src/HOL/Analysis/Polytope.thy
 author nipkow Mon Oct 17 11:46:22 2016 +0200 (2016-10-17) changeset 64267 b9a1486e79be parent 64240 eabf80376aab child 65680 378a2f11bec9 permissions -rw-r--r--
setsum -> sum
```     1 section \<open>Faces, Extreme Points, Polytopes, Polyhedra etc.\<close>
```
```     2
```
```     3 text\<open>Ported from HOL Light by L C Paulson\<close>
```
```     4
```
```     5 theory Polytope
```
```     6 imports Cartesian_Euclidean_Space
```
```     7 begin
```
```     8
```
```     9 subsection \<open>Faces of a (usually convex) set\<close>
```
```    10
```
```    11 definition face_of :: "['a::real_vector set, 'a set] \<Rightarrow> bool" (infixr "(face'_of)" 50)
```
```    12   where
```
```    13   "T face_of S \<longleftrightarrow>
```
```    14         T \<subseteq> S \<and> convex T \<and>
```
```    15         (\<forall>a \<in> S. \<forall>b \<in> S. \<forall>x \<in> T. x \<in> open_segment a b \<longrightarrow> a \<in> T \<and> b \<in> T)"
```
```    16
```
```    17 lemma face_ofD: "\<lbrakk>T face_of S; x \<in> open_segment a b; a \<in> S; b \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> a \<in> T \<and> b \<in> T"
```
```    18   unfolding face_of_def by blast
```
```    19
```
```    20 lemma face_of_translation_eq [simp]:
```
```    21     "(op + a ` T face_of op + a ` S) \<longleftrightarrow> T face_of S"
```
```    22 proof -
```
```    23   have *: "\<And>a T S. T face_of S \<Longrightarrow> (op + a ` T face_of op + a ` S)"
```
```    24     apply (simp add: face_of_def Ball_def, clarify)
```
```    25     apply (drule open_segment_translation_eq [THEN iffD1])
```
```    26     using inj_image_mem_iff inj_add_left apply metis
```
```    27     done
```
```    28   show ?thesis
```
```    29     apply (rule iffI)
```
```    30     apply (force simp: image_comp o_def dest: * [where a = "-a"])
```
```    31     apply (blast intro: *)
```
```    32     done
```
```    33 qed
```
```    34
```
```    35 lemma face_of_linear_image:
```
```    36   assumes "linear f" "inj f"
```
```    37     shows "(f ` c face_of f ` S) \<longleftrightarrow> c face_of S"
```
```    38 by (simp add: face_of_def inj_image_subset_iff inj_image_mem_iff open_segment_linear_image assms)
```
```    39
```
```    40 lemma face_of_refl: "convex S \<Longrightarrow> S face_of S"
```
```    41   by (auto simp: face_of_def)
```
```    42
```
```    43 lemma face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
```
```    44   by (auto simp: face_of_def)
```
```    45
```
```    46 lemma empty_face_of [iff]: "{} face_of S"
```
```    47   by (simp add: face_of_def)
```
```    48
```
```    49 lemma face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
```
```    50   by (meson empty_face_of face_of_def subset_empty)
```
```    51
```
```    52 lemma face_of_trans [trans]: "\<lbrakk>S face_of T; T face_of u\<rbrakk> \<Longrightarrow> S face_of u"
```
```    53   unfolding face_of_def by (safe; blast)
```
```    54
```
```    55 lemma face_of_face: "T face_of S \<Longrightarrow> (f face_of T \<longleftrightarrow> f face_of S \<and> f \<subseteq> T)"
```
```    56   unfolding face_of_def by (safe; blast)
```
```    57
```
```    58 lemma face_of_subset: "\<lbrakk>F face_of S; F \<subseteq> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> F face_of T"
```
```    59   unfolding face_of_def by (safe; blast)
```
```    60
```
```    61 lemma face_of_slice: "\<lbrakk>F face_of S; convex T\<rbrakk> \<Longrightarrow> (F \<inter> T) face_of (S \<inter> T)"
```
```    62   unfolding face_of_def by (blast intro: convex_Int)
```
```    63
```
```    64 lemma face_of_Int: "\<lbrakk>t1 face_of S; t2 face_of S\<rbrakk> \<Longrightarrow> (t1 \<inter> t2) face_of S"
```
```    65   unfolding face_of_def by (blast intro: convex_Int)
```
```    66
```
```    67 lemma face_of_Inter: "\<lbrakk>A \<noteq> {}; \<And>T. T \<in> A \<Longrightarrow> T face_of S\<rbrakk> \<Longrightarrow> (\<Inter> A) face_of S"
```
```    68   unfolding face_of_def by (blast intro: convex_Inter)
```
```    69
```
```    70 lemma face_of_Int_Int: "\<lbrakk>F face_of T; F' face_of t'\<rbrakk> \<Longrightarrow> (F \<inter> F') face_of (T \<inter> t')"
```
```    71   unfolding face_of_def by (blast intro: convex_Int)
```
```    72
```
```    73 lemma face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
```
```    74   unfolding face_of_def by blast
```
```    75
```
```    76 lemma face_of_imp_eq_affine_Int:
```
```    77      fixes S :: "'a::euclidean_space set"
```
```    78      assumes S: "convex S" "closed S" and T: "T face_of S"
```
```    79      shows "T = (affine hull T) \<inter> S"
```
```    80 proof -
```
```    81   have "convex T" using T by (simp add: face_of_def)
```
```    82   have *: False if x: "x \<in> affine hull T" and "x \<in> S" "x \<notin> T" and y: "y \<in> rel_interior T" for x y
```
```    83   proof -
```
```    84     obtain e where "e>0" and e: "cball y e \<inter> affine hull T \<subseteq> T"
```
```    85       using y by (auto simp: rel_interior_cball)
```
```    86     have "y \<noteq> x" "y \<in> S" "y \<in> T"
```
```    87       using face_of_imp_subset rel_interior_subset T that by blast+
```
```    88     then have zne: "\<And>u. \<lbrakk>u \<in> {0<..<1}; (1 - u) *\<^sub>R y + u *\<^sub>R x \<in> T\<rbrakk> \<Longrightarrow>  False"
```
```    89       using \<open>x \<in> S\<close> \<open>x \<notin> T\<close> \<open>T face_of S\<close> unfolding face_of_def
```
```    90       apply clarify
```
```    91       apply (drule_tac x=x in bspec, assumption)
```
```    92       apply (drule_tac x=y in bspec, assumption)
```
```    93       apply (subst (asm) open_segment_commute)
```
```    94       apply (force simp: open_segment_image_interval image_def)
```
```    95       done
```
```    96     have in01: "min (1/2) (e / norm (x - y)) \<in> {0<..<1}"
```
```    97       using \<open>y \<noteq> x\<close> \<open>e > 0\<close> by simp
```
```    98     show ?thesis
```
```    99       apply (rule zne [OF in01])
```
```   100       apply (rule e [THEN subsetD])
```
```   101       apply (rule IntI)
```
```   102         using \<open>y \<noteq> x\<close> \<open>e > 0\<close>
```
```   103         apply (simp add: cball_def dist_norm algebra_simps)
```
```   104         apply (simp add: Real_Vector_Spaces.scaleR_diff_right [symmetric] norm_minus_commute min_mult_distrib_right)
```
```   105       apply (rule mem_affine [OF affine_affine_hull _ x])
```
```   106       using \<open>y \<in> T\<close>  apply (auto simp: hull_inc)
```
```   107       done
```
```   108   qed
```
```   109   show ?thesis
```
```   110     apply (rule subset_antisym)
```
```   111     using assms apply (simp add: hull_subset face_of_imp_subset)
```
```   112     apply (cases "T={}", simp)
```
```   113     apply (force simp: rel_interior_eq_empty [symmetric] \<open>convex T\<close> intro: *)
```
```   114     done
```
```   115 qed
```
```   116
```
```   117 lemma face_of_imp_closed:
```
```   118      fixes S :: "'a::euclidean_space set"
```
```   119      assumes "convex S" "closed S" "T face_of S" shows "closed T"
```
```   120   by (metis affine_affine_hull affine_closed closed_Int face_of_imp_eq_affine_Int assms)
```
```   121
```
```   122 lemma face_of_Int_supporting_hyperplane_le_strong:
```
```   123     assumes "convex(S \<inter> {x. a \<bullet> x = b})" and aleb: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b"
```
```   124       shows "(S \<inter> {x. a \<bullet> x = b}) face_of S"
```
```   125 proof -
```
```   126   have *: "a \<bullet> u = a \<bullet> x" if "x \<in> open_segment u v" "u \<in> S" "v \<in> S" and b: "b = a \<bullet> x"
```
```   127           for u v x
```
```   128   proof (rule antisym)
```
```   129     show "a \<bullet> u \<le> a \<bullet> x"
```
```   130       using aleb \<open>u \<in> S\<close> \<open>b = a \<bullet> x\<close> by blast
```
```   131   next
```
```   132     obtain \<xi> where "b = a \<bullet> ((1 - \<xi>) *\<^sub>R u + \<xi> *\<^sub>R v)" "0 < \<xi>" "\<xi> < 1"
```
```   133       using \<open>b = a \<bullet> x\<close> \<open>x \<in> open_segment u v\<close> in_segment
```
```   134       by (auto simp: open_segment_image_interval split: if_split_asm)
```
```   135     then have "b + \<xi> * (a \<bullet> u) \<le> a \<bullet> u + \<xi> * b"
```
```   136       using aleb [OF \<open>v \<in> S\<close>] by (simp add: algebra_simps)
```
```   137     then have "(1 - \<xi>) * b \<le> (1 - \<xi>) * (a \<bullet> u)"
```
```   138       by (simp add: algebra_simps)
```
```   139     then have "b \<le> a \<bullet> u"
```
```   140       using \<open>\<xi> < 1\<close> by auto
```
```   141     with b show "a \<bullet> x \<le> a \<bullet> u" by simp
```
```   142   qed
```
```   143   show ?thesis
```
```   144     apply (simp add: face_of_def assms)
```
```   145     using "*" open_segment_commute by blast
```
```   146 qed
```
```   147
```
```   148 lemma face_of_Int_supporting_hyperplane_ge_strong:
```
```   149    "\<lbrakk>convex(S \<inter> {x. a \<bullet> x = b}); \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk>
```
```   150     \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
```
```   151   using face_of_Int_supporting_hyperplane_le_strong [of S "-a" "-b"] by simp
```
```   152
```
```   153 lemma face_of_Int_supporting_hyperplane_le:
```
```   154     "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
```
```   155   by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_le_strong)
```
```   156
```
```   157 lemma face_of_Int_supporting_hyperplane_ge:
```
```   158     "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
```
```   159   by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_ge_strong)
```
```   160
```
```   161 lemma face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
```
```   162   using face_of_def by blast
```
```   163
```
```   164 lemma face_of_imp_compact:
```
```   165     fixes S :: "'a::euclidean_space set"
```
```   166     shows "\<lbrakk>convex S; compact S; T face_of S\<rbrakk> \<Longrightarrow> compact T"
```
```   167   by (meson bounded_subset compact_eq_bounded_closed face_of_imp_closed face_of_imp_subset)
```
```   168
```
```   169 lemma face_of_Int_subface:
```
```   170      "\<lbrakk>A \<inter> B face_of A; A \<inter> B face_of B; C face_of A; D face_of B\<rbrakk>
```
```   171       \<Longrightarrow> (C \<inter> D) face_of C \<and> (C \<inter> D) face_of D"
```
```   172   by (meson face_of_Int_Int face_of_face inf_le1 inf_le2)
```
```   173
```
```   174 lemma subset_of_face_of:
```
```   175     fixes S :: "'a::real_normed_vector set"
```
```   176     assumes "T face_of S" "u \<subseteq> S" "T \<inter> (rel_interior u) \<noteq> {}"
```
```   177       shows "u \<subseteq> T"
```
```   178 proof
```
```   179   fix c
```
```   180   assume "c \<in> u"
```
```   181   obtain b where "b \<in> T" "b \<in> rel_interior u" using assms by auto
```
```   182   then obtain e where "e>0" "b \<in> u" and e: "cball b e \<inter> affine hull u \<subseteq> u"
```
```   183     by (auto simp: rel_interior_cball)
```
```   184   show "c \<in> T"
```
```   185   proof (cases "b=c")
```
```   186     case True with \<open>b \<in> T\<close> show ?thesis by blast
```
```   187   next
```
```   188     case False
```
```   189     define d where "d = b + (e / norm(b - c)) *\<^sub>R (b - c)"
```
```   190     have "d \<in> cball b e \<inter> affine hull u"
```
```   191       using \<open>e > 0\<close> \<open>b \<in> u\<close> \<open>c \<in> u\<close>
```
```   192       by (simp add: d_def dist_norm hull_inc mem_affine_3_minus False)
```
```   193     with e have "d \<in> u" by blast
```
```   194     have nbc: "norm (b - c) + e > 0" using \<open>e > 0\<close>
```
```   195       by (metis add.commute le_less_trans less_add_same_cancel2 norm_ge_zero)
```
```   196     then have [simp]: "d \<noteq> c" using False scaleR_cancel_left [of "1 + (e / norm (b - c))" b c]
```
```   197       by (simp add: algebra_simps d_def) (simp add: divide_simps)
```
```   198     have [simp]: "((e - e * e / (e + norm (b - c))) / norm (b - c)) = (e / (e + norm (b - c)))"
```
```   199       using False nbc
```
```   200       apply (simp add: algebra_simps divide_simps)
```
```   201       by (metis mult_eq_0_iff norm_eq_zero norm_imp_pos_and_ge norm_pths(2) real_scaleR_def scaleR_left.add zero_less_norm_iff)
```
```   202     have "b \<in> open_segment d c"
```
```   203       apply (simp add: open_segment_image_interval)
```
```   204       apply (simp add: d_def algebra_simps image_def)
```
```   205       apply (rule_tac x="e / (e + norm (b - c))" in bexI)
```
```   206       using False nbc \<open>0 < e\<close>
```
```   207       apply (auto simp: algebra_simps)
```
```   208       done
```
```   209     then have "d \<in> T \<and> c \<in> T"
```
```   210       apply (rule face_ofD [OF \<open>T face_of S\<close>])
```
```   211       using \<open>d \<in> u\<close>  \<open>c \<in> u\<close> \<open>u \<subseteq> S\<close>  \<open>b \<in> T\<close>  apply auto
```
```   212       done
```
```   213     then show ?thesis ..
```
```   214   qed
```
```   215 qed
```
```   216
```
```   217 lemma face_of_eq:
```
```   218     fixes S :: "'a::real_normed_vector set"
```
```   219     assumes "T face_of S" "u face_of S" "(rel_interior T) \<inter> (rel_interior u) \<noteq> {}"
```
```   220       shows "T = u"
```
```   221   apply (rule subset_antisym)
```
```   222   apply (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subsetCE subset_of_face_of)
```
```   223   by (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subset_iff subset_of_face_of)
```
```   224
```
```   225 lemma face_of_disjoint_rel_interior:
```
```   226       fixes S :: "'a::real_normed_vector set"
```
```   227       assumes "T face_of S" "T \<noteq> S"
```
```   228         shows "T \<inter> rel_interior S = {}"
```
```   229   by (meson assms subset_of_face_of face_of_imp_subset order_refl subset_antisym)
```
```   230
```
```   231 lemma face_of_disjoint_interior:
```
```   232       fixes S :: "'a::real_normed_vector set"
```
```   233       assumes "T face_of S" "T \<noteq> S"
```
```   234         shows "T \<inter> interior S = {}"
```
```   235 proof -
```
```   236   have "T \<inter> interior S \<subseteq> rel_interior S"
```
```   237     by (meson inf_sup_ord(2) interior_subset_rel_interior order.trans)
```
```   238   thus ?thesis
```
```   239     by (metis (no_types) Int_greatest assms face_of_disjoint_rel_interior inf_sup_ord(1) subset_empty)
```
```   240 qed
```
```   241
```
```   242 lemma face_of_subset_rel_boundary:
```
```   243   fixes S :: "'a::real_normed_vector set"
```
```   244   assumes "T face_of S" "T \<noteq> S"
```
```   245     shows "T \<subseteq> (S - rel_interior S)"
```
```   246 by (meson DiffI assms disjoint_iff_not_equal face_of_disjoint_rel_interior face_of_imp_subset rev_subsetD subsetI)
```
```   247
```
```   248 lemma face_of_subset_rel_frontier:
```
```   249     fixes S :: "'a::real_normed_vector set"
```
```   250     assumes "T face_of S" "T \<noteq> S"
```
```   251       shows "T \<subseteq> rel_frontier S"
```
```   252   using assms closure_subset face_of_disjoint_rel_interior face_of_imp_subset rel_frontier_def by fastforce
```
```   253
```
```   254 lemma face_of_aff_dim_lt:
```
```   255   fixes S :: "'a::euclidean_space set"
```
```   256   assumes "convex S" "T face_of S" "T \<noteq> S"
```
```   257     shows "aff_dim T < aff_dim S"
```
```   258 proof -
```
```   259   have "aff_dim T \<le> aff_dim S"
```
```   260     by (simp add: face_of_imp_subset aff_dim_subset assms)
```
```   261   moreover have "aff_dim T \<noteq> aff_dim S"
```
```   262   proof (cases "T = {}")
```
```   263     case True then show ?thesis
```
```   264       by (metis aff_dim_empty \<open>T \<noteq> S\<close>)
```
```   265   next case False then show ?thesis
```
```   266     by (metis Set.set_insert assms convex_rel_frontier_aff_dim dual_order.irrefl face_of_imp_convex face_of_subset_rel_frontier insert_not_empty subsetI)
```
```   267   qed
```
```   268   ultimately show ?thesis
```
```   269     by simp
```
```   270 qed
```
```   271
```
```   272
```
```   273 lemma affine_diff_divide:
```
```   274     assumes "affine S" "k \<noteq> 0" "k \<noteq> 1" and xy: "x \<in> S" "y /\<^sub>R (1 - k) \<in> S"
```
```   275       shows "(x - y) /\<^sub>R k \<in> S"
```
```   276 proof -
```
```   277   have "inverse(k) *\<^sub>R (x - y) = (1 - inverse k) *\<^sub>R inverse(1 - k) *\<^sub>R y + inverse(k) *\<^sub>R x"
```
```   278     using assms
```
```   279     by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] divide_simps)
```
```   280   then show ?thesis
```
```   281     using \<open>affine S\<close> xy by (auto simp: affine_alt)
```
```   282 qed
```
```   283
```
```   284 lemma face_of_convex_hulls:
```
```   285       assumes S: "finite S" "T \<subseteq> S" and disj: "affine hull T \<inter> convex hull (S - T) = {}"
```
```   286       shows  "(convex hull T) face_of (convex hull S)"
```
```   287 proof -
```
```   288   have fin: "finite T" "finite (S - T)" using assms
```
```   289     by (auto simp: finite_subset)
```
```   290   have *: "x \<in> convex hull T"
```
```   291           if x: "x \<in> convex hull S" and y: "y \<in> convex hull S" and w: "w \<in> convex hull T" "w \<in> open_segment x y"
```
```   292           for x y w
```
```   293   proof -
```
```   294     have waff: "w \<in> affine hull T"
```
```   295       using convex_hull_subset_affine_hull w by blast
```
```   296     obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "sum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
```
```   297                  and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "sum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
```
```   298       using x y by (auto simp: assms convex_hull_finite)
```
```   299     obtain u where "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> convex hull T" "x \<noteq> y" and weq: "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
```
```   300                and u01: "0 < u" "u < 1"
```
```   301       using w by (auto simp: open_segment_image_interval split: if_split_asm)
```
```   302     define c where "c i = (1 - u) * a i + u * b i" for i
```
```   303     have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
```
```   304       using a b u01 by (simp add: c_def)
```
```   305     have sumc1: "sum c S = 1"
```
```   306       by (simp add: c_def sum.distrib sum_distrib_left [symmetric] asum bsum)
```
```   307     have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
```
```   308       apply (simp add: c_def sum.distrib scaleR_left_distrib)
```
```   309       by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.sum [symmetric] aeqx beqy)
```
```   310     show ?thesis
```
```   311     proof (cases "sum c (S - T) = 0")
```
```   312       case True
```
```   313       have ci0: "\<And>i. i \<in> (S - T) \<Longrightarrow> c i = 0"
```
```   314         using True cge0 by (simp add: \<open>finite S\<close> sum_nonneg_eq_0_iff)
```
```   315       have a0: "a i = 0" if "i \<in> (S - T)" for i
```
```   316         using ci0 [OF that] u01 a [of i] b [of i] that
```
```   317         by (simp add: c_def Groups.ordered_comm_monoid_add_class.add_nonneg_eq_0_iff)
```
```   318       have [simp]: "sum a T = 1"
```
```   319         using assms by (metis sum.mono_neutral_cong_right a0 asum)
```
```   320       show ?thesis
```
```   321         apply (simp add: convex_hull_finite \<open>finite T\<close>)
```
```   322         apply (rule_tac x=a in exI)
```
```   323         using a0 assms
```
```   324         apply (auto simp: cge0 a aeqx [symmetric] sum.mono_neutral_right)
```
```   325         done
```
```   326     next
```
```   327       case False
```
```   328       define k where "k = sum c (S - T)"
```
```   329       have "k > 0" using False
```
```   330         unfolding k_def by (metis DiffD1 antisym_conv cge0 sum_nonneg not_less)
```
```   331       have weq_sumsum: "w = sum (\<lambda>x. c x *\<^sub>R x) T + sum (\<lambda>x. c x *\<^sub>R x) (S - T)"
```
```   332         by (metis (no_types) add.commute S(1) S(2) sum.subset_diff sumci_xy weq)
```
```   333       show ?thesis
```
```   334       proof (cases "k = 1")
```
```   335         case True
```
```   336         then have "sum c T = 0"
```
```   337           by (simp add: S k_def sum_diff sumc1)
```
```   338         then have [simp]: "sum c (S - T) = 1"
```
```   339           by (simp add: S sum_diff sumc1)
```
```   340         have ci0: "\<And>i. i \<in> T \<Longrightarrow> c i = 0"
```
```   341           by (meson \<open>finite T\<close> \<open>sum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 sum_nonneg_eq_0_iff subsetCE)
```
```   342         then have [simp]: "(\<Sum>i\<in>S-T. c i *\<^sub>R i) = w"
```
```   343           by (simp add: weq_sumsum)
```
```   344         have "w \<in> convex hull (S - T)"
```
```   345           apply (simp add: convex_hull_finite fin)
```
```   346           apply (rule_tac x=c in exI)
```
```   347           apply (auto simp: cge0 weq True k_def)
```
```   348           done
```
```   349         then show ?thesis
```
```   350           using disj waff by blast
```
```   351       next
```
```   352         case False
```
```   353         then have sumcf: "sum c T = 1 - k"
```
```   354           by (simp add: S k_def sum_diff sumc1)
```
```   355         have "(\<Sum>i\<in>T. c i *\<^sub>R i) /\<^sub>R (1 - k) \<in> convex hull T"
```
```   356           apply (simp add: convex_hull_finite fin)
```
```   357           apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
```
```   358           apply auto
```
```   359           apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) sum_nonneg subsetCE)
```
```   360           apply (metis False mult.commute right_inverse right_minus_eq sum_distrib_left sumcf)
```
```   361           by (metis (mono_tags, lifting) scaleR_right.sum scaleR_scaleR sum.cong)
```
```   362         with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - sum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
```
```   363           by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
```
```   364         moreover have "inverse(k) *\<^sub>R (w - sum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
```
```   365           apply (simp add: weq_sumsum convex_hull_finite fin)
```
```   366           apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
```
```   367           using \<open>k > 0\<close> cge0
```
```   368           apply (auto simp: scaleR_right.sum sum_distrib_left [symmetric] k_def [symmetric])
```
```   369           done
```
```   370         ultimately show ?thesis
```
```   371           using disj by blast
```
```   372       qed
```
```   373     qed
```
```   374   qed
```
```   375   have [simp]: "convex hull T \<subseteq> convex hull S"
```
```   376     by (simp add: \<open>T \<subseteq> S\<close> hull_mono)
```
```   377   show ?thesis
```
```   378     using open_segment_commute by (auto simp: face_of_def intro: *)
```
```   379 qed
```
```   380
```
```   381 proposition face_of_convex_hull_insert:
```
```   382    "\<lbrakk>finite S; a \<notin> affine hull S; T face_of convex hull S\<rbrakk> \<Longrightarrow> T face_of convex hull insert a S"
```
```   383   apply (rule face_of_trans, blast)
```
```   384   apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
```
```   385   done
```
```   386
```
```   387 proposition face_of_affine_trivial:
```
```   388     assumes "affine S" "T face_of S"
```
```   389     shows "T = {} \<or> T = S"
```
```   390 proof (rule ccontr, clarsimp)
```
```   391   assume "T \<noteq> {}" "T \<noteq> S"
```
```   392   then obtain a where "a \<in> T" by auto
```
```   393   then have "a \<in> S"
```
```   394     using \<open>T face_of S\<close> face_of_imp_subset by blast
```
```   395   have "S \<subseteq> T"
```
```   396   proof
```
```   397     fix b  assume "b \<in> S"
```
```   398     show "b \<in> T"
```
```   399     proof (cases "a = b")
```
```   400       case True with \<open>a \<in> T\<close> show ?thesis by auto
```
```   401     next
```
```   402       case False
```
```   403       then have "a \<in> open_segment (2 *\<^sub>R a - b) b"
```
```   404         apply (auto simp: open_segment_def closed_segment_def)
```
```   405         apply (rule_tac x="1/2" in exI)
```
```   406         apply (simp add: algebra_simps)
```
```   407         by (simp add: scaleR_2)
```
```   408       moreover have "2 *\<^sub>R a - b \<in> S"
```
```   409         by (rule mem_affine [OF \<open>affine S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>, of 2 "-1", simplified])
```
```   410       moreover note \<open>b \<in> S\<close> \<open>a \<in> T\<close>
```
```   411       ultimately show ?thesis
```
```   412         by (rule face_ofD [OF \<open>T face_of S\<close>, THEN conjunct2])
```
```   413     qed
```
```   414   qed
```
```   415   then show False
```
```   416     using \<open>T \<noteq> S\<close> \<open>T face_of S\<close> face_of_imp_subset by blast
```
```   417 qed
```
```   418
```
```   419
```
```   420 lemma face_of_affine_eq:
```
```   421    "affine S \<Longrightarrow> (T face_of S \<longleftrightarrow> T = {} \<or> T = S)"
```
```   422 using affine_imp_convex face_of_affine_trivial face_of_refl by auto
```
```   423
```
```   424
```
```   425 lemma Inter_faces_finite_altbound:
```
```   426     fixes T :: "'a::euclidean_space set set"
```
```   427     assumes cfaI: "\<And>c. c \<in> T \<Longrightarrow> c face_of S"
```
```   428     shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<and> \<Inter>F' = \<Inter>T"
```
```   429 proof (cases "\<forall>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<longrightarrow> (\<exists>c. c \<in> T \<and> c \<inter> (\<Inter>F') \<subset> (\<Inter>F'))")
```
```   430   case True
```
```   431   then obtain c where c:
```
```   432        "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> T; card F' \<le> DIM('a) + 2\<rbrakk> \<Longrightarrow> c F' \<in> T \<and> c F' \<inter> (\<Inter>F') \<subset> (\<Inter>F')"
```
```   433     by metis
```
```   434   define d where "d = rec_nat {c{}} (\<lambda>n r. insert (c r) r)"
```
```   435   have [simp]: "d 0 = {c {}}"
```
```   436     by (simp add: d_def)
```
```   437   have dSuc [simp]: "\<And>n. d (Suc n) = insert (c (d n)) (d n)"
```
```   438     by (simp add: d_def)
```
```   439   have dn_notempty: "d n \<noteq> {}" for n
```
```   440     by (induction n) auto
```
```   441   have dn_le_Suc: "d n \<subseteq> T \<and> finite(d n) \<and> card(d n) \<le> Suc n" if "n \<le> DIM('a) + 2" for n
```
```   442   using that
```
```   443   proof (induction n)
```
```   444     case 0
```
```   445     then show ?case by (simp add: c)
```
```   446   next
```
```   447     case (Suc n)
```
```   448     then show ?case by (auto simp: c card_insert_if)
```
```   449   qed
```
```   450   have aff_dim_le: "aff_dim(\<Inter>(d n)) \<le> DIM('a) - int n" if "n \<le> DIM('a) + 2" for n
```
```   451   using that
```
```   452   proof (induction n)
```
```   453     case 0
```
```   454     then show ?case
```
```   455       by (simp add: aff_dim_le_DIM)
```
```   456   next
```
```   457     case (Suc n)
```
```   458     have fs: "\<Inter>d (Suc n) face_of S"
```
```   459       by (meson Suc.prems cfaI dn_le_Suc dn_notempty face_of_Inter subsetCE)
```
```   460     have condn: "convex (\<Inter>d n)"
```
```   461       using Suc.prems nat_le_linear not_less_eq_eq
```
```   462       by (blast intro: face_of_imp_convex cfaI convex_Inter dest: dn_le_Suc)
```
```   463     have fdn: "\<Inter>d (Suc n) face_of \<Inter>d n"
```
```   464       by (metis (no_types, lifting) Inter_anti_mono Suc.prems dSuc cfaI dn_le_Suc dn_notempty face_of_Inter face_of_imp_subset face_of_subset subset_iff subset_insertI)
```
```   465     have ne: "\<Inter>d (Suc n) \<noteq> \<Inter>d n"
```
```   466       by (metis (no_types, lifting) Suc.prems Suc_leD c complete_lattice_class.Inf_insert dSuc dn_le_Suc less_irrefl order.trans)
```
```   467     have *: "\<And>m::int. \<And>d. \<And>d'::int. d < d' \<and> d' \<le> m - n \<Longrightarrow> d \<le> m - of_nat(n+1)"
```
```   468       by arith
```
```   469     have "aff_dim (\<Inter>d (Suc n)) < aff_dim (\<Inter>d n)"
```
```   470       by (rule face_of_aff_dim_lt [OF condn fdn ne])
```
```   471     moreover have "aff_dim (\<Inter>d n) \<le> int (DIM('a)) - int n"
```
```   472       using Suc by auto
```
```   473     ultimately
```
```   474     have "aff_dim (\<Inter>d (Suc n)) \<le> int (DIM('a)) - (n+1)" by arith
```
```   475     then show ?case by linarith
```
```   476   qed
```
```   477   have "aff_dim (\<Inter>d (DIM('a) + 2)) \<le> -2"
```
```   478       using aff_dim_le [OF order_refl] by simp
```
```   479   with aff_dim_geq [of "\<Inter>d (DIM('a) + 2)"] show ?thesis
```
```   480     using order.trans by fastforce
```
```   481 next
```
```   482   case False
```
```   483   then show ?thesis
```
```   484     apply simp
```
```   485     apply (erule ex_forward)
```
```   486     by blast
```
```   487 qed
```
```   488
```
```   489 lemma faces_of_translation:
```
```   490    "{F. F face_of image (\<lambda>x. a + x) S} = image (image (\<lambda>x. a + x)) {F. F face_of S}"
```
```   491 apply (rule subset_antisym, clarify)
```
```   492 apply (auto simp: image_iff)
```
```   493 apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
```
```   494 done
```
```   495
```
```   496 proposition face_of_Times:
```
```   497   assumes "F face_of S" and "F' face_of S'"
```
```   498     shows "(F \<times> F') face_of (S \<times> S')"
```
```   499 proof -
```
```   500   have "F \<times> F' \<subseteq> S \<times> S'"
```
```   501     using assms [unfolded face_of_def] by blast
```
```   502   moreover
```
```   503   have "convex (F \<times> F')"
```
```   504     using assms [unfolded face_of_def] by (blast intro: convex_Times)
```
```   505   moreover
```
```   506     have "a \<in> F \<and> a' \<in> F' \<and> b \<in> F \<and> b' \<in> F'"
```
```   507        if "a \<in> S" "b \<in> S" "a' \<in> S'" "b' \<in> S'" "x \<in> F \<times> F'" "x \<in> open_segment (a,a') (b,b')"
```
```   508        for a b a' b' x
```
```   509   proof (cases "b=a \<or> b'=a'")
```
```   510     case True with that show ?thesis
```
```   511       using assms
```
```   512       by (force simp: in_segment dest: face_ofD)
```
```   513   next
```
```   514     case False with assms [unfolded face_of_def] that show ?thesis
```
```   515       by (blast dest!: open_segment_PairD)
```
```   516   qed
```
```   517   ultimately show ?thesis
```
```   518     unfolding face_of_def by blast
```
```   519 qed
```
```   520
```
```   521 corollary face_of_Times_decomp:
```
```   522     fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
```
```   523     shows "c face_of (S \<times> S') \<longleftrightarrow> (\<exists>F F'. F face_of S \<and> F' face_of S' \<and> c = F \<times> F')"
```
```   524      (is "?lhs = ?rhs")
```
```   525 proof
```
```   526   assume c: ?lhs
```
```   527   show ?rhs
```
```   528   proof (cases "c = {}")
```
```   529     case True then show ?thesis by auto
```
```   530   next
```
```   531     case False
```
```   532     have 1: "fst ` c \<subseteq> S" "snd ` c \<subseteq> S'"
```
```   533       using c face_of_imp_subset by fastforce+
```
```   534     have "convex c"
```
```   535       using c by (metis face_of_imp_convex)
```
```   536     have conv: "convex (fst ` c)" "convex (snd ` c)"
```
```   537       by (simp_all add: \<open>convex c\<close> convex_linear_image fst_linear snd_linear)
```
```   538     have fstab: "a \<in> fst ` c \<and> b \<in> fst ` c"
```
```   539             if "a \<in> S" "b \<in> S" "x \<in> open_segment a b" "(x,x') \<in> c" for a b x x'
```
```   540     proof -
```
```   541       have *: "(x,x') \<in> open_segment (a,x') (b,x')"
```
```   542         using that by (auto simp: in_segment)
```
```   543       show ?thesis
```
```   544         using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
```
```   545     qed
```
```   546     have fst: "fst ` c face_of S"
```
```   547       by (force simp: face_of_def 1 conv fstab)
```
```   548     have sndab: "a' \<in> snd ` c \<and> b' \<in> snd ` c"
```
```   549             if "a' \<in> S'" "b' \<in> S'" "x' \<in> open_segment a' b'" "(x,x') \<in> c" for a' b' x x'
```
```   550     proof -
```
```   551       have *: "(x,x') \<in> open_segment (x,a') (x,b')"
```
```   552         using that by (auto simp: in_segment)
```
```   553       show ?thesis
```
```   554         using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
```
```   555     qed
```
```   556     have snd: "snd ` c face_of S'"
```
```   557       by (force simp: face_of_def 1 conv sndab)
```
```   558     have cc: "rel_interior c \<subseteq> rel_interior (fst ` c) \<times> rel_interior (snd ` c)"
```
```   559       by (force simp: face_of_Times rel_interior_Times conv fst snd \<open>convex c\<close> fst_linear snd_linear rel_interior_convex_linear_image [symmetric])
```
```   560     have "c = fst ` c \<times> snd ` c"
```
```   561       apply (rule face_of_eq [OF c])
```
```   562       apply (simp_all add: face_of_Times rel_interior_Times conv fst snd)
```
```   563       using False rel_interior_eq_empty \<open>convex c\<close> cc
```
```   564       apply blast
```
```   565       done
```
```   566     with fst snd show ?thesis by metis
```
```   567   qed
```
```   568 next
```
```   569   assume ?rhs with face_of_Times show ?lhs by auto
```
```   570 qed
```
```   571
```
```   572 lemma face_of_Times_eq:
```
```   573     fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
```
```   574     shows "(F \<times> F') face_of (S \<times> S') \<longleftrightarrow>
```
```   575            F = {} \<or> F' = {} \<or> F face_of S \<and> F' face_of S'"
```
```   576 by (auto simp: face_of_Times_decomp times_eq_iff)
```
```   577
```
```   578 lemma hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
```
```   579 proof -
```
```   580   have "{x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
```
```   581     by auto
```
```   582   with face_of_Int_supporting_hyperplane_le [OF convex_halfspace_le [of a b], of a b]
```
```   583   show ?thesis by auto
```
```   584 qed
```
```   585
```
```   586 lemma hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
```
```   587 proof -
```
```   588   have "{x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
```
```   589     by auto
```
```   590   with face_of_Int_supporting_hyperplane_ge [OF convex_halfspace_ge [of b a], of b a]
```
```   591   show ?thesis by auto
```
```   592 qed
```
```   593
```
```   594 lemma face_of_halfspace_le:
```
```   595   fixes a :: "'n::euclidean_space"
```
```   596   shows "F face_of {x. a \<bullet> x \<le> b} \<longleftrightarrow>
```
```   597          F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b}"
```
```   598      (is "?lhs = ?rhs")
```
```   599 proof (cases "a = 0")
```
```   600   case True then show ?thesis
```
```   601     using face_of_affine_eq affine_UNIV by auto
```
```   602 next
```
```   603   case False
```
```   604   then have ine: "interior {x. a \<bullet> x \<le> b} \<noteq> {}"
```
```   605     using halfspace_eq_empty_lt interior_halfspace_le by blast
```
```   606   show ?thesis
```
```   607   proof
```
```   608     assume L: ?lhs
```
```   609     have "F \<noteq> {x. a \<bullet> x \<le> b} \<Longrightarrow> F face_of {x. a \<bullet> x = b}"
```
```   610       using False
```
```   611       apply (simp add: frontier_halfspace_le [symmetric] rel_frontier_nonempty_interior [OF ine, symmetric])
```
```   612       apply (rule face_of_subset [OF L])
```
```   613       apply (simp add: face_of_subset_rel_frontier [OF L])
```
```   614       apply (force simp: rel_frontier_def closed_halfspace_le)
```
```   615       done
```
```   616     with L show ?rhs
```
```   617       using affine_hyperplane face_of_affine_eq by blast
```
```   618   next
```
```   619     assume ?rhs
```
```   620     then show ?lhs
```
```   621       by (metis convex_halfspace_le empty_face_of face_of_refl hyperplane_face_of_halfspace_le)
```
```   622   qed
```
```   623 qed
```
```   624
```
```   625 lemma face_of_halfspace_ge:
```
```   626   fixes a :: "'n::euclidean_space"
```
```   627   shows "F face_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow>
```
```   628          F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<ge> b}"
```
```   629 using face_of_halfspace_le [of F "-a" "-b"] by simp
```
```   630
```
```   631 subsection\<open>Exposed faces\<close>
```
```   632
```
```   633 text\<open>That is, faces that are intersection with supporting hyperplane\<close>
```
```   634
```
```   635 definition exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
```
```   636                                (infixr "(exposed'_face'_of)" 50)
```
```   637   where "T exposed_face_of S \<longleftrightarrow>
```
```   638          T face_of S \<and> (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b})"
```
```   639
```
```   640 lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
```
```   641   apply (simp add: exposed_face_of_def)
```
```   642   apply (rule_tac x=0 in exI)
```
```   643   apply (rule_tac x=1 in exI, force)
```
```   644   done
```
```   645
```
```   646 lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
```
```   647   apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
```
```   648   apply (rule_tac x=0 in exI)+
```
```   649   apply force
```
```   650   done
```
```   651
```
```   652 lemma exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
```
```   653   by simp
```
```   654
```
```   655 lemma exposed_face_of:
```
```   656     "T exposed_face_of S \<longleftrightarrow>
```
```   657      T face_of S \<and>
```
```   658      (T = {} \<or> T = S \<or>
```
```   659       (\<exists>a b. a \<noteq> 0 \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b}))"
```
```   660 proof (cases "T = {}")
```
```   661   case True then show ?thesis
```
```   662     by simp
```
```   663 next
```
```   664   case False
```
```   665   show ?thesis
```
```   666   proof (cases "T = S")
```
```   667     case True then show ?thesis
```
```   668       by (simp add: face_of_refl_eq)
```
```   669   next
```
```   670     case False
```
```   671     with \<open>T \<noteq> {}\<close> show ?thesis
```
```   672       apply (auto simp: exposed_face_of_def)
```
```   673       apply (metis inner_zero_left)
```
```   674       done
```
```   675   qed
```
```   676 qed
```
```   677
```
```   678 lemma exposed_face_of_Int_supporting_hyperplane_le:
```
```   679    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
```
```   680 by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)
```
```   681
```
```   682 lemma exposed_face_of_Int_supporting_hyperplane_ge:
```
```   683    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
```
```   684 using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp
```
```   685
```
```   686 proposition exposed_face_of_Int:
```
```   687   assumes "T exposed_face_of S"
```
```   688       and "u exposed_face_of S"
```
```   689     shows "(T \<inter> u) exposed_face_of S"
```
```   690 proof -
```
```   691   obtain a b where T: "S \<inter> {x. a \<bullet> x = b} face_of S"
```
```   692                and S: "S \<subseteq> {x. a \<bullet> x \<le> b}"
```
```   693                and teq: "T = S \<inter> {x. a \<bullet> x = b}"
```
```   694     using assms by (auto simp: exposed_face_of_def)
```
```   695   obtain a' b' where u: "S \<inter> {x. a' \<bullet> x = b'} face_of S"
```
```   696                  and s': "S \<subseteq> {x. a' \<bullet> x \<le> b'}"
```
```   697                  and ueq: "u = S \<inter> {x. a' \<bullet> x = b'}"
```
```   698     using assms by (auto simp: exposed_face_of_def)
```
```   699   have tu: "T \<inter> u face_of S"
```
```   700     using T teq u ueq by (simp add: face_of_Int)
```
```   701   have ss: "S \<subseteq> {x. (a + a') \<bullet> x \<le> b + b'}"
```
```   702     using S s' by (force simp: inner_left_distrib)
```
```   703   show ?thesis
```
```   704     apply (simp add: exposed_face_of_def tu)
```
```   705     apply (rule_tac x="a+a'" in exI)
```
```   706     apply (rule_tac x="b+b'" in exI)
```
```   707     using S s'
```
```   708     apply (fastforce simp: ss inner_left_distrib teq ueq)
```
```   709     done
```
```   710 qed
```
```   711
```
```   712 proposition exposed_face_of_Inter:
```
```   713     fixes P :: "'a::euclidean_space set set"
```
```   714   assumes "P \<noteq> {}"
```
```   715       and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S"
```
```   716     shows "\<Inter>P exposed_face_of S"
```
```   717 proof -
```
```   718   obtain Q where "finite Q" and QsubP: "Q \<subseteq> P" "card Q \<le> DIM('a) + 2" and IntQ: "\<Inter>Q = \<Inter>P"
```
```   719     using Inter_faces_finite_altbound [of P S] assms [unfolded exposed_face_of]
```
```   720     by force
```
```   721   show ?thesis
```
```   722   proof (cases "Q = {}")
```
```   723     case True then show ?thesis
```
```   724       by (metis Inf_empty Inf_lower IntQ assms ex_in_conv subset_antisym top_greatest)
```
```   725   next
```
```   726     case False
```
```   727     have "Q \<subseteq> {T. T exposed_face_of S}"
```
```   728       using QsubP assms by blast
```
```   729     moreover have "Q \<subseteq> {T. T exposed_face_of S} \<Longrightarrow> \<Inter>Q exposed_face_of S"
```
```   730       using \<open>finite Q\<close> False
```
```   731       apply (induction Q rule: finite_induct)
```
```   732       using exposed_face_of_Int apply fastforce+
```
```   733       done
```
```   734     ultimately show ?thesis
```
```   735       by (simp add: IntQ)
```
```   736   qed
```
```   737 qed
```
```   738
```
```   739 proposition exposed_face_of_sums:
```
```   740   assumes "convex S" and "convex T"
```
```   741       and "F exposed_face_of {x + y | x y. x \<in> S \<and> y \<in> T}"
```
```   742           (is "F exposed_face_of ?ST")
```
```   743   obtains k l
```
```   744     where "k exposed_face_of S" "l exposed_face_of T"
```
```   745           "F = {x + y | x y. x \<in> k \<and> y \<in> l}"
```
```   746 proof (cases "F = {}")
```
```   747   case True then show ?thesis
```
```   748     using that by blast
```
```   749 next
```
```   750   case False
```
```   751   show ?thesis
```
```   752   proof (cases "F = ?ST")
```
```   753     case True then show ?thesis
```
```   754       using assms exposed_face_of_refl_eq that by blast
```
```   755   next
```
```   756     case False
```
```   757     obtain p where "p \<in> F" using \<open>F \<noteq> {}\<close> by blast
```
```   758     moreover
```
```   759     obtain u z where T: "?ST \<inter> {x. u \<bullet> x = z} face_of ?ST"
```
```   760                  and S: "?ST \<subseteq> {x. u \<bullet> x \<le> z}"
```
```   761                  and feq: "F = ?ST \<inter> {x. u \<bullet> x = z}"
```
```   762       using assms by (auto simp: exposed_face_of_def)
```
```   763     ultimately obtain a0 b0
```
```   764             where p: "p = a0 + b0" and "a0 \<in> S" "b0 \<in> T" and z: "u \<bullet> p = z"
```
```   765       by auto
```
```   766     have lez: "u \<bullet> (x + y) \<le> z" if "x \<in> S" "y \<in> T" for x y
```
```   767       using S that by auto
```
```   768     have sef: "S \<inter> {x. u \<bullet> x = u \<bullet> a0} exposed_face_of S"
```
```   769       apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex S\<close>])
```
```   770       apply (metis p z add_le_cancel_right inner_right_distrib lez [OF _ \<open>b0 \<in> T\<close>])
```
```   771       done
```
```   772     have tef: "T \<inter> {x. u \<bullet> x = u \<bullet> b0} exposed_face_of T"
```
```   773       apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex T\<close>])
```
```   774       apply (metis p z add.commute add_le_cancel_right inner_right_distrib lez [OF \<open>a0 \<in> S\<close>])
```
```   775       done
```
```   776     have "{x + y |x y. x \<in> S \<and> u \<bullet> x = u \<bullet> a0 \<and> y \<in> T \<and> u \<bullet> y = u \<bullet> b0} \<subseteq> F"
```
```   777       by (auto simp: feq) (metis inner_right_distrib p z)
```
```   778     moreover have "F \<subseteq> {x + y |x y. x \<in> S \<and> u \<bullet> x = u \<bullet> a0 \<and> y \<in> T \<and> u \<bullet> y = u \<bullet> b0}"
```
```   779       apply (auto simp: feq)
```
```   780       apply (rename_tac x y)
```
```   781       apply (rule_tac x=x in exI)
```
```   782       apply (rule_tac x=y in exI, simp)
```
```   783       using z p \<open>a0 \<in> S\<close> \<open>b0 \<in> T\<close>
```
```   784       apply clarify
```
```   785       apply (simp add: inner_right_distrib)
```
```   786       apply (metis add_le_cancel_right antisym lez [unfolded inner_right_distrib] add.commute)
```
```   787       done
```
```   788     ultimately have "F = {x + y |x y. x \<in> S \<inter> {x. u \<bullet> x = u \<bullet> a0} \<and> y \<in> T \<inter> {x. u \<bullet> x = u \<bullet> b0}}"
```
```   789       by blast
```
```   790     then show ?thesis
```
```   791       by (rule that [OF sef tef])
```
```   792   qed
```
```   793 qed
```
```   794
```
```   795 subsection\<open>Extreme points of a set: its singleton faces\<close>
```
```   796
```
```   797 definition extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
```
```   798                                (infixr "(extreme'_point'_of)" 50)
```
```   799   where "x extreme_point_of S \<longleftrightarrow>
```
```   800          x \<in> S \<and> (\<forall>a \<in> S. \<forall>b \<in> S. x \<notin> open_segment a b)"
```
```   801
```
```   802 lemma extreme_point_of_stillconvex:
```
```   803    "convex S \<Longrightarrow> (x extreme_point_of S \<longleftrightarrow> x \<in> S \<and> convex(S - {x}))"
```
```   804   by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)
```
```   805
```
```   806 lemma face_of_singleton:
```
```   807    "{x} face_of S \<longleftrightarrow> x extreme_point_of S"
```
```   808 by (fastforce simp add: extreme_point_of_def face_of_def)
```
```   809
```
```   810 lemma extreme_point_not_in_REL_INTERIOR:
```
```   811     fixes S :: "'a::real_normed_vector set"
```
```   812     shows "\<lbrakk>x extreme_point_of S; S \<noteq> {x}\<rbrakk> \<Longrightarrow> x \<notin> rel_interior S"
```
```   813 apply (simp add: face_of_singleton [symmetric])
```
```   814 apply (blast dest: face_of_disjoint_rel_interior)
```
```   815 done
```
```   816
```
```   817 lemma extreme_point_not_in_interior:
```
```   818     fixes S :: "'a::{real_normed_vector, perfect_space} set"
```
```   819     shows "x extreme_point_of S \<Longrightarrow> x \<notin> interior S"
```
```   820 apply (case_tac "S = {x}")
```
```   821 apply (simp add: empty_interior_finite)
```
```   822 by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)
```
```   823
```
```   824 lemma extreme_point_of_face:
```
```   825      "F face_of S \<Longrightarrow> v extreme_point_of F \<longleftrightarrow> v extreme_point_of S \<and> v \<in> F"
```
```   826   by (meson empty_subsetI face_of_face face_of_singleton insert_subset)
```
```   827
```
```   828 lemma extreme_point_of_convex_hull:
```
```   829    "x extreme_point_of (convex hull S) \<Longrightarrow> x \<in> S"
```
```   830 apply (simp add: extreme_point_of_stillconvex)
```
```   831 using hull_minimal [of S "(convex hull S) - {x}" convex]
```
```   832 using hull_subset [of S convex]
```
```   833 apply blast
```
```   834 done
```
```   835
```
```   836 lemma extreme_points_of_convex_hull:
```
```   837    "{x. x extreme_point_of (convex hull S)} \<subseteq> S"
```
```   838 using extreme_point_of_convex_hull by auto
```
```   839
```
```   840 lemma extreme_point_of_empty [simp]: "~ (x extreme_point_of {})"
```
```   841   by (simp add: extreme_point_of_def)
```
```   842
```
```   843 lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
```
```   844   using extreme_point_of_stillconvex by auto
```
```   845
```
```   846 lemma extreme_point_of_translation_eq:
```
```   847    "(a + x) extreme_point_of (image (\<lambda>x. a + x) S) \<longleftrightarrow> x extreme_point_of S"
```
```   848 by (auto simp: extreme_point_of_def)
```
```   849
```
```   850 lemma extreme_points_of_translation:
```
```   851    "{x. x extreme_point_of (image (\<lambda>x. a + x) S)} =
```
```   852     (\<lambda>x. a + x) ` {x. x extreme_point_of S}"
```
```   853 using extreme_point_of_translation_eq
```
```   854 by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
```
```   855
```
```   856 lemma extreme_point_of_Int:
```
```   857    "\<lbrakk>x extreme_point_of S; x extreme_point_of T\<rbrakk> \<Longrightarrow> x extreme_point_of (S \<inter> T)"
```
```   858 by (simp add: extreme_point_of_def)
```
```   859
```
```   860 lemma extreme_point_of_Int_supporting_hyperplane_le:
```
```   861    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
```
```   862 apply (simp add: face_of_singleton [symmetric])
```
```   863 by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)
```
```   864
```
```   865 lemma extreme_point_of_Int_supporting_hyperplane_ge:
```
```   866    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
```
```   867 apply (simp add: face_of_singleton [symmetric])
```
```   868 by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)
```
```   869
```
```   870 lemma exposed_point_of_Int_supporting_hyperplane_le:
```
```   871    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
```
```   872 apply (simp add: exposed_face_of_def face_of_singleton)
```
```   873 apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
```
```   874 done
```
```   875
```
```   876 lemma exposed_point_of_Int_supporting_hyperplane_ge:
```
```   877     "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
```
```   878 using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
```
```   879 by simp
```
```   880
```
```   881 lemma extreme_point_of_convex_hull_insert:
```
```   882    "\<lbrakk>finite S; a \<notin> convex hull S\<rbrakk> \<Longrightarrow> a extreme_point_of (convex hull (insert a S))"
```
```   883 apply (case_tac "a \<in> S")
```
```   884 apply (simp add: hull_inc)
```
```   885 using face_of_convex_hulls [of "insert a S" "{a}"]
```
```   886 apply (auto simp: face_of_singleton hull_same)
```
```   887 done
```
```   888
```
```   889 subsection\<open>Facets\<close>
```
```   890
```
```   891 definition facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
```
```   892                     (infixr "(facet'_of)" 50)
```
```   893   where "F facet_of S \<longleftrightarrow> F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1"
```
```   894
```
```   895 lemma facet_of_empty [simp]: "~ S facet_of {}"
```
```   896   by (simp add: facet_of_def)
```
```   897
```
```   898 lemma facet_of_irrefl [simp]: "~ S facet_of S "
```
```   899   by (simp add: facet_of_def)
```
```   900
```
```   901 lemma facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
```
```   902   by (simp add: facet_of_def)
```
```   903
```
```   904 lemma facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
```
```   905   by (simp add: face_of_imp_subset facet_of_def)
```
```   906
```
```   907 lemma hyperplane_facet_of_halfspace_le:
```
```   908    "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
```
```   909 unfolding facet_of_def hyperplane_eq_empty
```
```   910 by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
```
```   911            DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)
```
```   912
```
```   913 lemma hyperplane_facet_of_halfspace_ge:
```
```   914     "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<ge> b}"
```
```   915 unfolding facet_of_def hyperplane_eq_empty
```
```   916 by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
```
```   917            DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)
```
```   918
```
```   919 lemma facet_of_halfspace_le:
```
```   920     "F facet_of {x. a \<bullet> x \<le> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
```
```   921     (is "?lhs = ?rhs")
```
```   922 proof
```
```   923   assume c: ?lhs
```
```   924   with c facet_of_irrefl show ?rhs
```
```   925     by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
```
```   926 next
```
```   927   assume ?rhs then show ?lhs
```
```   928     by (simp add: hyperplane_facet_of_halfspace_le)
```
```   929 qed
```
```   930
```
```   931 lemma facet_of_halfspace_ge:
```
```   932     "F facet_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
```
```   933 using facet_of_halfspace_le [of F "-a" "-b"] by simp
```
```   934
```
```   935 subsection \<open>Edges: faces of affine dimension 1\<close>
```
```   936
```
```   937 definition edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
```
```   938   where "e edge_of S \<longleftrightarrow> e face_of S \<and> aff_dim e = 1"
```
```   939
```
```   940 lemma edge_of_imp_subset:
```
```   941    "S edge_of T \<Longrightarrow> S \<subseteq> T"
```
```   942 by (simp add: edge_of_def face_of_imp_subset)
```
```   943
```
```   944 subsection\<open>Existence of extreme points\<close>
```
```   945
```
```   946 lemma different_norm_3_collinear_points:
```
```   947   fixes a :: "'a::euclidean_space"
```
```   948   assumes "x \<in> open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
```
```   949   shows False
```
```   950 proof -
```
```   951   obtain u where "norm ((1 - u) *\<^sub>R a + u *\<^sub>R b) = norm b"
```
```   952              and "a \<noteq> b"
```
```   953              and u01: "0 < u" "u < 1"
```
```   954     using assms by (auto simp: open_segment_image_interval if_splits)
```
```   955   then have "(1 - u) *\<^sub>R a \<bullet> (1 - u) *\<^sub>R a + ((1 - u) * 2) *\<^sub>R a \<bullet> u *\<^sub>R b =
```
```   956              (1 - u * u) *\<^sub>R (a \<bullet> a)"
```
```   957     using assms by (simp add: norm_eq algebra_simps inner_commute)
```
```   958   then have "(1 - u) *\<^sub>R ((1 - u) *\<^sub>R a \<bullet> a + (2 * u) *\<^sub>R  a \<bullet> b) =
```
```   959              (1 - u) *\<^sub>R ((1 + u) *\<^sub>R (a \<bullet> a))"
```
```   960     by (simp add: algebra_simps)
```
```   961   then have "(1 - u) *\<^sub>R (a \<bullet> a) + (2 * u) *\<^sub>R (a \<bullet> b) = (1 + u) *\<^sub>R (a \<bullet> a)"
```
```   962     using u01 by auto
```
```   963   then have "a \<bullet> b = a \<bullet> a"
```
```   964     using u01 by (simp add: algebra_simps)
```
```   965   then have "a = b"
```
```   966     using \<open>norm(a) = norm(b)\<close> norm_eq vector_eq by fastforce
```
```   967   then show ?thesis
```
```   968     using \<open>a \<noteq> b\<close> by force
```
```   969 qed
```
```   970
```
```   971 proposition extreme_point_exists_convex:
```
```   972   fixes S :: "'a::euclidean_space set"
```
```   973   assumes "compact S" "convex S" "S \<noteq> {}"
```
```   974   obtains x where "x extreme_point_of S"
```
```   975 proof -
```
```   976   obtain x where "x \<in> S" and xsup: "\<And>y. y \<in> S \<Longrightarrow> norm y \<le> norm x"
```
```   977     using distance_attains_sup [of S 0] assms by auto
```
```   978   have False if "a \<in> S" "b \<in> S" and x: "x \<in> open_segment a b" for a b
```
```   979   proof -
```
```   980     have noax: "norm a \<le> norm x" and nobx: "norm b \<le> norm x" using xsup that by auto
```
```   981     have "a \<noteq> b"
```
```   982       using empty_iff open_segment_idem x by auto
```
```   983     have *: "(1 - u) * na + u * nb < norm x" if "na < norm x"  "nb \<le> norm x" "0 < u" "u < 1" for na nb u
```
```   984     proof -
```
```   985       have "(1 - u) * na + u * nb < (1 - u) * norm x + u * nb"
```
```   986         by (simp add: that)
```
```   987       also have "... \<le> (1 - u) * norm x + u * norm x"
```
```   988         by (simp add: that)
```
```   989       finally have "(1 - u) * na + u * nb < (1 - u) * norm x + u * norm x" .
```
```   990       then show ?thesis
```
```   991       using scaleR_collapse [symmetric, of "norm x" u] by auto
```
```   992     qed
```
```   993     have "norm x < norm x" if "norm a < norm x"
```
```   994       using x
```
```   995       apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
```
```   996       apply (rule norm_triangle_lt)
```
```   997       apply (simp add: norm_mult)
```
```   998       using * [of "norm a" "norm b"] nobx that
```
```   999         apply blast
```
```  1000       done
```
```  1001     moreover have "norm x < norm x" if "norm b < norm x"
```
```  1002       using x
```
```  1003       apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
```
```  1004       apply (rule norm_triangle_lt)
```
```  1005       apply (simp add: norm_mult)
```
```  1006       using * [of "norm b" "norm a" "1-u" for u] noax that
```
```  1007         apply (simp add: add.commute)
```
```  1008       done
```
```  1009     ultimately have "~ (norm a < norm x) \<and> ~ (norm b < norm x)"
```
```  1010       by auto
```
```  1011     then show ?thesis
```
```  1012       using different_norm_3_collinear_points noax nobx that(3) by fastforce
```
```  1013   qed
```
```  1014   then show ?thesis
```
```  1015     apply (rule_tac x=x in that)
```
```  1016     apply (force simp: extreme_point_of_def \<open>x \<in> S\<close>)
```
```  1017     done
```
```  1018 qed
```
```  1019
```
```  1020 subsection\<open>Krein-Milman, the weaker form\<close>
```
```  1021
```
```  1022 proposition Krein_Milman:
```
```  1023   fixes S :: "'a::euclidean_space set"
```
```  1024   assumes "compact S" "convex S"
```
```  1025     shows "S = closure(convex hull {x. x extreme_point_of S})"
```
```  1026 proof (cases "S = {}")
```
```  1027   case True then show ?thesis   by simp
```
```  1028 next
```
```  1029   case False
```
```  1030   have "closed S"
```
```  1031     by (simp add: \<open>compact S\<close> compact_imp_closed)
```
```  1032   have "closure (convex hull {x. x extreme_point_of S}) \<subseteq> S"
```
```  1033     apply (rule closure_minimal [OF hull_minimal \<open>closed S\<close>])
```
```  1034     using assms
```
```  1035     apply (auto simp: extreme_point_of_def)
```
```  1036     done
```
```  1037   moreover have "u \<in> closure (convex hull {x. x extreme_point_of S})"
```
```  1038                 if "u \<in> S" for u
```
```  1039   proof (rule ccontr)
```
```  1040     assume unot: "u \<notin> closure(convex hull {x. x extreme_point_of S})"
```
```  1041     then obtain a b where "a \<bullet> u < b"
```
```  1042           and ab: "\<And>x. x \<in> closure(convex hull {x. x extreme_point_of S}) \<Longrightarrow> b < a \<bullet> x"
```
```  1043       using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
```
```  1044       by blast
```
```  1045     have "continuous_on S (op \<bullet> a)"
```
```  1046       by (rule continuous_intros)+
```
```  1047     then obtain m where "m \<in> S" and m: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> m \<le> a \<bullet> y"
```
```  1048       using continuous_attains_inf [of S "\<lambda>x. a \<bullet> x"] \<open>compact S\<close> \<open>u \<in> S\<close>
```
```  1049       by auto
```
```  1050     define T where "T = S \<inter> {x. a \<bullet> x = a \<bullet> m}"
```
```  1051     have "m \<in> T"
```
```  1052       by (simp add: T_def \<open>m \<in> S\<close>)
```
```  1053     moreover have "compact T"
```
```  1054       by (simp add: T_def compact_Int_closed [OF \<open>compact S\<close> closed_hyperplane])
```
```  1055     moreover have "convex T"
```
```  1056       by (simp add: T_def convex_Int [OF \<open>convex S\<close> convex_hyperplane])
```
```  1057     ultimately obtain v where v: "v extreme_point_of T"
```
```  1058       using extreme_point_exists_convex [of T] by auto
```
```  1059     then have "{v} face_of T"
```
```  1060       by (simp add: face_of_singleton)
```
```  1061     also have "T face_of S"
```
```  1062       by (simp add: T_def m face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
```
```  1063     finally have "v extreme_point_of S"
```
```  1064       by (simp add: face_of_singleton)
```
```  1065     then have "b < a \<bullet> v"
```
```  1066       using closure_subset by (simp add: closure_hull hull_inc ab)
```
```  1067     then show False
```
```  1068       using \<open>a \<bullet> u < b\<close> \<open>{v} face_of T\<close> face_of_imp_subset m T_def that by fastforce
```
```  1069   qed
```
```  1070   ultimately show ?thesis
```
```  1071     by blast
```
```  1072 qed
```
```  1073
```
```  1074 text\<open>Now the sharper form.\<close>
```
```  1075
```
```  1076 lemma Krein_Milman_Minkowski_aux:
```
```  1077   fixes S :: "'a::euclidean_space set"
```
```  1078   assumes n: "dim S = n" and S: "compact S" "convex S" "0 \<in> S"
```
```  1079     shows "0 \<in> convex hull {x. x extreme_point_of S}"
```
```  1080 using n S
```
```  1081 proof (induction n arbitrary: S rule: less_induct)
```
```  1082   case (less n S) show ?case
```
```  1083   proof (cases "0 \<in> rel_interior S")
```
```  1084     case True with Krein_Milman show ?thesis
```
```  1085       by (metis subsetD convex_convex_hull convex_rel_interior_closure less.prems(2) less.prems(3) rel_interior_subset)
```
```  1086   next
```
```  1087     case False
```
```  1088     have "rel_interior S \<noteq> {}"
```
```  1089       by (simp add: rel_interior_convex_nonempty_aux less)
```
```  1090     then obtain c where c: "c \<in> rel_interior S" by blast
```
```  1091     obtain a where "a \<noteq> 0"
```
```  1092               and le_ay: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> 0 \<le> a \<bullet> y"
```
```  1093               and less_ay: "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> 0 < a \<bullet> y"
```
```  1094       by (blast intro: supporting_hyperplane_rel_boundary intro!: less False)
```
```  1095     have face: "S \<inter> {x. a \<bullet> x = 0} face_of S"
```
```  1096       apply (rule face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
```
```  1097       using le_ay by auto
```
```  1098     then have co: "compact (S \<inter> {x. a \<bullet> x = 0})" "convex (S \<inter> {x. a \<bullet> x = 0})"
```
```  1099       using less.prems by (blast intro: face_of_imp_compact face_of_imp_convex)+
```
```  1100     have "a \<bullet> y = 0" if "y \<in> span (S \<inter> {x. a \<bullet> x = 0})" for y
```
```  1101     proof -
```
```  1102       have "y \<in> span {x. a \<bullet> x = 0}"
```
```  1103         by (metis inf.cobounded2 span_mono subsetCE that)
```
```  1104       then show ?thesis
```
```  1105         by (blast intro: span_induct [OF _ subspace_hyperplane])
```
```  1106     qed
```
```  1107     then have "dim (S \<inter> {x. a \<bullet> x = 0}) < n"
```
```  1108       by (metis (no_types) less_ay c subsetD dim_eq_span inf.strict_order_iff
```
```  1109            inf_le1 \<open>dim S = n\<close> not_le rel_interior_subset span_0 span_clauses(1))
```
```  1110     then have "0 \<in> convex hull {x. x extreme_point_of (S \<inter> {x. a \<bullet> x = 0})}"
```
```  1111       by (rule less.IH) (auto simp: co less.prems)
```
```  1112     then show ?thesis
```
```  1113       by (metis (mono_tags, lifting) Collect_mono_iff \<open>S \<inter> {x. a \<bullet> x = 0} face_of S\<close> extreme_point_of_face hull_mono subset_iff)
```
```  1114   qed
```
```  1115 qed
```
```  1116
```
```  1117
```
```  1118 theorem Krein_Milman_Minkowski:
```
```  1119   fixes S :: "'a::euclidean_space set"
```
```  1120   assumes "compact S" "convex S"
```
```  1121     shows "S = convex hull {x. x extreme_point_of S}"
```
```  1122 proof
```
```  1123   show "S \<subseteq> convex hull {x. x extreme_point_of S}"
```
```  1124   proof
```
```  1125     fix a assume [simp]: "a \<in> S"
```
```  1126     have 1: "compact (op + (- a) ` S)"
```
```  1127       by (simp add: \<open>compact S\<close> compact_translation)
```
```  1128     have 2: "convex (op + (- a) ` S)"
```
```  1129       by (simp add: \<open>convex S\<close> convex_translation)
```
```  1130     show a_invex: "a \<in> convex hull {x. x extreme_point_of S}"
```
```  1131       using Krein_Milman_Minkowski_aux [OF refl 1 2]
```
```  1132             convex_hull_translation [of "-a"]
```
```  1133       by (auto simp: extreme_points_of_translation translation_assoc)
```
```  1134     qed
```
```  1135 next
```
```  1136   show "convex hull {x. x extreme_point_of S} \<subseteq> S"
```
```  1137   proof -
```
```  1138     have "{a. a extreme_point_of S} \<subseteq> S"
```
```  1139       using extreme_point_of_def by blast
```
```  1140     then show ?thesis
```
```  1141       by (simp add: \<open>convex S\<close> hull_minimal)
```
```  1142   qed
```
```  1143 qed
```
```  1144
```
```  1145
```
```  1146 subsection\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
```
```  1147
```
```  1148 lemma Krein_Milman_polytope:
```
```  1149   fixes S :: "'a::euclidean_space set"
```
```  1150   shows
```
```  1151    "finite S
```
```  1152        \<Longrightarrow> convex hull S =
```
```  1153            convex hull {x. x extreme_point_of (convex hull S)}"
```
```  1154 by (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
```
```  1155
```
```  1156 lemma extreme_points_of_convex_hull_eq:
```
```  1157   fixes S :: "'a::euclidean_space set"
```
```  1158   shows
```
```  1159    "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
```
```  1160         \<Longrightarrow> {x. x extreme_point_of (convex hull S)} = S"
```
```  1161 by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)
```
```  1162
```
```  1163
```
```  1164 lemma extreme_point_of_convex_hull_eq:
```
```  1165   fixes S :: "'a::euclidean_space set"
```
```  1166   shows
```
```  1167    "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
```
```  1168     \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
```
```  1169 using extreme_points_of_convex_hull_eq by auto
```
```  1170
```
```  1171 lemma extreme_point_of_convex_hull_convex_independent:
```
```  1172   fixes S :: "'a::euclidean_space set"
```
```  1173   assumes "compact S" and S: "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})"
```
```  1174   shows "(x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
```
```  1175 proof -
```
```  1176   have "convex hull T \<noteq> convex hull S" if "T \<subset> S" for T
```
```  1177   proof -
```
```  1178     obtain a where  "T \<subseteq> S" "a \<in> S" "a \<notin> T" using \<open>T \<subset> S\<close> by blast
```
```  1179     then show ?thesis
```
```  1180       by (metis (full_types) Diff_eq_empty_iff Diff_insert0 S hull_mono hull_subset insert_Diff_single subsetCE)
```
```  1181   qed
```
```  1182   then show ?thesis
```
```  1183     by (rule extreme_point_of_convex_hull_eq [OF \<open>compact S\<close>])
```
```  1184 qed
```
```  1185
```
```  1186 lemma extreme_point_of_convex_hull_affine_independent:
```
```  1187   fixes S :: "'a::euclidean_space set"
```
```  1188   shows
```
```  1189    "~ affine_dependent S
```
```  1190          \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
```
```  1191 by (metis aff_independent_finite affine_dependent_def affine_hull_convex_hull extreme_point_of_convex_hull_convex_independent finite_imp_compact hull_inc)
```
```  1192
```
```  1193 text\<open>Elementary proofs exist, not requiring Euclidean spaces and all this development\<close>
```
```  1194 lemma extreme_point_of_convex_hull_2:
```
```  1195   fixes x :: "'a::euclidean_space"
```
```  1196   shows "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x = a \<or> x = b"
```
```  1197 proof -
```
```  1198   have "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x \<in> {a,b}"
```
```  1199     by (intro extreme_point_of_convex_hull_affine_independent affine_independent_2)
```
```  1200   then show ?thesis
```
```  1201     by simp
```
```  1202 qed
```
```  1203
```
```  1204 lemma extreme_point_of_segment:
```
```  1205   fixes x :: "'a::euclidean_space"
```
```  1206   shows
```
```  1207    "x extreme_point_of closed_segment a b \<longleftrightarrow> x = a \<or> x = b"
```
```  1208 by (simp add: extreme_point_of_convex_hull_2 segment_convex_hull)
```
```  1209
```
```  1210 lemma face_of_convex_hull_subset:
```
```  1211   fixes S :: "'a::euclidean_space set"
```
```  1212   assumes "compact S" and T: "T face_of (convex hull S)"
```
```  1213   obtains s' where "s' \<subseteq> S" "T = convex hull s'"
```
```  1214 apply (rule_tac s' = "{x. x extreme_point_of T}" in that)
```
```  1215 using T extreme_point_of_convex_hull extreme_point_of_face apply blast
```
```  1216 by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)
```
```  1217
```
```  1218
```
```  1219 proposition face_of_convex_hull_affine_independent:
```
```  1220   fixes S :: "'a::euclidean_space set"
```
```  1221   assumes "~ affine_dependent S"
```
```  1222     shows "(T face_of (convex hull S) \<longleftrightarrow> (\<exists>c. c \<subseteq> S \<and> T = convex hull c))"
```
```  1223           (is "?lhs = ?rhs")
```
```  1224 proof
```
```  1225   assume ?lhs
```
```  1226   then show ?rhs
```
```  1227     by (meson \<open>T face_of convex hull S\<close> aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
```
```  1228 next
```
```  1229   assume ?rhs
```
```  1230   then obtain c where "c \<subseteq> S" and T: "T = convex hull c"
```
```  1231     by blast
```
```  1232   have "affine hull c \<inter> affine hull (S - c) = {}"
```
```  1233     apply (rule disjoint_affine_hull [OF assms \<open>c \<subseteq> S\<close>], auto)
```
```  1234     done
```
```  1235   then have "affine hull c \<inter> convex hull (S - c) = {}"
```
```  1236     using convex_hull_subset_affine_hull by fastforce
```
```  1237   then show ?lhs
```
```  1238     by (metis face_of_convex_hulls \<open>c \<subseteq> S\<close> aff_independent_finite assms T)
```
```  1239 qed
```
```  1240
```
```  1241 lemma facet_of_convex_hull_affine_independent:
```
```  1242   fixes S :: "'a::euclidean_space set"
```
```  1243   assumes "~ affine_dependent S"
```
```  1244     shows "T facet_of (convex hull S) \<longleftrightarrow>
```
```  1245            T \<noteq> {} \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u}))"
```
```  1246           (is "?lhs = ?rhs")
```
```  1247 proof
```
```  1248   assume ?lhs
```
```  1249   then have "T face_of (convex hull S)" "T \<noteq> {}"
```
```  1250         and afft: "aff_dim T = aff_dim (convex hull S) - 1"
```
```  1251     by (auto simp: facet_of_def)
```
```  1252   then obtain c where "c \<subseteq> S" and c: "T = convex hull c"
```
```  1253     by (auto simp: face_of_convex_hull_affine_independent [OF assms])
```
```  1254   then have affs: "aff_dim S = aff_dim c + 1"
```
```  1255     by (metis aff_dim_convex_hull afft eq_diff_eq)
```
```  1256   have "~ affine_dependent c"
```
```  1257     using \<open>c \<subseteq> S\<close> affine_dependent_subset assms by blast
```
```  1258   with affs have "card (S - c) = 1"
```
```  1259     apply (simp add: aff_dim_affine_independent [symmetric] aff_dim_convex_hull)
```
```  1260     by (metis aff_dim_affine_independent aff_independent_finite One_nat_def \<open>c \<subseteq> S\<close> add.commute
```
```  1261                 add_diff_cancel_right' assms card_Diff_subset card_mono of_nat_1 of_nat_diff of_nat_eq_iff)
```
```  1262   then obtain u where u: "u \<in> S - c"
```
```  1263     by (metis DiffI \<open>c \<subseteq> S\<close> aff_independent_finite assms cancel_comm_monoid_add_class.diff_cancel
```
```  1264                 card_Diff_subset subsetI subset_antisym zero_neq_one)
```
```  1265   then have u: "S = insert u c"
```
```  1266     by (metis Diff_subset \<open>c \<subseteq> S\<close> \<open>card (S - c) = 1\<close> card_1_singletonE double_diff insert_Diff insert_subset singletonD)
```
```  1267   have "T = convex hull (c - {u})"
```
```  1268     by (metis Diff_empty Diff_insert0 \<open>T facet_of convex hull S\<close> c facet_of_irrefl insert_absorb u)
```
```  1269   with \<open>T \<noteq> {}\<close> show ?rhs
```
```  1270     using c u by auto
```
```  1271 next
```
```  1272   assume ?rhs
```
```  1273   then obtain u where "T \<noteq> {}" "u \<in> S" and u: "T = convex hull (S - {u})"
```
```  1274     by (force simp: facet_of_def)
```
```  1275   then have "\<not> S \<subseteq> {u}"
```
```  1276     using \<open>T \<noteq> {}\<close> u by auto
```
```  1277   have [simp]: "aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1"
```
```  1278     using assms \<open>u \<in> S\<close>
```
```  1279     apply (simp add: aff_dim_convex_hull affine_dependent_def)
```
```  1280     apply (drule bspec, assumption)
```
```  1281     by (metis add_diff_cancel_right' aff_dim_insert insert_Diff [of u S])
```
```  1282   show ?lhs
```
```  1283     apply (subst u)
```
```  1284     apply (simp add: \<open>\<not> S \<subseteq> {u}\<close> facet_of_def face_of_convex_hull_affine_independent [OF assms], blast)
```
```  1285     done
```
```  1286 qed
```
```  1287
```
```  1288 lemma facet_of_convex_hull_affine_independent_alt:
```
```  1289   fixes S :: "'a::euclidean_space set"
```
```  1290   shows
```
```  1291    "~affine_dependent S
```
```  1292         \<Longrightarrow> (T facet_of (convex hull S) \<longleftrightarrow>
```
```  1293              2 \<le> card S \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
```
```  1294 apply (simp add: facet_of_convex_hull_affine_independent)
```
```  1295 apply (auto simp: Set.subset_singleton_iff)
```
```  1296 apply (metis Diff_cancel Int_empty_right Int_insert_right_if1  aff_independent_finite card_eq_0_iff card_insert_if card_mono card_subset_eq convex_hull_eq_empty eq_iff equals0D finite_insert finite_subset inf.absorb_iff2 insert_absorb insert_not_empty  not_less_eq_eq numeral_2_eq_2)
```
```  1297 done
```
```  1298
```
```  1299 lemma segment_face_of:
```
```  1300   assumes "(closed_segment a b) face_of S"
```
```  1301   shows "a extreme_point_of S" "b extreme_point_of S"
```
```  1302 proof -
```
```  1303   have as: "{a} face_of S"
```
```  1304     by (metis (no_types) assms convex_hull_singleton empty_iff extreme_point_of_convex_hull_insert face_of_face face_of_singleton finite.emptyI finite.insertI insert_absorb insert_iff segment_convex_hull)
```
```  1305   moreover have "{b} face_of S"
```
```  1306   proof -
```
```  1307     have "b \<in> convex hull {a} \<or> b extreme_point_of convex hull {b, a}"
```
```  1308       by (meson extreme_point_of_convex_hull_insert finite.emptyI finite.insertI)
```
```  1309     moreover have "closed_segment a b = convex hull {b, a}"
```
```  1310       using closed_segment_commute segment_convex_hull by blast
```
```  1311     ultimately show ?thesis
```
```  1312       by (metis as assms face_of_face convex_hull_singleton empty_iff face_of_singleton insertE)
```
```  1313     qed
```
```  1314   ultimately show "a extreme_point_of S" "b extreme_point_of S"
```
```  1315     using face_of_singleton by blast+
```
```  1316 qed
```
```  1317
```
```  1318
```
```  1319 lemma Krein_Milman_frontier:
```
```  1320   fixes S :: "'a::euclidean_space set"
```
```  1321   assumes "convex S" "compact S"
```
```  1322     shows "S = convex hull (frontier S)"
```
```  1323           (is "?lhs = ?rhs")
```
```  1324 proof
```
```  1325   have "?lhs \<subseteq> convex hull {x. x extreme_point_of S}"
```
```  1326     using Krein_Milman_Minkowski assms by blast
```
```  1327   also have "... \<subseteq> ?rhs"
```
```  1328     apply (rule hull_mono)
```
```  1329     apply (auto simp: frontier_def extreme_point_not_in_interior)
```
```  1330     using closure_subset apply (force simp: extreme_point_of_def)
```
```  1331     done
```
```  1332   finally show "?lhs \<subseteq> ?rhs" .
```
```  1333 next
```
```  1334   have "?rhs \<subseteq> convex hull S"
```
```  1335     by (metis Diff_subset \<open>compact S\<close> closure_closed compact_eq_bounded_closed frontier_def hull_mono)
```
```  1336   also have "... \<subseteq> ?lhs"
```
```  1337     by (simp add: \<open>convex S\<close> hull_same)
```
```  1338   finally show "?rhs \<subseteq> ?lhs" .
```
```  1339 qed
```
```  1340
```
```  1341 subsection\<open>Polytopes\<close>
```
```  1342
```
```  1343 definition polytope where
```
```  1344  "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
```
```  1345
```
```  1346 lemma polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
```
```  1347 apply (simp add: polytope_def, safe)
```
```  1348 apply (metis convex_hull_translation finite_imageI translation_galois)
```
```  1349 by (metis convex_hull_translation finite_imageI)
```
```  1350
```
```  1351 lemma polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
```
```  1352   unfolding polytope_def using convex_hull_linear_image by blast
```
```  1353
```
```  1354 lemma polytope_empty: "polytope {}"
```
```  1355   using convex_hull_empty polytope_def by blast
```
```  1356
```
```  1357 lemma polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
```
```  1358   using polytope_def by auto
```
```  1359
```
```  1360 lemma polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
```
```  1361   unfolding polytope_def
```
```  1362   by (metis finite_cartesian_product convex_hull_Times)
```
```  1363
```
```  1364 lemma face_of_polytope_polytope:
```
```  1365   fixes S :: "'a::euclidean_space set"
```
```  1366   shows "\<lbrakk>polytope S; F face_of S\<rbrakk> \<Longrightarrow> polytope F"
```
```  1367 unfolding polytope_def
```
```  1368 by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)
```
```  1369
```
```  1370 lemma finite_polytope_faces:
```
```  1371   fixes S :: "'a::euclidean_space set"
```
```  1372   assumes "polytope S"
```
```  1373   shows "finite {F. F face_of S}"
```
```  1374 proof -
```
```  1375   obtain v where "finite v" "S = convex hull v"
```
```  1376     using assms polytope_def by auto
```
```  1377   have "finite (op hull convex ` {T. T \<subseteq> v})"
```
```  1378     by (simp add: \<open>finite v\<close>)
```
```  1379   moreover have "{F. F face_of S} \<subseteq> (op hull convex ` {T. T \<subseteq> v})"
```
```  1380     by (metis (no_types, lifting) \<open>finite v\<close> \<open>S = convex hull v\<close> face_of_convex_hull_subset finite_imp_compact image_eqI mem_Collect_eq subsetI)
```
```  1381   ultimately show ?thesis
```
```  1382     by (blast intro: finite_subset)
```
```  1383 qed
```
```  1384
```
```  1385 lemma finite_polytope_facets:
```
```  1386   assumes "polytope S"
```
```  1387   shows "finite {T. T facet_of S}"
```
```  1388 by (simp add: assms facet_of_def finite_polytope_faces)
```
```  1389
```
```  1390 lemma polytope_scaling:
```
```  1391   assumes "polytope S"  shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
```
```  1392 by (simp add: assms polytope_linear_image)
```
```  1393
```
```  1394 lemma polytope_imp_compact:
```
```  1395   fixes S :: "'a::real_normed_vector set"
```
```  1396   shows "polytope S \<Longrightarrow> compact S"
```
```  1397 by (metis finite_imp_compact_convex_hull polytope_def)
```
```  1398
```
```  1399 lemma polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
```
```  1400   by (metis convex_convex_hull polytope_def)
```
```  1401
```
```  1402 lemma polytope_imp_closed:
```
```  1403   fixes S :: "'a::real_normed_vector set"
```
```  1404   shows "polytope S \<Longrightarrow> closed S"
```
```  1405 by (simp add: compact_imp_closed polytope_imp_compact)
```
```  1406
```
```  1407 lemma polytope_imp_bounded:
```
```  1408   fixes S :: "'a::real_normed_vector set"
```
```  1409   shows "polytope S \<Longrightarrow> bounded S"
```
```  1410 by (simp add: compact_imp_bounded polytope_imp_compact)
```
```  1411
```
```  1412 lemma polytope_interval: "polytope(cbox a b)"
```
```  1413   unfolding polytope_def by (meson closed_interval_as_convex_hull)
```
```  1414
```
```  1415 lemma polytope_sing: "polytope {a}"
```
```  1416   using polytope_def by force
```
```  1417
```
```  1418
```
```  1419 subsection\<open>Polyhedra\<close>
```
```  1420
```
```  1421 definition polyhedron where
```
```  1422  "polyhedron S \<equiv>
```
```  1423         \<exists>F. finite F \<and>
```
```  1424             S = \<Inter> F \<and>
```
```  1425             (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b})"
```
```  1426
```
```  1427 lemma polyhedron_Int [intro,simp]:
```
```  1428    "\<lbrakk>polyhedron S; polyhedron T\<rbrakk> \<Longrightarrow> polyhedron (S \<inter> T)"
```
```  1429   apply (simp add: polyhedron_def, clarify)
```
```  1430   apply (rename_tac F G)
```
```  1431   apply (rule_tac x="F \<union> G" in exI, auto)
```
```  1432   done
```
```  1433
```
```  1434 lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
```
```  1435   unfolding polyhedron_def
```
```  1436   by (rule_tac x="{}" in exI) auto
```
```  1437
```
```  1438 lemma polyhedron_Inter [intro,simp]:
```
```  1439    "\<lbrakk>finite F; \<And>S. S \<in> F \<Longrightarrow> polyhedron S\<rbrakk> \<Longrightarrow> polyhedron(\<Inter>F)"
```
```  1440 by (induction F rule: finite_induct) auto
```
```  1441
```
```  1442
```
```  1443 lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
```
```  1444 proof -
```
```  1445   have "\<exists>a. a \<noteq> 0 \<and>
```
```  1446              (\<exists>b. {x. (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b})"
```
```  1447     by (rule_tac x="(SOME i. i \<in> Basis)" in exI) (force simp: SOME_Basis nonzero_Basis)
```
```  1448   moreover have "\<exists>a b. a \<noteq> 0 \<and>
```
```  1449                        {x. - (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b}"
```
```  1450       apply (rule_tac x="-(SOME i. i \<in> Basis)" in exI)
```
```  1451       apply (rule_tac x="-1" in exI)
```
```  1452       apply (simp add: SOME_Basis nonzero_Basis)
```
```  1453       done
```
```  1454   ultimately show ?thesis
```
```  1455     unfolding polyhedron_def
```
```  1456     apply (rule_tac x="{{x. (SOME i. i \<in> Basis) \<bullet> x \<le> -1},
```
```  1457                         {x. -(SOME i. i \<in> Basis) \<bullet> x \<le> -1}}" in exI)
```
```  1458     apply force
```
```  1459     done
```
```  1460 qed
```
```  1461
```
```  1462 lemma polyhedron_halfspace_le:
```
```  1463   fixes a :: "'a :: euclidean_space"
```
```  1464   shows "polyhedron {x. a \<bullet> x \<le> b}"
```
```  1465 proof (cases "a = 0")
```
```  1466   case True then show ?thesis by auto
```
```  1467 next
```
```  1468   case False
```
```  1469   then show ?thesis
```
```  1470     unfolding polyhedron_def
```
```  1471     by (rule_tac x="{{x. a \<bullet> x \<le> b}}" in exI) auto
```
```  1472 qed
```
```  1473
```
```  1474 lemma polyhedron_halfspace_ge:
```
```  1475   fixes a :: "'a :: euclidean_space"
```
```  1476   shows "polyhedron {x. a \<bullet> x \<ge> b}"
```
```  1477 using polyhedron_halfspace_le [of "-a" "-b"] by simp
```
```  1478
```
```  1479 lemma polyhedron_hyperplane:
```
```  1480   fixes a :: "'a :: euclidean_space"
```
```  1481   shows "polyhedron {x. a \<bullet> x = b}"
```
```  1482 proof -
```
```  1483   have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
```
```  1484     by force
```
```  1485   then show ?thesis
```
```  1486     by (simp add: polyhedron_halfspace_ge polyhedron_halfspace_le)
```
```  1487 qed
```
```  1488
```
```  1489 lemma affine_imp_polyhedron:
```
```  1490   fixes S :: "'a :: euclidean_space set"
```
```  1491   shows "affine S \<Longrightarrow> polyhedron S"
```
```  1492 by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])
```
```  1493
```
```  1494 lemma polyhedron_imp_closed:
```
```  1495   fixes S :: "'a :: euclidean_space set"
```
```  1496   shows "polyhedron S \<Longrightarrow> closed S"
```
```  1497 apply (simp add: polyhedron_def)
```
```  1498 using closed_halfspace_le by fastforce
```
```  1499
```
```  1500 lemma polyhedron_imp_convex:
```
```  1501   fixes S :: "'a :: euclidean_space set"
```
```  1502   shows "polyhedron S \<Longrightarrow> convex S"
```
```  1503 apply (simp add: polyhedron_def)
```
```  1504 using convex_Inter convex_halfspace_le by fastforce
```
```  1505
```
```  1506 lemma polyhedron_affine_hull:
```
```  1507   fixes S :: "'a :: euclidean_space set"
```
```  1508   shows "polyhedron(affine hull S)"
```
```  1509 by (simp add: affine_imp_polyhedron)
```
```  1510
```
```  1511
```
```  1512 subsection\<open>Canonical polyhedron representation making facial structure explicit\<close>
```
```  1513
```
```  1514 lemma polyhedron_Int_affine:
```
```  1515   fixes S :: "'a :: euclidean_space set"
```
```  1516   shows "polyhedron S \<longleftrightarrow>
```
```  1517            (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
```
```  1518                 (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}))"
```
```  1519         (is "?lhs = ?rhs")
```
```  1520 proof
```
```  1521   assume ?lhs then show ?rhs
```
```  1522     apply (simp add: polyhedron_def)
```
```  1523     apply (erule ex_forward)
```
```  1524     using hull_subset apply force
```
```  1525     done
```
```  1526 next
```
```  1527   assume ?rhs then show ?lhs
```
```  1528     apply clarify
```
```  1529     apply (erule ssubst)
```
```  1530     apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le)
```
```  1531     done
```
```  1532 qed
```
```  1533
```
```  1534 proposition rel_interior_polyhedron_explicit:
```
```  1535   assumes "finite F"
```
```  1536       and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  1537       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  1538       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
```
```  1539     shows "rel_interior S = {x \<in> S. \<forall>h \<in> F. a h \<bullet> x < b h}"
```
```  1540 proof -
```
```  1541   have rels: "\<And>x. x \<in> rel_interior S \<Longrightarrow> x \<in> S"
```
```  1542     by (meson IntE mem_rel_interior)
```
```  1543   moreover have "a i \<bullet> x < b i" if x: "x \<in> rel_interior S" and "i \<in> F" for x i
```
```  1544   proof -
```
```  1545     have fif: "F - {i} \<subset> F"
```
```  1546       using \<open>i \<in> F\<close> Diff_insert_absorb Diff_subset set_insert psubsetI by blast
```
```  1547     then have "S \<subset> affine hull S \<inter> \<Inter>(F - {i})"
```
```  1548       by (rule psub)
```
```  1549     then obtain z where ssub: "S \<subseteq> \<Inter>(F - {i})" and zint: "z \<in> \<Inter>(F - {i})"
```
```  1550                     and "z \<notin> S" and zaff: "z \<in> affine hull S"
```
```  1551       by auto
```
```  1552     have "z \<noteq> x"
```
```  1553       using \<open>z \<notin> S\<close> rels x by blast
```
```  1554     have "z \<notin> affine hull S \<inter> \<Inter>F"
```
```  1555       using \<open>z \<notin> S\<close> seq by auto
```
```  1556     then have aiz: "a i \<bullet> z > b i"
```
```  1557       using faceq zint zaff by fastforce
```
```  1558     obtain e where "e > 0" "x \<in> S" and e: "ball x e \<inter> affine hull S \<subseteq> S"
```
```  1559       using x by (auto simp: mem_rel_interior_ball)
```
```  1560     then have ins: "\<And>y. \<lbrakk>norm (x - y) < e; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
```
```  1561       by (metis IntI subsetD dist_norm mem_ball)
```
```  1562     define \<xi> where "\<xi> = min (1/2) (e / 2 / norm(z - x))"
```
```  1563     have "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) = norm (\<xi> *\<^sub>R (x - z))"
```
```  1564       by (simp add: \<xi>_def algebra_simps norm_mult)
```
```  1565     also have "... = \<xi> * norm (x - z)"
```
```  1566       using \<open>e > 0\<close> by (simp add: \<xi>_def)
```
```  1567     also have "... < e"
```
```  1568       using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def divide_simps norm_minus_commute)
```
```  1569     finally have lte: "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) < e" .
```
```  1570     have \<xi>_aff: "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> affine hull S"
```
```  1571       by (metis \<open>x \<in> S\<close> add.commute affine_affine_hull diff_add_cancel hull_inc mem_affine zaff)
```
```  1572     have "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> S"
```
```  1573       apply (rule ins [OF _ \<xi>_aff])
```
```  1574       apply (simp add: algebra_simps lte)
```
```  1575       done
```
```  1576     then obtain l where l: "0 < l" "l < 1" and ls: "(l *\<^sub>R z + (1 - l) *\<^sub>R x) \<in> S"
```
```  1577       apply (rule_tac l = \<xi> in that)
```
```  1578       using \<open>e > 0\<close> \<open>z \<noteq> x\<close>  apply (auto simp: \<xi>_def)
```
```  1579       done
```
```  1580     then have i: "l *\<^sub>R z + (1 - l) *\<^sub>R x \<in> i"
```
```  1581       using seq \<open>i \<in> F\<close> by auto
```
```  1582     have "b i * l + (a i \<bullet> x) * (1 - l) < a i \<bullet> (l *\<^sub>R z + (1 - l) *\<^sub>R x)"
```
```  1583       using l by (simp add: algebra_simps aiz)
```
```  1584     also have "\<dots> \<le> b i" using i l
```
```  1585       using faceq mem_Collect_eq \<open>i \<in> F\<close> by blast
```
```  1586     finally have "(a i \<bullet> x) * (1 - l) < b i * (1 - l)"
```
```  1587       by (simp add: algebra_simps)
```
```  1588     with l show ?thesis
```
```  1589       by simp
```
```  1590   qed
```
```  1591   moreover have "x \<in> rel_interior S"
```
```  1592            if "x \<in> S" and less: "\<And>h. h \<in> F \<Longrightarrow> a h \<bullet> x < b h" for x
```
```  1593   proof -
```
```  1594     have 1: "\<And>h. h \<in> F \<Longrightarrow> x \<in> interior h"
```
```  1595       by (metis interior_halfspace_le mem_Collect_eq less faceq)
```
```  1596     have 2: "\<And>y. \<lbrakk>\<forall>h\<in>F. y \<in> interior h; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
```
```  1597       by (metis IntI Inter_iff contra_subsetD interior_subset seq)
```
```  1598     show ?thesis
```
```  1599       apply (simp add: rel_interior \<open>x \<in> S\<close>)
```
```  1600       apply (rule_tac x="\<Inter>h\<in>F. interior h" in exI)
```
```  1601       apply (auto simp: \<open>finite F\<close> open_INT 1 2)
```
```  1602       done
```
```  1603   qed
```
```  1604   ultimately show ?thesis by blast
```
```  1605 qed
```
```  1606
```
```  1607
```
```  1608 lemma polyhedron_Int_affine_parallel:
```
```  1609   fixes S :: "'a :: euclidean_space set"
```
```  1610   shows "polyhedron S \<longleftrightarrow>
```
```  1611          (\<exists>F. finite F \<and>
```
```  1612               S = (affine hull S) \<inter> (\<Inter>F) \<and>
```
```  1613               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
```
```  1614                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)))"
```
```  1615     (is "?lhs = ?rhs")
```
```  1616 proof
```
```  1617   assume ?lhs
```
```  1618   then obtain F where "finite F" and seq: "S = (affine hull S) \<inter> \<Inter>F"
```
```  1619                   and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
```
```  1620     by (fastforce simp add: polyhedron_Int_affine)
```
```  1621   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  1622     by metis
```
```  1623   show ?rhs
```
```  1624   proof -
```
```  1625     have "\<exists>a' b'. a' \<noteq> 0 \<and>
```
```  1626                   affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> h \<and>
```
```  1627                   (\<forall>w \<in> affine hull S. (w + a') \<in> affine hull S)"
```
```  1628         if "h \<in> F" "~(affine hull S \<subseteq> h)" for h
```
```  1629     proof -
```
```  1630       have "a h \<noteq> 0" and "h = {x. a h \<bullet> x \<le> b h}" "h \<inter> \<Inter>F = \<Inter>F"
```
```  1631         using \<open>h \<in> F\<close> ab by auto
```
```  1632       then have "(affine hull S) \<inter> {x. a h \<bullet> x \<le> b h} \<noteq> {}"
```
```  1633         by (metis (no_types) affine_hull_eq_empty inf.absorb_iff2 inf_assoc inf_bot_right inf_commute seq that(2))
```
```  1634       moreover have "~ (affine hull S \<subseteq> {x. a h \<bullet> x \<le> b h})"
```
```  1635         using \<open>h = {x. a h \<bullet> x \<le> b h}\<close> that(2) by blast
```
```  1636       ultimately show ?thesis
```
```  1637         using affine_parallel_slice [of "affine hull S"]
```
```  1638         by (metis \<open>h = {x. a h \<bullet> x \<le> b h}\<close> affine_affine_hull)
```
```  1639     qed
```
```  1640     then obtain a b
```
```  1641          where ab: "\<And>h. \<lbrakk>h \<in> F; ~ (affine hull S \<subseteq> h)\<rbrakk>
```
```  1642              \<Longrightarrow> a h \<noteq> 0 \<and>
```
```  1643                   affine hull S \<inter> {x. a h \<bullet> x \<le> b h} = affine hull S \<inter> h \<and>
```
```  1644                   (\<forall>w \<in> affine hull S. (w + a h) \<in> affine hull S)"
```
```  1645       by metis
```
```  1646     have seq2: "S = affine hull S \<inter> (\<Inter>h\<in>{h \<in> F. \<not> affine hull S \<subseteq> h}. {x. a h \<bullet> x \<le> b h})"
```
```  1647       by (subst seq) (auto simp: ab INT_extend_simps)
```
```  1648     show ?thesis
```
```  1649       apply (rule_tac x="(\<lambda>h. {x. a h \<bullet> x \<le> b h}) ` {h. h \<in> F \<and> ~(affine hull S \<subseteq> h)}" in exI)
```
```  1650       apply (intro conjI seq2)
```
```  1651         using \<open>finite F\<close> apply force
```
```  1652        using ab apply blast
```
```  1653        done
```
```  1654   qed
```
```  1655 next
```
```  1656   assume ?rhs then show ?lhs
```
```  1657     apply (simp add: polyhedron_Int_affine)
```
```  1658     by metis
```
```  1659 qed
```
```  1660
```
```  1661
```
```  1662 proposition polyhedron_Int_affine_parallel_minimal:
```
```  1663   fixes S :: "'a :: euclidean_space set"
```
```  1664   shows "polyhedron S \<longleftrightarrow>
```
```  1665          (\<exists>F. finite F \<and>
```
```  1666               S = (affine hull S) \<inter> (\<Inter>F) \<and>
```
```  1667               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
```
```  1668                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)) \<and>
```
```  1669               (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> (\<Inter>F')))"
```
```  1670     (is "?lhs = ?rhs")
```
```  1671 proof
```
```  1672   assume ?lhs
```
```  1673   then obtain f0
```
```  1674            where f0: "finite f0"
```
```  1675                  "S = (affine hull S) \<inter> (\<Inter>f0)"
```
```  1676                    (is "?P f0")
```
```  1677                  "\<forall>h \<in> f0. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
```
```  1678                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)"
```
```  1679                    (is "?Q f0")
```
```  1680     by (force simp: polyhedron_Int_affine_parallel)
```
```  1681   define n where "n = (LEAST n. \<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F)"
```
```  1682   have nf: "\<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F"
```
```  1683     apply (simp add: n_def)
```
```  1684     apply (rule LeastI [where k = "card f0"])
```
```  1685     using f0 apply auto
```
```  1686     done
```
```  1687   then obtain F where F: "card F = n" "finite F" and seq: "?P F" and aff: "?Q F"
```
```  1688     by blast
```
```  1689   then have "~ (finite g \<and> ?P g \<and> ?Q g)" if "card g < n" for g
```
```  1690     using that by (auto simp: n_def dest!: not_less_Least)
```
```  1691   then have *: "~ (?P g \<and> ?Q g)" if "g \<subset> F" for g
```
```  1692     using that \<open>finite F\<close> psubset_card_mono \<open>card F = n\<close>
```
```  1693     by (metis finite_Int inf.strict_order_iff)
```
```  1694   have 1: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subseteq> affine hull S \<inter> \<Inter>F'"
```
```  1695     by (subst seq) blast
```
```  1696   have 2: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<noteq> affine hull S \<inter> \<Inter>F'"
```
```  1697     apply (frule *)
```
```  1698     by (metis aff subsetCE subset_iff_psubset_eq)
```
```  1699   show ?rhs
```
```  1700     by (metis \<open>finite F\<close> seq aff psubsetI 1 2)
```
```  1701 next
```
```  1702   assume ?rhs then show ?lhs
```
```  1703     by (auto simp: polyhedron_Int_affine_parallel)
```
```  1704 qed
```
```  1705
```
```  1706
```
```  1707 lemma polyhedron_Int_affine_minimal:
```
```  1708   fixes S :: "'a :: euclidean_space set"
```
```  1709   shows "polyhedron S \<longleftrightarrow>
```
```  1710          (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
```
```  1711               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}) \<and>
```
```  1712               (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'))"
```
```  1713 apply (rule iffI)
```
```  1714  apply (force simp: polyhedron_Int_affine_parallel_minimal elim!: ex_forward)
```
```  1715 apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
```
```  1716 done
```
```  1717
```
```  1718 proposition facet_of_polyhedron_explicit:
```
```  1719   assumes "finite F"
```
```  1720       and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  1721       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  1722       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
```
```  1723     shows "c facet_of S \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
```
```  1724 proof (cases "S = {}")
```
```  1725   case True with psub show ?thesis by force
```
```  1726 next
```
```  1727   case False
```
```  1728   have "polyhedron S"
```
```  1729     apply (simp add: polyhedron_Int_affine)
```
```  1730     apply (rule_tac x=F in exI)
```
```  1731     using assms  apply force
```
```  1732     done
```
```  1733   then have "convex S"
```
```  1734     by (rule polyhedron_imp_convex)
```
```  1735   with False rel_interior_eq_empty have "rel_interior S \<noteq> {}" by blast
```
```  1736   then obtain x where "x \<in> rel_interior S" by auto
```
```  1737   then obtain T where "open T" "x \<in> T" "x \<in> S" "T \<inter> affine hull S \<subseteq> S"
```
```  1738     by (force simp: mem_rel_interior)
```
```  1739   then have xaff: "x \<in> affine hull S" and xint: "x \<in> \<Inter>F"
```
```  1740     using seq hull_inc by auto
```
```  1741   have "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
```
```  1742     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
```
```  1743   with \<open>x \<in> rel_interior S\<close>
```
```  1744   have [simp]: "\<And>h. h\<in>F \<Longrightarrow> a h \<bullet> x < b h" by blast
```
```  1745   have *: "(S \<inter> {x. a h \<bullet> x = b h}) facet_of S" if "h \<in> F" for h
```
```  1746   proof -
```
```  1747     have "S \<subset> affine hull S \<inter> \<Inter>(F - {h})"
```
```  1748       using psub that by (metis Diff_disjoint Diff_subset insert_disjoint(2) psubsetI)
```
```  1749     then obtain z where zaff: "z \<in> affine hull S" and zint: "z \<in> \<Inter>(F - {h})" and "z \<notin> S"
```
```  1750       by force
```
```  1751     then have "z \<noteq> x" "z \<notin> h" using seq \<open>x \<in> S\<close> by auto
```
```  1752     have "x \<in> h" using that xint by auto
```
```  1753     then have able: "a h \<bullet> x \<le> b h"
```
```  1754       using faceq that by blast
```
```  1755     also have "... < a h \<bullet> z" using \<open>z \<notin> h\<close> faceq [OF that] xint by auto
```
```  1756     finally have xltz: "a h \<bullet> x < a h \<bullet> z" .
```
```  1757     define l where "l = (b h - a h \<bullet> x) / (a h \<bullet> z - a h \<bullet> x)"
```
```  1758     define w where "w = (1 - l) *\<^sub>R x + l *\<^sub>R z"
```
```  1759     have "0 < l" "l < 1"
```
```  1760       using able xltz \<open>b h < a h \<bullet> z\<close> \<open>h \<in> F\<close>
```
```  1761       by (auto simp: l_def divide_simps)
```
```  1762     have awlt: "a i \<bullet> w < b i" if "i \<in> F" "i \<noteq> h" for i
```
```  1763     proof -
```
```  1764       have "(1 - l) * (a i \<bullet> x) < (1 - l) * b i"
```
```  1765         by (simp add: \<open>l < 1\<close> \<open>i \<in> F\<close>)
```
```  1766       moreover have "l * (a i \<bullet> z) \<le> l * b i"
```
```  1767         apply (rule mult_left_mono)
```
```  1768         apply (metis Diff_insert_absorb Inter_iff Set.set_insert \<open>h \<in> F\<close> faceq insertE mem_Collect_eq that zint)
```
```  1769         using \<open>0 < l\<close>
```
```  1770         apply simp
```
```  1771         done
```
```  1772       ultimately show ?thesis by (simp add: w_def algebra_simps)
```
```  1773     qed
```
```  1774     have weq: "a h \<bullet> w = b h"
```
```  1775       using xltz unfolding w_def l_def
```
```  1776       by (simp add: algebra_simps) (simp add: field_simps)
```
```  1777     have "w \<in> affine hull S"
```
```  1778       by (simp add: w_def mem_affine xaff zaff)
```
```  1779     moreover have "w \<in> \<Inter>F"
```
```  1780       using \<open>a h \<bullet> w = b h\<close> awlt faceq less_eq_real_def by blast
```
```  1781     ultimately have "w \<in> S"
```
```  1782       using seq by blast
```
```  1783     with weq have "S \<inter> {x. a h \<bullet> x = b h} \<noteq> {}" by blast
```
```  1784     moreover have "S \<inter> {x. a h \<bullet> x = b h} face_of S"
```
```  1785       apply (rule face_of_Int_supporting_hyperplane_le)
```
```  1786       apply (rule \<open>convex S\<close>)
```
```  1787       apply (subst (asm) seq)
```
```  1788       using faceq that apply fastforce
```
```  1789       done
```
```  1790     moreover have "affine hull (S \<inter> {x. a h \<bullet> x = b h}) =
```
```  1791                    (affine hull S) \<inter> {x. a h \<bullet> x = b h}"
```
```  1792     proof
```
```  1793       show "affine hull (S \<inter> {x. a h \<bullet> x = b h}) \<subseteq> affine hull S \<inter> {x. a h \<bullet> x = b h}"
```
```  1794         apply (intro Int_greatest hull_mono Int_lower1)
```
```  1795         apply (metis affine_hull_eq affine_hyperplane hull_mono inf_le2)
```
```  1796         done
```
```  1797     next
```
```  1798       show "affine hull S \<inter> {x. a h \<bullet> x = b h} \<subseteq> affine hull (S \<inter> {x. a h \<bullet> x = b h})"
```
```  1799       proof
```
```  1800         fix y
```
```  1801         assume yaff: "y \<in> affine hull S \<inter> {y. a h \<bullet> y = b h}"
```
```  1802         obtain T where "0 < T"
```
```  1803                  and T: "\<And>j. \<lbrakk>j \<in> F; j \<noteq> h\<rbrakk> \<Longrightarrow> T * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
```
```  1804         proof (cases "F - {h} = {}")
```
```  1805           case True then show ?thesis
```
```  1806             by (rule_tac T=1 in that) auto
```
```  1807         next
```
```  1808           case False
```
```  1809           then obtain h' where h': "h' \<in> F - {h}" by auto
```
```  1810           define inff where "inff =
```
```  1811             (INF j:F - {h}.
```
```  1812               if 0 < a j \<bullet> y - a j \<bullet> w
```
```  1813               then (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)
```
```  1814               else 1)"
```
```  1815           have "0 < inff"
```
```  1816             apply (simp add: inff_def)
```
```  1817             apply (rule finite_imp_less_Inf)
```
```  1818               using \<open>finite F\<close> apply blast
```
```  1819              using h' apply blast
```
```  1820             apply simp
```
```  1821             using awlt apply (force simp: divide_simps)
```
```  1822             done
```
```  1823           moreover have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
```
```  1824                         if "j \<in> F" "j \<noteq> h" for j
```
```  1825           proof (cases "a j \<bullet> w < a j \<bullet> y")
```
```  1826             case True
```
```  1827             then have "inff \<le> (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)"
```
```  1828               apply (simp add: inff_def)
```
```  1829               apply (rule cInf_le_finite)
```
```  1830               using \<open>finite F\<close> apply blast
```
```  1831               apply (simp add: that split: if_split_asm)
```
```  1832               done
```
```  1833             then show ?thesis
```
```  1834               using \<open>0 < inff\<close> awlt [OF that] mult_strict_left_mono
```
```  1835               by (fastforce simp add: algebra_simps divide_simps split: if_split_asm)
```
```  1836           next
```
```  1837             case False
```
```  1838             with \<open>0 < inff\<close> have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> 0"
```
```  1839               by (simp add: mult_le_0_iff)
```
```  1840             also have "... < b j - a j \<bullet> w"
```
```  1841               by (simp add: awlt that)
```
```  1842             finally show ?thesis by simp
```
```  1843           qed
```
```  1844           ultimately show ?thesis
```
```  1845             by (blast intro: that)
```
```  1846         qed
```
```  1847         define c where "c = (1 - T) *\<^sub>R w + T *\<^sub>R y"
```
```  1848         have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> j" if "j \<in> F" for j
```
```  1849         proof (cases "j = h")
```
```  1850           case True
```
```  1851           have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a h \<bullet> x \<le> b h}"
```
```  1852             using weq yaff by (auto simp: algebra_simps)
```
```  1853           with True faceq [OF that] show ?thesis by metis
```
```  1854         next
```
```  1855           case False
```
```  1856           with T that have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a j \<bullet> x \<le> b j}"
```
```  1857             by (simp add: algebra_simps)
```
```  1858           with faceq [OF that] show ?thesis by simp
```
```  1859         qed
```
```  1860         moreover have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> affine hull S"
```
```  1861           apply (rule affine_affine_hull [simplified affine_alt, rule_format])
```
```  1862           apply (simp add: \<open>w \<in> affine hull S\<close>)
```
```  1863           using yaff apply blast
```
```  1864           done
```
```  1865         ultimately have "c \<in> S"
```
```  1866           using seq by (force simp: c_def)
```
```  1867         moreover have "a h \<bullet> c = b h"
```
```  1868           using yaff by (force simp: c_def algebra_simps weq)
```
```  1869         ultimately have caff: "c \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
```
```  1870           by (simp add: hull_inc)
```
```  1871         have waff: "w \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
```
```  1872           using \<open>w \<in> S\<close> weq by (blast intro: hull_inc)
```
```  1873         have yeq: "y = (1 - inverse T) *\<^sub>R w + c /\<^sub>R T"
```
```  1874           using \<open>0 < T\<close> by (simp add: c_def algebra_simps)
```
```  1875         show "y \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
```
```  1876           by (metis yeq affine_affine_hull [simplified affine_alt, rule_format, OF waff caff])
```
```  1877       qed
```
```  1878     qed
```
```  1879     ultimately show ?thesis
```
```  1880       apply (simp add: facet_of_def)
```
```  1881       apply (subst aff_dim_affine_hull [symmetric])
```
```  1882       using  \<open>b h < a h \<bullet> z\<close> zaff
```
```  1883       apply (force simp: aff_dim_affine_Int_hyperplane)
```
```  1884       done
```
```  1885   qed
```
```  1886   show ?thesis
```
```  1887   proof
```
```  1888     show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h} \<Longrightarrow> c facet_of S"
```
```  1889       using * by blast
```
```  1890   next
```
```  1891     assume "c facet_of S"
```
```  1892     then have "c face_of S" "convex c" "c \<noteq> {}" and affc: "aff_dim c = aff_dim S - 1"
```
```  1893       by (auto simp: facet_of_def face_of_imp_convex)
```
```  1894     then obtain x where x: "x \<in> rel_interior c"
```
```  1895       by (force simp: rel_interior_eq_empty)
```
```  1896     then have "x \<in> c"
```
```  1897       by (meson subsetD rel_interior_subset)
```
```  1898     then have "x \<in> S"
```
```  1899       using \<open>c facet_of S\<close> facet_of_imp_subset by blast
```
```  1900     have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
```
```  1901       by (rule rel_interior_polyhedron_explicit [OF assms])
```
```  1902     have "c \<noteq> S"
```
```  1903       using \<open>c facet_of S\<close> facet_of_irrefl by blast
```
```  1904     then have "x \<notin> rel_interior S"
```
```  1905       by (metis IntI empty_iff \<open>x \<in> c\<close> \<open>c \<noteq> S\<close> \<open>c face_of S\<close> face_of_disjoint_rel_interior)
```
```  1906     with rels \<open>x \<in> S\<close> obtain i where "i \<in> F" and i: "a i \<bullet> x \<ge> b i"
```
```  1907       by force
```
```  1908     have "x \<in> {u. a i \<bullet> u \<le> b i}"
```
```  1909       by (metis IntD2 InterE \<open>i \<in> F\<close> \<open>x \<in> S\<close> faceq seq)
```
```  1910     then have "a i \<bullet> x \<le> b i" by simp
```
```  1911     then have "a i \<bullet> x = b i" using i by auto
```
```  1912     have "c \<subseteq> S \<inter> {x. a i \<bullet> x = b i}"
```
```  1913       apply (rule subset_of_face_of [of _ S])
```
```  1914         apply (simp add: "*" \<open>i \<in> F\<close> facet_of_imp_face_of)
```
```  1915        apply (simp add: \<open>c face_of S\<close> face_of_imp_subset)
```
```  1916       using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> S\<close> x by blast
```
```  1917     then have cface: "c face_of (S \<inter> {x. a i \<bullet> x = b i})"
```
```  1918       by (meson \<open>c face_of S\<close> face_of_subset inf_le1)
```
```  1919     have con: "convex (S \<inter> {x. a i \<bullet> x = b i})"
```
```  1920       by (simp add: \<open>convex S\<close> convex_Int convex_hyperplane)
```
```  1921     show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h}"
```
```  1922       apply (rule_tac x=i in exI)
```
```  1923       apply (simp add: \<open>i \<in> F\<close>)
```
```  1924       by (metis (no_types) * \<open>i \<in> F\<close> affc facet_of_def less_irrefl face_of_aff_dim_lt [OF con cface])
```
```  1925   qed
```
```  1926 qed
```
```  1927
```
```  1928
```
```  1929 lemma face_of_polyhedron_subset_explicit:
```
```  1930   fixes S :: "'a :: euclidean_space set"
```
```  1931   assumes "finite F"
```
```  1932       and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  1933       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  1934       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
```
```  1935       and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
```
```  1936    obtains h where "h \<in> F" "c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
```
```  1937 proof -
```
```  1938   have "c \<subseteq> S" using \<open>c face_of S\<close>
```
```  1939     by (simp add: face_of_imp_subset)
```
```  1940   have "polyhedron S"
```
```  1941     apply (simp add: polyhedron_Int_affine)
```
```  1942     by (metis \<open>finite F\<close> faceq seq)
```
```  1943   then have "convex S"
```
```  1944     by (simp add: polyhedron_imp_convex)
```
```  1945   then have *: "(S \<inter> {x. a h \<bullet> x = b h}) face_of S" if "h \<in> F" for h
```
```  1946     apply (rule face_of_Int_supporting_hyperplane_le)
```
```  1947     using faceq seq that by fastforce
```
```  1948   have "rel_interior c \<noteq> {}"
```
```  1949     using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
```
```  1950   then obtain x where "x \<in> rel_interior c" by auto
```
```  1951   have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
```
```  1952     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
```
```  1953   then have xnot: "x \<notin> rel_interior S"
```
```  1954     by (metis IntI \<open>x \<in> rel_interior c\<close> c \<open>c \<noteq> S\<close> contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
```
```  1955   then have "x \<in> S"
```
```  1956     using \<open>c \<subseteq> S\<close> \<open>x \<in> rel_interior c\<close> rel_interior_subset by auto
```
```  1957   then have xint: "x \<in> \<Inter>F"
```
```  1958     using seq by blast
```
```  1959   have "F \<noteq> {}" using assms
```
```  1960     by (metis affine_Int affine_Inter affine_affine_hull ex_in_conv face_of_affine_trivial)
```
```  1961   then obtain i where "i \<in> F" "~ (a i \<bullet> x < b i)"
```
```  1962     using \<open>x \<in> S\<close> rels xnot by auto
```
```  1963   with xint have "a i \<bullet> x = b i"
```
```  1964     by (metis eq_iff mem_Collect_eq not_le Inter_iff faceq)
```
```  1965   have face: "S \<inter> {x. a i \<bullet> x = b i} face_of S"
```
```  1966     by (simp add: "*" \<open>i \<in> F\<close>)
```
```  1967   show ?thesis
```
```  1968     apply (rule_tac h = i in that)
```
```  1969      apply (rule \<open>i \<in> F\<close>)
```
```  1970     apply (rule subset_of_face_of [OF face \<open>c \<subseteq> S\<close>])
```
```  1971     using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> rel_interior c\<close> \<open>x \<in> S\<close> apply blast
```
```  1972     done
```
```  1973 qed
```
```  1974
```
```  1975 text\<open>Initial part of proof duplicates that above\<close>
```
```  1976 proposition face_of_polyhedron_explicit:
```
```  1977   fixes S :: "'a :: euclidean_space set"
```
```  1978   assumes "finite F"
```
```  1979       and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  1980       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  1981       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
```
```  1982       and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
```
```  1983     shows "c = \<Inter>{S \<inter> {x. a h \<bullet> x = b h} | h. h \<in> F \<and> c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}"
```
```  1984 proof -
```
```  1985   let ?ab = "\<lambda>h. {x. a h \<bullet> x = b h}"
```
```  1986   have "c \<subseteq> S" using \<open>c face_of S\<close>
```
```  1987     by (simp add: face_of_imp_subset)
```
```  1988   have "polyhedron S"
```
```  1989     apply (simp add: polyhedron_Int_affine)
```
```  1990     by (metis \<open>finite F\<close> faceq seq)
```
```  1991   then have "convex S"
```
```  1992     by (simp add: polyhedron_imp_convex)
```
```  1993   then have *: "(S \<inter> ?ab h) face_of S" if "h \<in> F" for h
```
```  1994     apply (rule face_of_Int_supporting_hyperplane_le)
```
```  1995     using faceq seq that by fastforce
```
```  1996   have "rel_interior c \<noteq> {}"
```
```  1997     using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
```
```  1998   then obtain z where z: "z \<in> rel_interior c" by auto
```
```  1999   have rels: "rel_interior S = {z \<in> S. \<forall>h\<in>F. a h \<bullet> z < b h}"
```
```  2000     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
```
```  2001   then have xnot: "z \<notin> rel_interior S"
```
```  2002     by (metis IntI \<open>z \<in> rel_interior c\<close> c \<open>c \<noteq> S\<close> contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
```
```  2003   then have "z \<in> S"
```
```  2004     using \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close> rel_interior_subset by auto
```
```  2005   with seq have xint: "z \<in> \<Inter>F" by blast
```
```  2006   have "open (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
```
```  2007     by (auto simp: \<open>finite F\<close> open_halfspace_lt open_INT)
```
```  2008   then obtain e where "0 < e"
```
```  2009                  "ball z e \<subseteq> (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
```
```  2010     by (auto intro: openE [of _ z])
```
```  2011   then have e: "\<And>h. \<lbrakk>h \<in> F; a h \<bullet> z < b h\<rbrakk> \<Longrightarrow> ball z e \<subseteq> {w. a h \<bullet> w < b h}"
```
```  2012     by blast
```
```  2013   have "c \<subseteq> (S \<inter> ?ab h) \<longleftrightarrow> z \<in> S \<inter> ?ab h" if "h \<in> F" for h
```
```  2014   proof
```
```  2015     show "z \<in> S \<inter> ?ab h \<Longrightarrow> c \<subseteq> S \<inter> ?ab h"
```
```  2016       apply (rule subset_of_face_of [of _ S])
```
```  2017       using that \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close>
```
```  2018       using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub]
```
```  2019             unfolding facet_of_def
```
```  2020       apply auto
```
```  2021       done
```
```  2022   next
```
```  2023     show "c \<subseteq> S \<inter> ?ab h \<Longrightarrow> z \<in> S \<inter> ?ab h"
```
```  2024       using \<open>z \<in> rel_interior c\<close> rel_interior_subset by force
```
```  2025   qed
```
```  2026   then have **: "{S \<inter> ?ab h | h. h \<in> F \<and> c \<subseteq> S \<and> c \<subseteq> ?ab h} =
```
```  2027                  {S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<inter> ?ab h}"
```
```  2028     by blast
```
```  2029   have bsub: "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
```
```  2030              \<subseteq> affine hull S \<inter> \<Inter>F \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
```
```  2031             if "i \<in> F" and i: "a i \<bullet> z = b i" for i
```
```  2032   proof -
```
```  2033     have sub: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> j"
```
```  2034              if "j \<in> F" for j
```
```  2035     proof -
```
```  2036       have "a j \<bullet> z \<le> b j" using faceq that xint by auto
```
```  2037       then consider "a j \<bullet> z < b j" | "a j \<bullet> z = b j" by linarith
```
```  2038       then have "\<exists>G. G \<in> {?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<and> ball z e \<inter> G \<subseteq> j"
```
```  2039       proof cases
```
```  2040         assume "a j \<bullet> z < b j"
```
```  2041         then have "ball z e \<inter> {x. a i \<bullet> x = b i} \<subseteq> j"
```
```  2042           using e [OF \<open>j \<in> F\<close>] faceq that
```
```  2043           by (fastforce simp: ball_def)
```
```  2044         then show ?thesis
```
```  2045           by (rule_tac x="{x. a i \<bullet> x = b i}" in exI) (force simp: \<open>i \<in> F\<close> i)
```
```  2046       next
```
```  2047         assume eq: "a j \<bullet> z = b j"
```
```  2048         with faceq that show ?thesis
```
```  2049           by (rule_tac x="{x. a j \<bullet> x = b j}" in exI) (fastforce simp add: \<open>j \<in> F\<close>)
```
```  2050       qed
```
```  2051       then show ?thesis  by blast
```
```  2052     qed
```
```  2053     have 1: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> affine hull S"
```
```  2054       apply (rule hull_mono)
```
```  2055       using that \<open>z \<in> S\<close> by auto
```
```  2056     have 2: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
```
```  2057           \<subseteq> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
```
```  2058       by (rule hull_minimal) (auto intro: affine_hyperplane)
```
```  2059     have 3: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> \<Inter>F"
```
```  2060       by (iprover intro: sub Inter_greatest)
```
```  2061     have *: "\<lbrakk>A \<subseteq> (B :: 'a set); A \<subseteq> C; E \<inter> C \<subseteq> D\<rbrakk> \<Longrightarrow> E \<inter> A \<subseteq> (B \<inter> D) \<inter> C"
```
```  2062              for A B C D E  by blast
```
```  2063     show ?thesis by (intro * 1 2 3)
```
```  2064   qed
```
```  2065   have "\<exists>h. h \<in> F \<and> c \<subseteq> ?ab h"
```
```  2066     apply (rule face_of_polyhedron_subset_explicit [OF \<open>finite F\<close> seq faceq psub])
```
```  2067     using assms by auto
```
```  2068   then have fac: "\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> c \<subseteq> S \<inter> ?ab h} face_of S"
```
```  2069     using * by (force simp: \<open>c \<subseteq> S\<close> intro: face_of_Inter)
```
```  2070   have red:
```
```  2071      "(\<And>a. P a \<Longrightarrow> T \<subseteq> S \<inter> \<Inter>{F x |x. P x}) \<Longrightarrow> T \<subseteq> \<Inter>{S \<inter> F x |x. P x}"
```
```  2072      for P T F   by blast
```
```  2073   have "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
```
```  2074         \<subseteq> \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
```
```  2075     apply (rule red)
```
```  2076     apply (metis seq bsub)
```
```  2077     done
```
```  2078   with \<open>0 < e\<close> have zinrel: "z \<in> rel_interior
```
```  2079                     (\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<and> a h \<bullet> z = b h})"
```
```  2080     by (auto simp: mem_rel_interior_ball \<open>z \<in> S\<close>)
```
```  2081   show ?thesis
```
```  2082     apply (rule face_of_eq [OF c fac])
```
```  2083     using z zinrel apply (force simp: **)
```
```  2084     done
```
```  2085 qed
```
```  2086
```
```  2087
```
```  2088 subsection\<open>More general corollaries from the explicit representation\<close>
```
```  2089
```
```  2090 corollary facet_of_polyhedron:
```
```  2091   assumes "polyhedron S" and "c facet_of S"
```
```  2092   obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x \<le> b}" "c = S \<inter> {x. a \<bullet> x = b}"
```
```  2093 proof -
```
```  2094   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  2095              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
```
```  2096              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
```
```  2097     using assms by (simp add: polyhedron_Int_affine_minimal) meson
```
```  2098   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  2099     by metis
```
```  2100   obtain i where "i \<in> F" and c: "c = S \<inter> {x. a i \<bullet> x = b i}"
```
```  2101     using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] assms
```
```  2102     by force
```
```  2103   moreover have ssub: "S \<subseteq> {x. a i \<bullet> x \<le> b i}"
```
```  2104      apply (subst seq)
```
```  2105      using \<open>i \<in> F\<close> ab by auto
```
```  2106   ultimately show ?thesis
```
```  2107     by (rule_tac a = "a i" and b = "b i" in that) (simp_all add: ab)
```
```  2108 qed
```
```  2109
```
```  2110 corollary face_of_polyhedron:
```
```  2111   assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
```
```  2112     shows "c = \<Inter>{F. F facet_of S \<and> c \<subseteq> F}"
```
```  2113 proof -
```
```  2114   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  2115              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
```
```  2116              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
```
```  2117     using assms by (simp add: polyhedron_Int_affine_minimal) meson
```
```  2118   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  2119     by metis
```
```  2120   show ?thesis
```
```  2121     apply (subst face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
```
```  2122     apply (auto simp: assms facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] cong: Collect_cong)
```
```  2123     done
```
```  2124 qed
```
```  2125
```
```  2126 lemma face_of_polyhedron_subset_facet:
```
```  2127   assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
```
```  2128   obtains F where "F facet_of S" "c \<subseteq> F"
```
```  2129 using face_of_polyhedron assms
```
```  2130 by (metis (no_types, lifting) Inf_greatest antisym_conv face_of_imp_subset mem_Collect_eq)
```
```  2131
```
```  2132
```
```  2133 lemma exposed_face_of_polyhedron:
```
```  2134   assumes "polyhedron S"
```
```  2135     shows "F exposed_face_of S \<longleftrightarrow> F face_of S"
```
```  2136 proof
```
```  2137   show "F exposed_face_of S \<Longrightarrow> F face_of S"
```
```  2138     by (simp add: exposed_face_of_def)
```
```  2139 next
```
```  2140   assume "F face_of S"
```
```  2141   show "F exposed_face_of S"
```
```  2142   proof (cases "F = {} \<or> F = S")
```
```  2143     case True then show ?thesis
```
```  2144       using \<open>F face_of S\<close> exposed_face_of by blast
```
```  2145   next
```
```  2146     case False
```
```  2147     then have "{g. g facet_of S \<and> F \<subseteq> g} \<noteq> {}"
```
```  2148       by (metis Collect_empty_eq_bot \<open>F face_of S\<close> assms empty_def face_of_polyhedron_subset_facet)
```
```  2149     moreover have "\<And>T. \<lbrakk>T facet_of S; F \<subseteq> T\<rbrakk> \<Longrightarrow> T exposed_face_of S"
```
```  2150       by (metis assms exposed_face_of facet_of_imp_face_of facet_of_polyhedron)
```
```  2151     ultimately have "\<Inter>{fa.
```
```  2152        fa facet_of S \<and> F \<subseteq> fa} exposed_face_of S"
```
```  2153       by (metis (no_types, lifting) mem_Collect_eq exposed_face_of_Inter)
```
```  2154     then show ?thesis
```
```  2155       using False
```
```  2156       apply (subst face_of_polyhedron [OF assms \<open>F face_of S\<close>], auto)
```
```  2157       done
```
```  2158   qed
```
```  2159 qed
```
```  2160
```
```  2161 lemma face_of_polyhedron_polyhedron:
```
```  2162   fixes S :: "'a :: euclidean_space set"
```
```  2163   assumes "polyhedron S" "c face_of S"
```
```  2164     shows "polyhedron c"
```
```  2165 by (metis assms face_of_imp_eq_affine_Int polyhedron_Int polyhedron_affine_hull polyhedron_imp_closed polyhedron_imp_convex)
```
```  2166
```
```  2167 lemma finite_polyhedron_faces:
```
```  2168   fixes S :: "'a :: euclidean_space set"
```
```  2169   assumes "polyhedron S"
```
```  2170     shows "finite {F. F face_of S}"
```
```  2171 proof -
```
```  2172   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  2173              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
```
```  2174              and min:   "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
```
```  2175     using assms by (simp add: polyhedron_Int_affine_minimal) meson
```
```  2176   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  2177     by metis
```
```  2178   have "finite {\<Inter>{S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F'}| F'. F' \<in> Pow F}"
```
```  2179     by (simp add: \<open>finite F\<close>)
```
```  2180   moreover have "{F. F face_of S} - {{}, S} \<subseteq> {\<Inter>{S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F'}| F'. F' \<in> Pow F}"
```
```  2181     apply clarify
```
```  2182     apply (rename_tac c)
```
```  2183     apply (drule face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min, simplified], simp_all)
```
```  2184     apply (erule ssubst)
```
```  2185     apply (rule_tac x="{h \<in> F. c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}" in exI, auto)
```
```  2186     done
```
```  2187   ultimately show ?thesis
```
```  2188     by (meson finite.emptyI finite.insertI finite_Diff2 finite_subset)
```
```  2189 qed
```
```  2190
```
```  2191 lemma finite_polyhedron_exposed_faces:
```
```  2192    "polyhedron S \<Longrightarrow> finite {F. F exposed_face_of S}"
```
```  2193 using exposed_face_of_polyhedron finite_polyhedron_faces by fastforce
```
```  2194
```
```  2195 lemma finite_polyhedron_extreme_points:
```
```  2196   fixes S :: "'a :: euclidean_space set"
```
```  2197   shows "polyhedron S \<Longrightarrow> finite {v. v extreme_point_of S}"
```
```  2198 apply (simp add: face_of_singleton [symmetric])
```
```  2199 apply (rule finite_subset [OF _ finite_vimageI [OF finite_polyhedron_faces]], auto)
```
```  2200 done
```
```  2201
```
```  2202 lemma finite_polyhedron_facets:
```
```  2203   fixes S :: "'a :: euclidean_space set"
```
```  2204   shows "polyhedron S \<Longrightarrow> finite {F. F facet_of S}"
```
```  2205 unfolding facet_of_def
```
```  2206 by (blast intro: finite_subset [OF _ finite_polyhedron_faces])
```
```  2207
```
```  2208
```
```  2209 proposition rel_interior_of_polyhedron:
```
```  2210   fixes S :: "'a :: euclidean_space set"
```
```  2211   assumes "polyhedron S"
```
```  2212     shows "rel_interior S = S - \<Union>{F. F facet_of S}"
```
```  2213 proof -
```
```  2214   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
```
```  2215              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
```
```  2216              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
```
```  2217     using assms by (simp add: polyhedron_Int_affine_minimal) meson
```
```  2218   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
```
```  2219     by metis
```
```  2220   have facet: "(c facet_of S) \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})" for c
```
```  2221     by (rule facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
```
```  2222   have rel: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
```
```  2223     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
```
```  2224   have "a h \<bullet> x < b h" if "x \<in> S" "h \<in> F" and xnot: "x \<notin> \<Union>{F. F facet_of S}" for x h
```
```  2225   proof -
```
```  2226     have "x \<in> \<Inter>F" using seq that by force
```
```  2227     with \<open>h \<in> F\<close> ab have "a h \<bullet> x \<le> b h" by auto
```
```  2228     then consider "a h \<bullet> x < b h" | "a h \<bullet> x = b h" by linarith
```
```  2229     then show ?thesis
```
```  2230     proof cases
```
```  2231       case 1 then show ?thesis .
```
```  2232     next
```
```  2233       case 2
```
```  2234       have "Collect (op \<in> x) \<notin> Collect (op \<in> (\<Union>{A. A facet_of S}))"
```
```  2235         using xnot by fastforce
```
```  2236       then have "F \<notin> Collect (op \<in> h)"
```
```  2237         using 2 \<open>x \<in> S\<close> facet by blast
```
```  2238       with \<open>h \<in> F\<close> have "\<Inter>F \<subseteq> S \<inter> {x. a h \<bullet> x = b h}" by blast
```
```  2239       with 2 that \<open>x \<in> \<Inter>F\<close> show ?thesis
```
```  2240         apply simp
```
```  2241         apply (drule_tac x="\<Inter>F" in spec)
```
```  2242         apply (simp add: facet)
```
```  2243         apply (drule_tac x=h in spec)
```
```  2244         using seq by auto
```
```  2245       qed
```
```  2246   qed
```
```  2247   moreover have "\<exists>h\<in>F. a h \<bullet> x \<ge> b h" if "x \<in> \<Union>{F. F facet_of S}" for x
```
```  2248     using that by (force simp: facet)
```
```  2249   ultimately show ?thesis
```
```  2250     by (force simp: rel)
```
```  2251 qed
```
```  2252
```
```  2253 lemma rel_boundary_of_polyhedron:
```
```  2254   fixes S :: "'a :: euclidean_space set"
```
```  2255   assumes "polyhedron S"
```
```  2256     shows "S - rel_interior S = \<Union> {F. F facet_of S}"
```
```  2257 using facet_of_imp_subset by (fastforce simp add: rel_interior_of_polyhedron assms)
```
```  2258
```
```  2259 lemma rel_frontier_of_polyhedron:
```
```  2260   fixes S :: "'a :: euclidean_space set"
```
```  2261   assumes "polyhedron S"
```
```  2262     shows "rel_frontier S = \<Union> {F. F facet_of S}"
```
```  2263 by (simp add: assms rel_frontier_def polyhedron_imp_closed rel_boundary_of_polyhedron)
```
```  2264
```
```  2265 lemma rel_frontier_of_polyhedron_alt:
```
```  2266   fixes S :: "'a :: euclidean_space set"
```
```  2267   assumes "polyhedron S"
```
```  2268     shows "rel_frontier S = \<Union> {F. F face_of S \<and> (F \<noteq> S)}"
```
```  2269 apply (rule subset_antisym)
```
```  2270   apply (force simp: rel_frontier_of_polyhedron facet_of_def assms)
```
```  2271 using face_of_subset_rel_frontier by fastforce
```
```  2272
```
```  2273
```
```  2274 text\<open>A characterization of polyhedra as having finitely many faces\<close>
```
```  2275
```
```  2276 proposition polyhedron_eq_finite_exposed_faces:
```
```  2277   fixes S :: "'a :: euclidean_space set"
```
```  2278   shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F exposed_face_of S}"
```
```  2279          (is "?lhs = ?rhs")
```
```  2280 proof
```
```  2281   assume ?lhs
```
```  2282   then show ?rhs
```
```  2283     by (auto simp: polyhedron_imp_closed polyhedron_imp_convex finite_polyhedron_exposed_faces)
```
```  2284 next
```
```  2285   assume ?rhs
```
```  2286   then have "closed S" "convex S" and fin: "finite {F. F exposed_face_of S}" by auto
```
```  2287   show ?lhs
```
```  2288   proof (cases "S = {}")
```
```  2289     case True then show ?thesis by auto
```
```  2290   next
```
```  2291     case False
```
```  2292     define F where "F = {h. h exposed_face_of S \<and> h \<noteq> {} \<and> h \<noteq> S}"
```
```  2293     have "finite F" by (simp add: fin F_def)
```
```  2294     have hface: "h face_of S"
```
```  2295       and "\<exists>a b. a \<noteq> 0 \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> h = S \<inter> {x. a \<bullet> x = b}"
```
```  2296       if "h \<in> F" for h
```
```  2297       using exposed_face_of F_def that by simp_all auto
```
```  2298     then obtain a b where ab:
```
```  2299       "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> S \<subseteq> {x. a h \<bullet> x \<le> b h} \<and> h = S \<inter> {x. a h \<bullet> x = b h}"
```
```  2300       by metis
```
```  2301     have *: "False"
```
```  2302       if paff: "p \<in> affine hull S" and "p \<notin> S"
```
```  2303       and pint: "p \<in> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F}" for p
```
```  2304     proof -
```
```  2305       have "rel_interior S \<noteq> {}"
```
```  2306         by (simp add: \<open>S \<noteq> {}\<close> \<open>convex S\<close> rel_interior_eq_empty)
```
```  2307       then obtain c where c: "c \<in> rel_interior S" by auto
```
```  2308       with rel_interior_subset have "c \<in> S"  by blast
```
```  2309       have ccp: "closed_segment c p \<subseteq> affine hull S"
```
```  2310         by (meson affine_affine_hull affine_imp_convex c closed_segment_subset hull_subset paff rel_interior_subset subsetCE)
```
```  2311       obtain x where xcl: "x \<in> closed_segment c p" and "x \<in> S" and xnot: "x \<notin> rel_interior S"
```
```  2312         using connected_openin [of "closed_segment c p"]
```
```  2313         apply simp
```
```  2314         apply (drule_tac x="closed_segment c p \<inter> rel_interior S" in spec)
```
```  2315         apply (erule impE)
```
```  2316          apply (force simp: openin_rel_interior openin_Int intro: openin_subtopology_Int_subset [OF _ ccp])
```
```  2317         apply (drule_tac x="closed_segment c p \<inter> (- S)" in spec)
```
```  2318         using rel_interior_subset \<open>closed S\<close> c \<open>p \<notin> S\<close> apply blast
```
```  2319         done
```
```  2320       then obtain \<mu> where "0 \<le> \<mu>" "\<mu> \<le> 1" and xeq: "x = (1 - \<mu>) *\<^sub>R c + \<mu> *\<^sub>R p"
```
```  2321         by (auto simp: in_segment)
```
```  2322       show False
```
```  2323       proof (cases "\<mu>=0 \<or> \<mu>=1")
```
```  2324         case True with xeq c xnot \<open>x \<in> S\<close> \<open>p \<notin> S\<close>
```
```  2325         show False by auto
```
```  2326       next
```
```  2327         case False
```
```  2328         then have xos: "x \<in> open_segment c p"
```
```  2329           using \<open>x \<in> S\<close> c open_segment_def that(2) xcl xnot by auto
```
```  2330         have xclo: "x \<in> closure S"
```
```  2331           using \<open>x \<in> S\<close> closure_subset by blast
```
```  2332         obtain d where "d \<noteq> 0"
```
```  2333               and dle: "\<And>y. y \<in> closure S \<Longrightarrow> d \<bullet> x \<le> d \<bullet> y"
```
```  2334               and dless: "\<And>y. y \<in> rel_interior S \<Longrightarrow> d \<bullet> x < d \<bullet> y"
```
```  2335           by (metis supporting_hyperplane_relative_frontier [OF \<open>convex S\<close> xclo xnot])
```
```  2336         have sex: "S \<inter> {y. d \<bullet> y = d \<bullet> x} exposed_face_of S"
```
```  2337           by (simp add: \<open>closed S\<close> dle exposed_face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
```
```  2338         have sne: "S \<inter> {y. d \<bullet> y = d \<bullet> x} \<noteq> {}"
```
```  2339           using \<open>x \<in> S\<close> by blast
```
```  2340         have sns: "S \<inter> {y. d \<bullet> y = d \<bullet> x} \<noteq> S"
```
```  2341           by (metis (mono_tags) Int_Collect c subsetD dless not_le order_refl rel_interior_subset)
```
```  2342         obtain h where "h \<in> F" "x \<in> h"
```
```  2343           apply (rule_tac h="S \<inter> {y. d \<bullet> y = d \<bullet> x}" in that)
```
```  2344           apply (simp_all add: F_def sex sne sns \<open>x \<in> S\<close>)
```
```  2345           done
```
```  2346         have abface: "{y. a h \<bullet> y = b h} face_of {y. a h \<bullet> y \<le> b h}"
```
```  2347           using hyperplane_face_of_halfspace_le by blast
```
```  2348         then have "c \<in> h"
```
```  2349           using face_ofD [OF abface xos] \<open>c \<in> S\<close> \<open>h \<in> F\<close> ab pint \<open>x \<in> h\<close> by blast
```
```  2350         with c have "h \<inter> rel_interior S \<noteq> {}" by blast
```
```  2351         then show False
```
```  2352           using \<open>h \<in> F\<close> F_def face_of_disjoint_rel_interior hface by auto
```
```  2353       qed
```
```  2354     qed
```
```  2355     have "S \<subseteq> affine hull S \<inter> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F}"
```
```  2356       using ab by (auto simp: hull_subset)
```
```  2357     moreover have "affine hull S \<inter> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F} \<subseteq> S"
```
```  2358       using * by blast
```
```  2359     ultimately have "S = affine hull S \<inter> \<Inter> {{x. a h \<bullet> x \<le> b h} |h. h \<in> F}" ..
```
```  2360     then show ?thesis
```
```  2361       apply (rule ssubst)
```
```  2362       apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le simp: \<open>finite F\<close>)
```
```  2363       done
```
```  2364   qed
```
```  2365 qed
```
```  2366
```
```  2367 corollary polyhedron_eq_finite_faces:
```
```  2368   fixes S :: "'a :: euclidean_space set"
```
```  2369   shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F face_of S}"
```
```  2370          (is "?lhs = ?rhs")
```
```  2371 proof
```
```  2372   assume ?lhs
```
```  2373   then show ?rhs
```
```  2374     by (simp add: finite_polyhedron_faces polyhedron_imp_closed polyhedron_imp_convex)
```
```  2375 next
```
```  2376   assume ?rhs
```
```  2377   then show ?lhs
```
```  2378     by (force simp: polyhedron_eq_finite_exposed_faces exposed_face_of intro: finite_subset)
```
```  2379 qed
```
```  2380
```
```  2381 lemma polyhedron_linear_image_eq:
```
```  2382   fixes h :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  2383   assumes "linear h" "bij h"
```
```  2384     shows "polyhedron (h ` S) \<longleftrightarrow> polyhedron S"
```
```  2385 proof -
```
```  2386   have *: "{f. P f} = (image h) ` {f. P (h ` f)}" for P
```
```  2387     apply safe
```
```  2388     apply (rule_tac x="inv h ` x" in image_eqI)
```
```  2389     apply (auto simp: \<open>bij h\<close> bij_is_surj image_f_inv_f)
```
```  2390     done
```
```  2391   have "inj h" using bij_is_inj assms by blast
```
```  2392   then have injim: "inj_on (op ` h) A" for A
```
```  2393     by (simp add: inj_on_def inj_image_eq_iff)
```
```  2394   show ?thesis
```
```  2395     using \<open>linear h\<close> \<open>inj h\<close>
```
```  2396     apply (simp add: polyhedron_eq_finite_faces closed_injective_linear_image_eq)
```
```  2397     apply (simp add: * face_of_linear_image [of h _ S, symmetric] finite_image_iff injim)
```
```  2398     done
```
```  2399 qed
```
```  2400
```
```  2401 lemma polyhedron_negations:
```
```  2402   fixes S :: "'a :: euclidean_space set"
```
```  2403   shows   "polyhedron S \<Longrightarrow> polyhedron(image uminus S)"
```
```  2404 by (auto simp: polyhedron_linear_image_eq linear_uminus bij_uminus)
```
```  2405
```
```  2406 subsection\<open>Relation between polytopes and polyhedra\<close>
```
```  2407
```
```  2408 lemma polytope_eq_bounded_polyhedron:
```
```  2409   fixes S :: "'a :: euclidean_space set"
```
```  2410   shows "polytope S \<longleftrightarrow> polyhedron S \<and> bounded S"
```
```  2411          (is "?lhs = ?rhs")
```
```  2412 proof
```
```  2413   assume ?lhs
```
```  2414   then show ?rhs
```
```  2415     by (simp add: finite_polytope_faces polyhedron_eq_finite_faces
```
```  2416                   polytope_imp_closed polytope_imp_convex polytope_imp_bounded)
```
```  2417 next
```
```  2418   assume ?rhs then show ?lhs
```
```  2419     unfolding polytope_def
```
```  2420     apply (rule_tac x="{v. v extreme_point_of S}" in exI)
```
```  2421     apply (simp add: finite_polyhedron_extreme_points Krein_Milman_Minkowski compact_eq_bounded_closed polyhedron_imp_closed polyhedron_imp_convex)
```
```  2422     done
```
```  2423 qed
```
```  2424
```
```  2425 lemma polytope_Int:
```
```  2426   fixes S :: "'a :: euclidean_space set"
```
```  2427   shows "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
```
```  2428 by (simp add: polytope_eq_bounded_polyhedron bounded_Int)
```
```  2429
```
```  2430
```
```  2431 lemma polytope_Int_polyhedron:
```
```  2432   fixes S :: "'a :: euclidean_space set"
```
```  2433   shows "\<lbrakk>polytope S; polyhedron T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
```
```  2434 by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
```
```  2435
```
```  2436 lemma polyhedron_Int_polytope:
```
```  2437   fixes S :: "'a :: euclidean_space set"
```
```  2438   shows "\<lbrakk>polyhedron S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
```
```  2439 by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
```
```  2440
```
```  2441 lemma polytope_imp_polyhedron:
```
```  2442   fixes S :: "'a :: euclidean_space set"
```
```  2443   shows "polytope S \<Longrightarrow> polyhedron S"
```
```  2444 by (simp add: polytope_eq_bounded_polyhedron)
```
```  2445
```
```  2446 lemma polytope_facet_exists:
```
```  2447   fixes p :: "'a :: euclidean_space set"
```
```  2448   assumes "polytope p" "0 < aff_dim p"
```
```  2449   obtains F where "F facet_of p"
```
```  2450 proof (cases "p = {}")
```
```  2451   case True with assms show ?thesis by auto
```
```  2452 next
```
```  2453   case False
```
```  2454   then obtain v where "v extreme_point_of p"
```
```  2455     using extreme_point_exists_convex
```
```  2456     by (blast intro: \<open>polytope p\<close> polytope_imp_compact polytope_imp_convex)
```
```  2457   then
```
```  2458   show ?thesis
```
```  2459     by (metis face_of_polyhedron_subset_facet polytope_imp_polyhedron aff_dim_sing
```
```  2460        all_not_in_conv assms face_of_singleton less_irrefl singletonI that)
```
```  2461 qed
```
```  2462
```
```  2463 lemma polyhedron_interval [iff]: "polyhedron(cbox a b)"
```
```  2464 by (metis polytope_imp_polyhedron polytope_interval)
```
```  2465
```
```  2466 lemma polyhedron_convex_hull:
```
```  2467   fixes S :: "'a :: euclidean_space set"
```
```  2468   shows "finite S \<Longrightarrow> polyhedron(convex hull S)"
```
```  2469 by (simp add: polytope_convex_hull polytope_imp_polyhedron)
```
```  2470
```
```  2471
```
```  2472 subsection\<open>Relative and absolute frontier of a polytope\<close>
```
```  2473
```
```  2474 lemma rel_boundary_of_convex_hull:
```
```  2475     fixes S :: "'a::euclidean_space set"
```
```  2476     assumes "~ affine_dependent S"
```
```  2477       shows "(convex hull S) - rel_interior(convex hull S) = (\<Union>a\<in>S. convex hull (S - {a}))"
```
```  2478 proof -
```
```  2479   have "finite S" by (metis assms aff_independent_finite)
```
```  2480   then consider "card S = 0" | "card S = 1" | "2 \<le> card S" by arith
```
```  2481   then show ?thesis
```
```  2482   proof cases
```
```  2483     case 1 then have "S = {}" by (simp add: \<open>finite S\<close>)
```
```  2484     then show ?thesis by simp
```
```  2485   next
```
```  2486     case 2 show ?thesis
```
```  2487       by (auto intro: card_1_singletonE [OF \<open>card S = 1\<close>])
```
```  2488   next
```
```  2489     case 3
```
```  2490     with assms show ?thesis
```
```  2491       by (auto simp: polyhedron_convex_hull rel_boundary_of_polyhedron facet_of_convex_hull_affine_independent_alt \<open>finite S\<close>)
```
```  2492   qed
```
```  2493 qed
```
```  2494
```
```  2495 proposition frontier_of_convex_hull:
```
```  2496     fixes S :: "'a::euclidean_space set"
```
```  2497     assumes "card S = Suc (DIM('a))"
```
```  2498       shows "frontier(convex hull S) = \<Union> {convex hull (S - {a}) | a. a \<in> S}"
```
```  2499 proof (cases "affine_dependent S")
```
```  2500   case True
```
```  2501     have [iff]: "finite S"
```
```  2502       using assms using card_infinite by force
```
```  2503     then have ccs: "closed (convex hull S)"
```
```  2504       by (simp add: compact_imp_closed finite_imp_compact_convex_hull)
```
```  2505     { fix x T
```
```  2506       assume "finite T" "T \<subseteq> S" "int (card T) \<le> aff_dim S + 1" "x \<in> convex hull T"
```
```  2507       then have "S \<noteq> T"
```
```  2508         using True \<open>finite S\<close> aff_dim_le_card affine_independent_iff_card by fastforce
```
```  2509       then obtain a where "a \<in> S" "a \<notin> T"
```
```  2510         using \<open>T \<subseteq> S\<close> by blast
```
```  2511       then have "x \<in> (\<Union>a\<in>S. convex hull (S - {a}))"
```
```  2512         using True affine_independent_iff_card [of S]
```
```  2513         apply simp
```
```  2514         apply (metis (no_types, hide_lams) Diff_eq_empty_iff Diff_insert0 \<open>a \<notin> T\<close> \<open>T \<subseteq> S\<close> \<open>x \<in> convex hull T\<close>  hull_mono insert_Diff_single   subsetCE)
```
```  2515         done
```
```  2516     } note * = this
```
```  2517     have 1: "convex hull S \<subseteq> (\<Union> a\<in>S. convex hull (S - {a}))"
```
```  2518       apply (subst caratheodory_aff_dim)
```
```  2519       apply (blast intro: *)
```
```  2520       done
```
```  2521     have 2: "\<Union>((\<lambda>a. convex hull (S - {a})) ` S) \<subseteq> convex hull S"
```
```  2522       by (rule Union_least) (metis (no_types, lifting)  Diff_subset hull_mono imageE)
```
```  2523     show ?thesis using True
```
```  2524       apply (simp add: segment_convex_hull frontier_def)
```
```  2525       using interior_convex_hull_eq_empty [OF assms]
```
```  2526       apply (simp add: closure_closed [OF ccs])
```
```  2527       apply (rule subset_antisym)
```
```  2528       using 1 apply blast
```
```  2529       using 2 apply blast
```
```  2530       done
```
```  2531 next
```
```  2532   case False
```
```  2533   then have "frontier (convex hull S) = (convex hull S) - rel_interior(convex hull S)"
```
```  2534     apply (simp add: rel_boundary_of_convex_hull [symmetric] frontier_def)
```
```  2535     by (metis aff_independent_finite assms closure_convex_hull finite_imp_compact_convex_hull hull_hull interior_convex_hull_eq_empty rel_interior_nonempty_interior)
```
```  2536   also have "... = \<Union>{convex hull (S - {a}) |a. a \<in> S}"
```
```  2537   proof -
```
```  2538     have "convex hull S - rel_interior (convex hull S) = rel_frontier (convex hull S)"
```
```  2539       by (simp add: False aff_independent_finite polyhedron_convex_hull rel_boundary_of_polyhedron rel_frontier_of_polyhedron)
```
```  2540     then show ?thesis
```
```  2541       by (simp add: False rel_frontier_convex_hull_cases)
```
```  2542   qed
```
```  2543   finally show ?thesis .
```
```  2544 qed
```
```  2545
```
```  2546 subsection\<open>Special case of a triangle\<close>
```
```  2547
```
```  2548 proposition frontier_of_triangle:
```
```  2549     fixes a :: "'a::euclidean_space"
```
```  2550     assumes "DIM('a) = 2"
```
```  2551     shows "frontier(convex hull {a,b,c}) = closed_segment a b \<union> closed_segment b c \<union> closed_segment c a"
```
```  2552           (is "?lhs = ?rhs")
```
```  2553 proof (cases "b = a \<or> c = a \<or> c = b")
```
```  2554   case True then show ?thesis
```
```  2555     by (auto simp: assms segment_convex_hull frontier_def empty_interior_convex_hull insert_commute card_insert_le_m1 hull_inc insert_absorb)
```
```  2556 next
```
```  2557   case False then have [simp]: "card {a, b, c} = Suc (DIM('a))"
```
```  2558     by (simp add: card_insert Set.insert_Diff_if assms)
```
```  2559   show ?thesis
```
```  2560   proof
```
```  2561     show "?lhs \<subseteq> ?rhs"
```
```  2562       using False
```
```  2563       by (force simp: segment_convex_hull frontier_of_convex_hull insert_Diff_if insert_commute split: if_split_asm)
```
```  2564     show "?rhs \<subseteq> ?lhs"
```
```  2565       using False
```
```  2566       apply (simp add: frontier_of_convex_hull segment_convex_hull)
```
```  2567       apply (intro conjI subsetI)
```
```  2568         apply (rule_tac X="convex hull {a,b}" in UnionI; force simp: Set.insert_Diff_if)
```
```  2569        apply (rule_tac X="convex hull {b,c}" in UnionI; force)
```
```  2570       apply (rule_tac X="convex hull {a,c}" in UnionI; force simp: insert_commute Set.insert_Diff_if)
```
```  2571       done
```
```  2572   qed
```
```  2573 qed
```
```  2574
```
```  2575 corollary inside_of_triangle:
```
```  2576     fixes a :: "'a::euclidean_space"
```
```  2577     assumes "DIM('a) = 2"
```
```  2578     shows "inside (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a) = interior(convex hull {a,b,c})"
```
```  2579 by (metis assms frontier_of_triangle bounded_empty bounded_insert convex_convex_hull inside_frontier_eq_interior bounded_convex_hull)
```
```  2580
```
```  2581 corollary interior_of_triangle:
```
```  2582     fixes a :: "'a::euclidean_space"
```
```  2583     assumes "DIM('a) = 2"
```
```  2584     shows "interior(convex hull {a,b,c}) =
```
```  2585            convex hull {a,b,c} - (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a)"
```
```  2586   using interior_subset
```
```  2587   by (force simp: frontier_of_triangle [OF assms, symmetric] frontier_def Diff_Diff_Int)
```
```  2588
```
```  2589 subsection\<open>Subdividing a cell complex\<close>
```
```  2590
```
```  2591 lemma subdivide_interval:
```
```  2592   fixes x::real
```
```  2593   assumes "a < \<bar>x - y\<bar>" "0 < a"
```
```  2594   obtains n where "n \<in> \<int>" "x < n * a \<and> n * a < y \<or> y <  n * a \<and> n * a < x"
```
```  2595 proof -
```
```  2596   consider "a + x < y" | "a + y < x"
```
```  2597     using assms by linarith
```
```  2598   then show ?thesis
```
```  2599   proof cases
```
```  2600     case 1
```
```  2601     let ?n = "of_int (floor (x/a)) + 1"
```
```  2602     have x: "x < ?n * a"
```
```  2603       by (meson \<open>0 < a\<close> divide_less_eq floor_unique_iff)
```
```  2604     have "?n * a \<le> a + x"
```
```  2605       apply (simp add: algebra_simps)
```
```  2606       by (metis \<open>0 < a\<close> floor_correct less_irrefl nonzero_mult_div_cancel_left real_mult_le_cancel_iff2 times_divide_eq_right)
```
```  2607     also have "... < y"
```
```  2608       by (rule 1)
```
```  2609     finally have "?n * a < y" .
```
```  2610     with x show ?thesis
```
```  2611       using Ints_1 Ints_add Ints_of_int that by blast
```
```  2612   next
```
```  2613     case 2
```
```  2614     let ?n = "of_int (floor (y/a)) + 1"
```
```  2615     have y: "y < ?n * a"
```
```  2616       by (meson \<open>0 < a\<close> divide_less_eq floor_unique_iff)
```
```  2617     have "?n * a \<le> a + y"
```
```  2618       apply (simp add: algebra_simps)
```
```  2619       by (metis \<open>0 < a\<close> floor_correct less_irrefl nonzero_mult_div_cancel_left real_mult_le_cancel_iff2 times_divide_eq_right)
```
```  2620     also have "... < x"
```
```  2621       by (rule 2)
```
```  2622     finally have "?n * a < x" .
```
```  2623     then show ?thesis
```
```  2624       using Ints_1 Ints_add Ints_of_int that y by blast
```
```  2625   qed
```
```  2626 qed
```
```  2627
```
```  2628
```
```  2629 lemma cell_subdivision_lemma:
```
```  2630   assumes "finite \<F>"
```
```  2631       and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
```
```  2632       and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"
```
```  2633       and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
```
```  2634       and "finite I"
```
```  2635     shows "\<exists>\<F>'. \<Union>\<F>' = \<Union>\<F> \<and>
```
```  2636                  finite \<F>' \<and>
```
```  2637                  (\<forall>X \<in> \<F>'. polytope X) \<and>
```
```  2638                  (\<forall>X \<in> \<F>'. aff_dim X \<le> d) \<and>
```
```  2639                  (\<forall>X \<in> \<F>'. \<forall>Y \<in> \<F>'. X \<inter> Y face_of X \<and> X \<inter> Y face_of Y) \<and>
```
```  2640                  (\<forall>X \<in> \<F>'. \<forall>x \<in> X. \<forall>y \<in> X. \<forall>a b.
```
```  2641                           (a,b) \<in> I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or>
```
```  2642                                         a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b)"
```
```  2643   using \<open>finite I\<close>
```
```  2644 proof induction
```
```  2645   case empty
```
```  2646   then show ?case
```
```  2647     by (rule_tac x="\<F>" in exI) (simp add: assms)
```
```  2648 next
```
```  2649   case (insert ab I)
```
```  2650   then obtain \<F>' where eq: "\<Union>\<F>' = \<Union>\<F>" and "finite \<F>'"
```
```  2651                    and poly: "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X"
```
```  2652                    and aff: "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
```
```  2653                    and face: "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
```
```  2654                    and I: "\<And>X x y a b.  \<lbrakk>X \<in> \<F>'; x \<in> X; y \<in> X; (a,b) \<in> I\<rbrakk> \<Longrightarrow>
```
```  2655                                     a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
```
```  2656     by (auto simp: that)
```
```  2657   obtain a b where "ab = (a,b)"
```
```  2658     by fastforce
```
```  2659   let ?\<G> = "(\<lambda>X. X \<inter> {x. a \<bullet> x \<le> b}) ` \<F>' \<union> (\<lambda>X. X \<inter> {x. a \<bullet> x \<ge> b}) ` \<F>'"
```
```  2660   have eqInt: "(S \<inter> Collect P) \<inter> (T \<inter> Collect Q) = (S \<inter> T) \<inter> (Collect P \<inter> Collect Q)" for S T::"'a set" and P Q
```
```  2661     by blast
```
```  2662   show ?case
```
```  2663   proof (intro conjI exI)
```
```  2664     show "\<Union>?\<G> = \<Union>\<F>"
```
```  2665       by (force simp: eq [symmetric])
```
```  2666     show "finite ?\<G>"
```
```  2667       using \<open>finite \<F>'\<close> by force
```
```  2668     show "\<forall>X \<in> ?\<G>. polytope X"
```
```  2669       by (force simp: poly polytope_Int_polyhedron polyhedron_halfspace_le polyhedron_halfspace_ge)
```
```  2670     show "\<forall>X \<in> ?\<G>. aff_dim X \<le> d"
```
```  2671       by (auto; metis order_trans aff aff_dim_subset inf_le1)
```
```  2672     show "\<forall>X \<in> ?\<G>. \<forall>x \<in> X. \<forall>y \<in> X. \<forall>a b.
```
```  2673                           (a,b) \<in> insert ab I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or>
```
```  2674                                                   a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
```
```  2675       using \<open>ab = (a, b)\<close> I by fastforce
```
```  2676     show "\<forall>X \<in> ?\<G>. \<forall>Y \<in> ?\<G>. X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
```
```  2677       by (auto simp: eqInt halfspace_Int_eq face_of_Int_Int face face_of_halfspace_le face_of_halfspace_ge)
```
```  2678   qed
```
```  2679 qed
```
```  2680
```
```  2681
```
```  2682 proposition cell_complex_subdivision_exists:
```
```  2683   fixes \<F> :: "'a::euclidean_space set set"
```
```  2684   assumes "0 < e" "finite \<F>"
```
```  2685       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
```
```  2686       and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"
```
```  2687       and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
```
```  2688   obtains "\<F>'" where "finite \<F>'" "\<Union>\<F>' = \<Union>\<F>" "\<And>X. X \<in> \<F>' \<Longrightarrow> diameter X < e"
```
```  2689                 "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X" "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
```
```  2690                 "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
```
```  2691 proof -
```
```  2692   have "bounded(\<Union>\<F>)"
```
```  2693     by (simp add: \<open>finite \<F>\<close> poly bounded_Union polytope_imp_bounded)
```
```  2694   then obtain B where "B > 0" and B: "\<And>x. x \<in> \<Union>\<F> \<Longrightarrow> norm x < B"
```
```  2695     by (meson bounded_pos_less)
```
```  2696   define C where "C \<equiv> {z \<in> \<int>. \<bar>z * e / 2 / real DIM('a)\<bar> \<le> B}"
```
```  2697   define I where "I \<equiv> \<Union>i \<in> Basis. \<Union>j \<in> C. { (i::'a, j * e / 2 / DIM('a)) }"
```
```  2698   have "finite C"
```
```  2699     using finite_int_segment [of "-B / (e / 2 / DIM('a))" "B / (e / 2 / DIM('a))"]
```
```  2700     apply (simp add: C_def)
```
```  2701     apply (erule rev_finite_subset)
```
```  2702     using \<open>0 < e\<close>
```
```  2703     apply (auto simp: divide_simps)
```
```  2704     done
```
```  2705   then have "finite I"
```
```  2706     by (simp add: I_def)
```
```  2707   obtain \<F>' where eq: "\<Union>\<F>' = \<Union>\<F>" and "finite \<F>'"
```
```  2708               and poly: "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X"
```
```  2709               and aff: "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
```
```  2710               and face: "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
```
```  2711               and I: "\<And>X x y a b.  \<lbrakk>X \<in> \<F>'; x \<in> X; y \<in> X; (a,b) \<in> I\<rbrakk> \<Longrightarrow>
```
```  2712                                      a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
```
```  2713     apply (rule exE [OF cell_subdivision_lemma])
```
```  2714          apply (rule assms \<open>finite I\<close> | assumption)+
```
```  2715     apply (auto intro: that)
```
```  2716     done
```
```  2717   show ?thesis
```
```  2718   proof (rule_tac \<F>'="\<F>'" in that)
```
```  2719     show "diameter X < e" if "X \<in> \<F>'" for X
```
```  2720     proof -
```
```  2721       have "diameter X \<le> e/2"
```
```  2722       proof (rule diameter_le)
```
```  2723         show "norm (x - y) \<le> e / 2" if "x \<in> X" "y \<in> X" for x y
```
```  2724         proof -
```
```  2725           have "norm x < B" "norm y < B"
```
```  2726             using B \<open>X \<in> \<F>'\<close> eq that by fastforce+
```
```  2727           have "norm (x - y) \<le> (\<Sum>b\<in>Basis. \<bar>(x-y) \<bullet> b\<bar>)"
```
```  2728             by (rule norm_le_l1)
```
```  2729           also have "... \<le> of_nat (DIM('a)) * (e / 2 / DIM('a))"
```
```  2730           proof (rule sum_bounded_above)
```
```  2731             fix i::'a
```
```  2732             assume "i \<in> Basis"
```
```  2733             then have I': "\<And>z b. \<lbrakk>z \<in> C; b = z * e / (2 * real DIM('a))\<rbrakk> \<Longrightarrow> i \<bullet> x \<le> b \<and> i \<bullet> y \<le> b \<or> i \<bullet> x \<ge> b \<and> i \<bullet> y \<ge> b"
```
```  2734               using I \<open>X \<in> \<F>'\<close> that
```
```  2735               by (fastforce simp: I_def)
```
```  2736             show "\<bar>(x - y) \<bullet> i\<bar> \<le> e / 2 / real DIM('a)"
```
```  2737             proof (rule ccontr)
```
```  2738               assume "\<not> \<bar>(x - y) \<bullet> i\<bar> \<le> e / 2 / real DIM('a)"
```
```  2739               then have xyi: "\<bar>i \<bullet> x - i \<bullet> y\<bar> > e / 2 / real DIM('a)"
```
```  2740                 by (simp add: inner_commute inner_diff_right)
```
```  2741               obtain n where "n \<in> \<int>" and n: "i \<bullet> x < n * (e / 2 / real DIM('a)) \<and> n * (e / 2 / real DIM('a)) < i \<bullet> y \<or> i \<bullet> y < n * (e / 2 / real DIM('a)) \<and> n * (e / 2 / real DIM('a)) < i \<bullet> x"
```
```  2742                 using subdivide_interval [OF xyi] DIM_positive \<open>0 < e\<close>
```
```  2743                 by (auto simp: zero_less_divide_iff)
```
```  2744               have "\<bar>i \<bullet> x\<bar> < B"
```
```  2745                 by (metis \<open>i \<in> Basis\<close> \<open>norm x < B\<close> inner_commute norm_bound_Basis_lt)
```
```  2746               have "\<bar>i \<bullet> y\<bar> < B"
```
```  2747                 by (metis \<open>i \<in> Basis\<close> \<open>norm y < B\<close> inner_commute norm_bound_Basis_lt)
```
```  2748               have *: "\<bar>n * e\<bar> \<le> B * (2 * real DIM('a))"
```
```  2749                       if "\<bar>ix\<bar> < B" "\<bar>iy\<bar> < B"
```
```  2750                          and ix: "ix * (2 * real DIM('a)) < n * e"
```
```  2751                          and iy: "n * e < iy * (2 * real DIM('a))" for ix iy
```
```  2752               proof (rule abs_leI)
```
```  2753                 have "iy * (2 * real DIM('a)) \<le> B * (2 * real DIM('a))"
```
```  2754                   by (rule mult_right_mono) (use \<open>\<bar>iy\<bar> < B\<close> in linarith)+
```
```  2755                 then show "n * e \<le> B * (2 * real DIM('a))"
```
```  2756                   using iy by linarith
```
```  2757               next
```
```  2758                 have "- ix * (2 * real DIM('a)) \<le> B * (2 * real DIM('a))"
```
```  2759                   by (rule mult_right_mono) (use \<open>\<bar>ix\<bar> < B\<close> in linarith)+
```
```  2760                 then show "- (n * e) \<le> B * (2 * real DIM('a))"
```
```  2761                   using ix by linarith
```
```  2762               qed
```
```  2763               have "n \<in> C"
```
```  2764                 using \<open>n \<in> \<int>\<close> n  by (auto simp: C_def divide_simps intro: * \<open>\<bar>i \<bullet> x\<bar> < B\<close> \<open>\<bar>i \<bullet> y\<bar> < B\<close>)
```
```  2765               show False
```
```  2766                 using  I' [OF \<open>n \<in> C\<close> refl] n  by auto
```
```  2767             qed
```
```  2768           qed
```
```  2769           also have "... = e / 2"
```
```  2770             by simp
```
```  2771           finally show ?thesis .
```
```  2772         qed
```
```  2773       qed (use \<open>0 < e\<close> in force)
```
```  2774       also have "... < e"
```
```  2775         by (simp add: \<open>0 < e\<close>)
```
```  2776       finally show ?thesis .
```
```  2777     qed
```
```  2778   qed (auto simp: eq poly aff face  \<open>finite \<F>'\<close>)
```
```  2779 qed
```
```  2780
```
```  2781 end
```