have "z3 \<in> C"
```
```   953       using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
```
```   954       by fastforce
```
```   955     then have B': "f'' z3 \<ge> 0"
```
```   956       using assms by auto
```
```   957     from A' B' have "(y - z1) * f'' z3 \<ge> 0"
```
```   958       by auto
```
```   959     from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
```
```   960       by auto
```
```   961     from mult_right_mono_neg[OF this le(2)]
```
```   962     have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
```
```   963       by (simp add: algebra_simps)
```
```   964     then have "f' y * (x - y) - (f x - f y) \<le> 0"
```
```   965       using le by auto
```
```   966     then have res: "f' y * (x - y) \<le> f x - f y"
```
```   967       by auto
```
```   968     have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
```
```   969       using * z1 by auto
```
```   970     also have "\<dots> = (z1 - x) * f'' z2"
```
```   971       using z2 by auto
```
```   972     finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
```
```   973       by simp
```
```   974     have A: "z1 - x \<ge> 0"
```
```   975       using z1 by auto
```
```   976     have "z2 \<in> C"
```
```   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>
```
```   978       by fastforce
```
```   979     then have B: "f'' z2 \<ge> 0"
```
```   980       using assms by auto
```
```   981     from A B have "(z1 - x) * f'' z2 \<ge> 0"
```
```   982       by auto
```
```   983     with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
```
```   984       by auto
```
```   985     from mult_right_mono[OF this ge(2)]
```
```   986     have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
```
```   987       by (simp add: algebra_simps)
```
```   988     then have "f y - f x - f' x * (y - x) \<ge> 0"
```
```   989       using ge by auto
```
```   990     then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
```
```   991       using res by auto
```
```   992   qed
```
```   993   show ?thesis
```
```   994   proof (cases "x = y")
```
```   995     case True
```
```   996     with x y show ?thesis by auto
```
```   997   next
```
```   998     case False
```
```   999     with less_imp x y show ?thesis
```
```  1000       by (auto simp: neq_iff)
```
```  1001   qed
```
```  1002 qed
```
```  1003
```
```  1004 lemma f''_ge0_imp_convex:
```
```  1005   fixes f :: "real \<Rightarrow> real"
```
```  1006   assumes conv: "convex C"
```
```  1007     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
```
```  1008     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
```
```  1009     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
```
```  1010   shows "convex_on C f"
```
```  1011   using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
```
```  1012   by fastforce
```
```  1013
```
```  1014 lemma minus_log_convex:
```
```  1015   fixes b :: real
```
```  1016   assumes "b > 1"
```
```  1017   shows "convex_on {0 <..} (\<lambda> x. - log b x)"
```
```  1018 proof -
```
```  1019   have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
```
```  1020     using DERIV_log by auto
```
```  1021   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
```
```  1022     by (auto simp: DERIV_minus)
```
```  1023   have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
```
```  1024     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
```
```  1025   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
```
```  1026   have "\<And>z::real. z > 0 \<Longrightarrow>
```
```  1027     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
```
```  1028     by auto
```
```  1029   then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
```
```  1030     DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
```
```  1031     unfolding inverse_eq_divide by (auto simp: mult.assoc)
```
```  1032   have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
```
```  1033     using \<open>b > 1\<close> by (auto intro!: less_imp_le)
```
```  1034   from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
```
```  1035   show ?thesis
```
```  1036     by auto
```
```  1037 qed
```
```  1038
```
```  1039
```
```  1040 subsection%unimportant \<open>Convexity of real functions\<close>
```
```  1041
```
```  1042 lemma convex_on_realI:
```
```  1043   assumes "connected A"
```
```  1044     and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
```
```  1045     and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
```
```  1046   shows "convex_on A f"
```
```  1047 proof (rule convex_on_linorderI)
```
```  1048   fix t x y :: real
```
```  1049   assume t: "t > 0" "t < 1"
```
```  1050   assume xy: "x \<in> A" "y \<in> A" "x < y"
```
```  1051   define z where "z = (1 - t) * x + t * y"
```
```  1052   with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
```
```  1053     using connected_contains_Icc by blast
```
```  1054
```
```  1055   from xy t have xz: "z > x"
```
```  1056     by (simp add: z_def algebra_simps)
```
```  1057   have "y - z = (1 - t) * (y - x)"
```
```  1058     by (simp add: z_def algebra_simps)
```
```  1059   also from xy t have "\<dots> > 0"
```
```  1060     by (intro mult_pos_pos) simp_all
```
```  1061   finally have yz: "z < y"
```
```  1062     by simp
```
```  1063
```
```  1064   from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
```
```  1065     by (intro MVT2) (auto intro!: assms(2))
```
```  1066   then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
```
```  1067     by auto
```
```  1068   from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
```
```  1069     by (intro MVT2) (auto intro!: assms(2))
```
```  1070   then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
```
```  1071     by auto
```
```  1072
```
```  1073   from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
```
```  1074   also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
```
```  1075     by auto
```
```  1076   with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
```
```  1077     by (intro assms(3)) auto
```
```  1078   also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
```
```  1079   finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
```
```  1080     using xz yz by (simp add: field_simps)
```
```  1081   also have "z - x = t * (y - x)"
```
```  1082     by (simp add: z_def algebra_simps)
```
```  1083   also have "y - z = (1 - t) * (y - x)"
```
```  1084     by (simp add: z_def algebra_simps)
```
```  1085   finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
```
```  1086     using xy by simp
```
```  1087   then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
```
```  1088     by (simp add: z_def algebra_simps)
```
```  1089 qed
```
```  1090
```
```  1091 lemma convex_on_inverse:
```
```  1092   assumes "A \<subseteq> {0<..}"
```
```  1093   shows "convex_on A (inverse :: real \<Rightarrow> real)"
```
```  1094 proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
```
```  1095   fix u v :: real
```
```  1096   assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
```
```  1097   with assms show "-inverse (u^2) \<le> -inverse (v^2)"
```
```  1098     by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
```
```  1099 qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
```
```  1100
```
```  1101 lemma convex_onD_Icc':
```
```  1102   assumes "convex_on {x..y} f" "c \<in> {x..y}"
```
```  1103   defines "d \<equiv> y - x"
```
```  1104   shows "f c \<le> (f y - f x) / d * (c - x) + f x"
```
```  1105 proof (cases x y rule: linorder_cases)
```
```  1106   case less
```
```  1107   then have d: "d > 0"
```
```  1108     by (simp add: d_def)
```
```  1109   from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
```
```  1110     by (simp_all add: d_def divide_simps)
```
```  1111   have "f c = f (x + (c - x) * 1)"
```
```  1112     by simp
```
```  1113   also from less have "1 = ((y - x) / d)"
```
```  1114     by (simp add: d_def)
```
```  1115   also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
```
```  1116     by (simp add: field_simps)
```
```  1117   also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
```
```  1118     using assms less by (intro convex_onD_Icc) simp_all
```
```  1119   also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
```
```  1120     by (simp add: field_simps)
```
```  1121   finally show ?thesis .
```
```  1122 qed (insert assms(2), simp_all)
```
```  1123
```
```  1124 lemma convex_onD_Icc'':
```
```  1125   assumes "convex_on {x..y} f" "c \<in> {x..y}"
```
```  1126   defines "d \<equiv> y - x"
```
```  1127   shows "f c \<le> (f x - f y) / d * (y - c) + f y"
```
```  1128 proof (cases x y rule: linorder_cases)
```
```  1129   case less
```
```  1130   then have d: "d > 0"
```
```  1131     by (simp add: d_def)
```
```  1132   from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
```
```  1133     by (simp_all add: d_def divide_simps)
```
```  1134   have "f c = f (y - (y - c) * 1)"
```
```  1135     by simp
```
```  1136   also from less have "1 = ((y - x) / d)"
```
```  1137     by (simp add: d_def)
```
```  1138   also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
```
```  1139     by (simp add: field_simps)
```
```  1140   also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
```
```  1141     using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
```
```  1142   also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
```
```  1143     by (simp add: field_simps)
```
```  1144   finally show ?thesis .
```
```  1145 qed (insert assms(2), simp_all)
```
```  1146
```
```  1147 lemma convex_supp_sum:
```
```  1148   assumes "convex S" and 1: "supp_sum u I = 1"
```
```  1149       and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
```
```  1150     shows "supp_sum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
```
```  1151 proof -
```
```  1152   have fin: "finite {i \<in> I. u i \<noteq> 0}"
```
```  1153     using 1 sum.infinite by (force simp: supp_sum_def support_on_def)
```
```  1154   then have eq: "supp_sum (\<lambda>i. u i *\<^sub>R f i) I = sum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
```
```  1155     by (force intro: sum.mono_neutral_left simp: supp_sum_def support_on_def)
```
```  1156   show ?thesis
```
```  1157     apply (simp add: eq)
```
```  1158     apply (rule convex_sum [OF fin \<open>convex S\<close>])
```
```  1159     using 1 assms apply (auto simp: supp_sum_def support_on_def)
```
```  1160     done
```
```  1161 qed
```
```  1162
```
```  1163 lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
```
```  1164   by (metis convex_translation translation_galois)
```
```  1165
```
```  1166 lemma convex_linear_image_eq [simp]:
```
```  1167     fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
```
```  1168     shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
```
```  1169     by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
```
```  1170
```
```  1171 lemma closure_bounded_linear_image_subset:
```
```  1172   assumes f: "bounded_linear f"
```
```  1173   shows "f ` closure S \<subseteq> closure (f ` S)"
```
```  1174   using linear_continuous_on [OF f] closed_closure closure_subset
```
```  1175   by (rule image_closure_subset)
```
```  1176
```
```  1177 lemma closure_linear_image_subset:
```
```  1178   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::real_normed_vector"
```
```  1179   assumes "linear f"
```
```  1180   shows "f ` (closure S) \<subseteq> closure (f ` S)"
```
```  1181   using assms unfolding linear_conv_bounded_linear
```
```  1182   by (rule closure_bounded_linear_image_subset)
```
```  1183
```
```  1184 lemma closed_injective_linear_image:
```
```  1185     fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1186     assumes S: "closed S" and f: "linear f" "inj f"
```
```  1187     shows "closed (f ` S)"
```
```  1188 proof -
```
```  1189   obtain g where g: "linear g" "g \<circ> f = id"
```
```  1190     using linear_injective_left_inverse [OF f] by blast
```
```  1191   then have confg: "continuous_on (range f) g"
```
```  1192     using linear_continuous_on linear_conv_bounded_linear by blast
```
```  1193   have [simp]: "g ` f ` S = S"
```
```  1194     using g by (simp add: image_comp)
```
```  1195   have cgf: "closed (g ` f ` S)"
```
```  1196     by (simp add: \<open>g \<circ> f = id\<close> S image_comp)
```
```  1197   have [simp]: "(range f \<inter> g -` S) = f ` S"
```
```  1198     using g unfolding o_def id_def image_def by auto metis+
```
```  1199   show ?thesis
```
```  1200   proof (rule closedin_closed_trans [of "range f"])
```
```  1201     show "closedin (subtopology euclidean (range f)) (f ` S)"
```
```  1202       using continuous_closedin_preimage [OF confg cgf] by simp
```
```  1203     show "closed (range f)"
```
```  1204       apply (rule closed_injective_image_subspace)
```
```  1205       using f apply (auto simp: linear_linear linear_injective_0)
```
```  1206       done
```
```  1207   qed
```
```  1208 qed
```
```  1209
```
```  1210 lemma closed_injective_linear_image_eq:
```
```  1211     fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1212     assumes f: "linear f" "inj f"
```
```  1213       shows "(closed(image f s) \<longleftrightarrow> closed s)"
```
```  1214   by (metis closed_injective_linear_image closure_eq closure_linear_image_subset closure_subset_eq f(1) f(2) inj_image_subset_iff)
```
```  1215
```
```  1216 lemma closure_injective_linear_image:
```
```  1217     fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1218     shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
```
```  1219   apply (rule subset_antisym)
```
```  1220   apply (simp add: closure_linear_image_subset)
```
```  1221   by (simp add: closure_minimal closed_injective_linear_image closure_subset image_mono)
```
```  1222
```
```  1223 lemma closure_bounded_linear_image:
```
```  1224     fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1225     shows "\<lbrakk>linear f; bounded S\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
```
```  1226   apply (rule subset_antisym, simp add: closure_linear_image_subset)
```
```  1227   apply (rule closure_minimal, simp add: closure_subset image_mono)
```
```  1228   by (meson bounded_closure closed_closure compact_continuous_image compact_eq_bounded_closed linear_continuous_on linear_conv_bounded_linear)
```
```  1229
```
```  1230 lemma closure_scaleR:
```
```  1231   fixes S :: "'a::real_normed_vector set"
```
```  1232   shows "(( *\<^sub>R) c) ` (closure S) = closure ((( *\<^sub>R) c) ` S)"
```
```  1233 proof
```
```  1234   show "(( *\<^sub>R) c) ` (closure S) \<subseteq> closure ((( *\<^sub>R) c) ` S)"
```
```  1235     using bounded_linear_scaleR_right
```
```  1236     by (rule closure_bounded_linear_image_subset)
```
```  1237   show "closure ((( *\<^sub>R) c) ` S) \<subseteq> (( *\<^sub>R) c) ` (closure S)"
```
```  1238     by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
```
```  1239 qed
```
```  1240
```
```  1241 lemma fst_linear: "linear fst"
```
```  1242   unfolding linear_iff by (simp add: algebra_simps)
```
```  1243
```
```  1244 lemma snd_linear: "linear snd"
```
```  1245   unfolding linear_iff by (simp add: algebra_simps)
```
```  1246
```
```  1247 lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
```
```  1248   unfolding linear_iff by (simp add: algebra_simps)
```
```  1249
```
```  1250 lemma vector_choose_size:
```
```  1251   assumes "0 \<le> c"
```
```  1252   obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
```
```  1253 proof -
```
```  1254   obtain a::'a where "a \<noteq> 0"
```
```  1255     using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
```
```  1256   then show ?thesis
```
```  1257     by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
```
```  1258 qed
```
```  1259
```
```  1260 lemma vector_choose_dist:
```
```  1261   assumes "0 \<le> c"
```
```  1262   obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
```
```  1263 by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
```
```  1264
```
```  1265 lemma sphere_eq_empty [simp]:
```
```  1266   fixes a :: "'a::{real_normed_vector, perfect_space}"
```
```  1267   shows "sphere a r = {} \<longleftrightarrow> r < 0"
```
```  1268 by (auto simp: sphere_def dist_norm) (metis dist_norm le_less_linear vector_choose_dist)
```
```  1269
```
```  1270 lemma sum_delta_notmem:
```
```  1271   assumes "x \<notin> s"
```
```  1272   shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
```
```  1273     and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
```
```  1274     and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
```
```  1275     and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
```
```  1276   apply (rule_tac [!] sum.cong)
```
```  1277   using assms
```
```  1278   apply auto
```
```  1279   done
```
```  1280
```
```  1281 lemma sum_delta'':
```
```  1282   fixes s::"'a::real_vector set"
```
```  1283   assumes "finite s"
```
```  1284   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)"
```
```  1285 proof -
```
```  1286   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)"
```
```  1287     by auto
```
```  1288   show ?thesis
```
```  1289     unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
```
```  1290 qed
```
```  1291
```
```  1292 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)"
```
```  1293   by (fact if_distrib)
```
```  1294
```
```  1295 lemma dist_triangle_eq:
```
```  1296   fixes x y z :: "'a::real_inner"
```
```  1297   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
```
```  1298     norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
```
```  1299 proof -
```
```  1300   have *: "x - y + (y - z) = x - z" by auto
```
```  1301   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
```
```  1302     by (auto simp:norm_minus_commute)
```
```  1303 qed
```
```  1304
```
```  1305
```
```  1306 subsection \<open>Affine set and affine hull\<close>
```
```  1307
```
```  1308 definition%important affine :: "'a::real_vector set \<Rightarrow> bool"
```
```  1309   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)"
```
```  1310
```
```  1311 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)"
```
```  1312   unfolding affine_def by (metis eq_diff_eq')
```
```  1313
```
```  1314 lemma affine_empty [iff]: "affine {}"
```
```  1315   unfolding affine_def by auto
```
```  1316
```
```  1317 lemma affine_sing [iff]: "affine {x}"
```
```  1318   unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
```
```  1319
```
```  1320 lemma affine_UNIV [iff]: "affine UNIV"
```
```  1321   unfolding affine_def by auto
```
```  1322
```
```  1323 lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
```
```  1324   unfolding affine_def by auto
```
```  1325
```
```  1326 lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
```
```  1327   unfolding affine_def by auto
```
```  1328
```
```  1329 lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
```
```  1330   apply (clarsimp simp add: affine_def)
```
```  1331   apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
```
```  1332   apply (auto simp: algebra_simps)
```
```  1333   done
```
```  1334
```
```  1335 lemma affine_affine_hull [simp]: "affine(affine hull s)"
```
```  1336   unfolding hull_def
```
```  1337   using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
```
```  1338
```
```  1339 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
```
```  1340   by (metis affine_affine_hull hull_same)
```
```  1341
```
```  1342 lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
```
```  1343   by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
```
```  1344
```
```  1345
```
```  1346 subsubsection%unimportant \<open>Some explicit formulations (from Lars Schewe)\<close>
```
```  1347
```
```  1348 lemma affine:
```
```  1349   fixes V::"'a::real_vector set"
```
```  1350   shows "affine V \<longleftrightarrow>
```
```  1351          (\<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)"
```
```  1352 proof -
```
```  1353   have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
```
```  1354     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
```
```  1355   proof (cases "x = y")
```
```  1356     case True
```
```  1357     then show ?thesis
```
```  1358       using that by (metis scaleR_add_left scaleR_one)
```
```  1359   next
```
```  1360     case False
```
```  1361     then show ?thesis
```
```  1362       using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
```
```  1363   qed
```
```  1364   moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
```
```  1365                 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"
```
```  1366                   and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
```
```  1367   proof -
```
```  1368     define n where "n = card S"
```
```  1369     consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
```
```  1370     then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
```
```  1371     proof cases
```
```  1372       assume "card S = 1"
```
```  1373       then obtain a where "S={a}"
```
```  1374         by (auto simp: card_Suc_eq)
```
```  1375       then show ?thesis
```
```  1376         using that by simp
```
```  1377     next
```
```  1378       assume "card S = 2"
```
```  1379       then obtain a b where "S = {a, b}"
```
```  1380         by (metis Suc_1 card_1_singletonE card_Suc_eq)
```
```  1381       then show ?thesis
```
```  1382         using *[of a b] that
```
```  1383         by (auto simp: sum_clauses(2))
```
```  1384     next
```
```  1385       assume "card S > 2"
```
```  1386       then show ?thesis using that n_def
```
```  1387       proof (induct n arbitrary: u S)
```
```  1388         case 0
```
```  1389         then show ?case by auto
```
```  1390       next
```
```  1391         case (Suc n u S)
```
```  1392         have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
```
```  1393           using that unfolding card_eq_sum by auto
```
```  1394         with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
```
```  1395         have c: "card (S - {x}) = card S - 1"
```
```  1396           by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
```
```  1397         have "sum u (S - {x}) = 1 - u x"
```
```  1398           by (simp add: Suc.prems sum_diff1_ring \<open>x \<in> S\<close>)
```
```  1399         with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
```
```  1400           by auto
```
```  1401         have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
```
```  1402         proof (cases "card (S - {x}) > 2")
```
```  1403           case True
```
```  1404           then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
```
```  1405             using Suc.prems c by force+
```
```  1406           show ?thesis
```
```  1407           proof (rule Suc.hyps)
```
```  1408             show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
```
```  1409               by (auto simp: eq1 sum_distrib_left[symmetric])
```
```  1410           qed (use S Suc.prems True in auto)
```
```  1411         next
```
```  1412           case False
```
```  1413           then have "card (S - {x}) = Suc (Suc 0)"
```
```  1414             using Suc.prems c by auto
```
```  1415           then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
```
```  1416             unfolding card_Suc_eq by auto
```
```  1417           then show ?thesis
```
```  1418             using eq1 \<open>S \<subseteq> V\<close>
```
```  1419             by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
```
```  1420         qed
```
```  1421         have "u x + (1 - u x) = 1 \<Longrightarrow>
```
```  1422           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"
```
```  1423           by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
```
```  1424         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)"
```
```  1425           by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
```
```  1426         ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
```
```  1427           by (simp add: x)
```
```  1428       qed
```
```  1429     qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
```
```  1430   qed
```
```  1431   ultimately show ?thesis
```
```  1432     unfolding affine_def by meson
```
```  1433 qed
```
```  1434
```
```  1435
```
```  1436 lemma affine_hull_explicit:
```
```  1437   "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}"
```
```  1438   (is "_ = ?rhs")
```
```  1439 proof (rule hull_unique)
```
```  1440   show "p \<subseteq> ?rhs"
```
```  1441   proof (intro subsetI CollectI exI conjI)
```
```  1442     show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
```
```  1443       by auto
```
```  1444   qed auto
```
```  1445   show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
```
```  1446     using that unfolding affine by blast
```
```  1447   show "affine ?rhs"
```
```  1448     unfolding affine_def
```
```  1449   proof clarify
```
```  1450     fix u v :: real and sx ux sy uy
```
```  1451     assume uv: "u + v = 1"
```
```  1452       and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
```
```  1453       and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
```
```  1454     have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
```
```  1455       by auto
```
```  1456     show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
```
```  1457         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)"
```
```  1458     proof (intro exI conjI)
```
```  1459       show "finite (sx \<union> sy)"
```
```  1460         using x y by auto
```
```  1461       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"
```
```  1462         using x y uv
```
```  1463         by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
```
```  1464       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)
```
```  1465           = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
```
```  1466         using x y
```
```  1467         unfolding scaleR_left_distrib scaleR_zero_left if_smult
```
```  1468         by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric]  **)
```
```  1469       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)"
```
```  1470         unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
```
```  1471       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)
```
```  1472                   = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
```
```  1473     qed (use x y in auto)
```
```  1474   qed
```
```  1475 qed
```
```  1476
```
```  1477 lemma affine_hull_finite:
```
```  1478   assumes "finite S"
```
```  1479   shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```  1480 proof -
```
```  1481   have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
```
```  1482     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
```
```  1483   proof -
```
```  1484     have "S \<inter> F = F"
```
```  1485       using that by auto
```
```  1486     show ?thesis
```
```  1487     proof (intro exI conjI)
```
```  1488       show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
```
```  1489         by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
```
```  1490       show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
```
```  1491         by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
```
```  1492     qed
```
```  1493   qed
```
```  1494   show ?thesis
```
```  1495     unfolding affine_hull_explicit using assms
```
```  1496     by (fastforce dest: *)
```
```  1497 qed
```
```  1498
```
```  1499
```
```  1500 subsubsection%unimportant \<open>Stepping theorems and hence small special cases\<close>
```
```  1501
```
```  1502 lemma affine_hull_empty[simp]: "affine hull {} = {}"
```
```  1503   by simp
```
```  1504
```
```  1505 lemma affine_hull_finite_step:
```
```  1506   fixes y :: "'a::real_vector"
```
```  1507   shows "finite S \<Longrightarrow>
```
```  1508       (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
```
```  1509       (\<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")
```
```  1510 proof -
```
```  1511   assume fin: "finite S"
```
```  1512   show "?lhs = ?rhs"
```
```  1513   proof
```
```  1514     assume ?lhs
```
```  1515     then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
```
```  1516       by auto
```
```  1517     show ?rhs
```
```  1518     proof (cases "a \<in> S")
```
```  1519       case True
```
```  1520       then show ?thesis
```
```  1521         using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
```
```  1522     next
```
```  1523       case False
```
```  1524       show ?thesis
```
```  1525         by (rule exI [where x="u a"]) (use u fin False in auto)
```
```  1526     qed
```
```  1527   next
```
```  1528     assume ?rhs
```
```  1529     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"
```
```  1530       by auto
```
```  1531     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)"
```
```  1532       by auto
```
```  1533     show ?lhs
```
```  1534     proof (cases "a \<in> S")
```
```  1535       case True
```
```  1536       show ?thesis
```
```  1537         by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
```
```  1538            (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
```
```  1539     next
```
```  1540       case False
```
```  1541       then show ?thesis
```
```  1542         apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
```
```  1543         apply (simp add: vu sum_clauses(2)[OF fin] *)
```
```  1544         by (simp add: sum_delta_notmem(3) vu)
```
```  1545     qed
```
```  1546   qed
```
```  1547 qed
```
```  1548
```
```  1549 lemma affine_hull_2:
```
```  1550   fixes a b :: "'a::real_vector"
```
```  1551   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
```
```  1552   (is "?lhs = ?rhs")
```
```  1553 proof -
```
```  1554   have *:
```
```  1555     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
```
```  1556     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
```
```  1557   have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
```
```  1558     using affine_hull_finite[of "{a,b}"] by auto
```
```  1559   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
```
```  1560     by (simp add: affine_hull_finite_step[of "{b}" a])
```
```  1561   also have "\<dots> = ?rhs" unfolding * by auto
```
```  1562   finally show ?thesis by auto
```
```  1563 qed
```
```  1564
```
```  1565 lemma affine_hull_3:
```
```  1566   fixes a b c :: "'a::real_vector"
```
```  1567   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}"
```
```  1568 proof -
```
```  1569   have *:
```
```  1570     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
```
```  1571     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
```
```  1572   show ?thesis
```
```  1573     apply (simp add: affine_hull_finite affine_hull_finite_step)
```
```  1574     unfolding *
```
```  1575     apply safe
```
```  1576      apply (metis add.assoc)
```
```  1577     apply (rule_tac x=u in exI, force)
```
```  1578     done
```
```  1579 qed
```
```  1580
```
```  1581 lemma mem_affine:
```
```  1582   assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
```
```  1583   shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
```
```  1584   using assms affine_def[of S] by auto
```
```  1585
```
```  1586 lemma mem_affine_3:
```
```  1587   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
```
```  1588   shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
```
```  1589 proof -
```
```  1590   have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
```
```  1591     using affine_hull_3[of x y z] assms by auto
```
```  1592   moreover
```
```  1593   have "affine hull {x, y, z} \<subseteq> affine hull S"
```
```  1594     using hull_mono[of "{x, y, z}" "S"] assms by auto
```
```  1595   moreover
```
```  1596   have "affine hull S = S"
```
```  1597     using assms affine_hull_eq[of S] by auto
```
```  1598   ultimately show ?thesis by auto
```
```  1599 qed
```
```  1600
```
```  1601 lemma mem_affine_3_minus:
```
```  1602   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
```
```  1603   shows "x + v *\<^sub>R (y-z) \<in> S"
```
```  1604   using mem_affine_3[of S x y z 1 v "-v"] assms
```
```  1605   by (simp add: algebra_simps)
```
```  1606
```
```  1607 corollary mem_affine_3_minus2:
```
```  1608     "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
```
```  1609   by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
```
```  1610
```
```  1611
```
```  1612 subsubsection%unimportant \<open>Some relations between affine hull and subspaces\<close>
```
```  1613
```
```  1614 lemma affine_hull_insert_subset_span:
```
```  1615   "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
```
```  1616 proof -
```
```  1617   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)"
```
```  1618     if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
```
```  1619     for x F u
```
```  1620   proof -
```
```  1621     have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
```
```  1622       using that by auto
```
```  1623     show ?thesis
```
```  1624     proof (intro exI conjI)
```
```  1625       show "finite ((\<lambda>x. x - a) ` (F - {a}))"
```
```  1626         by (simp add: that(1))
```
```  1627       show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
```
```  1628         by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
```
```  1629             sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
```
```  1630     qed (use \<open>F \<subseteq> insert a S\<close> in auto)
```
```  1631   qed
```
```  1632   then show ?thesis
```
```  1633     unfolding affine_hull_explicit span_explicit by blast
```
```  1634 qed
```
```  1635
```
```  1636 lemma affine_hull_insert_span:
```
```  1637   assumes "a \<notin> S"
```
```  1638   shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x.  x \<in> S}}"
```
```  1639 proof -
```
```  1640   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"
```
```  1641     if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
```
```  1642   proof -
```
```  1643     from that
```
```  1644     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"
```
```  1645       unfolding span_explicit by auto
```
```  1646     define F where "F = (\<lambda>x. x + a) ` T"
```
```  1647     have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
```
```  1648       unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
```
```  1649     have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
```
```  1650       using F assms by auto
```
```  1651     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"
```
```  1652       apply (rule_tac x = "insert a F" in exI)
```
```  1653       apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
```
```  1654       using assms F
```
```  1655       apply (auto simp:  sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
```
```  1656       done
```
```  1657   qed
```
```  1658   show ?thesis
```
```  1659     by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
```
```  1660 qed
```
```  1661
```
```  1662 lemma affine_hull_span:
```
```  1663   assumes "a \<in> S"
```
```  1664   shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
```
```  1665   using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
```
```  1666
```
```  1667
```
```  1668 subsubsection%unimportant \<open>Parallel affine sets\<close>
```
```  1669
```
```  1670 definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
```
```  1671   where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
```
```  1672
```
```  1673 lemma affine_parallel_expl_aux:
```
```  1674   fixes S T :: "'a::real_vector set"
```
```  1675   assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
```
```  1676   shows "T = (\<lambda>x. a + x) ` S"
```
```  1677 proof -
```
```  1678   have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
```
```  1679     using that
```
```  1680     by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
```
```  1681   moreover have "T \<ge> (\<lambda>x. a + x) ` S"
```
```  1682     using assms by auto
```
```  1683   ultimately show ?thesis by auto
```
```  1684 qed
```
```  1685
```
```  1686 lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
```
```  1687   unfolding affine_parallel_def
```
```  1688   using affine_parallel_expl_aux[of S _ T] by auto
```
```  1689
```
```  1690 lemma affine_parallel_reflex: "affine_parallel S S"
```
```  1691   unfolding affine_parallel_def
```
```  1692   using image_add_0 by blast
```
```  1693
```
```  1694 lemma affine_parallel_commut:
```
```  1695   assumes "affine_parallel A B"
```
```  1696   shows "affine_parallel B A"
```
```  1697 proof -
```
```  1698   from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
```
```  1699     unfolding affine_parallel_def by auto
```
```  1700   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
```
```  1701   from B show ?thesis
```
```  1702     using translation_galois [of B a A]
```
```  1703     unfolding affine_parallel_def by auto
```
```  1704 qed
```
```  1705
```
```  1706 lemma affine_parallel_assoc:
```
```  1707   assumes "affine_parallel A B"
```
```  1708     and "affine_parallel B C"
```
```  1709   shows "affine_parallel A C"
```
```  1710 proof -
```
```  1711   from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
```
```  1712     unfolding affine_parallel_def by auto
```
```  1713   moreover
```
```  1714   from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
```
```  1715     unfolding affine_parallel_def by auto
```
```  1716   ultimately show ?thesis
```
```  1717     using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
```
```  1718 qed
```
```  1719
```
```  1720 lemma affine_translation_aux:
```
```  1721   fixes a :: "'a::real_vector"
```
```  1722   assumes "affine ((\<lambda>x. a + x) ` S)"
```
```  1723   shows "affine S"
```
```  1724 proof -
```
```  1725   {
```
```  1726     fix x y u v
```
```  1727     assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
```
```  1728     then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
```
```  1729       by auto
```
```  1730     then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
```
```  1731       using xy assms unfolding affine_def by auto
```
```  1732     have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
```
```  1733       by (simp add: algebra_simps)
```
```  1734     also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
```
```  1735       using \<open>u + v = 1\<close> by auto
```
```  1736     ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
```
```  1737       using h1 by auto
```
```  1738     then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
```
```  1739   }
```
```  1740   then show ?thesis unfolding affine_def by auto
```
```  1741 qed
```
```  1742
```
```  1743 lemma affine_translation:
```
```  1744   fixes a :: "'a::real_vector"
```
```  1745   shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
```
```  1746 proof -
```
```  1747   have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
```
```  1748     using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
```
```  1749     using translation_assoc[of "-a" a S] by auto
```
```  1750   then show ?thesis using affine_translation_aux by auto
```
```  1751 qed
```
```  1752
```
```  1753 lemma parallel_is_affine:
```
```  1754   fixes S T :: "'a::real_vector set"
```
```  1755   assumes "affine S" "affine_parallel S T"
```
```  1756   shows "affine T"
```
```  1757 proof -
```
```  1758   from assms obtain a where "T = (\<lambda>x. a + x) ` S"
```
```  1759     unfolding affine_parallel_def by auto
```
```  1760   then show ?thesis
```
```  1761     using affine_translation assms by auto
```
```  1762 qed
```
```  1763
```
```  1764 lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
```
```  1765   unfolding subspace_def affine_def by auto
```
```  1766
```
```  1767
```
```  1768 subsubsection%unimportant \<open>Subspace parallel to an affine set\<close>
```
```  1769
```
```  1770 lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
```
```  1771 proof -
```
```  1772   have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
```
```  1773     using subspace_imp_affine[of S] subspace_0 by auto
```
```  1774   {
```
```  1775     assume assm: "affine S \<and> 0 \<in> S"
```
```  1776     {
```
```  1777       fix c :: real
```
```  1778       fix x
```
```  1779       assume x: "x \<in> S"
```
```  1780       have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
```
```  1781       moreover
```
```  1782       have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
```
```  1783         using affine_alt[of S] assm x by auto
```
```  1784       ultimately have "c *\<^sub>R x \<in> S" by auto
```
```  1785     }
```
```  1786     then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
```
```  1787
```
```  1788     {
```
```  1789       fix x y
```
```  1790       assume xy: "x \<in> S" "y \<in> S"
```
```  1791       define u where "u = (1 :: real)/2"
```
```  1792       have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
```
```  1793         by auto
```
```  1794       moreover
```
```  1795       have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
```
```  1796         by (simp add: algebra_simps)
```
```  1797       moreover
```
```  1798       have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
```
```  1799         using affine_alt[of S] assm xy by auto
```
```  1800       ultimately
```
```  1801       have "(1/2) *\<^sub>R (x+y) \<in> S"
```
```  1802         using u_def by auto
```
```  1803       moreover
```
```  1804       have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
```
```  1805         by auto
```
```  1806       ultimately
```
```  1807       have "x + y \<in> S"
```
```  1808         using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
```
```  1809     }
```
```  1810     then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
```
```  1811       by auto
```
```  1812     then have "subspace S"
```
```  1813       using h1 assm unfolding subspace_def by auto
```
```  1814   }
```
```  1815   then show ?thesis using h0 by metis
```
```  1816 qed
```
```  1817
```
```  1818 lemma affine_diffs_subspace:
```
```  1819   assumes "affine S" "a \<in> S"
```
```  1820   shows "subspace ((\<lambda>x. (-a)+x) ` S)"
```
```  1821 proof -
```
```  1822   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
```
```  1823   have "affine ((\<lambda>x. (-a)+x) ` S)"
```
```  1824     using  affine_translation assms by auto
```
```  1825   moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
```
```  1826     using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
```
```  1827   ultimately show ?thesis using subspace_affine by auto
```
```  1828 qed
```
```  1829
```
```  1830 lemma parallel_subspace_explicit:
```
```  1831   assumes "affine S"
```
```  1832     and "a \<in> S"
```
```  1833   assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
```
```  1834   shows "subspace L \<and> affine_parallel S L"
```
```  1835 proof -
```
```  1836   from assms have "L = plus (- a) ` S" by auto
```
```  1837   then have par: "affine_parallel S L"
```
```  1838     unfolding affine_parallel_def ..
```
```  1839   then have "affine L" using assms parallel_is_affine by auto
```
```  1840   moreover have "0 \<in> L"
```
```  1841     using assms by auto
```
```  1842   ultimately show ?thesis
```
```  1843     using subspace_affine par by auto
```
```  1844 qed
```
```  1845
```
```  1846 lemma parallel_subspace_aux:
```
```  1847   assumes "subspace A"
```
```  1848     and "subspace B"
```
```  1849     and "affine_parallel A B"
```
```  1850   shows "A \<supseteq> B"
```
```  1851 proof -
```
```  1852   from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
```
```  1853     using affine_parallel_expl[of A B] by auto
```
```  1854   then have "-a \<in> A"
```
```  1855     using assms subspace_0[of B] by auto
```
```  1856   then have "a \<in> A"
```
```  1857     using assms subspace_neg[of A "-a"] by auto
```
```  1858   then show ?thesis
```
```  1859     using assms a unfolding subspace_def by auto
```
```  1860 qed
```
```  1861
```
```  1862 lemma parallel_subspace:
```
```  1863   assumes "subspace A"
```
```  1864     and "subspace B"
```
```  1865     and "affine_parallel A B"
```
```  1866   shows "A = B"
```
```  1867 proof
```
```  1868   show "A \<supseteq> B"
```
```  1869     using assms parallel_subspace_aux by auto
```
```  1870   show "A \<subseteq> B"
```
```  1871     using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
```
```  1872 qed
```
```  1873
```
```  1874 lemma affine_parallel_subspace:
```
```  1875   assumes "affine S" "S \<noteq> {}"
```
```  1876   shows "\<exists>!L. subspace L \<and> affine_parallel S L"
```
```  1877 proof -
```
```  1878   have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
```
```  1879     using assms parallel_subspace_explicit by auto
```
```  1880   {
```
```  1881     fix L1 L2
```
```  1882     assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
```
```  1883     then have "affine_parallel L1 L2"
```
```  1884       using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
```
```  1885     then have "L1 = L2"
```
```  1886       using ass parallel_subspace by auto
```
```  1887   }
```
```  1888   then show ?thesis using ex by auto
```
```  1889 qed
```
```  1890
```
```  1891
```
```  1892 subsection \<open>Cones\<close>
```
```  1893
```
```  1894 definition%important cone :: "'a::real_vector set \<Rightarrow> bool"
```
```  1895   where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
```
```  1896
```
```  1897 lemma cone_empty[intro, simp]: "cone {}"
```
```  1898   unfolding cone_def by auto
```
```  1899
```
```  1900 lemma cone_univ[intro, simp]: "cone UNIV"
```
```  1901   unfolding cone_def by auto
```
```  1902
```
```  1903 lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
```
```  1904   unfolding cone_def by auto
```
```  1905
```
```  1906 lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
```
```  1907   by (simp add: cone_def subspace_scale)
```
```  1908
```
```  1909
```
```  1910 subsubsection \<open>Conic hull\<close>
```
```  1911
```
```  1912 lemma cone_cone_hull: "cone (cone hull s)"
```
```  1913   unfolding hull_def by auto
```
```  1914
```
```  1915 lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
```
```  1916   apply (rule hull_eq)
```
```  1917   using cone_Inter
```
```  1918   unfolding subset_eq
```
```  1919   apply auto
```
```  1920   done
```
```  1921
```
```  1922 lemma mem_cone:
```
```  1923   assumes "cone S" "x \<in> S" "c \<ge> 0"
```
```  1924   shows "c *\<^sub>R x \<in> S"
```
```  1925   using assms cone_def[of S] by auto
```
```  1926
```
```  1927 lemma cone_contains_0:
```
```  1928   assumes "cone S"
```
```  1929   shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
```
```  1930 proof -
```
```  1931   {
```
```  1932     assume "S \<noteq> {}"
```
```  1933     then obtain a where "a \<in> S" by auto
```
```  1934     then have "0 \<in> S"
```
```  1935       using assms mem_cone[of S a 0] by auto
```
```  1936   }
```
```  1937   then show ?thesis by auto
```
```  1938 qed
```
```  1939
```
```  1940 lemma cone_0: "cone {0}"
```
```  1941   unfolding cone_def by auto
```
```  1942
```
```  1943 lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
```
```  1944   unfolding cone_def by blast
```
```  1945
```
```  1946 lemma cone_iff:
```
```  1947   assumes "S \<noteq> {}"
```
```  1948   shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (( *\<^sub>R) c) ` S = S)"
```
```  1949 proof -
```
```  1950   {
```
```  1951     assume "cone S"
```
```  1952     {
```
```  1953       fix c :: real
```
```  1954       assume "c > 0"
```
```  1955       {
```
```  1956         fix x
```
```  1957         assume "x \<in> S"
```
```  1958         then have "x \<in> (( *\<^sub>R) c) ` S"
```
```  1959           unfolding image_def
```
```  1960           using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
```
```  1961             exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
```
```  1962           by auto
```
```  1963       }
```
```  1964       moreover
```
```  1965       {
```
```  1966         fix x
```
```  1967         assume "x \<in> (( *\<^sub>R) c) ` S"
```
```  1968         then have "x \<in> S"
```
```  1969           using \<open>cone S\<close> \<open>c > 0\<close>
```
```  1970           unfolding cone_def image_def \<open>c > 0\<close> by auto
```
```  1971       }
```
```  1972       ultimately have "(( *\<^sub>R) c) ` S = S" by auto
```
```  1973     }
```
```  1974     then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (( *\<^sub>R) c) ` S = S)"
```
```  1975       using \<open>cone S\<close> cone_contains_0[of S] assms by auto
```
```  1976   }
```
```  1977   moreover
```
```  1978   {
```
```  1979     assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (( *\<^sub>R) c) ` S = S)"
```
```  1980     {
```
```  1981       fix x
```
```  1982       assume "x \<in> S"
```
```  1983       fix c1 :: real
```
```  1984       assume "c1 \<ge> 0"
```
```  1985       then have "c1 = 0 \<or> c1 > 0" by auto
```
```  1986       then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
```
```  1987     }
```
```  1988     then have "cone S" unfolding cone_def by auto
```
```  1989   }
```
```  1990   ultimately show ?thesis by blast
```
```  1991 qed
```
```  1992
```
```  1993 lemma cone_hull_empty: "cone hull {} = {}"
```
```  1994   by (metis cone_empty cone_hull_eq)
```
```  1995
```
```  1996 lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
```
```  1997   by (metis bot_least cone_hull_empty hull_subset xtrans(5))
```
```  1998
```
```  1999 lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
```
```  2000   using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
```
```  2001   by auto
```
```  2002
```
```  2003 lemma mem_cone_hull:
```
```  2004   assumes "x \<in> S" "c \<ge> 0"
```
```  2005   shows "c *\<^sub>R x \<in> cone hull S"
```
```  2006   by (metis assms cone_cone_hull hull_inc mem_cone)
```
```  2007
```
```  2008 proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
```
```  2009   (is "?lhs = ?rhs")
```
```  2010 proof -
```
```  2011   {
```
```  2012     fix x
```
```  2013     assume "x \<in> ?rhs"
```
```  2014     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
```
```  2015       by auto
```
```  2016     fix c :: real
```
```  2017     assume c: "c \<ge> 0"
```
```  2018     then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
```
```  2019       using x by (simp add: algebra_simps)
```
```  2020     moreover
```
```  2021     have "c * cx \<ge> 0" using c x by auto
```
```  2022     ultimately
```
```  2023     have "c *\<^sub>R x \<in> ?rhs" using x by auto
```
```  2024   }
```
```  2025   then have "cone ?rhs"
```
```  2026     unfolding cone_def by auto
```
```  2027   then have "?rhs \<in> Collect cone"
```
```  2028     unfolding mem_Collect_eq by auto
```
```  2029   {
```
```  2030     fix x
```
```  2031     assume "x \<in> S"
```
```  2032     then have "1 *\<^sub>R x \<in> ?rhs"
```
```  2033       apply auto
```
```  2034       apply (rule_tac x = 1 in exI, auto)
```
```  2035       done
```
```  2036     then have "x \<in> ?rhs" by auto
```
```  2037   }
```
```  2038   then have "S \<subseteq> ?rhs" by auto
```
```  2039   then have "?lhs \<subseteq> ?rhs"
```
```  2040     using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
```
```  2041   moreover
```
```  2042   {
```
```  2043     fix x
```
```  2044     assume "x \<in> ?rhs"
```
```  2045     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
```
```  2046       by auto
```
```  2047     then have "xx \<in> cone hull S"
```
```  2048       using hull_subset[of S] by auto
```
```  2049     then have "x \<in> ?lhs"
```
```  2050       using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
```
```  2051   }
```
```  2052   ultimately show ?thesis by auto
```
```  2053 qed
```
```  2054
```
```  2055 lemma cone_closure:
```
```  2056   fixes S :: "'a::real_normed_vector set"
```
```  2057   assumes "cone S"
```
```  2058   shows "cone (closure S)"
```
```  2059 proof (cases "S = {}")
```
```  2060   case True
```
```  2061   then show ?thesis by auto
```
```  2062 next
```
```  2063   case False
```
```  2064   then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` S = S)"
```
```  2065     using cone_iff[of S] assms by auto
```
```  2066   then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` closure S = closure S)"
```
```  2067     using closure_subset by (auto simp: closure_scaleR)
```
```  2068   then show ?thesis
```
```  2069     using False cone_iff[of "closure S"] by auto
```
```  2070 qed
```
```  2071
```
```  2072
```
```  2073 subsection \<open>Affine dependence and consequential theorems (from Lars Schewe)\<close>
```
```  2074
```
```  2075 definition%important affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
```
```  2076   where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
```
```  2077
```
```  2078 lemma affine_dependent_subset:
```
```  2079    "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
```
```  2080 apply (simp add: affine_dependent_def Bex_def)
```
```  2081 apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
```
```  2082 done
```
```  2083
```
```  2084 lemma affine_independent_subset:
```
```  2085   shows "\<lbrakk>~ affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> ~ affine_dependent s"
```
```  2086 by (metis affine_dependent_subset)
```
```  2087
```
```  2088 lemma affine_independent_Diff:
```
```  2089    "~ affine_dependent s \<Longrightarrow> ~ affine_dependent(s - t)"
```
```  2090 by (meson Diff_subset affine_dependent_subset)
```
```  2091
```
```  2092 proposition affine_dependent_explicit:
```
```  2093   "affine_dependent p \<longleftrightarrow>
```
```  2094     (\<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)"
```
```  2095 proof -
```
```  2096   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"
```
```  2097     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
```
```  2098   proof (intro exI conjI)
```
```  2099     have "x \<notin> S"
```
```  2100       using that by auto
```
```  2101     then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
```
```  2102       using that by (simp add: sum_delta_notmem)
```
```  2103     show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
```
```  2104       using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
```
```  2105   qed (use that in auto)
```
```  2106   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"
```
```  2107     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
```
```  2108   proof (intro bexI exI conjI)
```
```  2109     have "S \<noteq> {v}"
```
```  2110       using that by auto
```
```  2111     then show "S - {v} \<noteq> {}"
```
```  2112       using that by auto
```
```  2113     show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
```
```  2114       unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
```
```  2115     show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
```
```  2116       unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
```
```  2117                 scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
```
```  2118       using that by auto
```
```  2119     show "S - {v} \<subseteq> p - {v}"
```
```  2120       using that by auto
```
```  2121   qed (use that in auto)
```
```  2122   ultimately show ?thesis
```
```  2123     unfolding affine_dependent_def affine_hull_explicit by auto
```
```  2124 qed
```
```  2125
```
```  2126 lemma affine_dependent_explicit_finite:
```
```  2127   fixes S :: "'a::real_vector set"
```
```  2128   assumes "finite S"
```
```  2129   shows "affine_dependent S \<longleftrightarrow>
```
```  2130     (\<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)"
```
```  2131   (is "?lhs = ?rhs")
```
```  2132 proof
```
```  2133   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)"
```
```  2134     by auto
```
```  2135   assume ?lhs
```
```  2136   then obtain t u v where
```
```  2137     "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"
```
```  2138     unfolding affine_dependent_explicit by auto
```
```  2139   then show ?rhs
```
```  2140     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
```
```  2141     apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
```
```  2142     done
```
```  2143 next
```
```  2144   assume ?rhs
```
```  2145   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"
```
```  2146     by auto
```
```  2147   then show ?lhs unfolding affine_dependent_explicit
```
```  2148     using assms by auto
```
```  2149 qed
```
```  2150
```
```  2151
```
```  2152 subsection%unimportant \<open>Connectedness of convex sets\<close>
```
```  2153
```
```  2154 lemma connectedD:
```
```  2155   "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 = {}"
```
```  2156   by (rule Topological_Spaces.topological_space_class.connectedD)
```
```  2157
```
```  2158 lemma convex_connected:
```
```  2159   fixes S :: "'a::real_normed_vector set"
```
```  2160   assumes "convex S"
```
```  2161   shows "connected S"
```
```  2162 proof (rule connectedI)
```
```  2163   fix A B
```
```  2164   assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
```
```  2165   moreover
```
```  2166   assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
```
```  2167   then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
```
```  2168   define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
```
```  2169   then have "continuous_on {0 .. 1} f"
```
```  2170     by (auto intro!: continuous_intros)
```
```  2171   then have "connected (f ` {0 .. 1})"
```
```  2172     by (auto intro!: connected_continuous_image)
```
```  2173   note connectedD[OF this, of A B]
```
```  2174   moreover have "a \<in> A \<inter> f ` {0 .. 1}"
```
```  2175     using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
```
```  2176   moreover have "b \<in> B \<inter> f ` {0 .. 1}"
```
```  2177     using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
```
```  2178   moreover have "f ` {0 .. 1} \<subseteq> S"
```
```  2179     using \<open>convex S\<close> a b unfolding convex_def f_def by auto
```
```  2180   ultimately show False by auto
```
```  2181 qed
```
```  2182
```
```  2183 corollary connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
```
```  2184   by (simp add: convex_connected)
```
```  2185
```
```  2186 corollary component_complement_connected:
```
```  2187   fixes S :: "'a::real_normed_vector set"
```
```  2188   assumes "connected S" "C \<in> components (-S)"
```
```  2189   shows "connected(-C)"
```
```  2190   using component_diff_connected [of S UNIV] assms
```
```  2191   by (auto simp: Compl_eq_Diff_UNIV)
```
```  2192
```
```  2193 proposition clopen:
```
```  2194   fixes S :: "'a :: real_normed_vector set"
```
```  2195   shows "closed S \<and> open S \<longleftrightarrow> S = {} \<or> S = UNIV"
```
```  2196     by (force intro!: connected_UNIV [unfolded connected_clopen, rule_format])
```
```  2197
```
```  2198 corollary compact_open:
```
```  2199   fixes S :: "'a :: euclidean_space set"
```
```  2200   shows "compact S \<and> open S \<longleftrightarrow> S = {}"
```
```  2201   by (auto simp: compact_eq_bounded_closed clopen)
```
```  2202
```
```  2203 corollary finite_imp_not_open:
```
```  2204     fixes S :: "'a::{real_normed_vector, perfect_space} set"
```
```  2205     shows "\<lbrakk>finite S; open S\<rbrakk> \<Longrightarrow> S={}"
```
```  2206   using clopen [of S] finite_imp_closed not_bounded_UNIV by blast
```
```  2207
```
```  2208 corollary empty_interior_finite:
```
```  2209     fixes S :: "'a::{real_normed_vector, perfect_space} set"
```
```  2210     shows "finite S \<Longrightarrow> interior S = {}"
```
```  2211   by (metis interior_subset finite_subset open_interior [of S] finite_imp_not_open)
```
```  2212
```
```  2213 text \<open>Balls, being convex, are connected.\<close>
```
```  2214
```
```  2215 lemma convex_prod:
```
```  2216   assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
```
```  2217   shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
```
```  2218   using assms unfolding convex_def
```
```  2219   by (auto simp: inner_add_left)
```
```  2220
```
```  2221 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
```
```  2222   by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
```
```  2223
```
```  2224 lemma convex_local_global_minimum:
```
```  2225   fixes s :: "'a::real_normed_vector set"
```
```  2226   assumes "e > 0"
```
```  2227     and "convex_on s f"
```
```  2228     and "ball x e \<subseteq> s"
```
```  2229     and "\<forall>y\<in>ball x e. f x \<le> f y"
```
```  2230   shows "\<forall>y\<in>s. f x \<le> f y"
```
```  2231 proof (rule ccontr)
```
```  2232   have "x \<in> s" using assms(1,3) by auto
```
```  2233   assume "\<not> ?thesis"
```
```  2234   then obtain y where "y\<in>s" and y: "f x > f y" by auto
```
```  2235   then have xy: "0 < dist x y"  by auto
```
```  2236   then obtain u where "0 < u" "u \<le> 1" and u: "u < e / dist x y"
```
```  2237     using field_lbound_gt_zero[of 1 "e / dist x y"] xy \<open>e>0\<close> by auto
```
```  2238   then have "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y"
```
```  2239     using \<open>x\<in>s\<close> \<open>y\<in>s\<close>
```
```  2240     using assms(2)[unfolded convex_on_def,
```
```  2241       THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
```
```  2242     by auto
```
```  2243   moreover
```
```  2244   have *: "x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)"
```
```  2245     by (simp add: algebra_simps)
```
```  2246   have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e"
```
```  2247     unfolding mem_ball dist_norm
```
```  2248     unfolding * and norm_scaleR and abs_of_pos[OF \<open>0<u\<close>]
```
```  2249     unfolding dist_norm[symmetric]
```
```  2250     using u
```
```  2251     unfolding pos_less_divide_eq[OF xy]
```
```  2252     by auto
```
```  2253   then have "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)"
```
```  2254     using assms(4) by auto
```
```  2255   ultimately show False
```
```  2256     using mult_strict_left_mono[OF y \<open>u>0\<close>]
```
```  2257     unfolding left_diff_distrib
```
```  2258     by auto
```
```  2259 qed
```
```  2260
```
```  2261 lemma convex_ball [iff]:
```
```  2262   fixes x :: "'a::real_normed_vector"
```
```  2263   shows "convex (ball x e)"
```
```  2264 proof (auto simp: convex_def)
```
```  2265   fix y z
```
```  2266   assume yz: "dist x y < e" "dist x z < e"
```
```  2267   fix u v :: real
```
```  2268   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  2269   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
```
```  2270     using uv yz
```
```  2271     using convex_on_dist [of "ball x e" x, unfolded convex_on_def,
```
```  2272       THEN bspec[where x=y], THEN bspec[where x=z]]
```
```  2273     by auto
```
```  2274   then show "dist x (u *\<^sub>R y + v *\<^sub>R z) < e"
```
```  2275     using convex_bound_lt[OF yz uv] by auto
```
```  2276 qed
```
```  2277
```
```  2278 lemma convex_cball [iff]:
```
```  2279   fixes x :: "'a::real_normed_vector"
```
```  2280   shows "convex (cball x e)"
```
```  2281 proof -
```
```  2282   {
```
```  2283     fix y z
```
```  2284     assume yz: "dist x y \<le> e" "dist x z \<le> e"
```
```  2285     fix u v :: real
```
```  2286     assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  2287     have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
```
```  2288       using uv yz
```
```  2289       using convex_on_dist [of "cball x e" x, unfolded convex_on_def,
```
```  2290         THEN bspec[where x=y], THEN bspec[where x=z]]
```
```  2291       by auto
```
```  2292     then have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e"
```
```  2293       using convex_bound_le[OF yz uv] by auto
```
```  2294   }
```
```  2295   then show ?thesis by (auto simp: convex_def Ball_def)
```
```  2296 qed
```
```  2297
```
```  2298 lemma connected_ball [iff]:
```
```  2299   fixes x :: "'a::real_normed_vector"
```
```  2300   shows "connected (ball x e)"
```
```  2301   using convex_connected convex_ball by auto
```
```  2302
```
```  2303 lemma connected_cball [iff]:
```
```  2304   fixes x :: "'a::real_normed_vector"
```
```  2305   shows "connected (cball x e)"
```
```  2306   using convex_connected convex_cball by auto
```
```  2307
```
```  2308
```
```  2309 subsection \<open>Convex hull\<close>
```
```  2310
```
```  2311 lemma convex_convex_hull [iff]: "convex (convex hull s)"
```
```  2312   unfolding hull_def
```
```  2313   using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
```
```  2314   by auto
```
```  2315
```
```  2316 lemma convex_hull_subset:
```
```  2317     "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
```
```  2318   by (simp add: convex_convex_hull subset_hull)
```
```  2319
```
```  2320 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
```
```  2321   by (metis convex_convex_hull hull_same)
```
```  2322
```
```  2323 lemma bounded_convex_hull:
```
```  2324   fixes s :: "'a::real_normed_vector set"
```
```  2325   assumes "bounded s"
```
```  2326   shows "bounded (convex hull s)"
```
```  2327 proof -
```
```  2328   from assms obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
```
```  2329     unfolding bounded_iff by auto
```
```  2330   show ?thesis
```
```  2331     apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
```
```  2332     unfolding subset_hull[of convex, OF convex_cball]
```
```  2333     unfolding subset_eq mem_cball dist_norm using B
```
```  2334     apply auto
```
```  2335     done
```
```  2336 qed
```
```  2337
```
```  2338 lemma finite_imp_bounded_convex_hull:
```
```  2339   fixes s :: "'a::real_normed_vector set"
```
```  2340   shows "finite s \<Longrightarrow> bounded (convex hull s)"
```
```  2341   using bounded_convex_hull finite_imp_bounded
```
```  2342   by auto
```
```  2343
```
```  2344
```
```  2345 subsubsection%unimportant \<open>Convex hull is "preserved" by a linear function\<close>
```
```  2346
```
```  2347 lemma convex_hull_linear_image:
```
```  2348   assumes f: "linear f"
```
```  2349   shows "f ` (convex hull s) = convex hull (f ` s)"
```
```  2350 proof
```
```  2351   show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
```
```  2352     by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
```
```  2353   show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
```
```  2354   proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
```
```  2355     show "s \<subseteq> f -` (convex hull (f ` s))"
```
```  2356       by (fast intro: hull_inc)
```
```  2357     show "convex (f -` (convex hull (f ` s)))"
```
```  2358       by (intro convex_linear_vimage [OF f] convex_convex_hull)
```
```  2359   qed
```
```  2360 qed
```
```  2361
```
```  2362 lemma in_convex_hull_linear_image:
```
```  2363   assumes "linear f"
```
```  2364     and "x \<in> convex hull s"
```
```  2365   shows "f x \<in> convex hull (f ` s)"
```
```  2366   using convex_hull_linear_image[OF assms(1)] assms(2) by auto
```
```  2367
```
```  2368 lemma convex_hull_Times:
```
```  2369   "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
```
```  2370 proof
```
```  2371   show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
```
```  2372     by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
```
```  2373   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
```
```  2374   proof (rule hull_induct [OF x], rule hull_induct [OF y])
```
```  2375     fix x y assume "x \<in> s" and "y \<in> t"
```
```  2376     then show "(x, y) \<in> convex hull (s \<times> t)"
```
```  2377       by (simp add: hull_inc)
```
```  2378   next
```
```  2379     fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
```
```  2380     have "convex ?S"
```
```  2381       by (intro convex_linear_vimage convex_translation convex_convex_hull,
```
```  2382         simp add: linear_iff)
```
```  2383     also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
```
```  2384       by (auto simp: image_def Bex_def)
```
```  2385     finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
```
```  2386   next
```
```  2387     show "convex {x. (x, y) \<in> convex hull s \<times> t}"
```
```  2388     proof -
```
```  2389       fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
```
```  2390       have "convex ?S"
```
```  2391       by (intro convex_linear_vimage convex_translation convex_convex_hull,
```
```  2392         simp add: linear_iff)
```
```  2393       also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
```
```  2394         by (auto simp: image_def Bex_def)
```
```  2395       finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
```
```  2396     qed
```
```  2397   qed
```
```  2398   then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
```
```  2399     unfolding subset_eq split_paired_Ball_Sigma by blast
```
```  2400 qed
```
```  2401
```
```  2402
```
```  2403 subsubsection%unimportant \<open>Stepping theorems for convex hulls of finite sets\<close>
```
```  2404
```
```  2405 lemma convex_hull_empty[simp]: "convex hull {} = {}"
```
```  2406   by (rule hull_unique) auto
```
```  2407
```
```  2408 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
```
```  2409   by (rule hull_unique) auto
```
```  2410
```
```  2411 lemma convex_hull_insert:
```
```  2412   fixes S :: "'a::real_vector set"
```
```  2413   assumes "S \<noteq> {}"
```
```  2414   shows "convex hull (insert a S) =
```
```  2415          {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)}"
```
```  2416   (is "_ = ?hull")
```
```  2417 proof (intro equalityI hull_minimal subsetI)
```
```  2418   fix x
```
```  2419   assume "x \<in> insert a S"
```
```  2420   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)"
```
```  2421   unfolding insert_iff
```
```  2422   proof
```
```  2423     assume "x = a"
```
```  2424     then show ?thesis
```
```  2425       by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
```
```  2426   next
```
```  2427     assume "x \<in> S"
```
```  2428     with hull_subset[of S convex] show ?thesis
```
```  2429       by force
```
```  2430   qed
```
```  2431   then show "x \<in> ?hull"
```
```  2432     by simp
```
```  2433 next
```
```  2434   fix x
```
```  2435   assume "x \<in> ?hull"
```
```  2436   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"
```
```  2437     by auto
```
```  2438   have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
```
```  2439     using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
```
```  2440     by auto
```
```  2441   then show "x \<in> convex hull insert a S"
```
```  2442     unfolding obt(5) using obt(1-3)
```
```  2443     by (rule convexD [OF convex_convex_hull])
```
```  2444 next
```
```  2445   show "convex ?hull"
```
```  2446   proof (rule convexI)
```
```  2447     fix x y u v
```
```  2448     assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
```
```  2449     from x obtain u1 v1 b1 where
```
```  2450       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"
```
```  2451       by auto
```
```  2452     from y obtain u2 v2 b2 where
```
```  2453       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"
```
```  2454       by auto
```
```  2455     have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
```
```  2456       by (auto simp: algebra_simps)
```
```  2457     have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
```
```  2458       (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
```
```  2459     proof (cases "u * v1 + v * v2 = 0")
```
```  2460       case True
```
```  2461       have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
```
```  2462         by (auto simp: algebra_simps)
```
```  2463       have eq0: "u * v1 = 0" "v * v2 = 0"
```
```  2464         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>]
```
```  2465         by arith+
```
```  2466       then have "u * u1 + v * u2 = 1"
```
```  2467         using as(3) obt1(3) obt2(3) by auto
```
```  2468       then show ?thesis
```
```  2469         using "*" eq0 as obt1(4) xeq yeq by auto
```
```  2470     next
```
```  2471       case False
```
```  2472       have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
```
```  2473         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
```
```  2474       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
```
```  2475         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
```
```  2476       also have "\<dots> = u * v1 + v * v2"
```
```  2477         by simp
```
```  2478       finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
```
```  2479       let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
```
```  2480       have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
```
```  2481         using as(1,2) obt1(1,2) obt2(1,2) by auto
```
```  2482       show ?thesis
```
```  2483       proof
```
```  2484         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)"
```
```  2485           unfolding xeq yeq * **
```
```  2486           using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
```
```  2487         show "?b \<in> convex hull S"
```
```  2488           using False zeroes obt1(4) obt2(4)
```
```  2489           by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib  add_divide_distrib[symmetric]  zero_le_divide_iff)
```
```  2490       qed
```
```  2491     qed
```
```  2492     then obtain b where b: "b \<in> convex hull S"
```
```  2493        "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)" ..
```
```  2494
```
```  2495     have u1: "u1 \<le> 1"
```
```  2496       unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
```
```  2497     have u2: "u2 \<le> 1"
```
```  2498       unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
```
```  2499     have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
```
```  2500     proof (rule add_mono)
```
```  2501       show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
```
```  2502         by (simp_all add: as mult_right_mono)
```
```  2503     qed
```
```  2504     also have "\<dots> \<le> 1"
```
```  2505       unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
```
```  2506     finally have le1: "u1 * u + u2 * v \<le> 1" .
```
```  2507     show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
```
```  2508     proof (intro CollectI exI conjI)
```
```  2509       show "0 \<le> u * u1 + v * u2"
```
```  2510         by (simp add: as(1) as(2) obt1(1) obt2(1))
```
```  2511       show "0 \<le> 1 - u * u1 - v * u2"
```
```  2512         by (simp add: le1 diff_diff_add mult.commute)
```
```  2513     qed (use b in \<open>auto simp: algebra_simps\<close>)
```
```  2514   qed
```
```  2515 qed
```
```  2516
```
```  2517 lemma convex_hull_insert_alt:
```
```  2518    "convex hull (insert a S) =
```
```  2519      (if S = {} then {a}
```
```  2520       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})"
```
```  2521   apply (auto simp: convex_hull_insert)
```
```  2522   using diff_eq_eq apply fastforce
```
```  2523   by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
```
```  2524
```
```  2525 subsubsection%unimportant \<open>Explicit expression for convex hull\<close>
```
```  2526
```
```  2527 proposition convex_hull_indexed:
```
```  2528   fixes S :: "'a::real_vector set"
```
```  2529   shows "convex hull S =
```
```  2530     {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
```
```  2531                 (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
```
```  2532     (is "?xyz = ?hull")
```
```  2533 proof (rule hull_unique [OF _ convexI])
```
```  2534   show "S \<subseteq> ?hull"
```
```  2535     by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
```
```  2536 next
```
```  2537   fix T
```
```  2538   assume "S \<subseteq> T" "convex T"
```
```  2539   then show "?hull \<subseteq> T"
```
```  2540     by (blast intro: convex_sum)
```
```  2541 next
```
```  2542   fix x y u v
```
```  2543   assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
```
```  2544   assume xy: "x \<in> ?hull" "y \<in> ?hull"
```
```  2545   from xy obtain k1 u1 x1 where
```
```  2546     x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S"
```
```  2547                       "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
```
```  2548     by auto
```
```  2549   from xy obtain k2 u2 x2 where
```
```  2550     y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S"
```
```  2551                      "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
```
```  2552     by auto
```
```  2553   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)"
```
```  2554           "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
```
```  2555     by auto
```
```  2556   have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
```
```  2557     unfolding inj_on_def by auto
```
```  2558   let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
```
```  2559   let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
```
```  2560   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
```
```  2561   proof (intro CollectI exI conjI ballI)
```
```  2562     show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
```
```  2563       using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
```
```  2564     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"
```
```  2565       unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
```
```  2566         sum.reindex[OF inj] Collect_mem_eq o_def
```
```  2567       unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
```
```  2568       by (simp_all add: sum_distrib_left[symmetric]  x(2,3) y(2,3) uv(3))
```
```  2569   qed
```
```  2570 qed
```
```  2571
```
```  2572 lemma convex_hull_finite:
```
```  2573   fixes S :: "'a::real_vector set"
```
```  2574   assumes "finite S"
```
```  2575   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}"
```
```  2576   (is "?HULL = _")
```
```  2577 proof (rule hull_unique [OF _ convexI]; clarify)
```
```  2578   fix x
```
```  2579   assume "x \<in> S"
```
```  2580   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"
```
```  2581     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])
```
```  2582 next
```
```  2583   fix u v :: real
```
```  2584   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  2585   fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
```
```  2586   fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
```
```  2587   have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
```
```  2588     by (simp add: that uv ux(1) uy(1))
```
```  2589   moreover
```
```  2590   have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
```
```  2591     unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
```
```  2592     using uv(3) by auto
```
```  2593   moreover
```
```  2594   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)"
```
```  2595     unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
```
```  2596     by auto
```
```  2597   ultimately
```
```  2598   show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
```
```  2599              (\<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)"
```
```  2600     by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
```
```  2601 qed (use assms in \<open>auto simp: convex_explicit\<close>)
```
```  2602
```
```  2603
```
```  2604 subsubsection%unimportant \<open>Another formulation from Lars Schewe\<close>
```
```  2605
```
```  2606 lemma convex_hull_explicit:
```
```  2607   fixes p :: "'a::real_vector set"
```
```  2608   shows "convex hull p =
```
```  2609     {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}"
```
```  2610   (is "?lhs = ?rhs")
```
```  2611 proof -
```
```  2612   {
```
```  2613     fix x
```
```  2614     assume "x\<in>?lhs"
```
```  2615     then obtain k u y where
```
```  2616         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"
```
```  2617       unfolding convex_hull_indexed by auto
```
```  2618
```
```  2619     have fin: "finite {1..k}" by auto
```
```  2620     have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
```
```  2621     {
```
```  2622       fix j
```
```  2623       assume "j\<in>{1..k}"
```
```  2624       then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
```
```  2625         using obt(1)[THEN bspec[where x=j]] and obt(2)
```
```  2626         apply simp
```
```  2627         apply (rule sum_nonneg)
```
```  2628         using obt(1)
```
```  2629         apply auto
```
```  2630         done
```
```  2631     }
```
```  2632     moreover
```
```  2633     have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
```
```  2634       unfolding sum_image_gen[OF fin, symmetric] using obt(2) by auto
```
```  2635     moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
```
```  2636       using sum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
```
```  2637       unfolding scaleR_left.sum using obt(3) by auto
```
```  2638     ultimately
```
```  2639     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"
```
```  2640       apply (rule_tac x="y ` {1..k}" in exI)
```
```  2641       apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
```
```  2642       done
```
```  2643     then have "x\<in>?rhs" by auto
```
```  2644   }
```
```  2645   moreover
```
```  2646   {
```
```  2647     fix y
```
```  2648     assume "y\<in>?rhs"
```
```  2649     then obtain S u where
```
```  2650       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"
```
```  2651       by auto
```
```  2652
```
```  2653     obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
```
```  2654       using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
```
```  2655
```
```  2656     {
```
```  2657       fix i :: nat
```
```  2658       assume "i\<in>{1..card S}"
```
```  2659       then have "f i \<in> S"
```
```  2660         using f(2) by blast
```
```  2661       then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
```
```  2662     }
```
```  2663     moreover have *: "finite {1..card S}" by auto
```
```  2664     {
```
```  2665       fix y
```
```  2666       assume "y\<in>S"
```
```  2667       then obtain i where "i\<in>{1..card S}" "f i = y"
```
```  2668         using f using image_iff[of y f "{1..card S}"]
```
```  2669         by auto
```
```  2670       then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
```
```  2671         apply auto
```
```  2672         using f(1)[unfolded inj_on_def]
```
```  2673         by (metis One_nat_def atLeastAtMost_iff)
```
```  2674       then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
```
```  2675       then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
```
```  2676           "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
```
```  2677         by (auto simp: sum_constant_scaleR)
```
```  2678     }
```
```  2679     then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
```
```  2680       unfolding sum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
```
```  2681         and sum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
```
```  2682       unfolding f
```
```  2683       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"]
```
```  2684       using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
```
```  2685       unfolding obt(4,5)
```
```  2686       by auto
```
```  2687     ultimately
```
```  2688     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>
```
```  2689         (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
```
```  2690       apply (rule_tac x="card S" in exI)
```
```  2691       apply (rule_tac x="u \<circ> f" in exI)
```
```  2692       apply (rule_tac x=f in exI, fastforce)
```
```  2693       done
```
```  2694     then have "y \<in> ?lhs"
```
```  2695       unfolding convex_hull_indexed by auto
```
```  2696   }
```
```  2697   ultimately show ?thesis
```
```  2698     unfolding set_eq_iff by blast
```
```  2699 qed
```
```  2700
```
```  2701
```
```  2702 subsubsection%unimportant \<open>A stepping theorem for that expansion\<close>
```
```  2703
```
```  2704 lemma convex_hull_finite_step:
```
```  2705   fixes S :: "'a::real_vector set"
```
```  2706   assumes "finite S"
```
```  2707   shows
```
```  2708     "(\<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)
```
```  2709       \<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)"
```
```  2710   (is "?lhs = ?rhs")
```
```  2711 proof (rule, case_tac[!] "a\<in>S")
```
```  2712   assume "a \<in> S"
```
```  2713   then have *: "insert a S = S" by auto
```
```  2714   assume ?lhs
```
```  2715   then show ?rhs
```
```  2716     unfolding *  by (rule_tac x=0 in exI, auto)
```
```  2717 next
```
```  2718   assume ?lhs
```
```  2719   then obtain u where
```
```  2720       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"
```
```  2721     by auto
```
```  2722   assume "a \<notin> S"
```
```  2723   then show ?rhs
```
```  2724     apply (rule_tac x="u a" in exI)
```
```  2725     using u(1)[THEN bspec[where x=a]]
```
```  2726     apply simp
```
```  2727     apply (rule_tac x=u in exI)
```
```  2728     using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
```
```  2729     apply auto
```
```  2730     done
```
```  2731 next
```
```  2732   assume "a \<in> S"
```
```  2733   then have *: "insert a S = S" by auto
```
```  2734   have fin: "finite (insert a S)" using assms by auto
```
```  2735   assume ?rhs
```
```  2736   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"
```
```  2737     by auto
```
```  2738   show ?lhs
```
```  2739     apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
```
```  2740     unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
```
```  2741     unfolding sum_clauses(2)[OF assms]
```
```  2742     using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
```
```  2743     apply auto
```
```  2744     done
```
```  2745 next
```
```  2746   assume ?rhs
```
```  2747   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"
```
```  2748     by auto
```
```  2749   moreover assume "a \<notin> S"
```
```  2750   moreover
```
```  2751   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)"
```
```  2752     using \<open>a \<notin> S\<close>
```
```  2753     by (auto simp: intro!: sum.cong)
```
```  2754   ultimately show ?lhs
```
```  2755     by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
```
```  2756 qed
```
```  2757
```
```  2758
```
```  2759 subsubsection%unimportant \<open>Hence some special cases\<close>
```
```  2760
```
```  2761 lemma convex_hull_2:
```
```  2762   "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}"
```
```  2763 proof -
```
```  2764   have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
```
```  2765     by auto
```
```  2766   have **: "finite {b}" by auto
```
```  2767   show ?thesis
```
```  2768     apply (simp add: convex_hull_finite)
```
```  2769     unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
```
```  2770     apply auto
```
```  2771     apply (rule_tac x=v in exI)
```
```  2772     apply (rule_tac x="1 - v" in exI, simp)
```
```  2773     apply (rule_tac x=u in exI, simp)
```
```  2774     apply (rule_tac x="\<lambda>x. v" in exI, simp)
```
```  2775     done
```
```  2776 qed
```
```  2777
```
```  2778 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
```
```  2779   unfolding convex_hull_2
```
```  2780 proof (rule Collect_cong)
```
```  2781   have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
```
```  2782     by auto
```
```  2783   fix x
```
```  2784   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>
```
```  2785     (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
```
```  2786     unfolding *
```
```  2787     apply auto
```
```  2788     apply (rule_tac[!] x=u in exI)
```
```  2789     apply (auto simp: algebra_simps)
```
```  2790     done
```
```  2791 qed
```
```  2792
```
```  2793 lemma convex_hull_3:
```
```  2794   "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}"
```
```  2795 proof -
```
```  2796   have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
```
```  2797     by auto
```
```  2798   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
```
```  2799     by (auto simp: field_simps)
```
```  2800   show ?thesis
```
```  2801     unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
```
```  2802     unfolding convex_hull_finite_step[OF fin(3)]
```
```  2803     apply (rule Collect_cong, simp)
```
```  2804     apply auto
```
```  2805     apply (rule_tac x=va in exI)
```
```  2806     apply (rule_tac x="u c" in exI, simp)
```
```  2807     apply (rule_tac x="1 - v - w" in exI, simp)
```
```  2808     apply (rule_tac x=v in exI, simp)
```
```  2809     apply (rule_tac x="\<lambda>x. w" in exI, simp)
```
```  2810     done
```
```  2811 qed
```
```  2812
```
```  2813 lemma convex_hull_3_alt:
```
```  2814   "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}"
```
```  2815 proof -
```
```  2816   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
```
```  2817     by auto
```
```  2818   show ?thesis
```
```  2819     unfolding convex_hull_3
```
```  2820     apply (auto simp: *)
```
```  2821     apply (rule_tac x=v in exI)
```
```  2822     apply (rule_tac x=w in exI)
```
```  2823     apply (simp add: algebra_simps)
```
```  2824     apply (rule_tac x=u in exI)
```
```  2825     apply (rule_tac x=v in exI)
```
```  2826     apply (simp add: algebra_simps)
```
```  2827     done
```
```  2828 qed
```
```  2829
```
```  2830
```
```  2831 subsection%unimportant \<open>Relations among closure notions and corresponding hulls\<close>
```
```  2832
```
```  2833 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
```
```  2834   unfolding affine_def convex_def by auto
```
```  2835
```
```  2836 lemma convex_affine_hull [simp]: "convex (affine hull S)"
```
```  2837   by (simp add: affine_imp_convex)
```
```  2838
```
```  2839 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
```
```  2840   using subspace_imp_affine affine_imp_convex by auto
```
```  2841
```
```  2842 lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
```
```  2843   by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
```
```  2844
```
```  2845 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
```
```  2846   by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
```
```  2847
```
```  2848 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
```
```  2849   by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
```
```  2850
```
```  2851 lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
```
```  2852   unfolding affine_dependent_def dependent_def
```
```  2853   using affine_hull_subset_span by auto
```
```  2854
```
```  2855 lemma dependent_imp_affine_dependent:
```
```  2856   assumes "dependent {x - a| x . x \<in> s}"
```
```  2857     and "a \<notin> s"
```
```  2858   shows "affine_dependent (insert a s)"
```
```  2859 proof -
```
```  2860   from assms(1)[unfolded dependent_explicit] obtain S u v
```
```  2861     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"
```
```  2862     by auto
```
```  2863   define t where "t = (\<lambda>x. x + a) ` S"
```
```  2864
```
```  2865   have inj: "inj_on (\<lambda>x. x + a) S"
```
```  2866     unfolding inj_on_def by auto
```
```  2867   have "0 \<notin> S"
```
```  2868     using obt(2) assms(2) unfolding subset_eq by auto
```
```  2869   have fin: "finite t" and "t \<subseteq> s"
```
```  2870     unfolding t_def using obt(1,2) by auto
```
```  2871   then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
```
```  2872     by auto
```
```  2873   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
```
```  2874     apply (rule sum.cong)
```
```  2875     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
```
```  2876     apply auto
```
```  2877     done
```
```  2878   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
```
```  2879     unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
```
```  2880   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"
```
```  2881     using obt(3,4) \<open>0\<notin>S\<close>
```
```  2882     by (rule_tac x="v + a" in bexI) (auto simp: t_def)
```
```  2883   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)"
```
```  2884     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
```
```  2885   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
```
```  2886     unfolding scaleR_left.sum
```
```  2887     unfolding t_def and sum.reindex[OF inj] and o_def
```
```  2888     using obt(5)
```
```  2889     by (auto simp: sum.distrib scaleR_right_distrib)
```
```  2890   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"
```
```  2891     unfolding sum_clauses(2)[OF fin]
```
```  2892     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
```
```  2893     by (auto simp: *)
```
```  2894   ultimately show ?thesis
```
```  2895     unfolding affine_dependent_explicit
```
```  2896     apply (rule_tac x="insert a t" in exI, auto)
```
```  2897     done
```
```  2898 qed
```
```  2899
```
```  2900 lemma convex_cone:
```
```  2901   "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)"
```
```  2902   (is "?lhs = ?rhs")
```
```  2903 proof -
```
```  2904   {
```
```  2905     fix x y
```
```  2906     assume "x\<in>s" "y\<in>s" and ?lhs
```
```  2907     then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
```
```  2908       unfolding cone_def by auto
```
```  2909     then have "x + y \<in> s"
```
```  2910       using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
```
```  2911       apply (erule_tac x="2*\<^sub>R x" in ballE)
```
```  2912       apply (erule_tac x="2*\<^sub>R y" in ballE)
```
```  2913       apply (erule_tac x="1/2" in allE, simp)
```
```  2914       apply (erule_tac x="1/2" in allE, auto)
```
```  2915       done
```
```  2916   }
```
```  2917   then show ?thesis
```
```  2918     unfolding convex_def cone_def by blast
```
```  2919 qed
```
```  2920
```
```  2921 lemma affine_dependent_biggerset:
```
```  2922   fixes s :: "'a::euclidean_space set"
```
```  2923   assumes "finite s" "card s \<ge> DIM('a) + 2"
```
```  2924   shows "affine_dependent s"
```
```  2925 proof -
```
```  2926   have "s \<noteq> {}" using assms by auto
```
```  2927   then obtain a where "a\<in>s" by auto
```
```  2928   have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
```
```  2929     by auto
```
```  2930   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
```
```  2931     unfolding * by (simp add: card_image inj_on_def)
```
```  2932   also have "\<dots> > DIM('a)" using assms(2)
```
```  2933     unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
```
```  2934   finally show ?thesis
```
```  2935     apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
```
```  2936     apply (rule dependent_imp_affine_dependent)
```
```  2937     apply (rule dependent_biggerset, auto)
```
```  2938     done
```
```  2939 qed
```
```  2940
```
```  2941 lemma affine_dependent_biggerset_general:
```
```  2942   assumes "finite (S :: 'a::euclidean_space set)"
```
```  2943     and "card S \<ge> dim S + 2"
```
```  2944   shows "affine_dependent S"
```
```  2945 proof -
```
```  2946   from assms(2) have "S \<noteq> {}" by auto
```
```  2947   then obtain a where "a\<in>S" by auto
```
```  2948   have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
```
```  2949     by auto
```
```  2950   have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
```
```  2951     by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
```
```  2952   have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
```
```  2953     using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
```
```  2954   also have "\<dots> < dim S + 1" by auto
```
```  2955   also have "\<dots> \<le> card (S - {a})"
```
```  2956     using assms
```
```  2957     using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
```
```  2958     by auto
```
```  2959   finally show ?thesis
```
```  2960     apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
```
```  2961     apply (rule dependent_imp_affine_dependent)
```
```  2962     apply (rule dependent_biggerset_general)
```
```  2963     unfolding **
```
```  2964     apply auto
```
```  2965     done
```
```  2966 qed
```
```  2967
```
```  2968
```
```  2969 subsection%unimportant \<open>Some Properties of Affine Dependent Sets\<close>
```
```  2970
```
```  2971 lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
```
```  2972   by (simp add: affine_dependent_def)
```
```  2973
```
```  2974 lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
```
```  2975   by (simp add: affine_dependent_def)
```
```  2976
```
```  2977 lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
```
```  2978   by (simp add: affine_dependent_def insert_Diff_if hull_same)
```
```  2979
```
```  2980 lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
```
```  2981 proof -
```
```  2982   have "affine ((\<lambda>x. a + x) ` (affine hull S))"
```
```  2983     using affine_translation affine_affine_hull by blast
```
```  2984   moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
```
```  2985     using hull_subset[of S] by auto
```
```  2986   ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
```
```  2987     by (metis hull_minimal)
```
```  2988   have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
```
```  2989     using affine_translation affine_affine_hull by blast
```
```  2990   moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
```
```  2991     using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
```
```  2992   moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
```
```  2993     using translation_assoc[of "-a" a] by auto
```
```  2994   ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
```
```  2995     by (metis hull_minimal)
```
```  2996   then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
```
```  2997     by auto
```
```  2998   then show ?thesis using h1 by auto
```
```  2999 qed
```
```  3000
```
```  3001 lemma affine_dependent_translation:
```
```  3002   assumes "affine_dependent S"
```
```  3003   shows "affine_dependent ((\<lambda>x. a + x) ` S)"
```
```  3004 proof -
```
```  3005   obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
```
```  3006     using assms affine_dependent_def by auto
```
```  3007   have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
```
```  3008     by auto
```
```  3009   then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
```
```  3010     using affine_hull_translation[of a "S - {x}"] x by auto
```
```  3011   moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
```
```  3012     using x by auto
```
```  3013   ultimately show ?thesis
```
```  3014     unfolding affine_dependent_def by auto
```
```  3015 qed
```
```  3016
```
```  3017 lemma affine_dependent_translation_eq:
```
```  3018   "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
```
```  3019 proof -
```
```  3020   {
```
```  3021     assume "affine_dependent ((\<lambda>x. a + x) ` S)"
```
```  3022     then have "affine_dependent S"
```
```  3023       using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
```
```  3024       by auto
```
```  3025   }
```
```  3026   then show ?thesis
```
```  3027     using affine_dependent_translation by auto
```
```  3028 qed
```
```  3029
```
```  3030 lemma affine_hull_0_dependent:
```
```  3031   assumes "0 \<in> affine hull S"
```
```  3032   shows "dependent S"
```
```  3033 proof -
```
```  3034   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"
```
```  3035     using assms affine_hull_explicit[of S] by auto
```
```  3036   then have "\<exists>v\<in>s. u v \<noteq> 0"
```
```  3037     using sum_not_0[of "u" "s"] by auto
```
```  3038   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)"
```
```  3039     using s_u by auto
```
```  3040   then show ?thesis
```
```  3041     unfolding dependent_explicit[of S] by auto
```
```  3042 qed
```
```  3043
```
```  3044 lemma affine_dependent_imp_dependent2:
```
```  3045   assumes "affine_dependent (insert 0 S)"
```
```  3046   shows "dependent S"
```
```  3047 proof -
```
```  3048   obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
```
```  3049     using affine_dependent_def[of "(insert 0 S)"] assms by blast
```
```  3050   then have "x \<in> span (insert 0 S - {x})"
```
```  3051     using affine_hull_subset_span by auto
```
```  3052   moreover have "span (insert 0 S - {x}) = span (S - {x})"
```
```  3053     using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
```
```  3054   ultimately have "x \<in> span (S - {x})" by auto
```
```  3055   then have "x \<noteq> 0 \<Longrightarrow> dependent S"
```
```  3056     using x dependent_def by auto
```
```  3057   moreover
```
```  3058   {
```
```  3059     assume "x = 0"
```
```  3060     then have "0 \<in> affine hull S"
```
```  3061       using x hull_mono[of "S - {0}" S] by auto
```
```  3062     then have "dependent S"
```
```  3063       using affine_hull_0_dependent by auto
```
```  3064   }
```
```  3065   ultimately show ?thesis by auto
```
```  3066 qed
```
```  3067
```
```  3068 lemma affine_dependent_iff_dependent:
```
```  3069   assumes "a \<notin> S"
```
```  3070   shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
```
```  3071 proof -
```
```  3072   have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
```
```  3073   then show ?thesis
```
```  3074     using affine_dependent_translation_eq[of "(insert a S)" "-a"]
```
```  3075       affine_dependent_imp_dependent2 assms
```
```  3076       dependent_imp_affine_dependent[of a S]
```
```  3077     by (auto simp del: uminus_add_conv_diff)
```
```  3078 qed
```
```  3079
```
```  3080 lemma affine_dependent_iff_dependent2:
```
```  3081   assumes "a \<in> S"
```
```  3082   shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
```
```  3083 proof -
```
```  3084   have "insert a (S - {a}) = S"
```
```  3085     using assms by auto
```
```  3086   then show ?thesis
```
```  3087     using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
```
```  3088 qed
```
```  3089
```
```  3090 lemma affine_hull_insert_span_gen:
```
```  3091   "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
```
```  3092 proof -
```
```  3093   have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
```
```  3094     by auto
```
```  3095   {
```
```  3096     assume "a \<notin> s"
```
```  3097     then have ?thesis
```
```  3098       using affine_hull_insert_span[of a s] h1 by auto
```
```  3099   }
```
```  3100   moreover
```
```  3101   {
```
```  3102     assume a1: "a \<in> s"
```
```  3103     have "\<exists>x. x \<in> s \<and> -a+x=0"
```
```  3104       apply (rule exI[of _ a])
```
```  3105       using a1
```
```  3106       apply auto
```
```  3107       done
```
```  3108     then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
```
```  3109       by auto
```
```  3110     then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
```
```  3111       using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
```
```  3112     moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
```
```  3113       by auto
```
```  3114     moreover have "insert a (s - {a}) = insert a s"
```
```  3115       by auto
```
```  3116     ultimately have ?thesis
```
```  3117       using affine_hull_insert_span[of "a" "s-{a}"] by auto
```
```  3118   }
```
```  3119   ultimately show ?thesis by auto
```
```  3120 qed
```
```  3121
```
```  3122 lemma affine_hull_span2:
```
```  3123   assumes "a \<in> s"
```
```  3124   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
```
```  3125   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
```
```  3126   by auto
```
```  3127
```
```  3128 lemma affine_hull_span_gen:
```
```  3129   assumes "a \<in> affine hull s"
```
```  3130   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
```
```  3131 proof -
```
```  3132   have "affine hull (insert a s) = affine hull s"
```
```  3133     using hull_redundant[of a affine s] assms by auto
```
```  3134   then show ?thesis
```
```  3135     using affine_hull_insert_span_gen[of a "s"] by auto
```
```  3136 qed
```
```  3137
```
```  3138 lemma affine_hull_span_0:
```
```  3139   assumes "0 \<in> affine hull S"
```
```  3140   shows "affine hull S = span S"
```
```  3141   using affine_hull_span_gen[of "0" S] assms by auto
```
```  3142
```
```  3143 lemma extend_to_affine_basis_nonempty:
```
```  3144   fixes S V :: "'n::euclidean_space set"
```
```  3145   assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
```
```  3146   shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
```
```  3147 proof -
```
```  3148   obtain a where a: "a \<in> S"
```
```  3149     using assms by auto
```
```  3150   then have h0: "independent  ((\<lambda>x. -a + x) ` (S-{a}))"
```
```  3151     using affine_dependent_iff_dependent2 assms by auto
```
```  3152   obtain B where B:
```
```  3153     "(\<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"
```
```  3154     using assms
```
```  3155     by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
```
```  3156   define T where "T = (\<lambda>x. a+x) ` insert 0 B"
```
```  3157   then have "T = insert a ((\<lambda>x. a+x) ` B)"
```
```  3158     by auto
```
```  3159   then have "affine hull T = (\<lambda>x. a+x) ` span B"
```
```  3160     using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
```
```  3161     by auto
```
```  3162   then have "V \<subseteq> affine hull T"
```
```  3163     using B assms translation_inverse_subset[of a V "span B"]
```
```  3164     by auto
```
```  3165   moreover have "T \<subseteq> V"
```
```  3166     using T_def B a assms by auto
```
```  3167   ultimately have "affine hull T = affine hull V"
```
```  3168     by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
```
```  3169   moreover have "S \<subseteq> T"
```
```  3170     using T_def B translation_inverse_subset[of a "S-{a}" B]
```
```  3171     by auto
```
```  3172   moreover have "\<not> affine_dependent T"
```
```  3173     using T_def affine_dependent_translation_eq[of "insert 0 B"]
```
```  3174       affine_dependent_imp_dependent2 B
```
```  3175     by auto
```
```  3176   ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
```
```  3177 qed
```
```  3178
```
```  3179 lemma affine_basis_exists:
```
```  3180   fixes V :: "'n::euclidean_space set"
```
```  3181   shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
```
```  3182 proof (cases "V = {}")
```
```  3183   case True
```
```  3184   then show ?thesis
```
```  3185     using affine_independent_0 by auto
```
```  3186 next
```
```  3187   case False
```
```  3188   then obtain x where "x \<in> V" by auto
```
```  3189   then show ?thesis
```
```  3190     using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
```
```  3191     by auto
```
```  3192 qed
```
```  3193
```
```  3194 proposition extend_to_affine_basis:
```
```  3195   fixes S V :: "'n::euclidean_space set"
```
```  3196   assumes "\<not> affine_dependent S" "S \<subseteq> V"
```
```  3197   obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
```
```  3198 proof (cases "S = {}")
```
```  3199   case True then show ?thesis
```
```  3200     using affine_basis_exists by (metis empty_subsetI that)
```
```  3201 next
```
```  3202   case False
```
```  3203   then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
```
```  3204 qed
```
```  3205
```
```  3206
```
```  3207 subsection \<open>Affine Dimension of a Set\<close>
```
```  3208
```
```  3209 definition%important aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
```
```  3210   where "aff_dim V =
```
```  3211   (SOME d :: int.
```
```  3212     \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
```
```  3213
```
```  3214 lemma aff_dim_basis_exists:
```
```  3215   fixes V :: "('n::euclidean_space) set"
```
```  3216   shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
```
```  3217 proof -
```
```  3218   obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
```
```  3219     using affine_basis_exists[of V] by auto
```
```  3220   then show ?thesis
```
```  3221     unfolding aff_dim_def
```
```  3222       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"]
```
```  3223     apply auto
```
```  3224     apply (rule exI[of _ "int (card B) - (1 :: int)"])
```
```  3225     apply (rule exI[of _ "B"], auto)
```
```  3226     done
```
```  3227 qed
```
```  3228
```
```  3229 lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
```
```  3230 proof -
```
```  3231   have "S = {} \<Longrightarrow> affine hull S = {}"
```
```  3232     using affine_hull_empty by auto
```
```  3233   moreover have "affine hull S = {} \<Longrightarrow> S = {}"
```
```  3234     unfolding hull_def by auto
```
```  3235   ultimately show ?thesis by blast
```
```  3236 qed
```
```  3237
```
```  3238 lemma aff_dim_parallel_subspace_aux:
```
```  3239   fixes B :: "'n::euclidean_space set"
```
```  3240   assumes "\<not> affine_dependent B" "a \<in> B"
```
```  3241   shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
```
```  3242 proof -
```
```  3243   have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
```
```  3244     using affine_dependent_iff_dependent2 assms by auto
```
```  3245   then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
```
```  3246     "finite ((\<lambda>x. -a + x) ` (B - {a}))"
```
```  3247     using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
```
```  3248   show ?thesis
```
```  3249   proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
```
```  3250     case True
```
```  3251     have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
```
```  3252       using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
```
```  3253     then have "B = {a}" using True by auto
```
```  3254     then show ?thesis using assms fin by auto
```
```  3255   next
```
```  3256     case False
```
```  3257     then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
```
```  3258       using fin by auto
```
```  3259     moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
```
```  3260       by (rule card_image) (use translate_inj_on in blast)
```
```  3261     ultimately have "card (B-{a}) > 0" by auto
```
```  3262     then have *: "finite (B - {a})"
```
```  3263       using card_gt_0_iff[of "(B - {a})"] by auto
```
```  3264     then have "card (B - {a}) = card B - 1"
```
```  3265       using card_Diff_singleton assms by auto
```
```  3266     with * show ?thesis using fin h1 by auto
```
```  3267   qed
```
```  3268 qed
```
```  3269
```
```  3270 lemma aff_dim_parallel_subspace:
```
```  3271   fixes V L :: "'n::euclidean_space set"
```
```  3272   assumes "V \<noteq> {}"
```
```  3273     and "subspace L"
```
```  3274     and "affine_parallel (affine hull V) L"
```
```  3275   shows "aff_dim V = int (dim L)"
```
```  3276 proof -
```
```  3277   obtain B where
```
```  3278     B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
```
```  3279     using aff_dim_basis_exists by auto
```
```  3280   then have "B \<noteq> {}"
```
```  3281     using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
```
```  3282     by auto
```
```  3283   then obtain a where a: "a \<in> B" by auto
```
```  3284   define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
```
```  3285   moreover have "affine_parallel (affine hull B) Lb"
```
```  3286     using Lb_def B assms affine_hull_span2[of a B] a
```
```  3287       affine_parallel_commut[of "Lb" "(affine hull B)"]
```
```  3288     unfolding affine_parallel_def
```
```  3289     by auto
```
```  3290   moreover have "subspace Lb"
```
```  3291     using Lb_def subspace_span by auto
```
```  3292   moreover have "affine hull B \<noteq> {}"
```
```  3293     using assms B affine_hull_nonempty[of V] by auto
```
```  3294   ultimately have "L = Lb"
```
```  3295     using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
```
```  3296     by auto
```
```  3297   then have "dim L = dim Lb"
```
```  3298     by auto
```
```  3299   moreover have "card B - 1 = dim Lb" and "finite B"
```
```  3300     using Lb_def aff_dim_parallel_subspace_aux a B by auto
```
```  3301   ultimately show ?thesis
```
```  3302     using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
```
```  3303 qed
```
```  3304
```
```  3305 lemma aff_independent_finite:
```
```  3306   fixes B :: "'n::euclidean_space set"
```
```  3307   assumes "\<not> affine_dependent B"
```
```  3308   shows "finite B"
```
```  3309 proof -
```
```  3310   {
```
```  3311     assume "B \<noteq> {}"
```
```  3312     then obtain a where "a \<in> B" by auto
```
```  3313     then have ?thesis
```
```  3314       using aff_dim_parallel_subspace_aux assms by auto
```
```  3315   }
```
```  3316   then show ?thesis by auto
```
```  3317 qed
```
```  3318
```
```  3319 lemmas independent_finite = independent_imp_finite
```
```  3320
```
```  3321 lemma span_substd_basis:
```
```  3322   assumes d: "d \<subseteq> Basis"
```
```  3323   shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
```
```  3324   (is "_ = ?B")
```
```  3325 proof -
```
```  3326   have "d \<subseteq> ?B"
```
```  3327     using d by (auto simp: inner_Basis)
```
```  3328   moreover have s: "subspace ?B"
```
```  3329     using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
```
```  3330   ultimately have "span d \<subseteq> ?B"
```
```  3331     using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
```
```  3332   moreover have *: "card d \<le> dim (span d)"
```
```  3333     using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
```
```  3334       span_superset[of d]
```
```  3335     by auto
```
```  3336   moreover from * have "dim ?B \<le> dim (span d)"
```
```  3337     using dim_substandard[OF assms] by auto
```
```  3338   ultimately show ?thesis
```
```  3339     using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
```
```  3340 qed
```
```  3341
```
```  3342 lemma basis_to_substdbasis_subspace_isomorphism:
```
```  3343   fixes B :: "'a::euclidean_space set"
```
```  3344   assumes "independent B"
```
```  3345   shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
```
```  3346     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"
```
```  3347 proof -
```
```  3348   have B: "card B = dim B"
```
```  3349     using dim_unique[of B B "card B"] assms span_superset[of B] by auto
```
```  3350   have "dim B \<le> card (Basis :: 'a set)"
```
```  3351     using dim_subset_UNIV[of B] by simp
```
```  3352   from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
```
```  3353     by auto
```
```  3354   let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
```
```  3355   have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
```
```  3356   proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
```
```  3357     show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
```
```  3358       using d inner_not_same_Basis by blast
```
```  3359   qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
```
```  3360   with t \<open>card B = dim B\<close> d show ?thesis by auto
```
```  3361 qed
```
```  3362
```
```  3363 lemma aff_dim_empty:
```
```  3364   fixes S :: "'n::euclidean_space set"
```
```  3365   shows "S = {} \<longleftrightarrow> aff_dim S = -1"
```
```  3366 proof -
```
```  3367   obtain B where *: "affine hull B = affine hull S"
```
```  3368     and "\<not> affine_dependent B"
```
```  3369     and "int (card B) = aff_dim S + 1"
```
```  3370     using aff_dim_basis_exists by auto
```
```  3371   moreover
```
```  3372   from * have "S = {} \<longleftrightarrow> B = {}"
```
```  3373     using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
```
```  3374   ultimately show ?thesis
```
```  3375     using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
```
```  3376 qed
```
```  3377
```
```  3378 lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
```
```  3379   by (simp add: aff_dim_empty [symmetric])
```
```  3380
```
```  3381 lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
```
```  3382   unfolding aff_dim_def using hull_hull[of _ S] by auto
```
```  3383
```
```  3384 lemma aff_dim_affine_hull2:
```
```  3385   assumes "affine hull S = affine hull T"
```
```  3386   shows "aff_dim S = aff_dim T"
```
```  3387   unfolding aff_dim_def using assms by auto
```
```  3388
```
```  3389 lemma aff_dim_unique:
```
```  3390   fixes B V :: "'n::euclidean_space set"
```
```  3391   assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
```
```  3392   shows "of_nat (card B) = aff_dim V + 1"
```
```  3393 proof (cases "B = {}")
```
```  3394   case True
```
```  3395   then have "V = {}"
```
```  3396     using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
```
```  3397     by auto
```
```  3398   then have "aff_dim V = (-1::int)"
```
```  3399     using aff_dim_empty by auto
```
```  3400   then show ?thesis
```
```  3401     using \<open>B = {}\<close> by auto
```
```  3402 next
```
```  3403   case False
```
```  3404   then obtain a where a: "a \<in> B" by auto
```
```  3405   define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
```
```  3406   have "affine_parallel (affine hull B) Lb"
```
```  3407     using Lb_def affine_hull_span2[of a B] a
```
```  3408       affine_parallel_commut[of "Lb" "(affine hull B)"]
```
```  3409     unfolding affine_parallel_def by auto
```
```  3410   moreover have "subspace Lb"
```
```  3411     using Lb_def subspace_span by auto
```
```  3412   ultimately have "aff_dim B = int(dim Lb)"
```
```  3413     using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
```
```  3414   moreover have "(card B) - 1 = dim Lb" "finite B"
```
```  3415     using Lb_def aff_dim_parallel_subspace_aux a assms by auto
```
```  3416   ultimately have "of_nat (card B) = aff_dim B + 1"
```
```  3417     using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
```
```  3418   then show ?thesis
```
```  3419     using aff_dim_affine_hull2 assms by auto
```
```  3420 qed
```
```  3421
```
```  3422 lemma aff_dim_affine_independent:
```
```  3423   fixes B :: "'n::euclidean_space set"
```
```  3424   assumes "\<not> affine_dependent B"
```
```  3425   shows "of_nat (card B) = aff_dim B + 1"
```
```  3426   using aff_dim_unique[of B B] assms by auto
```
```  3427
```
```  3428 lemma affine_independent_iff_card:
```
```  3429     fixes s :: "'a::euclidean_space set"
```
```  3430     shows "~ affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
```
```  3431   apply (rule iffI)
```
```  3432   apply (simp add: aff_dim_affine_independent aff_independent_finite)
```
```  3433   by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
```
```  3434
```
```  3435 lemma aff_dim_sing [simp]:
```
```  3436   fixes a :: "'n::euclidean_space"
```
```  3437   shows "aff_dim {a} = 0"
```
```  3438   using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
```
```  3439
```
```  3440 lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
```
```  3441 proof (clarsimp)
```
```  3442   assume "a \<noteq> b"
```
```  3443   then have "aff_dim{a,b} = card{a,b} - 1"
```
```  3444     using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
```
```  3445   also have "\<dots> = 1"
```
```  3446     using \<open>a \<noteq> b\<close> by simp
```
```  3447   finally show "aff_dim {a, b} = 1" .
```
```  3448 qed
```
```  3449
```
```  3450 lemma aff_dim_inner_basis_exists:
```
```  3451   fixes V :: "('n::euclidean_space) set"
```
```  3452   shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
```
```  3453     \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
```
```  3454 proof -
```
```  3455   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
```
```  3456     using affine_basis_exists[of V] by auto
```
```  3457   then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
```
```  3458   with B show ?thesis by auto
```
```  3459 qed
```
```  3460
```
```  3461 lemma aff_dim_le_card:
```
```  3462   fixes V :: "'n::euclidean_space set"
```
```  3463   assumes "finite V"
```
```  3464   shows "aff_dim V \<le> of_nat (card V) - 1"
```
```  3465 proof -
```
```  3466   obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
```
```  3467     using aff_dim_inner_basis_exists[of V] by auto
```
```  3468   then have "card B \<le> card V"
```
```  3469     using assms card_mono by auto
```
```  3470   with B show ?thesis by auto
```
```  3471 qed
```
```  3472
```
```  3473 lemma aff_dim_parallel_eq:
```
```  3474   fixes S T :: "'n::euclidean_space set"
```
```  3475   assumes "affine_parallel (affine hull S) (affine hull T)"
```
```  3476   shows "aff_dim S = aff_dim T"
```
```  3477 proof -
```
```  3478   {
```
```  3479     assume "T \<noteq> {}" "S \<noteq> {}"
```
```  3480     then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
```
```  3481       using affine_parallel_subspace[of "affine hull T"]
```
```  3482         affine_affine_hull[of T] affine_hull_nonempty
```
```  3483       by auto
```
```  3484     then have "aff_dim T = int (dim L)"
```
```  3485       using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
```
```  3486     moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
```
```  3487        using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
```
```  3488     moreover from * have "aff_dim S = int (dim L)"
```
```  3489       using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
```
```  3490     ultimately have ?thesis by auto
```
```  3491   }
```
```  3492   moreover
```
```  3493   {
```
```  3494     assume "S = {}"
```
```  3495     then have "S = {}" and "T = {}"
```
```  3496       using assms affine_hull_nonempty
```
```  3497       unfolding affine_parallel_def
```
```  3498       by auto
```
```  3499     then have ?thesis using aff_dim_empty by auto
```
```  3500   }
```
```  3501   moreover
```
```  3502   {
```
```  3503     assume "T = {}"
```
```  3504     then have "S = {}" and "T = {}"
```
```  3505       using assms affine_hull_nonempty
```
```  3506       unfolding affine_parallel_def
```
```  3507       by auto
```
```  3508     then have ?thesis
```
```  3509       using aff_dim_empty by auto
```
```  3510   }
```
```  3511   ultimately show ?thesis by blast
```
```  3512 qed
```
```  3513
```
```  3514 lemma aff_dim_translation_eq:
```
```  3515   fixes a :: "'n::euclidean_space"
```
```  3516   shows "aff_dim ((\<lambda>x. a + x) ` S) = aff_dim S"
```
```  3517 proof -
```
```  3518   have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
```
```  3519     unfolding affine_parallel_def
```
```  3520     apply (rule exI[of _ "a"])
```
```  3521     using affine_hull_translation[of a S]
```
```  3522     apply auto
```
```  3523     done
```
```  3524   then show ?thesis
```
```  3525     using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
```
```  3526 qed
```
```  3527
```
```  3528 lemma aff_dim_affine:
```
```  3529   fixes S L :: "'n::euclidean_space set"
```
```  3530   assumes "S \<noteq> {}"
```
```  3531     and "affine S"
```
```  3532     and "subspace L"
```
```  3533     and "affine_parallel S L"
```
```  3534   shows "aff_dim S = int (dim L)"
```
```  3535 proof -
```
```  3536   have *: "affine hull S = S"
```
```  3537     using assms affine_hull_eq[of S] by auto
```
```  3538   then have "affine_parallel (affine hull S) L"
```
```  3539     using assms by (simp add: *)
```
```  3540   then show ?thesis
```
```  3541     using assms aff_dim_parallel_subspace[of S L] by blast
```
```  3542 qed
```
```  3543
```
```  3544 lemma dim_affine_hull:
```
```  3545   fixes S :: "'n::euclidean_space set"
```
```  3546   shows "dim (affine hull S) = dim S"
```
```  3547 proof -
```
```  3548   have "dim (affine hull S) \<ge> dim S"
```
```  3549     using dim_subset by auto
```
```  3550   moreover have "dim (span S) \<ge> dim (affine hull S)"
```
```  3551     using dim_subset affine_hull_subset_span by blast
```
```  3552   moreover have "dim (span S) = dim S"
```
```  3553     using dim_span by auto
```
```  3554   ultimately show ?thesis by auto
```
```  3555 qed
```
```  3556
```
```  3557 lemma aff_dim_subspace:
```
```  3558   fixes S :: "'n::euclidean_space set"
```
```  3559   assumes "subspace S"
```
```  3560   shows "aff_dim S = int (dim S)"
```
```  3561 proof (cases "S={}")
```
```  3562   case True with assms show ?thesis
```
```  3563     by (simp add: subspace_affine)
```
```  3564 next
```
```  3565   case False
```
```  3566   with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
```
```  3567   show ?thesis by auto
```
```  3568 qed
```
```  3569
```
```  3570 lemma aff_dim_zero:
```
```  3571   fixes S :: "'n::euclidean_space set"
```
```  3572   assumes "0 \<in> affine hull S"
```
```  3573   shows "aff_dim S = int (dim S)"
```
```  3574 proof -
```
```  3575   have "subspace (affine hull S)"
```
```  3576     using subspace_affine[of "affine hull S"] affine_affine_hull assms
```
```  3577     by auto
```
```  3578   then have "aff_dim (affine hull S) = int (dim (affine hull S))"
```
```  3579     using assms aff_dim_subspace[of "affine hull S"] by auto
```
```  3580   then show ?thesis
```
```  3581     using aff_dim_affine_hull[of S] dim_affine_hull[of S]
```
```  3582     by auto
```
```  3583 qed
```
```  3584
```
```  3585 lemma aff_dim_eq_dim:
```
```  3586   fixes S :: "'n::euclidean_space set"
```
```  3587   assumes "a \<in> affine hull S"
```
```  3588   shows "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
```
```  3589 proof -
```
```  3590   have "0 \<in> affine hull ((\<lambda>x. -a+x) ` S)"
```
```  3591     unfolding Convex_Euclidean_Space.affine_hull_translation
```
```  3592     using assms by (simp add: ab_group_add_class.ab_left_minus image_iff)
```
```  3593   with aff_dim_zero show ?thesis
```
```  3594     by (metis aff_dim_translation_eq)
```
```  3595 qed
```
```  3596
```
```  3597 lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
```
```  3598   using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
```
```  3599     dim_UNIV[where 'a="'n::euclidean_space"]
```
```  3600   by auto
```
```  3601
```
```  3602 lemma aff_dim_geq:
```
```  3603   fixes V :: "'n::euclidean_space set"
```
```  3604   shows "aff_dim V \<ge> -1"
```
```  3605 proof -
```
```  3606   obtain B where "affine hull B = affine hull V"
```
```  3607     and "\<not> affine_dependent B"
```
```  3608     and "int (card B) = aff_dim V + 1"
```
```  3609     using aff_dim_basis_exists by auto
```
```  3610   then show ?thesis by auto
```
```  3611 qed
```
```  3612
```
```  3613 lemma aff_dim_negative_iff [simp]:
```
```  3614   fixes S :: "'n::euclidean_space set"
```
```  3615   shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
```
```  3616 by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
```
```  3617
```
```  3618 lemma aff_lowdim_subset_hyperplane:
```
```  3619   fixes S :: "'a::euclidean_space set"
```
```  3620   assumes "aff_dim S < DIM('a)"
```
```  3621   obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
```
```  3622 proof (cases "S={}")
```
```  3623   case True
```
```  3624   moreover
```
```  3625   have "(SOME b. b \<in> Basis) \<noteq> 0"
```
```  3626     by (metis norm_some_Basis norm_zero zero_neq_one)
```
```  3627   ultimately show ?thesis
```
```  3628     using that by blast
```
```  3629 next
```
```  3630   case False
```
```  3631   then obtain c S' where "c \<notin> S'" "S = insert c S'"
```
```  3632     by (meson equals0I mk_disjoint_insert)
```
```  3633   have "dim ((+) (-c) ` S) < DIM('a)"
```
```  3634     by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
```
```  3635   then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
```
```  3636     using lowdim_subset_hyperplane by blast
```
```  3637   moreover
```
```  3638   have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
```
```  3639   proof -
```
```  3640     have "w-c \<in> span ((+) (- c) ` S)"
```
```  3641       by (simp add: span_base \<open>w \<in> S\<close>)
```
```  3642     with that have "w-c \<in> {x. a \<bullet> x = 0}"
```
```  3643       by blast
```
```  3644     then show ?thesis
```
```  3645       by (auto simp: algebra_simps)
```
```  3646   qed
```
```  3647   ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
```
```  3648     by blast
```
```  3649   then show ?thesis
```
```  3650     by (rule that[OF \<open>a \<noteq> 0\<close>])
```
```  3651 qed
```
```  3652
```
```  3653 lemma affine_independent_card_dim_diffs:
```
```  3654   fixes S :: "'a :: euclidean_space set"
```
```  3655   assumes "~ affine_dependent S" "a \<in> S"
```
```  3656     shows "card S = dim {x - a|x. x \<in> S} + 1"
```
```  3657 proof -
```
```  3658   have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
```
```  3659   have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
```
```  3660   proof (cases "x = a")
```
```  3661     case True then show ?thesis by (simp add: span_clauses)
```
```  3662   next
```
```  3663     case False then show ?thesis
```
```  3664       using assms by (blast intro: span_base that)
```
```  3665   qed
```
```  3666   have "\<not> affine_dependent (insert a S)"
```
```  3667     by (simp add: assms insert_absorb)
```
```  3668   then have 3: "independent {b - a |b. b \<in> S - {a}}"
```
```  3669       using dependent_imp_affine_dependent by fastforce
```
```  3670   have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
```
```  3671     by blast
```
```  3672   then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
```
```  3673     by simp
```
```  3674   also have "\<dots> = card (S - {a})"
```
```  3675     by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
```
```  3676   also have "\<dots> = card S - 1"
```
```  3677     by (simp add: aff_independent_finite assms)
```
```  3678   finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
```
```  3679   have "finite S"
```
```  3680     by (meson assms aff_independent_finite)
```
```  3681   with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
```
```  3682   moreover have "dim {x - a |x. x \<in> S} = card S - 1"
```
```  3683     using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
```
```  3684   ultimately show ?thesis
```
```  3685     by auto
```
```  3686 qed
```
```  3687
```
```  3688 lemma independent_card_le_aff_dim:
```
```  3689   fixes B :: "'n::euclidean_space set"
```
```  3690   assumes "B \<subseteq> V"
```
```  3691   assumes "\<not> affine_dependent B"
```
```  3692   shows "int (card B) \<le> aff_dim V + 1"
```
```  3693 proof -
```
```  3694   obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
```
```  3695     by (metis assms extend_to_affine_basis[of B V])
```
```  3696   then have "of_nat (card T) = aff_dim V + 1"
```
```  3697     using aff_dim_unique by auto
```
```  3698   then show ?thesis
```
```  3699     using T card_mono[of T B] aff_independent_finite[of T] by auto
```
```  3700 qed
```
```  3701
```
```  3702 lemma aff_dim_subset:
```
```  3703   fixes S T :: "'n::euclidean_space set"
```
```  3704   assumes "S \<subseteq> T"
```
```  3705   shows "aff_dim S \<le> aff_dim T"
```
```  3706 proof -
```
```  3707   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
```
```  3708     "of_nat (card B) = aff_dim S + 1"
```
```  3709     using aff_dim_inner_basis_exists[of S] by auto
```
```  3710   then have "int (card B) \<le> aff_dim T + 1"
```
```  3711     using assms independent_card_le_aff_dim[of B T] by auto
```
```  3712   with B show ?thesis by auto
```
```  3713 qed
```
```  3714
```
```  3715 lemma aff_dim_le_DIM:
```
```  3716   fixes S :: "'n::euclidean_space set"
```
```  3717   shows "aff_dim S \<le> int (DIM('n))"
```
```  3718 proof -
```
```  3719   have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
```
```  3720     using aff_dim_UNIV by auto
```
```  3721   then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
```
```  3722     using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
```
```  3723 qed
```
```  3724
```
```  3725 lemma affine_dim_equal:
```
```  3726   fixes S :: "'n::euclidean_space set"
```
```  3727   assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
```
```  3728   shows "S = T"
```
```  3729 proof -
```
```  3730   obtain a where "a \<in> S" using assms by auto
```
```  3731   then have "a \<in> T" using assms by auto
```
```  3732   define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
```
```  3733   then have ls: "subspace LS" "affine_parallel S LS"
```
```  3734     using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
```
```  3735   then have h1: "int(dim LS) = aff_dim S"
```
```  3736     using assms aff_dim_affine[of S LS] by auto
```
```  3737   have "T \<noteq> {}" using assms by auto
```
```  3738   define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
```
```  3739   then have lt: "subspace LT \<and> affine_parallel T LT"
```
```  3740     using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
```
```  3741   then have "int(dim LT) = aff_dim T"
```
```  3742     using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
```
```  3743   then have "dim LS = dim LT"
```
```  3744     using h1 assms by auto
```
```  3745   moreover have "LS \<le> LT"
```
```  3746     using LS_def LT_def assms by auto
```
```  3747   ultimately have "LS = LT"
```
```  3748     using subspace_dim_equal[of LS LT] ls lt by auto
```
```  3749   moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
```
```  3750     using LS_def by auto
```
```  3751   moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
```
```  3752     using LT_def by auto
```
```  3753   ultimately show ?thesis by auto
```
```  3754 qed
```
```  3755
```
```  3756 lemma aff_dim_eq_0:
```
```  3757   fixes S :: "'a::euclidean_space set"
```
```  3758   shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
```
```  3759 proof (cases "S = {}")
```
```  3760   case True
```
```  3761   then show ?thesis
```
```  3762     by auto
```
```  3763 next
```
```  3764   case False
```
```  3765   then obtain a where "a \<in> S" by auto
```
```  3766   show ?thesis
```
```  3767   proof safe
```
```  3768     assume 0: "aff_dim S = 0"
```
```  3769     have "~ {a,b} \<subseteq> S" if "b \<noteq> a" for b
```
```  3770       by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
```
```  3771     then show "\<exists>a. S = {a}"
```
```  3772       using \<open>a \<in> S\<close> by blast
```
```  3773   qed auto
```
```  3774 qed
```
```  3775
```
```  3776 lemma affine_hull_UNIV:
```
```  3777   fixes S :: "'n::euclidean_space set"
```
```  3778   assumes "aff_dim S = int(DIM('n))"
```
```  3779   shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
```
```  3780 proof -
```
```  3781   have "S \<noteq> {}"
```
```  3782     using assms aff_dim_empty[of S] by auto
```
```  3783   have h0: "S \<subseteq> affine hull S"
```
```  3784     using hull_subset[of S _] by auto
```
```  3785   have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
```
```  3786     using aff_dim_UNIV assms by auto
```
```  3787   then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
```
```  3788     using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
```
```  3789   have h3: "aff_dim S \<le> aff_dim (affine hull S)"
```
```  3790     using h0 aff_dim_subset[of S "affine hull S"] assms by auto
```
```  3791   then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
```
```  3792     using h0 h1 h2 by auto
```
```  3793   then show ?thesis
```
```  3794     using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
```
```  3795       affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
```
```  3796     by auto
```
```  3797 qed
```
```  3798
```
```  3799 lemma disjoint_affine_hull:
```
```  3800   fixes s :: "'n::euclidean_space set"
```
```  3801   assumes "~ affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
```
```  3802     shows "(affine hull t) \<inter> (affine hull u) = {}"
```
```  3803 proof -
```
```  3804   have "finite s" using assms by (simp add: aff_independent_finite)
```
```  3805   then have "finite t" "finite u" using assms finite_subset by blast+
```
```  3806   { fix y
```
```  3807     assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
```
```  3808     then obtain a b
```
```  3809            where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
```
```  3810              and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
```
```  3811       by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
```
```  3812     define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
```
```  3813     have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
```
```  3814     have "sum c s = 0"
```
```  3815       by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
```
```  3816     moreover have "~ (\<forall>v\<in>s. c v = 0)"
```
```  3817       by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one)
```
```  3818     moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
```
```  3819       by (simp add: c_def if_smult sum_negf
```
```  3820              comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
```
```  3821     ultimately have False
```
```  3822       using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
```
```  3823   }
```
```  3824   then show ?thesis by blast
```
```  3825 qed
```
```  3826
```
```  3827 lemma aff_dim_convex_hull:
```
```  3828   fixes S :: "'n::euclidean_space set"
```
```  3829   shows "aff_dim (convex hull S) = aff_dim S"
```
```  3830   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
```
```  3831     hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
```
```  3832     aff_dim_subset[of "convex hull S" "affine hull S"]
```
```  3833   by auto
```
```  3834
```
```  3835 lemma aff_dim_cball:
```
```  3836   fixes a :: "'n::euclidean_space"
```
```  3837   assumes "e > 0"
```
```  3838   shows "aff_dim (cball a e) = int (DIM('n))"
```
```  3839 proof -
```
```  3840   have "(\<lambda>x. a + x) ` (cball 0 e) \<subseteq> cball a e"
```
```  3841     unfolding cball_def dist_norm by auto
```
```  3842   then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \<le> aff_dim (cball a e)"
```
```  3843     using aff_dim_translation_eq[of a "cball 0 e"]
```
```  3844           aff_dim_subset[of "(+) a ` cball 0 e" "cball a e"]
```
```  3845     by auto
```
```  3846   moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
```
```  3847     using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
```
```  3848       centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
```
```  3849     by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
```
```  3850   ultimately show ?thesis
```
```  3851     using aff_dim_le_DIM[of "cball a e"] by auto
```
```  3852 qed
```
```  3853
```
```  3854 lemma aff_dim_open:
```
```  3855   fixes S :: "'n::euclidean_space set"
```
```  3856   assumes "open S"
```
```  3857     and "S \<noteq> {}"
```
```  3858   shows "aff_dim S = int (DIM('n))"
```
```  3859 proof -
```
```  3860   obtain x where "x \<in> S"
```
```  3861     using assms by auto
```
```  3862   then obtain e where e: "e > 0" "cball x e \<subseteq> S"
```
```  3863     using open_contains_cball[of S] assms by auto
```
```  3864   then have "aff_dim (cball x e) \<le> aff_dim S"
```
```  3865     using aff_dim_subset by auto
```
```  3866   with e show ?thesis
```
```  3867     using aff_dim_cball[of e x] aff_dim_le_DIM[of S] by auto
```
```  3868 qed
```
```  3869
```
```  3870 lemma low_dim_interior:
```
```  3871   fixes S :: "'n::euclidean_space set"
```
```  3872   assumes "\<not> aff_dim S = int (DIM('n))"
```
```  3873   shows "interior S = {}"
```
```  3874 proof -
```
```  3875   have "aff_dim(interior S) \<le> aff_dim S"
```
```  3876     using interior_subset aff_dim_subset[of "interior S" S] by auto
```
```  3877   then show ?thesis
```
```  3878     using aff_dim_open[of "interior S"] aff_dim_le_DIM[of S] assms by auto
```
```  3879 qed
```
```  3880
```
```  3881 corollary empty_interior_lowdim:
```
```  3882   fixes S :: "'n::euclidean_space set"
```
```  3883   shows "dim S < DIM ('n) \<Longrightarrow> interior S = {}"
```
```  3884 by (metis low_dim_interior affine_hull_UNIV dim_affine_hull less_not_refl dim_UNIV)
```
```  3885
```
```  3886 corollary aff_dim_nonempty_interior:
```
```  3887   fixes S :: "'a::euclidean_space set"
```
```  3888   shows "interior S \<noteq> {} \<Longrightarrow> aff_dim S = DIM('a)"
```
```  3889 by (metis low_dim_interior)
```
```  3890
```
```  3891
```
```  3892 subsection \<open>Caratheodory's theorem\<close>
```
```  3893
```
```  3894 lemma convex_hull_caratheodory_aff_dim:
```
```  3895   fixes p :: "('a::euclidean_space) set"
```
```  3896   shows "convex hull p =
```
```  3897     {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
```
```  3898       (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
```
```  3899   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
```
```  3900 proof (intro allI iffI)
```
```  3901   fix y
```
```  3902   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>
```
```  3903     sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
```
```  3904   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"
```
```  3905   then obtain N where "?P N" by auto
```
```  3906   then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
```
```  3907     apply (rule_tac ex_least_nat_le, auto)
```
```  3908     done
```
```  3909   then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
```
```  3910     by blast
```
```  3911   then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
```
```  3912     "sum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
```
```  3913
```
```  3914   have "card s \<le> aff_dim p + 1"
```
```  3915   proof (rule ccontr, simp only: not_le)
```
```  3916     assume "aff_dim p + 1 < card s"
```
```  3917     then have "affine_dependent s"
```
```  3918       using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
```
```  3919       by blast
```
```  3920     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"
```
```  3921       using affine_dependent_explicit_finite[OF obt(1)] by auto
```
```  3922     define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
```
```  3923     define t where "t = Min i"
```
```  3924     have "\<exists>x\<in>s. w x < 0"
```
```  3925     proof (rule ccontr, simp add: not_less)
```
```  3926       assume as:"\<forall>x\<in>s. 0 \<le> w x"
```
```  3927       then have "sum w (s - {v}) \<ge> 0"
```
```  3928         apply (rule_tac sum_nonneg, auto)
```
```  3929         done
```
```  3930       then have "sum w s > 0"
```
```  3931         unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
```
```  3932         using as[THEN bspec[where x=v]]  \<open>v\<in>s\<close>  \<open>w v \<noteq> 0\<close> by auto
```
```  3933       then show False using wv(1) by auto
```
```  3934     qed
```
```  3935     then have "i \<noteq> {}" unfolding i_def by auto
```
```  3936     then have "t \<ge> 0"
```
```  3937       using Min_ge_iff[of i 0 ] and obt(1)
```
```  3938       unfolding t_def i_def
```
```  3939       using obt(4)[unfolded le_less]
```
```  3940       by (auto simp: divide_le_0_iff)
```
```  3941     have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
```
```  3942     proof
```
```  3943       fix v
```
```  3944       assume "v \<in> s"
```
```  3945       then have v: "0 \<le> u v"
```
```  3946         using obt(4)[THEN bspec[where x=v]] by auto
```
```  3947       show "0 \<le> u v + t * w v"
```
```  3948       proof (cases "w v < 0")
```
```  3949         case False
```
```  3950         thus ?thesis using v \<open>t\<ge>0\<close> by auto
```
```  3951       next
```
```  3952         case True
```
```  3953         then have "t \<le> u v / (- w v)"
```
```  3954           using \<open>v\<in>s\<close> unfolding t_def i_def
```
```  3955           apply (rule_tac Min_le)
```
```  3956           using obt(1) apply auto
```
```  3957           done
```
```  3958         then show ?thesis
```
```  3959           unfolding real_0_le_add_iff
```
```  3960           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
```
```  3961           by auto
```
```  3962       qed
```
```  3963     qed
```
```  3964     obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
```
```  3965       using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
```
```  3966     then have a: "a \<in> s" "u a + t * w a = 0" by auto
```
```  3967     have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)"
```
```  3968       unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
```
```  3969     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
```
```  3970       unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
```
```  3971     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"
```
```  3972       unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
```
```  3973       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
```
```  3974     ultimately have "?P (n - 1)"
```
```  3975       apply (rule_tac x="(s - {a})" in exI)
```
```  3976       apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
```
```  3977       using obt(1-3) and t and a
```
```  3978       apply (auto simp: * scaleR_left_distrib)
```
```  3979       done
```
```  3980     then show False
```
```  3981       using smallest[THEN spec[where x="n - 1"]] by auto
```
```  3982   qed
```
```  3983   then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
```
```  3984       (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
```
```  3985     using obt by auto
```
```  3986 qed auto
```
```  3987
```
```  3988 lemma caratheodory_aff_dim:
```
```  3989   fixes p :: "('a::euclidean_space) set"
```
```  3990   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}"
```
```  3991         (is "?lhs = ?rhs")
```
```  3992 proof
```
```  3993   show "?lhs \<subseteq> ?rhs"
```
```  3994     apply (subst convex_hull_caratheodory_aff_dim, clarify)
```
```  3995     apply (rule_tac x=s in exI)
```
```  3996     apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
```
```  3997     done
```
```  3998 next
```
```  3999   show "?rhs \<subseteq> ?lhs"
```
```  4000     using hull_mono by blast
```
```  4001 qed
```
```  4002
```
```  4003 lemma convex_hull_caratheodory:
```
```  4004   fixes p :: "('a::euclidean_space) set"
```
```  4005   shows "convex hull p =
```
```  4006             {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
```
```  4007               (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
```
```  4008         (is "?lhs = ?rhs")
```
```  4009 proof (intro set_eqI iffI)
```
```  4010   fix x
```
```  4011   assume "x \<in> ?lhs" then show "x \<in> ?rhs"
```
```  4012     apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
```
```  4013     apply (erule ex_forward)+
```
```  4014     using aff_dim_le_DIM [of p]
```
```  4015     apply simp
```
```  4016     done
```
```  4017 next
```
```  4018   fix x
```
```  4019   assume "x \<in> ?rhs" then show "x \<in> ?lhs"
```
```  4020     by (auto simp: convex_hull_explicit)
```
```  4021 qed
```
```  4022
```
```  4023 theorem caratheodory:
```
```  4024   "convex hull p =
```
```  4025     {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
```
```  4026       card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
```
```  4027 proof safe
```
```  4028   fix x
```
```  4029   assume "x \<in> convex hull p"
```
```  4030   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
```
```  4031     "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
```
```  4032     unfolding convex_hull_caratheodory by auto
```
```  4033   then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
```
```  4034     apply (rule_tac x=s in exI)
```
```  4035     using hull_subset[of s convex]
```
```  4036     using convex_convex_hull[simplified convex_explicit, of s,
```
```  4037       THEN spec[where x=s], THEN spec[where x=u]]
```
```  4038     apply auto
```
```  4039     done
```
```  4040 next
```
```  4041   fix x s
```
```  4042   assume  "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
```
```  4043   then show "x \<in> convex hull p"
```
```  4044     using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
```
```  4045 qed
```
```  4046
```
```  4047
```
```  4048 subsection \<open>Relative interior of a set\<close>
```
```  4049
```
```  4050 definition%important "rel_interior S =
```
```  4051   {x. \<exists>T. openin (subtopology euclidean (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
```
```  4052
```
```  4053 lemma rel_interior_mono:
```
```  4054    "\<lbrakk>S \<subseteq> T; affine hull S = affine hull T\<rbrakk>
```
```  4055    \<Longrightarrow> (rel_interior S) \<subseteq> (rel_interior T)"
```
```  4056   by (auto simp: rel_interior_def)
```
```  4057
```
```  4058 lemma rel_interior_maximal:
```
```  4059    "\<lbrakk>T \<subseteq> S; openin(subtopology euclidean (affine hull S)) T\<rbrakk> \<Longrightarrow> T \<subseteq> (rel_interior S)"
```
```  4060   by (auto simp: rel_interior_def)
```
```  4061
```
```  4062 lemma rel_interior:
```
```  4063   "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
```
```  4064   unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
```
```  4065   apply auto
```
```  4066 proof -
```
```  4067   fix x T
```
```  4068   assume *: "x \<in> S" "open T" "x \<in> T" "T \<inter> affine hull S \<subseteq> S"
```
```  4069   then have **: "x \<in> T \<inter> affine hull S"
```
```  4070     using hull_inc by auto
```
```  4071   show "\<exists>Tb. (\<exists>Ta. open Ta \<and> Tb = affine hull S \<inter> Ta) \<and> x \<in> Tb \<and> Tb \<subseteq> S"
```
```  4072     apply (rule_tac x = "T \<inter> (affine hull S)" in exI)
```
```  4073     using * **
```
```  4074     apply auto
```
```  4075     done
```
```  4076 qed
```
```  4077
```
```  4078 lemma mem_rel_interior: "x \<in> rel_interior S \<longleftrightarrow> (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
```
```  4079   by (auto simp: rel_interior)
```
```  4080
```
```  4081 lemma mem_rel_interior_ball:
```
```  4082   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S)"
```
```  4083   apply (simp add: rel_interior, safe)
```
```  4084   apply (force simp: open_contains_ball)
```
```  4085   apply (rule_tac x = "ball x e" in exI, simp)
```
```  4086   done
```
```  4087
```
```  4088 lemma rel_interior_ball:
```
```  4089   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S}"
```
```  4090   using mem_rel_interior_ball [of _ S] by auto
```
```  4091
```
```  4092 lemma mem_rel_interior_cball:
```
```  4093   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S)"
```
```  4094   apply (simp add: rel_interior, safe)
```
```  4095   apply (force simp: open_contains_cball)
```
```  4096   apply (rule_tac x = "ball x e" in exI)
```
```  4097   apply (simp add: subset_trans [OF ball_subset_cball], auto)
```
```  4098   done
```
```  4099
```
```  4100 lemma rel_interior_cball:
```
```  4101   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S}"
```
```  4102   using mem_rel_interior_cball [of _ S] by auto
```
```  4103
```
```  4104 lemma rel_interior_empty [simp]: "rel_interior {} = {}"
```
```  4105    by (auto simp: rel_interior_def)
```
```  4106
```
```  4107 lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
```
```  4108   by (metis affine_hull_eq affine_sing)
```
```  4109
```
```  4110 lemma rel_interior_sing [simp]:
```
```  4111     fixes a :: "'n::euclidean_space"  shows "rel_interior {a} = {a}"
```
```  4112   apply (auto simp: rel_interior_ball)
```
```  4113   apply (rule_tac x=1 in exI, force)
```
```  4114   done
```
```  4115
```
```  4116 lemma subset_rel_interior:
```
```  4117   fixes S T :: "'n::euclidean_space set"
```
```  4118   assumes "S \<subseteq> T"
```
```  4119     and "affine hull S = affine hull T"
```
```  4120   shows "rel_interior S \<subseteq> rel_interior T"
```
```  4121   using assms by (auto simp: rel_interior_def)
```
```  4122
```
```  4123 lemma rel_interior_subset: "rel_interior S \<subseteq> S"
```
```  4124   by (auto simp: rel_interior_def)
```
```  4125
```
```  4126 lemma rel_interior_subset_closure: "rel_interior S \<subseteq> closure S"
```
```  4127   using rel_interior_subset by (auto simp: closure_def)
```
```  4128
```
```  4129 lemma interior_subset_rel_interior: "interior S \<subseteq> rel_interior S"
```
```  4130   by (auto simp: rel_interior interior_def)
```
```  4131
```
```  4132 lemma interior_rel_interior:
```
```  4133   fixes S :: "'n::euclidean_space set"
```
```  4134   assumes "aff_dim S = int(DIM('n))"
```
```  4135   shows "rel_interior S = interior S"
```
```  4136 proof -
```
```  4137   have "affine hull S = UNIV"
```
```  4138     using assms affine_hull_UNIV[of S] by auto
```
```  4139   then show ?thesis
```
```  4140     unfolding rel_interior interior_def by auto
```
```  4141 qed
```
```  4142
```
```  4143 lemma rel_interior_interior:
```
```  4144   fixes S :: "'n::euclidean_space set"
```
```  4145   assumes "affine hull S = UNIV"
```
```  4146   shows "rel_interior S = interior S"
```
```  4147   using assms unfolding rel_interior interior_def by auto
```
```  4148
```
```  4149 lemma rel_interior_open:
```
```  4150   fixes S :: "'n::euclidean_space set"
```
```  4151   assumes "open S"
```
```  4152   shows "rel_interior S = S"
```
```  4153   by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
```
```  4154
```
```  4155 lemma interior_ball [simp]: "interior (ball x e) = ball x e"
```
```  4156   by (simp add: interior_open)
```
```  4157
```
```  4158 lemma interior_rel_interior_gen:
```
```  4159   fixes S :: "'n::euclidean_space set"
```
```  4160   shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
```
```  4161   by (metis interior_rel_interior low_dim_interior)
```
```  4162
```
```  4163 lemma rel_interior_nonempty_interior:
```
```  4164   fixes S :: "'n::euclidean_space set"
```
```  4165   shows "interior S \<noteq> {} \<Longrightarrow> rel_interior S = interior S"
```
```  4166 by (metis interior_rel_interior_gen)
```
```  4167
```
```  4168 lemma affine_hull_nonempty_interior:
```
```  4169   fixes S :: "'n::euclidean_space set"
```
```  4170   shows "interior S \<noteq> {} \<Longrightarrow> affine hull S = UNIV"
```
```  4171 by (metis affine_hull_UNIV interior_rel_interior_gen)
```
```  4172
```
```  4173 lemma rel_interior_affine_hull [simp]:
```
```  4174   fixes S :: "'n::euclidean_space set"
```
```  4175   shows "rel_interior (affine hull S) = affine hull S"
```
```  4176 proof -
```
```  4177   have *: "rel_interior (affine hull S) \<subseteq> affine hull S"
```
```  4178     using rel_interior_subset by auto
```
```  4179   {
```
```  4180     fix x
```
```  4181     assume x: "x \<in> affine hull S"
```
```  4182     define e :: real where "e = 1"
```
```  4183     then have "e > 0" "ball x e \<inter> affine hull (affine hull S) \<subseteq> affine hull S"
```
```  4184       using hull_hull[of _ S] by auto
```
```  4185     then have "x \<in> rel_interior (affine hull S)"
```
```  4186       using x rel_interior_ball[of "affine hull S"] by auto
```
```  4187   }
```
```  4188   then show ?thesis using * by auto
```
```  4189 qed
```
```  4190
```
```  4191 lemma rel_interior_UNIV [simp]: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
```
```  4192   by (metis open_UNIV rel_interior_open)
```
```  4193
```
```  4194 lemma rel_interior_convex_shrink:
```
```  4195   fixes S :: "'a::euclidean_space set"
```
```  4196   assumes "convex S"
```
```  4197     and "c \<in> rel_interior S"
```
```  4198     and "x \<in> S"
```
```  4199     and "0 < e"
```
```  4200     and "e \<le> 1"
```
```  4201   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
```
```  4202 proof -
```
```  4203   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
```
```  4204     using assms(2) unfolding  mem_rel_interior_ball by auto
```
```  4205   {
```
```  4206     fix y
```
```  4207     assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d" "y \<in> affine hull S"
```
```  4208     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
```
```  4209       using \<open>e > 0\<close> by (auto simp: scaleR_left_diff_distrib scaleR_right_diff_distrib)
```
```  4210     have "x \<in> affine hull S"
```
```  4211       using assms hull_subset[of S] by auto
```
```  4212     moreover have "1 / e + - ((1 - e) / e) = 1"
```
```  4213       using \<open>e > 0\<close> left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
```
```  4214     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \<in> affine hull S"
```
```  4215       using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
```
```  4216       by (simp add: algebra_simps)
```
```  4217     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
```
```  4218       unfolding dist_norm norm_scaleR[symmetric]
```
```  4219       apply (rule arg_cong[where f=norm])
```
```  4220       using \<open>e > 0\<close>
```
```  4221       apply (auto simp: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
```
```  4222       done
```
```  4223     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
```
```  4224       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
```
```  4225     also have "\<dots> < d"
```
```  4226       using as[unfolded dist_norm] and \<open>e > 0\<close>
```
```  4227       by (auto simp:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
```
```  4228     finally have "y \<in> S"
```
```  4229       apply (subst *)
```
```  4230       apply (rule assms(1)[unfolded convex_alt,rule_format])
```
```  4231       apply (rule d[THEN subsetD])
```
```  4232       unfolding mem_ball
```
```  4233       using assms(3-5) **
```
```  4234       apply auto
```
```  4235       done
```
```  4236   }
```
```  4237   then have "ball (x - e *\<^sub>R (x - c)) (e*d) \<inter> affine hull S \<subseteq> S"
```
```  4238     by auto
```
```  4239   moreover have "e * d > 0"
```
```  4240     using \<open>e > 0\<close> \<open>d > 0\<close> by simp
```
```  4241   moreover have c: "c \<in> S"
```
```  4242     using assms rel_interior_subset by auto
```
```  4243   moreover from c have "x - e *\<^sub>R (x - c) \<in> S"
```
```  4244     using convexD_alt[of S x c e]
```
```  4245     apply (simp add: algebra_simps)
```
```  4246     using assms
```
```  4247     apply auto
```
```  4248     done
```
```  4249   ultimately show ?thesis
```
```  4250     using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] \<open>e > 0\<close> by auto
```
```  4251 qed
```
```  4252
```
```  4253 lemma interior_real_semiline:
```
```  4254   fixes a :: real
```
```  4255   shows "interior {a..} = {a<..}"
```
```  4256 proof -
```
```  4257   {
```
```  4258     fix y
```
```  4259     assume "a < y"
```
```  4260     then have "y \<in> interior {a..}"
```
```  4261       apply (simp add: mem_interior)
```
```  4262       apply (rule_tac x="(y-a)" in exI)
```
```  4263       apply (auto simp: dist_norm)
```
```  4264       done
```
```  4265   }
```
```  4266   moreover
```
```  4267   {
```
```  4268     fix y
```
```  4269     assume "y \<in> interior {a..}"
```
```  4270     then obtain e where e: "e > 0" "cball y e \<subseteq> {a..}"
```
```  4271       using mem_interior_cball[of y "{a..}"] by auto
```
```  4272     moreover from e have "y - e \<in> cball y e"
```
```  4273       by (auto simp: cball_def dist_norm)
```
```  4274     ultimately have "a \<le> y - e" by blast
```
```  4275     then have "a < y" using e by auto
```
```  4276   }
```
```  4277   ultimately show ?thesis by auto
```
```  4278 qed
```
```  4279
```
```  4280 lemma continuous_ge_on_Ioo:
```
```  4281   assumes "continuous_on {c..d} g" "\<And>x. x \<in> {c<..<d} \<Longrightarrow> g x \<ge> a" "c < d" "x \<in> {c..d}"
```
```  4282   shows "g (x::real) \<ge> (a::real)"
```
```  4283 proof-
```
```  4284   from assms(3) have "{c..d} = closure {c<..<d}" by (rule closure_greaterThanLessThan[symmetric])
```
```  4285   also from assms(2) have "{c<..<d} \<subseteq> (g -` {a..} \<inter> {c..d})" by auto
```
```  4286   hence "closure {c<..<d} \<subseteq> closure (g -` {a..} \<inter> {c..d})" by (rule closure_mono)
```
```  4287   also from assms(1) have "closed (g -` {a..} \<inter> {c..d})"
```
```  4288     by (auto simp: continuous_on_closed_vimage)
```
```  4289   hence "closure (g -` {a..} \<inter> {c..d}) = g -` {a..} \<inter> {c..d}" by simp
```
```  4290   finally show ?thesis using \<open>x \<in> {c..d}\<close> by auto
```
```  4291 qed
```
```  4292
```
```  4293 lemma interior_real_semiline':
```
```  4294   fixes a :: real
```
```  4295   shows "interior {..a} = {..<a}"
```
```  4296 proof -
```
```  4297   {
```
```  4298     fix y
```
```  4299     assume "a > y"
```
```  4300     then have "y \<in> interior {..a}"
```
```  4301       apply (simp add: mem_interior)
```
```  4302       apply (rule_tac x="(a-y)" in exI)
```
```  4303       apply (auto simp: dist_norm)
```
```  4304       done
```
```  4305   }
```
```  4306   moreover
```
```  4307   {
```
```  4308     fix y
```
```  4309     assume "y \<in> interior {..a}"
```
```  4310     then obtain e where e: "e > 0" "cball y e \<subseteq> {..a}"
```
```  4311       using mem_interior_cball[of y "{..a}"] by auto
```
```  4312     moreover from e have "y + e \<in> cball y e"
```
```  4313       by (auto simp: cball_def dist_norm)
```
```  4314     ultimately have "a \<ge> y + e" by auto
```
```  4315     then have "a > y" using e by auto
```
```  4316   }
```
```  4317   ultimately show ?thesis by auto
```
```  4318 qed
```
```  4319
```
```  4320 lemma interior_atLeastAtMost_real [simp]: "interior {a..b} = {a<..<b :: real}"
```
```  4321 proof-
```
```  4322   have "{a..b} = {a..} \<inter> {..b}" by auto
```
```  4323   also have "interior \<dots> = {a<..} \<inter> {..<b}"
```
```  4324     by (simp add: interior_real_semiline interior_real_semiline')
```
```  4325   also have "\<dots> = {a<..<b}" by auto
```
```  4326   finally show ?thesis .
```
```  4327 qed
```
```  4328
```
```  4329 lemma interior_atLeastLessThan [simp]:
```
```  4330   fixes a::real shows "interior {a..<b} = {a<..<b}"
```
```  4331   by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost_real interior_Int interior_interior interior_real_semiline)
```
```  4332
```
```  4333 lemma interior_lessThanAtMost [simp]:
```
```  4334   fixes a::real shows "interior {a<..b} = {a<..<b}"
```
```  4335   by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost_real interior_Int
```
```  4336             interior_interior interior_real_semiline)
```
```  4337
```
```  4338 lemma interior_greaterThanLessThan_real [simp]: "interior {a<..<b} = {a<..<b :: real}"
```
```  4339   by (metis interior_atLeastAtMost_real interior_interior)
```
```  4340
```
```  4341 lemma frontier_real_Iic [simp]:
```
```  4342   fixes a :: real
```
```  4343   shows "frontier {..a} = {a}"
```
```  4344   unfolding frontier_def by (auto simp: interior_real_semiline')
```
```  4345
```
```  4346 lemma rel_interior_real_box [simp]:
```
```  4347   fixes a b :: real
```
```  4348   assumes "a < b"
```
```  4349   shows "rel_interior {a .. b} = {a <..< b}"
```
```  4350 proof -
```
```  4351   have "box a b \<noteq> {}"
```
```  4352     using assms
```
```  4353     unfolding set_eq_iff
```
```  4354     by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
```
```  4355   then show ?thesis
```
```  4356     using interior_rel_interior_gen[of "cbox a b", symmetric]
```
```  4357     by (simp split: if_split_asm del: box_real add: box_real[symmetric] interior_cbox)
```
```  4358 qed
```
```  4359
```
```  4360 lemma rel_interior_real_semiline [simp]:
```
```  4361   fixes a :: real
```
```  4362   shows "rel_interior {a..} = {a<..}"
```
```  4363 proof -
```
```  4364   have *: "{a<..} \<noteq> {}"
```
```  4365     unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
```
```  4366   then show ?thesis using interior_real_semiline interior_rel_interior_gen[of "{a..}"]
```
```  4367     by (auto split: if_split_asm)
```
```  4368 qed
```
```  4369
```
```  4370 subsubsection \<open>Relative open sets\<close>
```
```  4371
```
```  4372 definition%important "rel_open S \<longleftrightarrow> rel_interior S = S"
```
```  4373
```
```  4374 lemma rel_open: "rel_open S \<longleftrightarrow> openin (subtopology euclidean (affine hull S)) S"
```
```  4375   unfolding rel_open_def rel_interior_def
```
```  4376   apply auto
```
```  4377   using openin_subopen[of "subtopology euclidean (affine hull S)" S]
```
```  4378   apply auto
```
```  4379   done
```
```  4380
```
```  4381 lemma openin_rel_interior: "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
```
```  4382   apply (simp add: rel_interior_def)
```
```  4383   apply (subst openin_subopen, blast)
```
```  4384   done
```
```  4385
```
```  4386 lemma openin_set_rel_interior:
```
```  4387    "openin (subtopology euclidean S) (rel_interior S)"
```
```  4388 by (rule openin_subset_trans [OF openin_rel_interior rel_interior_subset hull_subset])
```
```  4389
```
```  4390 lemma affine_rel_open:
```
```  4391   fixes S :: "'n::euclidean_space set"
```
```  4392   assumes "affine S"
```
```  4393   shows "rel_open S"
```
```  4394   unfolding rel_open_def
```
```  4395   using assms rel_interior_affine_hull[of S] affine_hull_eq[of S]
```
```  4396   by metis
```
```  4397
```
```  4398 lemma affine_closed:
```
```  4399   fixes S :: "'n::euclidean_space set"
```
```  4400   assumes "affine S"
```
```  4401   shows "closed S"
```
```  4402 proof -
```
```  4403   {
```
```  4404     assume "S \<noteq> {}"
```
```  4405     then obtain L where L: "subspace L" "affine_parallel S L"
```
```  4406       using assms affine_parallel_subspace[of S] by auto
```
```  4407     then obtain a where a: "S = ((+) a ` L)"
```
```  4408       using affine_parallel_def[of L S] affine_parallel_commut by auto
```
```  4409     from L have "closed L" using closed_subspace by auto
```
```  4410     then have "closed S"
```
```  4411       using closed_translation a by auto
```
```  4412   }
```
```  4413   then show ?thesis by auto
```
```  4414 qed
```
```  4415
```
```  4416 lemma closure_affine_hull:
```
```  4417   fixes S :: "'n::euclidean_space set"
```
```  4418   shows "closure S \<subseteq> affine hull S"
```
```  4419   by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
```
```  4420
```
```  4421 lemma closure_same_affine_hull [simp]:
```
```  4422   fixes S :: "'n::euclidean_space set"
```
```  4423   shows "affine hull (closure S) = affine hull S"
```
```  4424 proof -
```
```  4425   have "affine hull (closure S) \<subseteq> affine hull S"
```
```  4426     using hull_mono[of "closure S" "affine hull S" "affine"]
```
```  4427       closure_affine_hull[of S] hull_hull[of "affine" S]
```
```  4428     by auto
```
```  4429   moreover have "affine hull (closure S) \<supseteq> affine hull S"
```
```  4430     using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
```
```  4431   ultimately show ?thesis by auto
```
```  4432 qed
```
```  4433
```
```  4434 lemma closure_aff_dim [simp]:
```
```  4435   fixes S :: "'n::euclidean_space set"
```
```  4436   shows "aff_dim (closure S) = aff_dim S"
```
```  4437 proof -
```
```  4438   have "aff_dim S \<le> aff_dim (closure S)"
```
```  4439     using aff_dim_subset closure_subset by auto
```
```  4440   moreover have "aff_dim (closure S) \<le> aff_dim (affine hull S)"
```
```  4441     using aff_dim_subset closure_affine_hull by blast
```
```  4442   moreover have "aff_dim (affine hull S) = aff_dim S"
```
```  4443     using aff_dim_affine_hull by auto
```
```  4444   ultimately show ?thesis by auto
```
```  4445 qed
```
```  4446
```
```  4447 lemma rel_interior_closure_convex_shrink:
```
```  4448   fixes S :: "_::euclidean_space set"
```
```  4449   assumes "convex S"
```
```  4450     and "c \<in> rel_interior S"
```
```  4451     and "x \<in> closure S"
```
```  4452     and "e > 0"
```
```  4453     and "e \<le> 1"
```
```  4454   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
```
```  4455 proof -
```
```  4456   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
```
```  4457     using assms(2) unfolding mem_rel_interior_ball by auto
```
```  4458   have "\<exists>y \<in> S. norm (y - x) * (1 - e) < e * d"
```
```  4459   proof (cases "x \<in> S")
```
```  4460     case True
```
```  4461     then show ?thesis using \<open>e > 0\<close> \<open>d > 0\<close>
```
```  4462       apply (rule_tac bexI[where x=x], auto)
```
```  4463       done
```
```  4464   next
```
```  4465     case False
```
```  4466     then have x: "x islimpt S"
```
```  4467       using assms(3)[unfolded closure_def] by auto
```
```  4468     show ?thesis
```
```  4469     proof (cases "e = 1")
```
```  4470       case True
```
```  4471       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
```
```  4472         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
```
```  4473       then show ?thesis
```
```  4474         apply (rule_tac x=y in bexI)
```
```  4475         unfolding True
```
```  4476         using \<open>d > 0\<close>
```
```  4477         apply auto
```
```  4478         done
```
```  4479     next
```
```  4480       case False
```
```  4481       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
```
```  4482         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
```
```  4483       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
```
```  4484         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
```
```  4485       then show ?thesis
```
```  4486         apply (rule_tac x=y in bexI)
```
```  4487         unfolding dist_norm
```
```  4488         using pos_less_divide_eq[OF *]
```
```  4489         apply auto
```
```  4490         done
```
```  4491     qed
```
```  4492   qed
```
```  4493   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
```
```  4494     by auto
```
```  4495   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
```
```  4496   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
```
```  4497     unfolding z_def using \<open>e > 0\<close>
```
```  4498     by (auto simp: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
```
```  4499   have zball: "z \<in> ball c d"
```
```  4500     using mem_ball z_def dist_norm[of c]
```
```  4501     using y and assms(4,5)
```
```  4502     by (auto simp:field_simps norm_minus_commute)
```
```  4503   have "x \<in> affine hull S"
```
```  4504     using closure_affine_hull assms by auto
```
```  4505   moreover have "y \<in> affine hull S"
```
```  4506     using \<open>y \<in> S\<close> hull_subset[of S] by auto
```
```  4507   moreover have "c \<in> affine hull S"
```
```  4508     using assms rel_interior_subset hull_subset[of S] by auto
```
```  4509   ultimately have "z \<in> affine hull S"
```
```  4510     using z_def affine_affine_hull[of S]
```
```  4511       mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
```
```  4512       assms
```
```  4513     by (auto simp: field_simps)
```
```  4514   then have "z \<in> S" using d zball by auto
```
```  4515   obtain d1 where "d1 > 0" and d1: "ball z d1 \<le> ball c d"
```
```  4516     using zball open_ball[of c d] openE[of "ball c d" z] by auto
```
```  4517   then have "ball z d1 \<inter> affine hull S \<subseteq> ball c d \<inter> affine hull S"
```
```  4518     by auto
```
```  4519   then have "ball z d1 \<inter> affine hull S \<subseteq> S"
```
```  4520     using d by auto
```
```  4521   then have "z \<in> rel_interior S"
```
```  4522     using mem_rel_interior_ball using \<open>d1 > 0\<close> \<open>z \<in> S\<close> by auto
```
```  4523   then have "y - e *\<^sub>R (y - z) \<in> rel_interior S"
```
```  4524     using rel_interior_convex_shrink[of S z y e] assms \<open>y \<in> S\<close> by auto
```
```  4525   then show ?thesis using * by auto
```
```  4526 qed
```
```  4527
```
```  4528 lemma rel_interior_eq:
```
```  4529    "rel_interior s = s \<longleftrightarrow> openin(subtopology euclidean (affine hull s)) s"
```
```  4530 using rel_open rel_open_def by blast
```
```  4531
```
```  4532 lemma rel_interior_openin:
```
```  4533    "openin(subtopology euclidean (affine hull s)) s \<Longrightarrow> rel_interior s = s"
```
```  4534 by (simp add: rel_interior_eq)
```
```  4535
```
```  4536 lemma rel_interior_affine:
```
```  4537   fixes S :: "'n::euclidean_space set"
```
```  4538   shows  "affine S \<Longrightarrow> rel_interior S = S"
```
```  4539 using affine_rel_open rel_open_def by auto
```
```  4540
```
```  4541 lemma rel_interior_eq_closure:
```
```  4542   fixes S :: "'n::euclidean_space set"
```
```  4543   shows "rel_interior S = closure S \<longleftrightarrow> affine S"
```
```  4544 proof (cases "S = {}")
```
```  4545   case True
```
```  4546  then show ?thesis
```
```  4547     by auto
```
```  4548 next
```
```  4549   case False show ?thesis
```
```  4550   proof
```
```  4551     assume eq: "rel_interior S = closure S"
```
```  4552     have "S = {} \<or> S = affine hull S"
```
```  4553       apply (rule connected_clopen [THEN iffD1, rule_format])
```
```  4554        apply (simp add: affine_imp_convex convex_connected)
```
```  4555       apply (rule conjI)
```
```  4556        apply (metis eq closure_subset openin_rel_interior rel_interior_subset subset_antisym)
```
```  4557       apply (metis closed_subset closure_subset_eq eq hull_subset rel_interior_subset)
```
```  4558       done
```
```  4559     with False have "affine hull S = S"
```
```  4560       by auto
```
```  4561     then show "affine S"
```
```  4562       by (metis affine_hull_eq)
```
```  4563   next
```
```  4564     assume "affine S"
```
```  4565     then show "rel_interior S = closure S"
```
```  4566       by (simp add: rel_interior_affine affine_closed)
```
```  4567   qed
```
```  4568 qed
```
```  4569
```
```  4570
```
```  4571 subsubsection%unimportant\<open>Relative interior preserves under linear transformations\<close>
```
```  4572
```
```  4573 lemma rel_interior_translation_aux:
```
```  4574   fixes a :: "'n::euclidean_space"
```
```  4575   shows "((\<lambda>x. a + x) ` rel_interior S) \<subseteq> rel_interior ((\<lambda>x. a + x) ` S)"
```
```  4576 proof -
```
```  4577   {
```
```  4578     fix x
```
```  4579     assume x: "x \<in> rel_interior S"
```
```  4580     then obtain T where "open T" "x \<in> T \<inter> S" "T \<inter> affine hull S \<subseteq> S"
```
```  4581       using mem_rel_interior[of x S] by auto
```
```  4582     then have "open ((\<lambda>x. a + x) ` T)"
```
```  4583       and "a + x \<in> ((\<lambda>x. a + x) ` T) \<inter> ((\<lambda>x. a + x) ` S)"
```
```  4584       and "((\<lambda>x. a + x) ` T) \<inter> affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` S"
```
```  4585       using affine_hull_translation[of a S] open_translation[of T a] x by auto
```
```  4586     then have "a + x \<in> rel_interior ((\<lambda>x. a + x) ` S)"
```
```  4587       using mem_rel_interior[of "a+x" "((\<lambda>x. a + x) ` S)"] by auto
```
```  4588   }
```
```  4589   then show ?thesis by auto
```
```  4590 qed
```
```  4591
```
```  4592 lemma rel_interior_translation:
```
```  4593   fixes a :: "'n::euclidean_space"
```
```  4594   shows "rel_interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` rel_interior S"
```
```  4595 proof -
```
```  4596   have "(\<lambda>x. (-a) + x) ` rel_interior ((\<lambda>x. a + x) ` S) \<subseteq> rel_interior S"
```
```  4597     using rel_interior_translation_aux[of "-a" "(\<lambda>x. a + x) ` S"]
```
```  4598       translation_assoc[of "-a" "a"]
```
```  4599     by auto
```
```  4600   then have "((\<lambda>x. a + x) ` rel_interior S) \<supseteq> rel_interior ((\<lambda>x. a + x) ` S)"
```
```  4601     using translation_inverse_subset[of a "rel_interior ((+) a ` S)" "rel_interior S"]
```
```  4602     by auto
```
```  4603   then show ?thesis
```
```  4604     using rel_interior_translation_aux[of a S] by auto
```
```  4605 qed
```
```  4606
```
```  4607
```
```  4608 lemma affine_hull_linear_image:
```
```  4609   assumes "bounded_linear f"
```
```  4610   shows "f ` (affine hull s) = affine hull f ` s"
```
```  4611 proof -
```
```  4612   interpret f: bounded_linear f by fact
```
```  4613   have "affine {x. f x \<in> affine hull f ` s}"
```
```  4614     unfolding affine_def
```
```  4615     by (auto simp: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
```
```  4616   moreover have "affine {x. x \<in> f ` (affine hull s)}"
```
```  4617     using affine_affine_hull[unfolded affine_def, of s]
```
```  4618     unfolding affine_def by (auto simp: f.scaleR [symmetric] f.add [symmetric])
```
```  4619   ultimately show ?thesis
```
```  4620     by (auto simp: hull_inc elim!: hull_induct)
```
```  4621 qed
```
```  4622
```
```  4623
```
```  4624 lemma rel_interior_injective_on_span_linear_image:
```
```  4625   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  4626     and S :: "'m::euclidean_space set"
```
```  4627   assumes "bounded_linear f"
```
```  4628     and "inj_on f (span S)"
```
```  4629   shows "rel_interior (f ` S) = f ` (rel_interior S)"
```
```  4630 proof -
```
```  4631   {
```
```  4632     fix z
```
```  4633     assume z: "z \<in> rel_interior (f ` S)"
```
```  4634     then have "z \<in> f ` S"
```
```  4635       using rel_interior_subset[of "f ` S"] by auto
```
```  4636     then obtain x where x: "x \<in> S" "f x = z" by auto
```
```  4637     obtain e2 where e2: "e2 > 0" "cball z e2 \<inter> affine hull (f ` S) \<subseteq> (f ` S)"
```
```  4638       using z rel_interior_cball[of "f ` S"] by auto
```
```  4639     obtain K where K: "K > 0" "\<And>x. norm (f x) \<le> norm x * K"
```
```  4640      using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
```
```  4641     define e1 where "e1 = 1 / K"
```
```  4642     then have e1: "e1 > 0" "\<And>x. e1 * norm (f x) \<le> norm x"
```
```  4643       using K pos_le_divide_eq[of e1] by auto
```
```  4644     define e where "e = e1 * e2"
```
```  4645     then have "e > 0" using e1 e2 by auto
```
```  4646     {
```
```  4647       fix y
```
```  4648       assume y: "y \<in> cball x e \<inter> affine hull S"
```
```  4649       then have h1: "f y \<in> affine hull (f ` S)"
```
```  4650         using affine_hull_linear_image[of f S] assms by auto
```
```  4651       from y have "norm (x-y) \<le> e1 * e2"
```
```  4652         using cball_def[of x e] dist_norm[of x y] e_def by auto
```
```  4653       moreover have "f x - f y = f (x - y)"
```
```  4654         using assms linear_diff[of f x y] linear_conv_bounded_linear[of f] by auto
```
```  4655       moreover have "e1 * norm (f (x-y)) \<le> norm (x - y)"
```
```  4656         using e1 by auto
```
```  4657       ultimately have "e1 * norm ((f x)-(f y)) \<le> e1 * e2"
```
```  4658         by auto
```
```  4659       then have "f y \<in> cball z e2"
```
```  4660         using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
```
```  4661       then have "f y \<in> f ` S"
```
```  4662         using y e2 h1 by auto
```
```  4663       then have "y \<in> S"
```
```  4664         using assms y hull_subset[of S] affine_hull_subset_span
```
```  4665           inj_on_image_mem_iff [OF \<open>inj_on f (span S)\<close>]
```
```  4666         by (metis Int_iff span_superset subsetCE)
```
```  4667     }
```
```  4668     then have "z \<in> f ` (rel_interior S)"
```
```  4669       using mem_rel_interior_cball[of x S] \<open>e > 0\<close> x by auto
```
```  4670   }
```
```  4671   moreover
```
```  4672   {
```
```  4673     fix x
```
```  4674     assume x: "x \<in> rel_interior S"
```
```  4675     then obtain e2 where e2: "e2 > 0" "cball x e2 \<inter> affine hull S \<subseteq> S"
```
```  4676       using rel_interior_cball[of S] by auto
```
```  4677     have "x \<in> S" using x rel_interior_subset by auto
```
```  4678     then have *: "f x \<in> f ` S" by auto
```
```  4679     have "\<forall>x\<in>span S. f x = 0 \<longrightarrow> x = 0"
```
```  4680       using assms subspace_span linear_conv_bounded_linear[of f]
```
```  4681         linear_injective_on_subspace_0[of f "span S"]
```
```  4682       by auto
```
```  4683     then obtain e1 where e1: "e1 > 0" "\<forall>x \<in> span S. e1 * norm x \<le> norm (f x)"
```
```  4684       using assms injective_imp_isometric[of "span S" f]
```
```  4685         subspace_span[of S] closed_subspace[of "span S"]
```
```  4686       by auto
```
```  4687     define e where "e = e1 * e2"
```
```  4688     hence "e > 0" using e1 e2 by auto
```
```  4689     {
```
```  4690       fix y
```
```  4691       assume y: "y \<in> cball (f x) e \<inter> affine hull (f ` S)"
```
```  4692       then have "y \<in> f ` (affine hull S)"
```
```  4693         using affine_hull_linear_image[of f S] assms by auto
```
```  4694       then obtain xy where xy: "xy \<in> affine hull S" "f xy = y" by auto
```
```  4695       with y have "norm (f x - f xy) \<le> e1 * e2"
```
```  4696         using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
```
```  4697       moreover have "f x - f xy = f (x - xy)"
```
```  4698         using assms linear_diff[of f x xy] linear_conv_bounded_linear[of f] by auto
```
```  4699       moreover have *: "x - xy \<in> span S"
```
```  4700         using subspace_diff[of "span S" x xy] subspace_span \<open>x \<in> S\<close> xy
```
```  4701           affine_hull_subset_span[of S] span_superset
```
```  4702         by auto
```
```  4703       moreover from * have "e1 * norm (x - xy) \<le> norm (f (x - xy))"
```
```  4704         using e1 by auto
```
```  4705       ultimately have "e1 * norm (x - xy) \<le> e1 * e2"
```
```  4706         by auto
```
```  4707       then have "xy \<in> cball x e2"
```
```  4708         using cball_def[of x e2] dist_norm[of x xy] e1 by auto
```
```  4709       then have "y \<in> f ` S"
```
```  4710         using xy e2 by auto
```
```  4711     }
```
```  4712     then have "f x \<in> rel_interior (f ` S)"
```
```  4713       using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * \<open>e > 0\<close> by auto
```
```  4714   }
```
```  4715   ultimately show ?thesis by auto
```
```  4716 qed
```
```  4717
```
```  4718 lemma rel_interior_injective_linear_image:
```
```  4719   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  4720   assumes "bounded_linear f"
```
```  4721     and "inj f"
```
```  4722   shows "rel_interior (f ` S) = f ` (rel_interior S)"
```
```  4723   using assms rel_interior_injective_on_span_linear_image[of f S]
```
```  4724     subset_inj_on[of f "UNIV" "span S"]
```
```  4725   by auto
```
```  4726
```
```  4727
```
```  4728 subsection%unimportant\<open>Some Properties of subset of standard basis\<close>
```
```  4729
```
```  4730 lemma affine_hull_substd_basis:
```
```  4731   assumes "d \<subseteq> Basis"
```
```  4732   shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
```
```  4733   (is "affine hull (insert 0 ?A) = ?B")
```
```  4734 proof -
```
```  4735   have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
```
```  4736     by auto
```
```  4737   show ?thesis
```
```  4738     unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
```
```  4739 qed
```
```  4740
```
```  4741 lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
```
```  4742   by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
```
```  4743
```
```  4744
```
```  4745 subsection%unimportant \<open>Openness and compactness are preserved by convex hull operation\<close>
```
```  4746
```
```  4747 lemma open_convex_hull[intro]:
```
```  4748   fixes S :: "'a::real_normed_vector set"
```
```  4749   assumes "open S"
```
```  4750   shows "open (convex hull S)"
```
```  4751 proof (clarsimp simp: open_contains_cball convex_hull_explicit)
```
```  4752   fix T and u :: "'a\<Rightarrow>real"
```
```  4753   assume obt: "finite T" "T\<subseteq>S" "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1"
```
```  4754
```
```  4755   from assms[unfolded open_contains_cball] obtain b
```
```  4756     where b: "\<And>x. x\<in>S \<Longrightarrow> 0 < b x \<and> cball x (b x) \<subseteq> S" by metis
```
```  4757   have "b ` T \<noteq> {}"
```
```  4758     using obt by auto
```
```  4759   define i where "i = b ` T"
```
```  4760   let ?\<Phi> = "\<lambda>y. \<exists>F. finite F \<and> F \<subseteq> S \<and> (\<exists>u. (\<forall>x\<in>F. 0 \<le> u x) \<and> sum u F = 1 \<and> (\<Sum>v\<in>F. u v *\<^sub>R v) = y)"
```
```  4761   let ?a = "\<Sum>v\<in>T. u v *\<^sub>R v"
```
```  4762   show "\<exists>e > 0. cball ?a e \<subseteq> {y. ?\<Phi> y}"
```
```  4763   proof (intro exI subsetI conjI)
```
```  4764     show "0 < Min i"
```
```  4765       unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] \<open>b ` T\<noteq>{}\<close>]
```
```  4766       using b \<open>T\<subseteq>S\<close> by auto
```
```  4767   next
```
```  4768     fix y
```
```  4769     assume "y \<in> cball ?a (Min i)"
```
```  4770     then have y: "norm (?a - y) \<le> Min i"
```
```  4771       unfolding dist_norm[symmetric] by auto
```
```  4772     { fix x
```
```  4773       assume "x \<in> T"
```
```  4774       then have "Min i \<le> b x"
```
```  4775         by (simp add: i_def obt(1))
```
```  4776       then have "x + (y - ?a) \<in> cball x (b x)"
```
```  4777         using y unfolding mem_cball dist_norm by auto
```
```  4778       moreover have "x \<in> S"
```
```  4779         using \<open>x\<in>T\<close> \<open>T\<subseteq>S\<close> by auto
```
```  4780       ultimately have "x + (y - ?a) \<in> S"
```
```  4781         using y b by blast
```
```  4782     }
```
```  4783     moreover
```
```  4784     have *: "inj_on (\<lambda>v. v + (y - ?a)) T"
```
```  4785       unfolding inj_on_def by auto
```
```  4786     have "(\<Sum>v\<in>(\<lambda>v. v + (y - ?a)) ` T. u (v - (y - ?a)) *\<^sub>R v) = y"
```
```  4787       unfolding sum.reindex[OF *] o_def using obt(4)
```
```  4788       by (simp add: sum.distrib sum_subtractf scaleR_left.sum[symmetric] scaleR_right_distrib)
```
```  4789     ultimately show "y \<in> {y. ?\<Phi> y}"
```
```  4790     proof (intro CollectI exI conjI)
```
```  4791       show "finite ((\<lambda>v. v + (y - ?a)) ` T)"
```
```  4792         by (simp add: obt(1))
```
```  4793       show "sum (\<lambda>v. u (v - (y - ?a))) ((\<lambda>v. v + (y - ?a)) ` T) = 1"
```
```  4794         unfolding sum.reindex[OF *] o_def using obt(4) by auto
```
```  4795     qed (use obt(1, 3) in auto)
```
```  4796   qed
```
```  4797 qed
```
```  4798
```
```  4799 lemma compact_convex_combinations:
```
```  4800   fixes S T :: "'a::real_normed_vector set"
```
```  4801   assumes "compact S" "compact T"
```
```  4802   shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T}"
```
```  4803 proof -
```
```  4804   let ?X = "{0..1} \<times> S \<times> T"
```
```  4805   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
```
```  4806   have *: "{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T} = ?h ` ?X"
```
```  4807     by force
```
```  4808   have "continuous_on ?X (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
```
```  4809     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
```
```  4810   with assms show ?thesis
```
```  4811     by (simp add: * compact_Times compact_continuous_image)
```
```  4812 qed
```
```  4813
```
```  4814 lemma finite_imp_compact_convex_hull:
```
```  4815   fixes S :: "'a::real_normed_vector set"
```
```  4816   assumes "finite S"
```
```  4817   shows "compact (convex hull S)"
```
```  4818 proof (cases "S = {}")
```
```  4819   case True
```
```  4820   then show ?thesis by simp
```
```  4821 next
```
```  4822   case False
```
```  4823   with assms show ?thesis
```
```  4824   proof (induct rule: finite_ne_induct)
```
```  4825     case (singleton x)
```
```  4826     show ?case by simp
```
```  4827   next
```
```  4828     case (insert x A)
```
```  4829     let ?f = "\<lambda>(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
```
```  4830     let ?T = "{0..1::real} \<times> (convex hull A)"
```
```  4831     have "continuous_on ?T ?f"
```
```  4832       unfolding split_def continuous_on by (intro ballI tendsto_intros)
```
```  4833     moreover have "compact ?T"
```
```  4834       by (intro compact_Times compact_Icc insert)
```
```  4835     ultimately have "compact (?f ` ?T)"
```
```  4836       by (rule compact_continuous_image)
```
```  4837     also have "?f ` ?T = convex hull (insert x A)"
```
```  4838       unfolding convex_hull_insert [OF \<open>A \<noteq> {}\<close>]
```
```  4839       apply safe
```
```  4840       apply (rule_tac x=a in exI, simp)
```
```  4841       apply (rule_tac x="1 - a" in exI, simp, fast)
```
```  4842       apply (rule_tac x="(u, b)" in image_eqI, simp_all)
```
```  4843       done
```
```  4844     finally show "compact (convex hull (insert x A))" .
```
```  4845   qed
```
```  4846 qed
```
```  4847
```
```  4848 lemma compact_convex_hull:
```
```  4849   fixes S :: "'a::euclidean_space set"
```
```  4850   assumes "compact S"
```
```  4851   shows "compact (convex hull S)"
```
```  4852 proof (cases "S = {}")
```
```  4853   case True
```
```  4854   then show ?thesis using compact_empty by simp
```
```  4855 next
```
```  4856   case False
```
```  4857   then obtain w where "w \<in> S" by auto
```
```  4858   show ?thesis
```
```  4859     unfolding caratheodory[of S]
```
```  4860   proof (induct ("DIM('a) + 1"))
```
```  4861     case 0
```
```  4862     have *: "{x.\<exists>sa. finite sa \<and> sa \<subseteq> S \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
```
```  4863       using compact_empty by auto
```
```  4864     from 0 show ?case unfolding * by simp
```
```  4865   next
```
```  4866     case (Suc n)
```
```  4867     show ?case
```
```  4868     proof (cases "n = 0")
```
```  4869       case True
```
```  4870       have "{x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T} = S"
```
```  4871         unfolding set_eq_iff and mem_Collect_eq
```
```  4872       proof (rule, rule)
```
```  4873         fix x
```
```  4874         assume "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
```
```  4875         then obtain T where T: "finite T" "T \<subseteq> S" "card T \<le> Suc n" "x \<in> convex hull T"
```
```  4876           by auto
```
```  4877         show "x \<in> S"
```
```  4878         proof (cases "card T = 0")
```
```  4879           case True
```
```  4880           then show ?thesis
```
```  4881             using T(4) unfolding card_0_eq[OF T(1)] by simp
```
```  4882         next
```
```  4883           case False
```
```  4884           then have "card T = Suc 0" using T(3) \<open>n=0\<close> by auto
```
```  4885           then obtain a where "T = {a}" unfolding card_Suc_eq by auto
```
```  4886           then show ?thesis using T(2,4) by simp
```
```  4887         qed
```
```  4888       next
```
```  4889         fix x assume "x\<in>S"
```
```  4890         then show "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
```
```  4891           apply (rule_tac x="{x}" in exI)
```
```  4892           unfolding convex_hull_singleton
```
```  4893           apply auto
```
```  4894           done
```
```  4895       qed
```
```  4896       then show ?thesis using assms by simp
```
```  4897     next
```
```  4898       case False
```
```  4899       have "{x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T} =
```
```  4900         {(1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
```
```  4901           0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> {x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> x \<in> convex hull T}}"
```
```  4902         unfolding set_eq_iff and mem_Collect_eq
```
```  4903       proof (rule, rule)
```
```  4904         fix x
```
```  4905         assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
```
```  4906           0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
```
```  4907         then obtain u v c T where obt: "x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
```
```  4908           "0 \<le> c \<and> c \<le> 1" "u \<in> S" "finite T" "T \<subseteq> S" "card T \<le> n"  "v \<in> convex hull T"
```
```  4909           by auto
```
```  4910         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u T"
```
```  4911           apply (rule convexD_alt)
```
```  4912           using obt(2) and convex_convex_hull and hull_subset[of "insert u T" convex]
```
```  4913           using obt(7) and hull_mono[of T "insert u T"]
```
```  4914           apply auto
```
```  4915           done
```
```  4916         ultimately show "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
```
```  4917           apply (rule_tac x="insert u T" in exI)
```
```  4918           apply (auto simp: card_insert_if)
```
```  4919           done
```
```  4920       next
```
```  4921         fix x
```
```  4922         assume "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
```
```  4923         then obtain T where T: "finite T" "T \<subseteq> S" "card T \<le> Suc n" "x \<in> convex hull T"
```
```  4924           by auto
```
```  4925         show "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
```
```  4926           0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
```
```  4927         proof (cases "card T = Suc n")
```
```  4928           case False
```
```  4929           then have "card T \<le> n" using T(3) by auto
```
```  4930           then show ?thesis
```
```  4931             apply (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI)
```
```  4932             using \<open>w\<in>S\<close> and T
```
```  4933             apply (auto intro!: exI[where x=T])
```
```  4934             done
```
```  4935         next
```
```  4936           case True
```
```  4937           then obtain a u where au: "T = insert a u" "a\<notin>u"
```
```  4938             apply (drule_tac card_eq_SucD, auto)
```
```  4939             done
```
```  4940           show ?thesis
```
```  4941           proof (cases "u = {}")
```
```  4942             case True
```
```  4943             then have "x = a" using T(4)[unfolded au] by auto
```
```  4944             show ?thesis unfolding \<open>x = a\<close>
```
```  4945               apply (rule_tac x=a in exI)
```
```  4946               apply (rule_tac x=a in exI)
```
```  4947               apply (rule_tac x=1 in exI)
```
```  4948               using T and \<open>n \<noteq> 0\<close>
```
```  4949               unfolding au
```
```  4950               apply (auto intro!: exI[where x="{a}"])
```
```  4951               done
```
```  4952           next
```
```  4953             case False
```
```  4954             obtain ux vx b where obt: "ux\<ge>0" "vx\<ge>0" "ux + vx = 1"
```
```  4955               "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
```
```  4956               using T(4)[unfolded au convex_hull_insert[OF False]]
```
```  4957               by auto
```
```  4958             have *: "1 - vx = ux" using obt(3) by auto
```
```  4959             show ?thesis
```
```  4960               apply (rule_tac x=a in exI)
```
```  4961               apply (rule_tac x=b in exI)
```
```  4962               apply (rule_tac x=vx in exI)
```
```  4963               using obt and T(1-3)
```
```  4964               unfolding au and * using card_insert_disjoint[OF _ au(2)]
```
```  4965               apply (auto intro!: exI[where x=u])
```
```  4966               done
```
```  4967           qed
```
```  4968         qed
```
```  4969       qed
```
```  4970       then show ?thesis
```
```  4971         using compact_convex_combinations[OF assms Suc] by simp
```
```  4972     qed
```
```  4973   qed
```
```  4974 qed
```
```  4975
```
```  4976
```
```  4977 subsection%unimportant \<open>Extremal points of a simplex are some vertices\<close>
```
```  4978
```
```  4979 lemma dist_increases_online:
```
```  4980   fixes a b d :: "'a::real_inner"
```
```  4981   assumes "d \<noteq> 0"
```
```  4982   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
```
```  4983 proof (cases "inner a d - inner b d > 0")
```
```  4984   case True
```
```  4985   then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
```
```  4986     apply (rule_tac add_pos_pos)
```
```  4987     using assms
```
```  4988     apply auto
```
```  4989     done
```
```  4990   then show ?thesis
```
```  4991     apply (rule_tac disjI2)
```
```  4992     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
```
```  4993     apply  (simp add: algebra_simps inner_commute)
```
```  4994     done
```
```  4995 next
```
```  4996   case False
```
```  4997   then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
```
```  4998     apply (rule_tac add_pos_nonneg)
```
```  4999     using assms
```
```  5000     apply auto
```
```  5001     done
```
```  5002   then show ?thesis
```
```  5003     apply (rule_tac disjI1)
```
```  5004     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
```
```  5005     apply (simp add: algebra_simps inner_commute)
```
```  5006     done
```
```  5007 qed
```
```  5008
```
```  5009 lemma norm_increases_online:
```
```  5010   fixes d :: "'a::real_inner"
```
```  5011   shows "d \<noteq> 0 \<Longrightarrow> norm (a + d) > norm a \<or> norm(a - d) > norm a"
```
```  5012   using dist_increases_online[of d a 0] unfolding dist_norm by auto
```
```  5013
```
```  5014 lemma simplex_furthest_lt:
```
```  5015   fixes S :: "'a::real_inner set"
```
```  5016   assumes "finite S"
```
```  5017   shows "\<forall>x \<in> convex hull S.  x \<notin> S \<longrightarrow> (\<exists>y \<in> convex hull S. norm (x - a) < norm(y - a))"
```
```  5018   using assms
```
```  5019 proof induct
```
```  5020   fix x S
```
```  5021   assume as: "finite S" "x\<notin>S" "\<forall>x\<in>convex hull S. x \<notin> S \<longrightarrow> (\<exists>y\<in>convex hull S. norm (x - a) < norm (y - a))"
```
```  5022   show "\<forall>xa\<in>convex hull insert x S. xa \<notin> insert x S \<longrightarrow>
```
```  5023     (\<exists>y\<in>convex hull insert x S. norm (xa - a) < norm (y - a))"
```
```  5024   proof (intro impI ballI, cases "S = {}")
```
```  5025     case False
```
```  5026     fix y
```
```  5027     assume y: "y \<in> convex hull insert x S" "y \<notin> insert x S"
```
```  5028     obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "y = u *\<^sub>R x + v *\<^sub>R b"
```
```  5029       using y(1)[unfolded convex_hull_insert[OF False]] by auto
```
```  5030     show "\<exists>z\<in>convex hull insert x S. norm (y - a) < norm (z - a)"
```
```  5031     proof (cases "y \<in> convex hull S")
```
```  5032       case True
```
```  5033       then obtain z where "z \<in> convex hull S" "norm (y - a) < norm (z - a)"
```
```  5034         using as(3)[THEN bspec[where x=y]] and y(2) by auto
```
```  5035       then show ?thesis
```
```  5036         apply (rule_tac x=z in bexI)
```
```  5037         unfolding convex_hull_insert[OF False]
```
```  5038         apply auto
```
```  5039         done
```
```  5040     next
```
```  5041       case False
```
```  5042       show ?thesis
```
```  5043         using obt(3)
```
```  5044       proof (cases "u = 0", case_tac[!] "v = 0")
```
```  5045         assume "u = 0" "v \<noteq> 0"
```
```  5046         then have "y = b" using obt by auto
```
```  5047         then show ?thesis using False and obt(4) by auto
```
```  5048       next
```
```  5049         assume "u \<noteq> 0" "v = 0"
```
```  5050         then have "y = x" using obt by auto
```
```  5051         then show ?thesis using y(2) by auto
```
```  5052       next
```
```  5053         assume "u \<noteq> 0" "v \<noteq> 0"
```
```  5054         then obtain w where w: "w>0" "w<u" "w<v"
```
```  5055           using field_lbound_gt_zero[of u v] and obt(1,2) by auto
```
```  5056         have "x \<noteq> b"
```
```  5057         proof
```
```  5058           assume "x = b"
```
```  5059           then have "y = b" unfolding obt(5)
```
```  5060             using obt(3) by (auto simp: scaleR_left_distrib[symmetric])
```
```  5061           then show False using obt(4) and False by simp
```
```  5062         qed
```
```  5063         then have *: "w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
```
```  5064         show ?thesis
```
```  5065           using dist_increases_online[OF *, of a y]
```
```  5066         proof (elim disjE)
```
```  5067           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
```
```  5068           then have "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
```
```  5069             unfolding dist_commute[of a]
```
```  5070             unfolding dist_norm obt(5)
```
```  5071             by (simp add: algebra_simps)
```
```  5072           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x S"
```
```  5073             unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
```
```  5074           proof (intro CollectI conjI exI)
```
```  5075             show "u + w \<ge> 0" "v - w \<ge> 0"
```
```  5076               using obt(1) w by auto
```
```  5077           qed (use obt in auto)
```
```  5078           ultimately show ?thesis by auto
```
```  5079         next
```
```  5080           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
```
```  5081           then have "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
```
```  5082             unfolding dist_commute[of a]
```
```  5083             unfolding dist_norm obt(5)
```
```  5084             by (simp add: algebra_simps)
```
```  5085           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x S"
```
```  5086             unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
```
```  5087           proof (intro CollectI conjI exI)
```
```  5088             show "u - w \<ge> 0" "v + w \<ge> 0"
```
```  5089               using obt(1) w by auto
```
```  5090           qed (use obt in auto)
```
```  5091           ultimately show ?thesis by auto
```
```  5092         qed
```
```  5093       qed auto
```
```  5094     qed
```
```  5095   qed auto
```
```  5096 qed (auto simp: assms)
```
```  5097
```
```  5098 lemma simplex_furthest_le:
```
```  5099   fixes S :: "'a::real_inner set"
```
```  5100   assumes "finite S"
```
```  5101     and "S \<noteq> {}"
```
```  5102   shows "\<exists>y\<in>S. \<forall>x\<in> convex hull S. norm (x - a) \<le> norm (y - a)"
```
```  5103 proof -
```
```  5104   have "convex hull S \<noteq> {}"
```
```  5105     using hull_subset[of S convex] and assms(2) by auto
```
```  5106   then obtain x where x: "x \<in> convex hull S" "\<forall>y\<in>convex hull S. norm (y - a) \<le> norm (x - a)"
```
```  5107     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF \<open>finite S\<close>], of a]
```
```  5108     unfolding dist_commute[of a]
```
```  5109     unfolding dist_norm
```
```  5110     by auto
```
```  5111   show ?thesis
```
```  5112   proof (cases "x \<in> S")
```
```  5113     case False
```
```  5114     then obtain y where "y \<in> convex hull S" "norm (x - a) < norm (y - a)"
```
```  5115       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
```
```  5116       by auto
```
```  5117     then show ?thesis
```
```  5118       using x(2)[THEN bspec[where x=y]] by auto
```
```  5119   next
```
```  5120     case True
```
```  5121     with x show ?thesis by auto
```
```  5122   qed
```
```  5123 qed
```
```  5124
```
```  5125 lemma simplex_furthest_le_exists:
```
```  5126   fixes S :: "('a::real_inner) set"
```
```  5127   shows "finite S \<Longrightarrow> \<forall>x\<in>(convex hull S). \<exists>y\<in>S. norm (x - a) \<le> norm (y - a)"
```
```  5128   using simplex_furthest_le[of S] by (cases "S = {}") auto
```
```  5129
```
```  5130 lemma simplex_extremal_le:
```
```  5131   fixes S :: "'a::real_inner set"
```
```  5132   assumes "finite S"
```
```  5133     and "S \<noteq> {}"
```
```  5134   shows "\<exists>u\<in>S. \<exists>v\<in>S. \<forall>x\<in>convex hull S. \<forall>y \<in> convex hull S. norm (x - y) \<le> norm (u - v)"
```
```  5135 proof -
```
```  5136   have "convex hull S \<noteq> {}"
```
```  5137     using hull_subset[of S convex] and assms(2) by auto
```
```  5138   then obtain u v where obt: "u \<in> convex hull S" "v \<in> convex hull S"
```
```  5139     "\<forall>x\<in>convex hull S. \<forall>y\<in>convex hull S. norm (x - y) \<le> norm (u - v)"
```
```  5140     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
```
```  5141     by (auto simp: dist_norm)
```
```  5142   then show ?thesis
```
```  5143   proof (cases "u\<notin>S \<or> v\<notin>S", elim disjE)
```
```  5144     assume "u \<notin> S"
```
```  5145     then obtain y where "y \<in> convex hull S" "norm (u - v) < norm (y - v)"
```
```  5146       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
```
```  5147       by auto
```
```  5148     then show ?thesis
```
```  5149       using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
```
```  5150       by auto
```
```  5151   next
```
```  5152     assume "v \<notin> S"
```
```  5153     then obtain y where "y \<in> convex hull S" "norm (v - u) < norm (y - u)"
```
```  5154       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
```
```  5155       by auto
```
```  5156     then show ?thesis
```
```  5157       using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
```
```  5158       by (auto simp: norm_minus_commute)
```
```  5159   qed auto
```
```  5160 qed
```
```  5161
```
```  5162 lemma simplex_extremal_le_exists:
```
```  5163   fixes S :: "'a::real_inner set"
```
```  5164   shows "finite S \<Longrightarrow> x \<in> convex hull S \<Longrightarrow> y \<in> convex hull S \<Longrightarrow>
```
```  5165     \<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"
```
```  5166   using convex_hull_empty simplex_extremal_le[of S]
```
```  5167   by(cases "S = {}") auto
```
```  5168
```
```  5169
```
```  5170 subsection \<open>Closest point of a convex set is unique, with a continuous projection\<close>
```
```  5171
```
```  5172 definition%important closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a"
```
```  5173   where "closest_point S a = (SOME x. x \<in> S \<and> (\<forall>y\<in>S. dist a x \<le> dist a y))"
```
```  5174
```
```  5175 lemma closest_point_exists:
```
```  5176   assumes "closed S"
```
```  5177     and "S \<noteq> {}"
```
```  5178   shows "closest_point S a \<in> S"
```
```  5179     and "\<forall>y\<in>S. dist a (closest_point S a) \<le> dist a y"
```
```  5180   unfolding closest_point_def
```
```  5181   apply(rule_tac[!] someI2_ex)
```
```  5182   apply (auto intro: distance_attains_inf[OF assms(1,2), of a])
```
```  5183   done
```
```  5184
```
```  5185 lemma closest_point_in_set: "closed S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> closest_point S a \<in> S"
```
```  5186   by (meson closest_point_exists)
```
```  5187
```
```  5188 lemma closest_point_le: "closed S \<Longrightarrow> x \<in> S \<Longrightarrow> dist a (closest_point S a) \<le> dist a x"
```
```  5189   using closest_point_exists[of S] by auto
```
```  5190
```
```  5191 lemma closest_point_self:
```
```  5192   assumes "x \<in> S"
```
```  5193   shows "closest_point S x = x"
```
```  5194   unfolding closest_point_def
```
```  5195   apply (rule some1_equality, rule ex1I[of _ x])
```
```  5196   using assms
```
```  5197   apply auto
```
```  5198   done
```
```  5199
```
```  5200 lemma closest_point_refl: "closed S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> closest_point S x = x \<longleftrightarrow> x \<in> S"
```
```  5201   using closest_point_in_set[of S x] closest_point_self[of x S]
```
```  5202   by auto
```
```  5203
```
```  5204 lemma closer_points_lemma:
```
```  5205   assumes "inner y z > 0"
```
```  5206   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
```
```  5207 proof -
```
```  5208   have z: "inner z z > 0"
```
```  5209     unfolding inner_gt_zero_iff using assms by auto
```
```  5210   have "norm (v *\<^sub>R z - y) < norm y"
```
```  5211     if "0 < v" and "v \<le> inner y z / inner z z" for v
```
```  5212     unfolding norm_lt using z assms that
```
```  5213     by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ \<open>0<v\<close>])
```
```  5214   then show ?thesis
```
```  5215     using assms z
```
```  5216     by (rule_tac x = "inner y z / inner z z" in exI) auto
```
```  5217 qed
```
```  5218
```
```  5219 lemma closer_point_lemma:
```
```  5220   assumes "inner (y - x) (z - x) > 0"
```
```  5221   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
```
```  5222 proof -
```
```  5223   obtain u where "u > 0"
```
```  5224     and u: "\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
```
```  5225     using closer_points_lemma[OF assms] by auto
```
```  5226   show ?thesis
```
```  5227     apply (rule_tac x="min u 1" in exI)
```
```  5228     using u[THEN spec[where x="min u 1"]] and \<open>u > 0\<close>
```
```  5229     unfolding dist_norm by (auto simp: norm_minus_commute field_simps)
```
```  5230 qed
```
```  5231
```
```  5232 lemma any_closest_point_dot:
```
```  5233   assumes "convex S" "closed S" "x \<in> S" "y \<in> S" "\<forall>z\<in>S. dist a x \<le> dist a z"
```
```  5234   shows "inner (a - x) (y - x) \<le> 0"
```
```  5235 proof (rule ccontr)
```
```  5236   assume "\<not> ?thesis"
```
```  5237   then obtain u where u: "u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a"
```
```  5238     using closer_point_lemma[of a x y] by auto
```
```  5239   let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y"
```
```  5240   have "?z \<in> S"
```
```  5241     using convexD_alt[OF assms(1,3,4), of u] using u by auto
```
```  5242   then show False
```
```  5243     using assms(5)[THEN bspec[where x="?z"]] and u(3)
```
```  5244     by (auto simp: dist_commute algebra_simps)
```
```  5245 qed
```
```  5246
```
```  5247 lemma any_closest_point_unique:
```
```  5248   fixes x :: "'a::real_inner"
```
```  5249   assumes "convex S" "closed S" "x \<in> S" "y \<in> S"
```
```  5250     "\<forall>z\<in>S. dist a x \<le> dist a z" "\<forall>z\<in>S. dist a y \<le> dist a z"
```
```  5251   shows "x = y"
```
```  5252   using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
```
```  5253   unfolding norm_pths(1) and norm_le_square
```
```  5254   by (auto simp: algebra_simps)
```
```  5255
```
```  5256 lemma closest_point_unique:
```
```  5257   assumes "convex S" "closed S" "x \<in> S" "\<forall>z\<in>S. dist a x \<le> dist a z"
```
```  5258   shows "x = closest_point S a"
```
```  5259   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point S a"]
```
```  5260   using closest_point_exists[OF assms(2)] and assms(3) by auto
```
```  5261
```
```  5262 lemma closest_point_dot:
```
```  5263   assumes "convex S" "closed S" "x \<in> S"
```
```  5264   shows "inner (a - closest_point S a) (x - closest_point S a) \<le> 0"
```
```  5265   apply (rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
```
```  5266   using closest_point_exists[OF assms(2)] and assms(3)
```
```  5267   apply auto
```
```  5268   done
```
```  5269
```
```  5270 lemma closest_point_lt:
```
```  5271   assumes "convex S" "closed S" "x \<in> S" "x \<noteq> closest_point S a"
```
```  5272   shows "dist a (closest_point S a) < dist a x"
```
```  5273   apply (rule ccontr)
```
```  5274   apply (rule_tac notE[OF assms(4)])
```
```  5275   apply (rule closest_point_unique[OF assms(1-3), of a])
```
```  5276   using closest_point_le[OF assms(2), of _ a]
```
```  5277   apply fastforce
```
```  5278   done
```
```  5279
```
```  5280 lemma closest_point_lipschitz:
```
```  5281   assumes "convex S"
```
```  5282     and "closed S" "S \<noteq> {}"
```
```  5283   shows "dist (closest_point S x) (closest_point S y) \<le> dist x y"
```
```  5284 proof -
```
```  5285   have "inner (x - closest_point S x) (closest_point S y - closest_point S x) \<le> 0"
```
```  5286     and "inner (y - closest_point S y) (closest_point S x - closest_point S y) \<le> 0"
```
```  5287     apply (rule_tac[!] any_closest_point_dot[OF assms(1-2)])
```
```  5288     using closest_point_exists[OF assms(2-3)]
```
```  5289     apply auto
```
```  5290     done
```
```  5291   then show ?thesis unfolding dist_norm and norm_le
```
```  5292     using inner_ge_zero[of "(x - closest_point S x) - (y - closest_point S y)"]
```
```  5293     by (simp add: inner_add inner_diff inner_commute)
```
```  5294 qed
```
```  5295
```
```  5296 lemma continuous_at_closest_point:
```
```  5297   assumes "convex S"
```
```  5298     and "closed S"
```
```  5299     and "S \<noteq> {}"
```
```  5300   shows "continuous (at x) (closest_point S)"
```
```  5301   unfolding continuous_at_eps_delta
```
```  5302   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
```
```  5303
```
```  5304 lemma continuous_on_closest_point:
```
```  5305   assumes "convex S"
```
```  5306     and "closed S"
```
```  5307     and "S \<noteq> {}"
```
```  5308   shows "continuous_on t (closest_point S)"
```
```  5309   by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
```
```  5310
```
```  5311 proposition closest_point_in_rel_interior:
```
```  5312   assumes "closed S" "S \<noteq> {}" and x: "x \<in> affine hull S"
```
```  5313     shows "closest_point S x \<in> rel_interior S \<longleftrightarrow> x \<in> rel_interior S"
```
```  5314 proof (cases "x \<in> S")
```
```  5315   case True
```
```  5316   then show ?thesis
```
```  5317     by (simp add: closest_point_self)
```
```  5318 next
```
```  5319   case False
```
```  5320   then have "False" if asm: "closest_point S x \<in> rel_interior S"
```
```  5321   proof -
```
```  5322     obtain e where "e > 0" and clox: "closest_point S x \<in> S"
```
```  5323                and e: "cball (closest_point S x) e \<inter> affine hull S \<subseteq> S"
```
```  5324       using asm mem_rel_interior_cball by blast
```
```  5325     then have clo_notx: "closest_point S x \<noteq> x"
```
```  5326       using \<open>x \<notin> S\<close> by auto
```
```  5327     define y where "y \<equiv> closest_point S x -
```
```  5328                         (min 1 (e / norm(closest_point S x - x))) *\<^sub>R (closest_point S x - x)"
```
```  5329     have "x - y = (1 - min 1 (e / norm (closest_point S x - x))) *\<^sub>R (x - closest_point S x)"
```
```  5330       by (simp add: y_def algebra_simps)
```
```  5331     then have "norm (x - y) = abs ((1 - min 1 (e / norm (closest_point S x - x)))) * norm(x - closest_point S x)"
```
```  5332       by simp
```
```  5333     also have "\<dots> < norm(x - closest_point S x)"
```
```  5334       using clo_notx \<open>e > 0\<close>
```
```  5335       by (auto simp: mult_less_cancel_right2 divide_simps)
```
```  5336     finally have no_less: "norm (x - y) < norm (x - closest_point S x)" .
```
```  5337     have "y \<in> affine hull S"
```
```  5338       unfolding y_def
```
```  5339       by (meson affine_affine_hull clox hull_subset mem_affine_3_minus2 subsetD x)
```
```  5340     moreover have "dist (closest_point S x) y \<le> e"
```
```  5341       using \<open>e > 0\<close> by (auto simp: y_def min_mult_distrib_right)
```
```  5342     ultimately have "y \<in> S"
```
```  5343       using subsetD [OF e] by simp
```
```  5344     then have "dist x (closest_point S x) \<le> dist x y"
```
```  5345       by (simp add: closest_point_le \<open>closed S\<close>)
```
```  5346     with no_less show False
```
```  5347       by (simp add: dist_norm)
```
```  5348   qed
```
```  5349   moreover have "x \<notin> rel_interior S"
```
```  5350     using rel_interior_subset False by blast
```
```  5351   ultimately show ?thesis by blast
```
```  5352 qed
```
```  5353
```
```  5354
```
```  5355 subsubsection%unimportant \<open>Various point-to-set separating/supporting hyperplane theorems\<close>
```
```  5356
```
```  5357 lemma supporting_hyperplane_closed_point:
```
```  5358   fixes z :: "'a::{real_inner,heine_borel}"
```
```  5359   assumes "convex S"
```
```  5360     and "closed S"
```
```  5361     and "S \<noteq> {}"
```
```  5362     and "z \<notin> S"
```
```  5363   shows "\<exists>a b. \<exists>y\<in>S. inner a z < b \<and> inner a y = b \<and> (\<forall>x\<in>S. inner a x \<ge> b)"
```
```  5364 proof -
```
```  5365   obtain y where "y \<in> S" and y: "\<forall>x\<in>S. dist z y \<le> dist z x"
```
```  5366     by (metis distance_attains_inf[OF assms(2-3)])
```
```  5367   show ?thesis
```
```  5368   proof (intro exI bexI conjI ballI)
```
```  5369     show "(y - z) \<bullet> z < (y - z) \<bullet> y"
```
```  5370       by (metis \<open>y \<in> S\<close> assms(4) diff_gt_0_iff_gt inner_commute inner_diff_left inner_gt_zero_iff right_minus_eq)
```
```  5371     show "(y - z) \<bullet> y \<le> (y - z) \<bullet> x" if "x \<in> S" for x
```
```  5372     proof (rule ccontr)
```
```  5373       have *: "\<And>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
```
```  5374         using assms(1)[unfolded convex_alt] and y and \<open>x\<in>S\<close> and \<open>y\<in>S\<close> by auto
```
```  5375       assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x"
```
```  5376       then obtain v where "v > 0" "v \<le> 1" "dist (y + v *\<^sub>R (x - y)) z < dist y z"
```
```  5377         using closer_point_lemma[of z y x] by (auto simp: inner_diff)
```
```  5378       then show False
```
```  5379         using *[of v] by (auto simp: dist_commute algebra_simps)
```
```  5380     qed
```
```  5381   qed (use \<open>y \<in> S\<close> in auto)
```
```  5382 qed
```
```  5383
```
```  5384 lemma separating_hyperplane_closed_point:
```
```  5385   fixes z :: "'a::{real_inner,heine_borel}"
```
```  5386   assumes "convex S"
```
```  5387     and "closed S"
```
```  5388     and "z \<notin> S"
```
```  5389   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>S. inner a x > b)"
```
```  5390 proof (cases "S = {}")
```
```  5391   case True
```
```  5392   then show ?thesis
```
```  5393     by (simp add: gt_ex)
```
```  5394 next
```
```  5395   case False
```
```  5396   obtain y where "y \<in> S" and y: "\<And>x. x \<in> S \<Longrightarrow> dist z y \<le> dist z x"
```
```  5397     by (metis distance_attains_inf[OF assms(2) False])
```
```  5398   show ?thesis
```
```  5399   proof (intro exI conjI ballI)
```
```  5400     show "(y - z) \<bullet> z < inner (y - z) z + (norm (y - z))\<^sup>2 / 2"
```
```  5401       using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
```
```  5402   next
```
```  5403     fix x
```
```  5404     assume "x \<in> S"
```
```  5405     have "False" if *: "0 < inner (z - y) (x - y)"
```
```  5406     proof -
```
```  5407       obtain u where "u > 0" "u \<le> 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
```
```  5408         using * closer_point_lemma by blast
```
```  5409       then show False using y[of "y + u *\<^sub>R (x - y)"] convexD_alt [OF \<open>convex S\<close>]
```
```  5410         using \<open>x\<in>S\<close> \<open>y\<in>S\<close> by (auto simp: dist_commute algebra_simps)
```
```  5411     qed
```
```  5412     moreover have "0 < (norm (y - z))\<^sup>2"
```
```  5413       using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
```
```  5414     then have "0 < inner (y - z) (y - z)"
```
```  5415       unfolding power2_norm_eq_inner by simp
```
```  5416     ultimately show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
```
```  5417       by (force simp: field_simps power2_norm_eq_inner inner_commute inner_diff)
```
```  5418   qed
```
```  5419 qed
```
```  5420
```
```  5421 lemma separating_hyperplane_closed_0:
```
```  5422   assumes "convex (S::('a::euclidean_space) set)"
```
```  5423     and "closed S"
```
```  5424     and "0 \<notin> S"
```
```  5425   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>S. inner a x > b)"
```
```  5426 proof (cases "S = {}")
```
```  5427   case True
```
```  5428   have "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
```
```  5429     by (metis Basis_zero SOME_Basis)
```
```  5430   then show ?thesis
```
```  5431     using True zero_less_one by blast
```
```  5432 next
```
```  5433   case False
```
```  5434   then show ?thesis
```
```  5435     using False using separating_hyperplane_closed_point[OF assms]
```
```  5436     by (metis all_not_in_conv inner_zero_left inner_zero_right less_eq_real_def not_le)
```
```  5437 qed
```
```  5438
```
```  5439
```
```  5440 subsubsection%unimportant \<open>Now set-to-set for closed/compact sets\<close>
```
```  5441
```
```  5442 lemma separating_hyperplane_closed_compact:
```
```  5443   fixes S :: "'a::euclidean_space set"
```
```  5444   assumes "convex S"
```
```  5445     and "closed S"
```
```  5446     and "convex T"
```
```  5447     and "compact T"
```
```  5448     and "T \<noteq> {}"
```
```  5449     and "S \<inter> T = {}"
```
```  5450   shows "\<exists>a b. (\<forall>x\<in>S. inner a x < b) \<and> (\<forall>x\<in>T. inner a x > b)"
```
```  5451 proof (cases "S = {}")
```
```  5452   case True
```
```  5453   obtain b where b: "b > 0" "\<forall>x\<in>T. norm x \<le> b"
```
```  5454     using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
```
```  5455   obtain z :: 'a where z: "norm z = b + 1"
```
```  5456     using vector_choose_size[of "b + 1"] and b(1) by auto
```
```  5457   then have "z \<notin> T" using b(2)[THEN bspec[where x=z]] by auto
```
```  5458   then obtain a b where ab: "inner a z < b" "\<forall>x\<in>T. b < inner a x"
```
```  5459     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
```
```  5460     by auto
```
```  5461   then show ?thesis
```
```  5462     using True by auto
```
```  5463 next
```
```  5464   case False
```
```  5465   then obtain y where "y \<in> S" by auto
```
```  5466   obtain a b where "0 < b" "\<forall>x \<in> (\<Union>x\<in> S. \<Union>y \<in> T. {x - y}). b < inner a x"
```
```  5467     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
```
```  5468     using closed_compact_differences[OF assms(2,4)]
```
```  5469     using assms(6) by auto
```
```  5470   then have ab: "\<forall>x\<in>S. \<forall>y\<in>T. b + inner a y < inner a x"
```
```  5471     apply -
```
```  5472     apply rule
```
```  5473     apply rule
```
```  5474     apply (erule_tac x="x - y" in ballE)
```
```  5475     apply (auto simp: inner_diff)
```
```  5476     done
```
```  5477   define k where "k = (SUP x:T. a \<bullet> x)"
```
```  5478   show ?thesis
```
```  5479     apply (rule_tac x="-a" in exI)
```
```  5480     apply (rule_tac x="-(k + b / 2)" in exI)
```
```  5481     apply (intro conjI ballI)
```
```  5482     unfolding inner_minus_left and neg_less_iff_less
```
```  5483   proof -
```
```  5484     fix x assume "x \<in> T"
```
```  5485     then have "inner a x - b / 2 < k"
```
```  5486       unfolding k_def
```
```  5487     proof (subst less_cSUP_iff)
```
```  5488       show "T \<noteq> {}" by fact
```
```  5489       show "bdd_above ((\<bullet>) a ` T)"
```
```  5490         using ab[rule_format, of y] \<open>y \<in> S\<close>
```
```  5491         by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
```
```  5492     qed (auto intro!: bexI[of _ x] \<open>0<b\<close>)
```
```  5493     then show "inner a x < k + b / 2"
```
```  5494       by auto
```
```  5495   next
```
```  5496     fix x
```
```  5497     assume "x \<in> S"
```
```  5498     then have "k \<le> inner a x - b"
```
```  5499       unfolding k_def
```
```  5500       apply (rule_tac cSUP_least)
```
```  5501       using assms(5)
```
```  5502       using ab[THEN bspec[where x=x]]
```
```  5503       apply auto
```
```  5504       done
```
```  5505     then show "k + b / 2 < inner a x"
```
```  5506       using \<open>0 < b\<close> by auto
```
```  5507   qed
```
```  5508 qed
```
```  5509
```
```  5510 lemma separating_hyperplane_compact_closed:
```
```  5511   fixes S :: "'a::euclidean_space set"
```
```  5512   assumes "convex S"
```
```  5513     and "compact S"
```
```  5514     and "S \<noteq> {}"
```
```  5515     and "convex T"
```
```  5516     and "closed T"
```
```  5517     and "S \<inter> T = {}"
```
```  5518   shows "\<exists>a b. (\<forall>x\<in>S. inner a x < b) \<and> (\<forall>x\<in>T. inner a x > b)"
```
```  5519 proof -
```
```  5520   obtain a b where "(\<forall>x\<in>T. inner a x < b) \<and> (\<forall>x\<in>S. b < inner a x)"
```
```  5521     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6)
```
```  5522     by auto
```
```  5523   then show ?thesis
```
```  5524     apply (rule_tac x="-a" in exI)
```
```  5525     apply (rule_tac x="-b" in exI, auto)
```
```  5526     done
```
```  5527 qed
```
```  5528
```
```  5529
```
```  5530 subsubsection%unimportant \<open>General case without assuming closure and getting non-strict separation\<close>
```
```  5531
```
```  5532 lemma separating_hyperplane_set_0:
```
```  5533   assumes "convex S" "(0::'a::euclidean_space) \<notin> S"
```
```  5534   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>S. 0 \<le> inner a x)"
```
```  5535 proof -
```
```  5536   let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
```
```  5537   have *: "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" if as: "f \<subseteq> ?k ` S" "finite f" for f
```
```  5538   proof -
```
```  5539     obtain c where c: "f = ?k ` c" "c \<subseteq> S" "finite c"
```
```  5540       using finite_subset_image[OF as(2,1)] by auto
```
```  5541     then obtain a b where ab: "a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < inner a x"
```
```  5542       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
```
```  5543       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
```
```  5544       using subset_hull[of convex, OF assms(1), symmetric, of c]
```
```  5545       by force
```
```  5546     then have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
```
```  5547       apply (rule_tac x = "inverse(norm a) *\<^sub>R a" in exI)
```
```  5548       using hull_subset[of c convex]
```
```  5549       unfolding subset_eq and inner_scaleR
```
```  5550       by (auto simp: inner_commute del: ballE elim!: ballE)
```
```  5551     then show "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}"
```
```  5552       unfolding c(1) frontier_cball sphere_def dist_norm by auto
```
```  5553   qed
```
```  5554   have "frontier (cball 0 1) \<inter> (\<Inter>(?k ` S)) \<noteq> {}"
```
```  5555     apply (rule compact_imp_fip)
```
```  5556     apply (rule compact_frontier[OF compact_cball])
```
```  5557     using * closed_halfspace_ge
```
```  5558     by auto
```
```  5559   then obtain x where "norm x = 1" "\<forall>y\<in>S. x\<in>?k y"
```
```  5560     unfolding frontier_cball dist_norm sphere_def by auto
```
```  5561   then show ?thesis
```
```  5562     by (metis inner_commute mem_Collect_eq norm_eq_zero zero_neq_one)
```
```  5563 qed
```
```  5564
```
```  5565 lemma separating_hyperplane_sets:
```
```  5566   fixes S T :: "'a::euclidean_space set"
```
```  5567   assumes "convex S"
```
```  5568     and "convex T"
```
```  5569     and "S \<noteq> {}"
```
```  5570     and "T \<noteq> {}"
```
```  5571     and "S \<inter> T = {}"
```
```  5572   shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>S. inner a x \<le> b) \<and> (\<forall>x\<in>T. inner a x \<ge> b)"
```
```  5573 proof -
```
```  5574   from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
```
```  5575   obtain a where "a \<noteq> 0" "\<forall>x\<in>{x - y |x y. x \<in> T \<and> y \<in> S}. 0 \<le> inner a x"
```
```  5576     using assms(3-5) by force
```
```  5577   then have *: "\<And>x y. x \<in> T \<Longrightarrow> y \<in> S \<Longrightarrow> inner a y \<le> inner a x"
```
```  5578     by (force simp: inner_diff)
```
```  5579   then have bdd: "bdd_above (((\<bullet>) a)`S)"
```
```  5580     using \<open>T \<noteq> {}\<close> by (auto intro: bdd_aboveI2[OF *])
```
```  5581   show ?thesis
```
```  5582     using \<open>a\<noteq>0\<close>
```
```  5583     by (intro exI[of _ a] exI[of _ "SUP x:S. a \<bullet> x"])
```
```  5584        (auto intro!: cSUP_upper bdd cSUP_least \<open>a \<noteq> 0\<close> \<open>S \<noteq> {}\<close> *)
```
```  5585 qed
```
```  5586
```
```  5587
```
```  5588 subsection%unimportant \<open>More convexity generalities\<close>
```
```  5589
```
```  5590 lemma convex_closure [intro,simp]:
```
```  5591   fixes S :: "'a::real_normed_vector set"
```
```  5592   assumes "convex S"
```
```  5593   shows "convex (closure S)"
```
```  5594   apply (rule convexI)
```
```  5595   apply (unfold closure_sequential, elim exE)
```
```  5596   apply (rule_tac x="\<lambda>n. u *\<^sub>R xa n + v *\<^sub>R xb n" in exI)
```
```  5597   apply (rule,rule)
```
```  5598   apply (rule convexD [OF assms])
```
```  5599   apply (auto del: tendsto_const intro!: tendsto_intros)
```
```  5600   done
```
```  5601
```
```  5602 lemma convex_interior [intro,simp]:
```
```  5603   fixes S :: "'a::real_normed_vector set"
```
```  5604   assumes "convex S"
```
```  5605   shows "convex (interior S)"
```
```  5606   unfolding convex_alt Ball_def mem_interior
```
```  5607 proof clarify
```
```  5608   fix x y u
```
```  5609   assume u: "0 \<le> u" "u \<le> (1::real)"
```
```  5610   fix e d
```
```  5611   assume ed: "ball x e \<subseteq> S" "ball y d \<subseteq> S" "0<d" "0<e"
```
```  5612   show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> S"
```
```  5613   proof (intro exI conjI subsetI)
```
```  5614     fix z
```
```  5615     assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
```
```  5616     then have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> S"
```
```  5617       apply (rule_tac assms[unfolded convex_alt, rule_format])
```
```  5618       using ed(1,2) and u
```
```  5619       unfolding subset_eq mem_ball Ball_def dist_norm
```
```  5620       apply (auto simp: algebra_simps)
```
```  5621       done
```
```  5622     then show "z \<in> S"
```
```  5623       using u by (auto simp: algebra_simps)
```
```  5624   qed(insert u ed(3-4), auto)
```
```  5625 qed
```
```  5626
```
```  5627 lemma convex_hull_eq_empty[simp]: "convex hull S = {} \<longleftrightarrow> S = {}"
```
```  5628   using hull_subset[of S convex] convex_hull_empty by auto
```
```  5629
```
```  5630
```
```  5631 subsection%unimportant \<open>Moving and scaling convex hulls\<close>
```
```  5632
```
```  5633 lemma convex_hull_set_plus:
```
```  5634   "convex hull (S + T) = convex hull S + convex hull T"
```
```  5635   unfolding set_plus_image
```
```  5636   apply (subst convex_hull_linear_image [symmetric])
```
```  5637   apply (simp add: linear_iff scaleR_right_distrib)
```
```  5638   apply (simp add: convex_hull_Times)
```
```  5639   done
```
```  5640
```
```  5641 lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
```
```  5642   unfolding set_plus_def by auto
```
```  5643
```
```  5644 lemma convex_hull_translation:
```
```  5645   "convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
```
```  5646   unfolding translation_eq_singleton_plus
```
```  5647   by (simp only: convex_hull_set_plus convex_hull_singleton)
```
```  5648
```
```  5649 lemma convex_hull_scaling:
```
```  5650   "convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
```
```  5651   using linear_scaleR by (rule convex_hull_linear_image [symmetric])
```
```  5652
```
```  5653 lemma convex_hull_affinity:
```
```  5654   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
```
```  5655   by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
```
```  5656
```
```  5657
```
```  5658 subsection%unimportant \<open>Convexity of cone hulls\<close>
```
```  5659
```
```  5660 lemma convex_cone_hull:
```
```  5661   assumes "convex S"
```
```  5662   shows "convex (cone hull S)"
```
```  5663 proof (rule convexI)
```
```  5664   fix x y
```
```  5665   assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
```
```  5666   then have "S \<noteq> {}"
```
```  5667     using cone_hull_empty_iff[of S] by auto
```
```  5668   fix u v :: real
```
```  5669   assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
```
```  5670   then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
```
```  5671     using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
```
```  5672   from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
```
```  5673     using cone_hull_expl[of S] by auto
```
```  5674   from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
```
```  5675     using cone_hull_expl[of S] by auto
```
```  5676   {
```
```  5677     assume "cx + cy \<le> 0"
```
```  5678     then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
```
```  5679       using x y by auto
```
```  5680     then have "u *\<^sub>R x + v *\<^sub>R y = 0"
```
```  5681       by auto
```
```  5682     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
```
```  5683       using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
```
```  5684   }
```
```  5685   moreover
```
```  5686   {
```
```  5687     assume "cx + cy > 0"
```
```  5688     then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
```
```  5689       using assms mem_convex_alt[of S xx yy cx cy] x y by auto
```
```  5690     then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
```
```  5691       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>
```
```  5692       by (auto simp: scaleR_right_distrib)
```
```  5693     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
```
```  5694       using x y by auto
```
```  5695   }
```
```  5696   moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
```
```  5697   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
```
```  5698 qed
```
```  5699
```
```  5700 lemma cone_convex_hull:
```
```  5701   assumes "cone S"
```
```  5702   shows "cone (convex hull S)"
```
```  5703 proof (cases "S = {}")
```
```  5704   case True
```
```  5705   then show ?thesis by auto
```
```  5706 next
```
```  5707   case False
```
```  5708   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` S = S)"
```
```  5709     using cone_iff[of S] assms by auto
```
```  5710   {
```
```  5711     fix c :: real
```
```  5712     assume "c > 0"
```
```  5713     then have "( *\<^sub>R) c ` (convex hull S) = convex hull (( *\<^sub>R) c ` S)"
```
```  5714       using convex_hull_scaling[of _ S] by auto
```
```  5715     also have "\<dots> = convex hull S"
```
```  5716       using * \<open>c > 0\<close> by auto
```
```  5717     finally have "( *\<^sub>R) c ` (convex hull S) = convex hull S"
```
```  5718       by auto
```
```  5719   }
```
```  5720   then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> (( *\<^sub>R) c ` (convex hull S)) = (convex hull S)"
```
```  5721     using * hull_subset[of S convex] by auto
```
```  5722   then show ?thesis
```
```  5723     using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
```
```  5724 qed
```
```  5725
```
```  5726 subsection%unimportant \<open>Convex set as intersection of halfspaces\<close>
```
```  5727
```
```  5728 lemma convex_halfspace_intersection:
```
```  5729   fixes s :: "('a::euclidean_space) set"
```
```  5730   assumes "closed s" "convex s"
```
```  5731   shows "s = \<Inter>{h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
```
```  5732   apply (rule set_eqI, rule)
```
```  5733   unfolding Inter_iff Ball_def mem_Collect_eq
```
```  5734   apply (rule,rule,erule conjE)
```
```  5735 proof -
```
```  5736   fix x
```
```  5737   assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
```
```  5738   then have "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}"
```
```  5739     by blast
```
```  5740   then show "x \<in> s"
```
```  5741     apply (rule_tac ccontr)
```
```  5742     apply (drule separating_hyperplane_closed_point[OF assms(2,1)])
```
```  5743     apply (erule exE)+
```
```  5744     apply (erule_tac x="-a" in allE)
```
```  5745     apply (erule_tac x="-b" in allE, auto)
```
```  5746     done
```
```  5747 qed auto
```
```  5748
```
```  5749
```
```  5750 subsection \<open>Radon's theorem (from Lars Schewe)\<close>
```
```  5751
```
```  5752 lemma radon_ex_lemma:
```
```  5753   assumes "finite c" "affine_dependent c"
```
```  5754   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"
```
```  5755 proof -
```
```  5756   from assms(2)[unfolded affine_dependent_explicit]
```
```  5757   obtain s u where
```
```  5758       "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"
```
```  5759     by blast
```
```  5760   then show ?thesis
```
```  5761     apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
```
```  5762     unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric]
```
```  5763     apply (auto simp: Int_absorb1)
```
```  5764     done
```
```  5765 qed
```
```  5766
```
```  5767 lemma radon_s_lemma:
```
```  5768   assumes "finite s"
```
```  5769     and "sum f s = (0::real)"
```
```  5770   shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}"
```
```  5771 proof -
```
```  5772   have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
```
```  5773     by auto
```
```  5774   show ?thesis
```
```  5775     unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
```
```  5776       and sum.distrib[symmetric] and *
```
```  5777     using assms(2)
```
```  5778     by assumption
```
```  5779 qed
```
```  5780
```
```  5781 lemma radon_v_lemma:
```
```  5782   assumes "finite s"
```
```  5783     and "sum f s = 0"
```
```  5784     and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
```
```  5785   shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}"
```
```  5786 proof -
```
```  5787   have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
```
```  5788     using assms(3) by auto
```
```  5789   show ?thesis
```
```  5790     unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
```
```  5791       and sum.distrib[symmetric] and *
```
```  5792     using assms(2)
```
```  5793     apply assumption
```
```  5794     done
```
```  5795 qed
```
```  5796
```
```  5797 lemma radon_partition:
```
```  5798   assumes "finite c" "affine_dependent c"
```
```  5799   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
```
```  5800 proof -
```
```  5801   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"
```
```  5802     using radon_ex_lemma[OF assms] by auto
```
```  5803   have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
```
```  5804     using assms(1) by auto
```
```  5805   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}"
```
```  5806   have "sum u {x \<in> c. 0 < u x} \<noteq> 0"
```
```  5807   proof (cases "u v \<ge> 0")
```
```  5808     case False
```
```  5809     then have "u v < 0" by auto
```
```  5810     then show ?thesis
```
```  5811     proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
```
```  5812       case True
```
```  5813       then show ?thesis
```
```  5814         using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
```
```  5815     next
```
```  5816       case False
```
```  5817       then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c"
```
```  5818         apply (rule_tac sum_mono, auto)
```
```  5819         done
```
```  5820       then show ?thesis
```
```  5821         unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
```
```  5822     qed
```
```  5823   qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
```
```  5824
```
```  5825   then have *: "sum u {x\<in>c. u x > 0} > 0"
```
```  5826     unfolding less_le
```
```  5827     apply (rule_tac conjI)
```
```  5828     apply (rule_tac sum_nonneg, auto)
```
```  5829     done
```
```  5830   moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c"
```
```  5831     "(\<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)"
```
```  5832     using assms(1)
```
```  5833     apply (rule_tac[!] sum.mono_neutral_left, auto)
```
```  5834     done
```
```  5835   then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}"
```
```  5836     "(\<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)"
```
```  5837     unfolding eq_neg_iff_add_eq_0
```
```  5838     using uv(1,4)
```
```  5839     by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
```
```  5840   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x"
```
```  5841     apply rule
```
```  5842     apply (rule mult_nonneg_nonneg)
```
```  5843     using *
```
```  5844     apply auto
```
```  5845     done
```
```  5846   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
```
```  5847     unfolding convex_hull_explicit mem_Collect_eq
```
```  5848     apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
```
```  5849     apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI)
```
```  5850     using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
```
```  5851     apply (auto simp: sum_negf sum_distrib_left[symmetric])
```
```  5852     done
```
```  5853   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
```
```  5854     apply rule
```
```  5855     apply (rule mult_nonneg_nonneg)
```
```  5856     using *
```
```  5857     apply auto
```
```  5858     done
```
```  5859   then have "z \<in> convex hull {v \<in> c. u v > 0}"
```
```  5860     unfolding convex_hull_explicit mem_Collect_eq
```
```  5861     apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
```
```  5862     apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI)
```
```  5863     using assms(1)
```
```  5864     unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
```
```  5865     using *
```
```  5866     apply (auto simp: sum_negf sum_distrib_left[symmetric])
```
```  5867     done
```
```  5868   ultimately show ?thesis
```
```  5869     apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
```
```  5870     apply (rule_tac x="{v\<in>c. u v > 0}" in exI, auto)
```
```  5871     done
```
```  5872 qed
```
```  5873
```
```  5874 theorem radon:
```
```  5875   assumes "affine_dependent c"
```
```  5876   obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
```
```  5877 proof -
```
```  5878   from assms[unfolded affine_dependent_explicit]
```
```  5879   obtain s u where
```
```  5880       "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"
```
```  5881     by blast
```
```  5882   then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
```
```  5883     unfolding affine_dependent_explicit by auto
```
```  5884   from radon_partition[OF *]
```
```  5885   obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
```
```  5886     by blast
```
```  5887   then show ?thesis
```
```  5888     apply (rule_tac that[of p m])
```
```  5889     using s
```
```  5890     apply auto
```
```  5891     done
```
```  5892 qed
```
```  5893
```
```  5894
```
```  5895 subsection \<open>Helly's theorem\<close>
```
```  5896
```
```  5897 lemma helly_induct:
```
```  5898   fixes f :: "'a::euclidean_space set set"
```
```  5899   assumes "card f = n"
```
```  5900     and "n \<ge> DIM('a) + 1"
```
```  5901     and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
```
```  5902   shows "\<Inter>f \<noteq> {}"
```
```  5903   using assms
```
```  5904 proof (induction n arbitrary: f)
```
```  5905   case 0
```
```  5906   then show ?case by auto
```
```  5907 next
```
```  5908   case (Suc n)
```
```  5909   have "finite f"
```
```  5910     using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
```
```  5911   show "\<Inter>f \<noteq> {}"
```
```  5912   proof (cases "n = DIM('a)")
```
```  5913     case True
```
```  5914     then show ?thesis
```
```  5915       by (simp add: Suc.prems(1) Suc.prems(4))
```
```  5916   next
```
```  5917     case False
```
```  5918     have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
```
```  5919     proof (rule Suc.IH[rule_format])
```
```  5920       show "card (f - {s}) = n"
```
```  5921         by (simp add: Suc.prems(1) \<open>finite f\<close> that)
```
```  5922       show "DIM('a) + 1 \<le> n"
```
```  5923         using False Suc.prems(2) by linarith
```
```  5924       show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
```
```  5925         by (simp add: Suc.prems(4) subset_Diff_insert)
```
```  5926     qed (use Suc in auto)
```
```  5927     then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
```
```  5928       by blast
```
```  5929     then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
```
```  5930       by metis
```
```  5931     show ?thesis
```
```  5932     proof (cases "inj_on X f")
```
```  5933       case False
```
```  5934       then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
```
```  5935         unfolding inj_on_def by auto
```
```  5936       then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
```
```  5937       show ?thesis
```
```  5938         by (metis "*" X disjoint_iff_not_equal st)
```
```  5939     next
```
```  5940       case True
```
```  5941       then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
```
```  5942         using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
```
```  5943         unfolding card_image[OF True] and \<open>card f = Suc n\<close>
```
```  5944         using Suc(3) \<open>finite f\<close> and False
```
```  5945         by auto
```
```  5946       have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
```
```  5947         using mp(2) by auto
```
```  5948       then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
```
```  5949         unfolding subset_image_iff by auto
```
```  5950       then have "f \<union> (g \<union> h) = f" by auto
```
```  5951       then have f: "f = g \<union> h"
```
```  5952         using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
```
```  5953         unfolding mp(2)[unfolded image_Un[symmetric] gh]
```
```  5954         by auto
```
```  5955       have *: "g \<inter> h = {}"
```
```  5956         using mp(1)
```
```  5957         unfolding gh
```
```  5958         using inj_on_image_Int[OF True gh(3,4)]
```
```  5959         by auto
```
```  5960       have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
```
```  5961         by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
```
```  5962       then show ?thesis
```
```  5963         unfolding f using mp(3)[unfolded gh] by blast
```
```  5964     qed
```
```  5965   qed
```
```  5966 qed
```
```  5967
```
```  5968 theorem helly:
```
```  5969   fixes f :: "'a::euclidean_space set set"
```
```  5970   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
```
```  5971     and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
```
```  5972   shows "\<Inter>f \<noteq> {}"
```
```  5973   apply (rule helly_induct)
```
```  5974   using assms
```
```  5975   apply auto
```
```  5976   done
```
```  5977
```
```  5978
```
```  5979 subsection \<open>Epigraphs of convex functions\<close>
```
```  5980
```
```  5981 definition%important "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
```
```  5982
```
```  5983 lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
```
```  5984   unfolding epigraph_def by auto
```
```  5985
```
```  5986 lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
```
```  5987 proof safe
```
```  5988   assume L: "convex (epigraph S f)"
```
```  5989   then show "convex_on S f"
```
```  5990     by (auto simp: convex_def convex_on_def epigraph_def)
```
```  5991   show "convex S"
```
```  5992     using L
```
```  5993     apply (clarsimp simp: convex_def convex_on_def epigraph_def)
```
```  5994     apply (erule_tac x=x in allE)
```
```  5995     apply (erule_tac x="f x" in allE, safe)
```
```  5996     apply (erule_tac x=y in allE)
```
```  5997     apply (erule_tac x="f y" in allE)
```
```  5998     apply (auto simp: )
```
```  5999     done
```
```  6000 next
```
```  6001   assume "convex_on S f" "convex S"
```
```  6002   then show "convex (epigraph S f)"
```
```  6003     unfolding convex_def convex_on_def epigraph_def
```
```  6004     apply safe
```
```  6005      apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
```