src/HOL/Analysis/Polytope.thy
author fleury <Mathias.Fleury@mpi-inf.mpg.de>
Mon Sep 19 20:06:21 2016 +0200 (2016-09-19)
changeset 63918 6bf55e6e0b75
parent 63807 5f77017055a3
child 63967 2aa42596edc3
permissions -rw-r--r--
left_distrib ~> distrib_right, right_distrib ~> distrib_left
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section \<open>Faces, Extreme Points, Polytopes, Polyhedra etc.\<close>
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text\<open>Ported from HOL Light by L C Paulson\<close>
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theory Polytope
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imports Cartesian_Euclidean_Space
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begin
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subsection \<open>Faces of a (usually convex) set\<close>
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definition face_of :: "['a::real_vector set, 'a set] \<Rightarrow> bool" (infixr "(face'_of)" 50)
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  where
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  "T face_of S \<longleftrightarrow>
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        T \<subseteq> S \<and> convex T \<and>
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        (\<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)"
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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"
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  unfolding face_of_def by blast
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lemma face_of_translation_eq [simp]:
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    "(op + a ` T face_of op + a ` S) \<longleftrightarrow> T face_of S"
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proof -
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  have *: "\<And>a T S. T face_of S \<Longrightarrow> (op + a ` T face_of op + a ` S)"
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    apply (simp add: face_of_def Ball_def, clarify)
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    apply (drule open_segment_translation_eq [THEN iffD1])
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    using inj_image_mem_iff inj_add_left apply metis
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    done
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  show ?thesis
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    apply (rule iffI)
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    apply (force simp: image_comp o_def dest: * [where a = "-a"])
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    apply (blast intro: *)
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    done
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qed
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lemma face_of_linear_image:
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  assumes "linear f" "inj f"
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    shows "(f ` c face_of f ` S) \<longleftrightarrow> c face_of S"
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by (simp add: face_of_def inj_image_subset_iff inj_image_mem_iff open_segment_linear_image assms)
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lemma face_of_refl: "convex S \<Longrightarrow> S face_of S"
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  by (auto simp: face_of_def)
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lemma face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
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  by (auto simp: face_of_def)
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lemma empty_face_of [iff]: "{} face_of S"
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  by (simp add: face_of_def)
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lemma face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
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  by (meson empty_face_of face_of_def subset_empty)
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lemma face_of_trans [trans]: "\<lbrakk>S face_of T; T face_of u\<rbrakk> \<Longrightarrow> S face_of u"
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  unfolding face_of_def by (safe; blast)
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lemma face_of_face: "T face_of S \<Longrightarrow> (f face_of T \<longleftrightarrow> f face_of S \<and> f \<subseteq> T)"
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  unfolding face_of_def by (safe; blast)
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lemma face_of_subset: "\<lbrakk>F face_of S; F \<subseteq> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> F face_of T"
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  unfolding face_of_def by (safe; blast)
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lemma face_of_slice: "\<lbrakk>F face_of S; convex T\<rbrakk> \<Longrightarrow> (F \<inter> T) face_of (S \<inter> T)"
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  unfolding face_of_def by (blast intro: convex_Int)
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lemma face_of_Int: "\<lbrakk>t1 face_of S; t2 face_of S\<rbrakk> \<Longrightarrow> (t1 \<inter> t2) face_of S"
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  unfolding face_of_def by (blast intro: convex_Int)
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lemma face_of_Inter: "\<lbrakk>A \<noteq> {}; \<And>T. T \<in> A \<Longrightarrow> T face_of S\<rbrakk> \<Longrightarrow> (\<Inter> A) face_of S"
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  unfolding face_of_def by (blast intro: convex_Inter)
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lemma face_of_Int_Int: "\<lbrakk>F face_of T; F' face_of t'\<rbrakk> \<Longrightarrow> (F \<inter> F') face_of (T \<inter> t')"
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  unfolding face_of_def by (blast intro: convex_Int)
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lemma face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
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  unfolding face_of_def by blast
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lemma face_of_imp_eq_affine_Int:
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     fixes S :: "'a::euclidean_space set"
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     assumes S: "convex S" "closed S" and T: "T face_of S"
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     shows "T = (affine hull T) \<inter> S"
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proof -
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  have "convex T" using T by (simp add: face_of_def)
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  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
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  proof -
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    obtain e where "e>0" and e: "cball y e \<inter> affine hull T \<subseteq> T"
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      using y by (auto simp: rel_interior_cball)
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    have "y \<noteq> x" "y \<in> S" "y \<in> T"
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      using face_of_imp_subset rel_interior_subset T that by blast+
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    then have zne: "\<And>u. \<lbrakk>u \<in> {0<..<1}; (1 - u) *\<^sub>R y + u *\<^sub>R x \<in> T\<rbrakk> \<Longrightarrow>  False"
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      using \<open>x \<in> S\<close> \<open>x \<notin> T\<close> \<open>T face_of S\<close> unfolding face_of_def
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      apply clarify
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      apply (drule_tac x=x in bspec, assumption)
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      apply (drule_tac x=y in bspec, assumption)
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      apply (subst (asm) open_segment_commute)
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      apply (force simp: open_segment_image_interval image_def)
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      done
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    have in01: "min (1/2) (e / norm (x - y)) \<in> {0<..<1}"
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      using \<open>y \<noteq> x\<close> \<open>e > 0\<close> by simp
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    show ?thesis
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      apply (rule zne [OF in01])
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      apply (rule e [THEN subsetD])
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      apply (rule IntI)
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        using \<open>y \<noteq> x\<close> \<open>e > 0\<close>
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        apply (simp add: cball_def dist_norm algebra_simps)
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        apply (simp add: Real_Vector_Spaces.scaleR_diff_right [symmetric] norm_minus_commute min_mult_distrib_right)
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      apply (rule mem_affine [OF affine_affine_hull _ x])
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      using \<open>y \<in> T\<close>  apply (auto simp: hull_inc)
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      done
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  qed
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  show ?thesis
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    apply (rule subset_antisym)
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    using assms apply (simp add: hull_subset face_of_imp_subset)
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    apply (cases "T={}", simp)
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    apply (force simp: rel_interior_eq_empty [symmetric] \<open>convex T\<close> intro: *)
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    done
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qed
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lemma face_of_imp_closed:
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     fixes S :: "'a::euclidean_space set"
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     assumes "convex S" "closed S" "T face_of S" shows "closed T"
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  by (metis affine_affine_hull affine_closed closed_Int face_of_imp_eq_affine_Int assms)
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lemma face_of_Int_supporting_hyperplane_le_strong:
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    assumes "convex(S \<inter> {x. a \<bullet> x = b})" and aleb: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b"
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      shows "(S \<inter> {x. a \<bullet> x = b}) face_of S"
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proof -
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  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"
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          for u v x
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  proof (rule antisym)
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    show "a \<bullet> u \<le> a \<bullet> x"
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      using aleb \<open>u \<in> S\<close> \<open>b = a \<bullet> x\<close> by blast
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  next
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    obtain \<xi> where "b = a \<bullet> ((1 - \<xi>) *\<^sub>R u + \<xi> *\<^sub>R v)" "0 < \<xi>" "\<xi> < 1"
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      using \<open>b = a \<bullet> x\<close> \<open>x \<in> open_segment u v\<close> in_segment
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      by (auto simp: open_segment_image_interval split: if_split_asm)
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    then have "b + \<xi> * (a \<bullet> u) \<le> a \<bullet> u + \<xi> * b"
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      using aleb [OF \<open>v \<in> S\<close>] by (simp add: algebra_simps)
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    then have "(1 - \<xi>) * b \<le> (1 - \<xi>) * (a \<bullet> u)"
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      by (simp add: algebra_simps)
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    then have "b \<le> a \<bullet> u"
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      using \<open>\<xi> < 1\<close> by auto
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    with b show "a \<bullet> x \<le> a \<bullet> u" by simp
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  qed
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  show ?thesis
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    apply (simp add: face_of_def assms)
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    using "*" open_segment_commute by blast
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qed
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lemma face_of_Int_supporting_hyperplane_ge_strong:
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   "\<lbrakk>convex(S \<inter> {x. a \<bullet> x = b}); \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk>
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    \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
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  using face_of_Int_supporting_hyperplane_le_strong [of S "-a" "-b"] by simp
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lemma face_of_Int_supporting_hyperplane_le:
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    "\<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"
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  by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_le_strong)
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lemma face_of_Int_supporting_hyperplane_ge:
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    "\<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"
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  by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_ge_strong)
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lemma face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
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  using face_of_def by blast
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lemma face_of_imp_compact:
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    fixes S :: "'a::euclidean_space set"
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    shows "\<lbrakk>convex S; compact S; T face_of S\<rbrakk> \<Longrightarrow> compact T"
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  by (meson bounded_subset compact_eq_bounded_closed face_of_imp_closed face_of_imp_subset)
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lemma face_of_Int_subface:
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     "\<lbrakk>A \<inter> B face_of A; A \<inter> B face_of B; C face_of A; D face_of B\<rbrakk>
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      \<Longrightarrow> (C \<inter> D) face_of C \<and> (C \<inter> D) face_of D"
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  by (meson face_of_Int_Int face_of_face inf_le1 inf_le2)
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lemma subset_of_face_of:
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    fixes S :: "'a::real_normed_vector set"
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    assumes "T face_of S" "u \<subseteq> S" "T \<inter> (rel_interior u) \<noteq> {}"
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      shows "u \<subseteq> T"
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proof
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  fix c
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  assume "c \<in> u"
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  obtain b where "b \<in> T" "b \<in> rel_interior u" using assms by auto
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  then obtain e where "e>0" "b \<in> u" and e: "cball b e \<inter> affine hull u \<subseteq> u"
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    by (auto simp: rel_interior_cball)
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  show "c \<in> T"
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  proof (cases "b=c")
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    case True with \<open>b \<in> T\<close> show ?thesis by blast
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  next
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    case False
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    define d where "d = b + (e / norm(b - c)) *\<^sub>R (b - c)"
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    have "d \<in> cball b e \<inter> affine hull u"
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      using \<open>e > 0\<close> \<open>b \<in> u\<close> \<open>c \<in> u\<close>
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      by (simp add: d_def dist_norm hull_inc mem_affine_3_minus False)
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    with e have "d \<in> u" by blast
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    have nbc: "norm (b - c) + e > 0" using \<open>e > 0\<close>
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      by (metis add.commute le_less_trans less_add_same_cancel2 norm_ge_zero)
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    then have [simp]: "d \<noteq> c" using False scaleR_cancel_left [of "1 + (e / norm (b - c))" b c]
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      by (simp add: algebra_simps d_def) (simp add: divide_simps)
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    have [simp]: "((e - e * e / (e + norm (b - c))) / norm (b - c)) = (e / (e + norm (b - c)))"
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      using False nbc
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      apply (simp add: algebra_simps divide_simps)
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      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)
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    have "b \<in> open_segment d c"
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      apply (simp add: open_segment_image_interval)
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      apply (simp add: d_def algebra_simps image_def)
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      apply (rule_tac x="e / (e + norm (b - c))" in bexI)
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      using False nbc \<open>0 < e\<close>
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      apply (auto simp: algebra_simps)
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      done
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    then have "d \<in> T \<and> c \<in> T"
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      apply (rule face_ofD [OF \<open>T face_of S\<close>])
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      using \<open>d \<in> u\<close>  \<open>c \<in> u\<close> \<open>u \<subseteq> S\<close>  \<open>b \<in> T\<close>  apply auto
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      done
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    then show ?thesis ..
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  qed
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qed
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lemma face_of_eq:
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    fixes S :: "'a::real_normed_vector set"
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    assumes "T face_of S" "u face_of S" "(rel_interior T) \<inter> (rel_interior u) \<noteq> {}"
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      shows "T = u"
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  apply (rule subset_antisym)
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  apply (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subsetCE subset_of_face_of)
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  by (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subset_iff subset_of_face_of)
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lemma face_of_disjoint_rel_interior:
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      fixes S :: "'a::real_normed_vector set"
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      assumes "T face_of S" "T \<noteq> S"
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        shows "T \<inter> rel_interior S = {}"
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  by (meson assms subset_of_face_of face_of_imp_subset order_refl subset_antisym)
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lemma face_of_disjoint_interior:
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      fixes S :: "'a::real_normed_vector set"
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      assumes "T face_of S" "T \<noteq> S"
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        shows "T \<inter> interior S = {}"
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proof -
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  have "T \<inter> interior S \<subseteq> rel_interior S"
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    by (meson inf_sup_ord(2) interior_subset_rel_interior order.trans)
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  thus ?thesis
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    by (metis (no_types) Int_greatest assms face_of_disjoint_rel_interior inf_sup_ord(1) subset_empty)
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qed
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lemma face_of_subset_rel_boundary:
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  fixes S :: "'a::real_normed_vector set"
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  assumes "T face_of S" "T \<noteq> S"
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    shows "T \<subseteq> (S - rel_interior S)"
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by (meson DiffI assms disjoint_iff_not_equal face_of_disjoint_rel_interior face_of_imp_subset rev_subsetD subsetI)
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lemma face_of_subset_rel_frontier:
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    fixes S :: "'a::real_normed_vector set"
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    assumes "T face_of S" "T \<noteq> S"
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      shows "T \<subseteq> rel_frontier S"
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  using assms closure_subset face_of_disjoint_rel_interior face_of_imp_subset rel_frontier_def by fastforce
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lemma face_of_aff_dim_lt:
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  fixes S :: "'a::euclidean_space set"
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  assumes "convex S" "T face_of S" "T \<noteq> S"
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    shows "aff_dim T < aff_dim S"
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proof -
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  have "aff_dim T \<le> aff_dim S"
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    by (simp add: face_of_imp_subset aff_dim_subset assms)
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  moreover have "aff_dim T \<noteq> aff_dim S"
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  proof (cases "T = {}")
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    case True then show ?thesis
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      by (metis aff_dim_empty \<open>T \<noteq> S\<close>)
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  next case False then show ?thesis
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    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)
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  qed
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  ultimately show ?thesis
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    by simp
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   270
qed
lp15@63078
   271
lp15@63078
   272
lp15@63078
   273
lemma affine_diff_divide:
lp15@63078
   274
    assumes "affine S" "k \<noteq> 0" "k \<noteq> 1" and xy: "x \<in> S" "y /\<^sub>R (1 - k) \<in> S"
lp15@63078
   275
      shows "(x - y) /\<^sub>R k \<in> S"
lp15@63078
   276
proof -
lp15@63078
   277
  have "inverse(k) *\<^sub>R (x - y) = (1 - inverse k) *\<^sub>R inverse(1 - k) *\<^sub>R y + inverse(k) *\<^sub>R x"
lp15@63078
   278
    using assms
lp15@63078
   279
    by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] divide_simps)
lp15@63078
   280
  then show ?thesis
lp15@63078
   281
    using \<open>affine S\<close> xy by (auto simp: affine_alt)
lp15@63078
   282
qed
lp15@63078
   283
lp15@63078
   284
lemma face_of_convex_hulls:
lp15@63078
   285
      assumes S: "finite S" "T \<subseteq> S" and disj: "affine hull T \<inter> convex hull (S - T) = {}"
lp15@63078
   286
      shows  "(convex hull T) face_of (convex hull S)"
lp15@63078
   287
proof -
lp15@63078
   288
  have fin: "finite T" "finite (S - T)" using assms
lp15@63078
   289
    by (auto simp: finite_subset)
lp15@63078
   290
  have *: "x \<in> convex hull T"
lp15@63078
   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"
lp15@63078
   292
          for x y w
lp15@63078
   293
  proof -
lp15@63078
   294
    have waff: "w \<in> affine hull T"
lp15@63078
   295
      using convex_hull_subset_affine_hull w by blast
lp15@63078
   296
    obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "setsum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
lp15@63078
   297
                 and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "setsum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
lp15@63078
   298
      using x y by (auto simp: assms convex_hull_finite)
lp15@63078
   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"
lp15@63078
   300
               and u01: "0 < u" "u < 1"
lp15@63078
   301
      using w by (auto simp: open_segment_image_interval split: if_split_asm)
wenzelm@63148
   302
    define c where "c i = (1 - u) * a i + u * b i" for i
lp15@63078
   303
    have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
lp15@63078
   304
      using a b u01 by (simp add: c_def)
lp15@63078
   305
    have sumc1: "setsum c S = 1"
Mathias@63918
   306
      by (simp add: c_def setsum.distrib setsum_distrib_left [symmetric] asum bsum)
lp15@63078
   307
    have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
lp15@63078
   308
      apply (simp add: c_def setsum.distrib scaleR_left_distrib)
lp15@63078
   309
      by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric] aeqx beqy)
lp15@63078
   310
    show ?thesis
lp15@63078
   311
    proof (cases "setsum c (S - T) = 0")
lp15@63078
   312
      case True
lp15@63078
   313
      have ci0: "\<And>i. i \<in> (S - T) \<Longrightarrow> c i = 0"
lp15@63078
   314
        using True cge0 by (simp add: \<open>finite S\<close> setsum_nonneg_eq_0_iff)
lp15@63078
   315
      have a0: "a i = 0" if "i \<in> (S - T)" for i
lp15@63078
   316
        using ci0 [OF that] u01 a [of i] b [of i] that
lp15@63078
   317
        by (simp add: c_def Groups.ordered_comm_monoid_add_class.add_nonneg_eq_0_iff)
lp15@63078
   318
      have [simp]: "setsum a T = 1"
lp15@63078
   319
        using assms by (metis setsum.mono_neutral_cong_right a0 asum)
lp15@63078
   320
      show ?thesis
lp15@63078
   321
        apply (simp add: convex_hull_finite \<open>finite T\<close>)
lp15@63078
   322
        apply (rule_tac x=a in exI)
lp15@63078
   323
        using a0 assms
lp15@63078
   324
        apply (auto simp: cge0 a aeqx [symmetric] setsum.mono_neutral_right)
lp15@63078
   325
        done
lp15@63078
   326
    next
lp15@63078
   327
      case False
wenzelm@63148
   328
      define k where "k = setsum c (S - T)"
lp15@63078
   329
      have "k > 0" using False
lp15@63078
   330
        unfolding k_def by (metis DiffD1 antisym_conv cge0 setsum_nonneg not_less)
lp15@63078
   331
      have weq_sumsum: "w = setsum (\<lambda>x. c x *\<^sub>R x) T + setsum (\<lambda>x. c x *\<^sub>R x) (S - T)"
lp15@63078
   332
        by (metis (no_types) add.commute S(1) S(2) setsum.subset_diff sumci_xy weq)
lp15@63078
   333
      show ?thesis
lp15@63078
   334
      proof (cases "k = 1")
lp15@63078
   335
        case True
lp15@63078
   336
        then have "setsum c T = 0"
lp15@63078
   337
          by (simp add: S k_def setsum_diff sumc1)
lp15@63078
   338
        then have [simp]: "setsum c (S - T) = 1"
lp15@63078
   339
          by (simp add: S setsum_diff sumc1)
lp15@63078
   340
        have ci0: "\<And>i. i \<in> T \<Longrightarrow> c i = 0"
wenzelm@63145
   341
          by (meson \<open>finite T\<close> \<open>setsum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 setsum_nonneg_eq_0_iff subsetCE)
lp15@63078
   342
        then have [simp]: "(\<Sum>i\<in>S-T. c i *\<^sub>R i) = w"
lp15@63078
   343
          by (simp add: weq_sumsum)
lp15@63078
   344
        have "w \<in> convex hull (S - T)"
lp15@63078
   345
          apply (simp add: convex_hull_finite fin)
lp15@63078
   346
          apply (rule_tac x=c in exI)
lp15@63078
   347
          apply (auto simp: cge0 weq True k_def)
lp15@63078
   348
          done
lp15@63078
   349
        then show ?thesis
lp15@63078
   350
          using disj waff by blast
lp15@63078
   351
      next
lp15@63078
   352
        case False
lp15@63078
   353
        then have sumcf: "setsum c T = 1 - k"
lp15@63078
   354
          by (simp add: S k_def setsum_diff sumc1)
lp15@63078
   355
        have "(\<Sum>i\<in>T. c i *\<^sub>R i) /\<^sub>R (1 - k) \<in> convex hull T"
lp15@63078
   356
          apply (simp add: convex_hull_finite fin)
lp15@63078
   357
          apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
lp15@63078
   358
          apply auto
lp15@63078
   359
          apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) setsum_nonneg subsetCE)
Mathias@63918
   360
          apply (metis False mult.commute right_inverse right_minus_eq setsum_distrib_left sumcf)
lp15@63078
   361
          by (metis (mono_tags, lifting) scaleR_right.setsum scaleR_scaleR setsum.cong)
wenzelm@63145
   362
        with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - setsum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
lp15@63078
   363
          by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
lp15@63078
   364
        moreover have "inverse(k) *\<^sub>R (w - setsum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
lp15@63078
   365
          apply (simp add: weq_sumsum convex_hull_finite fin)
lp15@63078
   366
          apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
lp15@63078
   367
          using \<open>k > 0\<close> cge0
Mathias@63918
   368
          apply (auto simp: scaleR_right.setsum setsum_distrib_left [symmetric] k_def [symmetric])
lp15@63078
   369
          done
lp15@63078
   370
        ultimately show ?thesis
lp15@63078
   371
          using disj by blast
lp15@63078
   372
      qed
lp15@63078
   373
    qed
lp15@63078
   374
  qed
lp15@63078
   375
  have [simp]: "convex hull T \<subseteq> convex hull S"
lp15@63078
   376
    by (simp add: \<open>T \<subseteq> S\<close> hull_mono)
lp15@63078
   377
  show ?thesis
lp15@63078
   378
    using open_segment_commute by (auto simp: face_of_def intro: *)
lp15@63078
   379
qed
lp15@63078
   380
lp15@63078
   381
proposition face_of_convex_hull_insert:
lp15@63078
   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"
lp15@63078
   383
  apply (rule face_of_trans, blast)
lp15@63078
   384
  apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
lp15@63078
   385
  done
lp15@63078
   386
lp15@63078
   387
proposition face_of_affine_trivial:
lp15@63078
   388
    assumes "affine S" "T face_of S"
lp15@63078
   389
    shows "T = {} \<or> T = S"
lp15@63078
   390
proof (rule ccontr, clarsimp)
lp15@63078
   391
  assume "T \<noteq> {}" "T \<noteq> S"
lp15@63078
   392
  then obtain a where "a \<in> T" by auto
lp15@63078
   393
  then have "a \<in> S"
lp15@63078
   394
    using \<open>T face_of S\<close> face_of_imp_subset by blast
lp15@63078
   395
  have "S \<subseteq> T"
lp15@63078
   396
  proof
lp15@63078
   397
    fix b  assume "b \<in> S"
lp15@63078
   398
    show "b \<in> T"
lp15@63078
   399
    proof (cases "a = b")
lp15@63078
   400
      case True with \<open>a \<in> T\<close> show ?thesis by auto
lp15@63078
   401
    next
lp15@63078
   402
      case False
lp15@63078
   403
      then have "a \<in> open_segment (2 *\<^sub>R a - b) b"
lp15@63078
   404
        apply (auto simp: open_segment_def closed_segment_def)
lp15@63078
   405
        apply (rule_tac x="1/2" in exI)
lp15@63078
   406
        apply (simp add: algebra_simps)
lp15@63078
   407
        by (simp add: scaleR_2)
lp15@63078
   408
      moreover have "2 *\<^sub>R a - b \<in> S"
lp15@63078
   409
        by (rule mem_affine [OF \<open>affine S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>, of 2 "-1", simplified])
lp15@63078
   410
      moreover note \<open>b \<in> S\<close> \<open>a \<in> T\<close>
lp15@63078
   411
      ultimately show ?thesis
lp15@63078
   412
        by (rule face_ofD [OF \<open>T face_of S\<close>, THEN conjunct2])
lp15@63078
   413
    qed
lp15@63078
   414
  qed
lp15@63078
   415
  then show False
wenzelm@63145
   416
    using \<open>T \<noteq> S\<close> \<open>T face_of S\<close> face_of_imp_subset by blast
lp15@63078
   417
qed
lp15@63078
   418
lp15@63078
   419
lp15@63078
   420
lemma face_of_affine_eq:
lp15@63078
   421
   "affine S \<Longrightarrow> (T face_of S \<longleftrightarrow> T = {} \<or> T = S)"
lp15@63078
   422
using affine_imp_convex face_of_affine_trivial face_of_refl by auto
lp15@63078
   423
lp15@63078
   424
lp15@63078
   425
lemma Inter_faces_finite_altbound:
lp15@63078
   426
    fixes T :: "'a::euclidean_space set set"
lp15@63078
   427
    assumes cfaI: "\<And>c. c \<in> T \<Longrightarrow> c face_of S"
lp15@63078
   428
    shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<and> \<Inter>F' = \<Inter>T"
lp15@63078
   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'))")
lp15@63078
   430
  case True
lp15@63078
   431
  then obtain c where c:
lp15@63078
   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')"
lp15@63078
   433
    by metis
wenzelm@63148
   434
  define d where "d = rec_nat {c{}} (\<lambda>n r. insert (c r) r)"
lp15@63078
   435
  have [simp]: "d 0 = {c {}}"
lp15@63078
   436
    by (simp add: d_def)
lp15@63078
   437
  have dSuc [simp]: "\<And>n. d (Suc n) = insert (c (d n)) (d n)"
lp15@63078
   438
    by (simp add: d_def)
lp15@63078
   439
  have dn_notempty: "d n \<noteq> {}" for n
lp15@63078
   440
    by (induction n) auto
lp15@63078
   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
lp15@63078
   442
  using that
lp15@63078
   443
  proof (induction n)
lp15@63078
   444
    case 0
lp15@63078
   445
    then show ?case by (simp add: c)
lp15@63078
   446
  next
lp15@63078
   447
    case (Suc n)
lp15@63078
   448
    then show ?case by (auto simp: c card_insert_if)
lp15@63078
   449
  qed
lp15@63078
   450
  have aff_dim_le: "aff_dim(\<Inter>(d n)) \<le> DIM('a) - int n" if "n \<le> DIM('a) + 2" for n
lp15@63078
   451
  using that
lp15@63078
   452
  proof (induction n)
lp15@63078
   453
    case 0
lp15@63078
   454
    then show ?case
lp15@63078
   455
      by (simp add: aff_dim_le_DIM)
lp15@63078
   456
  next
lp15@63078
   457
    case (Suc n)
lp15@63078
   458
    have fs: "\<Inter>d (Suc n) face_of S"
lp15@63078
   459
      by (meson Suc.prems cfaI dn_le_Suc dn_notempty face_of_Inter subsetCE)
lp15@63078
   460
    have condn: "convex (\<Inter>d n)"
lp15@63078
   461
      using Suc.prems nat_le_linear not_less_eq_eq
lp15@63078
   462
      by (blast intro: face_of_imp_convex cfaI convex_Inter dest: dn_le_Suc)
lp15@63078
   463
    have fdn: "\<Inter>d (Suc n) face_of \<Inter>d n"
lp15@63078
   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)
lp15@63078
   465
    have ne: "\<Inter>d (Suc n) \<noteq> \<Inter>d n"
lp15@63078
   466
      by (metis (no_types, lifting) Suc.prems Suc_leD c complete_lattice_class.Inf_insert dSuc dn_le_Suc less_irrefl order.trans)
lp15@63078
   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)"
lp15@63078
   468
      by arith
lp15@63078
   469
    have "aff_dim (\<Inter>d (Suc n)) < aff_dim (\<Inter>d n)"
lp15@63078
   470
      by (rule face_of_aff_dim_lt [OF condn fdn ne])
lp15@63078
   471
    moreover have "aff_dim (\<Inter>d n) \<le> int (DIM('a)) - int n"
lp15@63078
   472
      using Suc by auto
lp15@63078
   473
    ultimately
lp15@63078
   474
    have "aff_dim (\<Inter>d (Suc n)) \<le> int (DIM('a)) - (n+1)" by arith
lp15@63078
   475
    then show ?case by linarith
lp15@63078
   476
  qed
lp15@63078
   477
  have "aff_dim (\<Inter>d (DIM('a) + 2)) \<le> -2"
lp15@63078
   478
      using aff_dim_le [OF order_refl] by simp
lp15@63078
   479
  with aff_dim_geq [of "\<Inter>d (DIM('a) + 2)"] show ?thesis
lp15@63078
   480
    using order.trans by fastforce
lp15@63078
   481
next
lp15@63078
   482
  case False
lp15@63078
   483
  then show ?thesis
lp15@63078
   484
    apply simp
lp15@63078
   485
    apply (erule ex_forward)
lp15@63078
   486
    by blast
lp15@63078
   487
qed
lp15@63078
   488
lp15@63078
   489
lemma faces_of_translation:
lp15@63078
   490
   "{F. F face_of image (\<lambda>x. a + x) S} = image (image (\<lambda>x. a + x)) {F. F face_of S}"
lp15@63078
   491
apply (rule subset_antisym, clarify)
lp15@63078
   492
apply (auto simp: image_iff)
lp15@63078
   493
apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
lp15@63078
   494
done
lp15@63078
   495
lp15@63078
   496
proposition face_of_Times:
lp15@63078
   497
  assumes "F face_of S" and "F' face_of S'"
lp15@63078
   498
    shows "(F \<times> F') face_of (S \<times> S')"
lp15@63078
   499
proof -
lp15@63078
   500
  have "F \<times> F' \<subseteq> S \<times> S'"
lp15@63078
   501
    using assms [unfolded face_of_def] by blast
lp15@63078
   502
  moreover
lp15@63078
   503
  have "convex (F \<times> F')"
lp15@63078
   504
    using assms [unfolded face_of_def] by (blast intro: convex_Times)
lp15@63078
   505
  moreover
lp15@63078
   506
    have "a \<in> F \<and> a' \<in> F' \<and> b \<in> F \<and> b' \<in> F'"
lp15@63078
   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')"
lp15@63078
   508
       for a b a' b' x
lp15@63078
   509
  proof (cases "b=a \<or> b'=a'")
lp15@63078
   510
    case True with that show ?thesis
lp15@63078
   511
      using assms
lp15@63078
   512
      by (force simp: in_segment dest: face_ofD)
lp15@63078
   513
  next
lp15@63078
   514
    case False with assms [unfolded face_of_def] that show ?thesis
lp15@63078
   515
      by (blast dest!: open_segment_PairD)
lp15@63078
   516
  qed
lp15@63078
   517
  ultimately show ?thesis
lp15@63078
   518
    unfolding face_of_def by blast
lp15@63078
   519
qed
lp15@63078
   520
lp15@63078
   521
corollary face_of_Times_decomp:
lp15@63078
   522
    fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
lp15@63078
   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')"
lp15@63078
   524
     (is "?lhs = ?rhs")
lp15@63078
   525
proof
lp15@63078
   526
  assume c: ?lhs
lp15@63078
   527
  show ?rhs
lp15@63078
   528
  proof (cases "c = {}")
lp15@63078
   529
    case True then show ?thesis by auto
lp15@63078
   530
  next
lp15@63078
   531
    case False
lp15@63078
   532
    have 1: "fst ` c \<subseteq> S" "snd ` c \<subseteq> S'"
lp15@63078
   533
      using c face_of_imp_subset by fastforce+
lp15@63078
   534
    have "convex c"
lp15@63078
   535
      using c by (metis face_of_imp_convex)
lp15@63078
   536
    have conv: "convex (fst ` c)" "convex (snd ` c)"
lp15@63078
   537
      by (simp_all add: \<open>convex c\<close> convex_linear_image fst_linear snd_linear)
lp15@63078
   538
    have fstab: "a \<in> fst ` c \<and> b \<in> fst ` c"
lp15@63078
   539
            if "a \<in> S" "b \<in> S" "x \<in> open_segment a b" "(x,x') \<in> c" for a b x x'
lp15@63078
   540
    proof -
lp15@63078
   541
      have *: "(x,x') \<in> open_segment (a,x') (b,x')"
lp15@63078
   542
        using that by (auto simp: in_segment)
lp15@63078
   543
      show ?thesis
lp15@63078
   544
        using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
lp15@63078
   545
    qed
lp15@63078
   546
    have fst: "fst ` c face_of S"
lp15@63078
   547
      by (force simp: face_of_def 1 conv fstab)
lp15@63078
   548
    have sndab: "a' \<in> snd ` c \<and> b' \<in> snd ` c"
lp15@63078
   549
            if "a' \<in> S'" "b' \<in> S'" "x' \<in> open_segment a' b'" "(x,x') \<in> c" for a' b' x x'
lp15@63078
   550
    proof -
lp15@63078
   551
      have *: "(x,x') \<in> open_segment (x,a') (x,b')"
lp15@63078
   552
        using that by (auto simp: in_segment)
lp15@63078
   553
      show ?thesis
lp15@63078
   554
        using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
lp15@63078
   555
    qed
lp15@63078
   556
    have snd: "snd ` c face_of S'"
lp15@63078
   557
      by (force simp: face_of_def 1 conv sndab)
lp15@63078
   558
    have cc: "rel_interior c \<subseteq> rel_interior (fst ` c) \<times> rel_interior (snd ` c)"
lp15@63078
   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])
lp15@63078
   560
    have "c = fst ` c \<times> snd ` c"
lp15@63078
   561
      apply (rule face_of_eq [OF c])
lp15@63078
   562
      apply (simp_all add: face_of_Times rel_interior_Times conv fst snd)
lp15@63078
   563
      using False rel_interior_eq_empty \<open>convex c\<close> cc
lp15@63078
   564
      apply blast
lp15@63078
   565
      done
lp15@63078
   566
    with fst snd show ?thesis by metis
lp15@63078
   567
  qed
lp15@63078
   568
next
lp15@63078
   569
  assume ?rhs with face_of_Times show ?lhs by auto
lp15@63078
   570
qed
lp15@63078
   571
lp15@63078
   572
lemma face_of_Times_eq:
lp15@63078
   573
    fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
lp15@63078
   574
    shows "(F \<times> F') face_of (S \<times> S') \<longleftrightarrow>
lp15@63078
   575
           F = {} \<or> F' = {} \<or> F face_of S \<and> F' face_of S'"
lp15@63078
   576
by (auto simp: face_of_Times_decomp times_eq_iff)
lp15@63078
   577
lp15@63078
   578
lemma hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
lp15@63078
   579
proof -
lp15@63078
   580
  have "{x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
lp15@63078
   581
    by auto
lp15@63078
   582
  with face_of_Int_supporting_hyperplane_le [OF convex_halfspace_le [of a b], of a b]
lp15@63078
   583
  show ?thesis by auto
lp15@63078
   584
qed
lp15@63078
   585
lp15@63078
   586
lemma hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
lp15@63078
   587
proof -
lp15@63078
   588
  have "{x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
lp15@63078
   589
    by auto
lp15@63078
   590
  with face_of_Int_supporting_hyperplane_ge [OF convex_halfspace_ge [of b a], of b a]
lp15@63078
   591
  show ?thesis by auto
lp15@63078
   592
qed
lp15@63078
   593
lp15@63078
   594
lemma face_of_halfspace_le:
lp15@63078
   595
  fixes a :: "'n::euclidean_space"
lp15@63078
   596
  shows "F face_of {x. a \<bullet> x \<le> b} \<longleftrightarrow>
lp15@63078
   597
         F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b}"
lp15@63078
   598
     (is "?lhs = ?rhs")
lp15@63078
   599
proof (cases "a = 0")
lp15@63078
   600
  case True then show ?thesis
lp15@63078
   601
    using face_of_affine_eq affine_UNIV by auto
lp15@63078
   602
next
lp15@63078
   603
  case False
lp15@63078
   604
  then have ine: "interior {x. a \<bullet> x \<le> b} \<noteq> {}"
lp15@63078
   605
    using halfspace_eq_empty_lt interior_halfspace_le by blast
lp15@63078
   606
  show ?thesis
lp15@63078
   607
  proof
lp15@63078
   608
    assume L: ?lhs
lp15@63078
   609
    have "F \<noteq> {x. a \<bullet> x \<le> b} \<Longrightarrow> F face_of {x. a \<bullet> x = b}"
lp15@63078
   610
      using False
lp15@63078
   611
      apply (simp add: frontier_halfspace_le [symmetric] rel_frontier_nonempty_interior [OF ine, symmetric])
lp15@63078
   612
      apply (rule face_of_subset [OF L])
lp15@63078
   613
      apply (simp add: face_of_subset_rel_frontier [OF L])
lp15@63078
   614
      apply (force simp: rel_frontier_def closed_halfspace_le)
lp15@63078
   615
      done
lp15@63078
   616
    with L show ?rhs
lp15@63078
   617
      using affine_hyperplane face_of_affine_eq by blast
lp15@63078
   618
  next
lp15@63078
   619
    assume ?rhs
lp15@63078
   620
    then show ?lhs
lp15@63078
   621
      by (metis convex_halfspace_le empty_face_of face_of_refl hyperplane_face_of_halfspace_le)
lp15@63078
   622
  qed
lp15@63078
   623
qed
lp15@63078
   624
lp15@63078
   625
lemma face_of_halfspace_ge:
lp15@63078
   626
  fixes a :: "'n::euclidean_space"
lp15@63078
   627
  shows "F face_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow>
lp15@63078
   628
         F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<ge> b}"
lp15@63078
   629
using face_of_halfspace_le [of F "-a" "-b"] by simp
lp15@63078
   630
lp15@63078
   631
subsection\<open>Exposed faces\<close>
lp15@63078
   632
lp15@63078
   633
text\<open>That is, faces that are intersection with supporting hyperplane\<close>
lp15@63078
   634
lp15@63078
   635
definition exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
lp15@63078
   636
                               (infixr "(exposed'_face'_of)" 50)
lp15@63078
   637
  where "T exposed_face_of S \<longleftrightarrow>
lp15@63078
   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})"
lp15@63078
   639
lp15@63078
   640
lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
lp15@63078
   641
  apply (simp add: exposed_face_of_def)
lp15@63078
   642
  apply (rule_tac x=0 in exI)
lp15@63078
   643
  apply (rule_tac x=1 in exI, force)
lp15@63078
   644
  done
lp15@63078
   645
lp15@63078
   646
lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
lp15@63078
   647
  apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
lp15@63078
   648
  apply (rule_tac x=0 in exI)+
lp15@63078
   649
  apply force
lp15@63078
   650
  done
lp15@63078
   651
lp15@63078
   652
lemma exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
lp15@63078
   653
  by simp
lp15@63078
   654
lp15@63078
   655
lemma exposed_face_of:
lp15@63078
   656
    "T exposed_face_of S \<longleftrightarrow>
lp15@63078
   657
     T face_of S \<and>
lp15@63078
   658
     (T = {} \<or> T = S \<or>
lp15@63078
   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}))"
lp15@63078
   660
proof (cases "T = {}")
lp15@63078
   661
  case True then show ?thesis
lp15@63078
   662
    by simp
lp15@63078
   663
next
lp15@63078
   664
  case False
lp15@63078
   665
  show ?thesis
lp15@63078
   666
  proof (cases "T = S")
lp15@63078
   667
    case True then show ?thesis
lp15@63078
   668
      by (simp add: face_of_refl_eq)
lp15@63078
   669
  next
lp15@63078
   670
    case False
lp15@63078
   671
    with \<open>T \<noteq> {}\<close> show ?thesis
lp15@63078
   672
      apply (auto simp: exposed_face_of_def)
lp15@63078
   673
      apply (metis inner_zero_left)
lp15@63078
   674
      done
lp15@63078
   675
  qed
lp15@63078
   676
qed
lp15@63078
   677
lp15@63078
   678
lemma exposed_face_of_Int_supporting_hyperplane_le:
lp15@63078
   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"
lp15@63078
   680
by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)
lp15@63078
   681
lp15@63078
   682
lemma exposed_face_of_Int_supporting_hyperplane_ge:
lp15@63078
   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"
lp15@63078
   684
using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp
lp15@63078
   685
lp15@63078
   686
proposition exposed_face_of_Int:
lp15@63078
   687
  assumes "T exposed_face_of S"
lp15@63078
   688
      and "u exposed_face_of S"
lp15@63078
   689
    shows "(T \<inter> u) exposed_face_of S"
lp15@63078
   690
proof -
lp15@63078
   691
  obtain a b where T: "S \<inter> {x. a \<bullet> x = b} face_of S"
lp15@63078
   692
               and S: "S \<subseteq> {x. a \<bullet> x \<le> b}"
lp15@63078
   693
               and teq: "T = S \<inter> {x. a \<bullet> x = b}"
lp15@63078
   694
    using assms by (auto simp: exposed_face_of_def)
lp15@63078
   695
  obtain a' b' where u: "S \<inter> {x. a' \<bullet> x = b'} face_of S"
lp15@63078
   696
                 and s': "S \<subseteq> {x. a' \<bullet> x \<le> b'}"
lp15@63078
   697
                 and ueq: "u = S \<inter> {x. a' \<bullet> x = b'}"
lp15@63078
   698
    using assms by (auto simp: exposed_face_of_def)
lp15@63078
   699
  have tu: "T \<inter> u face_of S"
lp15@63078
   700
    using T teq u ueq by (simp add: face_of_Int)
lp15@63078
   701
  have ss: "S \<subseteq> {x. (a + a') \<bullet> x \<le> b + b'}"
lp15@63078
   702
    using S s' by (force simp: inner_left_distrib)
lp15@63078
   703
  show ?thesis
lp15@63078
   704
    apply (simp add: exposed_face_of_def tu)
lp15@63078
   705
    apply (rule_tac x="a+a'" in exI)
lp15@63078
   706
    apply (rule_tac x="b+b'" in exI)
lp15@63078
   707
    using S s'
lp15@63078
   708
    apply (fastforce simp: ss inner_left_distrib teq ueq)
lp15@63078
   709
    done
lp15@63078
   710
qed
lp15@63078
   711
lp15@63078
   712
proposition exposed_face_of_Inter:
lp15@63078
   713
    fixes P :: "'a::euclidean_space set set"
lp15@63078
   714
  assumes "P \<noteq> {}"
lp15@63078
   715
      and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S"
lp15@63078
   716
    shows "\<Inter>P exposed_face_of S"
lp15@63078
   717
proof -
lp15@63078
   718
  obtain Q where "finite Q" and QsubP: "Q \<subseteq> P" "card Q \<le> DIM('a) + 2" and IntQ: "\<Inter>Q = \<Inter>P"
lp15@63078
   719
    using Inter_faces_finite_altbound [of P S] assms [unfolded exposed_face_of]
lp15@63078
   720
    by force
lp15@63078
   721
  show ?thesis
lp15@63078
   722
  proof (cases "Q = {}")
lp15@63078
   723
    case True then show ?thesis
lp15@63078
   724
      by (metis Inf_empty Inf_lower IntQ assms ex_in_conv subset_antisym top_greatest)
lp15@63078
   725
  next
lp15@63078
   726
    case False
lp15@63078
   727
    have "Q \<subseteq> {T. T exposed_face_of S}"
lp15@63078
   728
      using QsubP assms by blast
lp15@63078
   729
    moreover have "Q \<subseteq> {T. T exposed_face_of S} \<Longrightarrow> \<Inter>Q exposed_face_of S"
lp15@63078
   730
      using \<open>finite Q\<close> False
lp15@63078
   731
      apply (induction Q rule: finite_induct)
lp15@63078
   732
      using exposed_face_of_Int apply fastforce+
lp15@63078
   733
      done
lp15@63078
   734
    ultimately show ?thesis
lp15@63078
   735
      by (simp add: IntQ)
lp15@63078
   736
  qed
lp15@63078
   737
qed
lp15@63078
   738
lp15@63078
   739
proposition exposed_face_of_sums:
lp15@63078
   740
  assumes "convex S" and "convex T"
lp15@63078
   741
      and "F exposed_face_of {x + y | x y. x \<in> S \<and> y \<in> T}"
lp15@63078
   742
          (is "F exposed_face_of ?ST")
lp15@63078
   743
  obtains k l
lp15@63078
   744
    where "k exposed_face_of S" "l exposed_face_of T"
lp15@63078
   745
          "F = {x + y | x y. x \<in> k \<and> y \<in> l}"
lp15@63078
   746
proof (cases "F = {}")
lp15@63078
   747
  case True then show ?thesis
lp15@63078
   748
    using that by blast
lp15@63078
   749
next
lp15@63078
   750
  case False
lp15@63078
   751
  show ?thesis
lp15@63078
   752
  proof (cases "F = ?ST")
lp15@63078
   753
    case True then show ?thesis
lp15@63078
   754
      using assms exposed_face_of_refl_eq that by blast
lp15@63078
   755
  next
lp15@63078
   756
    case False
lp15@63078
   757
    obtain p where "p \<in> F" using \<open>F \<noteq> {}\<close> by blast
lp15@63078
   758
    moreover
lp15@63078
   759
    obtain u z where T: "?ST \<inter> {x. u \<bullet> x = z} face_of ?ST"
lp15@63078
   760
                 and S: "?ST \<subseteq> {x. u \<bullet> x \<le> z}"
lp15@63078
   761
                 and feq: "F = ?ST \<inter> {x. u \<bullet> x = z}"
lp15@63078
   762
      using assms by (auto simp: exposed_face_of_def)
lp15@63078
   763
    ultimately obtain a0 b0
lp15@63078
   764
            where p: "p = a0 + b0" and "a0 \<in> S" "b0 \<in> T" and z: "u \<bullet> p = z"
lp15@63078
   765
      by auto
lp15@63078
   766
    have lez: "u \<bullet> (x + y) \<le> z" if "x \<in> S" "y \<in> T" for x y
lp15@63078
   767
      using S that by auto
lp15@63078
   768
    have sef: "S \<inter> {x. u \<bullet> x = u \<bullet> a0} exposed_face_of S"
lp15@63078
   769
      apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex S\<close>])
lp15@63078
   770
      apply (metis p z add_le_cancel_right inner_right_distrib lez [OF _ \<open>b0 \<in> T\<close>])
lp15@63078
   771
      done
lp15@63078
   772
    have tef: "T \<inter> {x. u \<bullet> x = u \<bullet> b0} exposed_face_of T"
lp15@63078
   773
      apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex T\<close>])
lp15@63078
   774
      apply (metis p z add.commute add_le_cancel_right inner_right_distrib lez [OF \<open>a0 \<in> S\<close>])
lp15@63078
   775
      done
lp15@63078
   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"
lp15@63078
   777
      by (auto simp: feq) (metis inner_right_distrib p z)
lp15@63078
   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}"
lp15@63078
   779
      apply (auto simp: feq)
lp15@63078
   780
      apply (rename_tac x y)
lp15@63078
   781
      apply (rule_tac x=x in exI)
lp15@63078
   782
      apply (rule_tac x=y in exI, simp)
lp15@63078
   783
      using z p \<open>a0 \<in> S\<close> \<open>b0 \<in> T\<close>
lp15@63078
   784
      apply clarify
lp15@63078
   785
      apply (simp add: inner_right_distrib)
lp15@63078
   786
      apply (metis add_le_cancel_right antisym lez [unfolded inner_right_distrib] add.commute)
lp15@63078
   787
      done
lp15@63078
   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}}"
lp15@63078
   789
      by blast
lp15@63078
   790
    then show ?thesis
lp15@63078
   791
      by (rule that [OF sef tef])
lp15@63078
   792
  qed
lp15@63078
   793
qed
lp15@63078
   794
lp15@63078
   795
subsection\<open>Extreme points of a set: its singleton faces\<close>
lp15@63078
   796
lp15@63078
   797
definition extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
lp15@63078
   798
                               (infixr "(extreme'_point'_of)" 50)
lp15@63078
   799
  where "x extreme_point_of S \<longleftrightarrow>
lp15@63078
   800
         x \<in> S \<and> (\<forall>a \<in> S. \<forall>b \<in> S. x \<notin> open_segment a b)"
lp15@63078
   801
lp15@63078
   802
lemma extreme_point_of_stillconvex:
lp15@63078
   803
   "convex S \<Longrightarrow> (x extreme_point_of S \<longleftrightarrow> x \<in> S \<and> convex(S - {x}))"
lp15@63078
   804
  by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)
lp15@63078
   805
lp15@63078
   806
lemma face_of_singleton:
lp15@63078
   807
   "{x} face_of S \<longleftrightarrow> x extreme_point_of S"
lp15@63078
   808
by (fastforce simp add: extreme_point_of_def face_of_def)
lp15@63078
   809
lp15@63078
   810
lemma extreme_point_not_in_REL_INTERIOR:
lp15@63078
   811
    fixes S :: "'a::real_normed_vector set"
lp15@63078
   812
    shows "\<lbrakk>x extreme_point_of S; S \<noteq> {x}\<rbrakk> \<Longrightarrow> x \<notin> rel_interior S"
lp15@63078
   813
apply (simp add: face_of_singleton [symmetric])
lp15@63078
   814
apply (blast dest: face_of_disjoint_rel_interior)
lp15@63078
   815
done
lp15@63078
   816
lp15@63078
   817
lemma extreme_point_not_in_interior:
lp15@63078
   818
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
lp15@63078
   819
    shows "x extreme_point_of S \<Longrightarrow> x \<notin> interior S"
lp15@63078
   820
apply (case_tac "S = {x}")
lp15@63078
   821
apply (simp add: empty_interior_finite)
lp15@63078
   822
by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)
lp15@63078
   823
lp15@63078
   824
lemma extreme_point_of_face:
lp15@63078
   825
     "F face_of S \<Longrightarrow> v extreme_point_of F \<longleftrightarrow> v extreme_point_of S \<and> v \<in> F"
lp15@63078
   826
  by (meson empty_subsetI face_of_face face_of_singleton insert_subset)
lp15@63078
   827
lp15@63078
   828
lemma extreme_point_of_convex_hull:
lp15@63078
   829
   "x extreme_point_of (convex hull S) \<Longrightarrow> x \<in> S"
lp15@63078
   830
apply (simp add: extreme_point_of_stillconvex)
lp15@63078
   831
using hull_minimal [of S "(convex hull S) - {x}" convex]
lp15@63078
   832
using hull_subset [of S convex]
lp15@63078
   833
apply blast
lp15@63078
   834
done
lp15@63078
   835
lp15@63078
   836
lemma extreme_points_of_convex_hull:
lp15@63078
   837
   "{x. x extreme_point_of (convex hull S)} \<subseteq> S"
lp15@63078
   838
using extreme_point_of_convex_hull by auto
lp15@63078
   839
lp15@63078
   840
lemma extreme_point_of_empty [simp]: "~ (x extreme_point_of {})"
lp15@63078
   841
  by (simp add: extreme_point_of_def)
lp15@63078
   842
lp15@63078
   843
lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
lp15@63078
   844
  using extreme_point_of_stillconvex by auto
lp15@63078
   845
lp15@63078
   846
lemma extreme_point_of_translation_eq:
lp15@63078
   847
   "(a + x) extreme_point_of (image (\<lambda>x. a + x) S) \<longleftrightarrow> x extreme_point_of S"
lp15@63078
   848
by (auto simp: extreme_point_of_def)
lp15@63078
   849
lp15@63078
   850
lemma extreme_points_of_translation:
lp15@63078
   851
   "{x. x extreme_point_of (image (\<lambda>x. a + x) S)} =
lp15@63078
   852
    (\<lambda>x. a + x) ` {x. x extreme_point_of S}"
lp15@63078
   853
using extreme_point_of_translation_eq
lp15@63078
   854
by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
lp15@63078
   855
lp15@63078
   856
lemma extreme_point_of_Int:
lp15@63078
   857
   "\<lbrakk>x extreme_point_of S; x extreme_point_of T\<rbrakk> \<Longrightarrow> x extreme_point_of (S \<inter> T)"
lp15@63078
   858
by (simp add: extreme_point_of_def)
lp15@63078
   859
lp15@63078
   860
lemma extreme_point_of_Int_supporting_hyperplane_le:
lp15@63078
   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"
lp15@63078
   862
apply (simp add: face_of_singleton [symmetric])
lp15@63078
   863
by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)
lp15@63078
   864
lp15@63078
   865
lemma extreme_point_of_Int_supporting_hyperplane_ge:
lp15@63078
   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"
lp15@63078
   867
apply (simp add: face_of_singleton [symmetric])
lp15@63078
   868
by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)
lp15@63078
   869
lp15@63078
   870
lemma exposed_point_of_Int_supporting_hyperplane_le:
lp15@63078
   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"
lp15@63078
   872
apply (simp add: exposed_face_of_def face_of_singleton)
lp15@63078
   873
apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
lp15@63078
   874
done
lp15@63078
   875
lp15@63078
   876
lemma exposed_point_of_Int_supporting_hyperplane_ge:
lp15@63078
   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"
lp15@63078
   878
using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
lp15@63078
   879
by simp
lp15@63078
   880
lp15@63078
   881
lemma extreme_point_of_convex_hull_insert:
lp15@63078
   882
   "\<lbrakk>finite S; a \<notin> convex hull S\<rbrakk> \<Longrightarrow> a extreme_point_of (convex hull (insert a S))"
lp15@63078
   883
apply (case_tac "a \<in> S")
lp15@63078
   884
apply (simp add: hull_inc)
lp15@63078
   885
using face_of_convex_hulls [of "insert a S" "{a}"]
lp15@63078
   886
apply (auto simp: face_of_singleton hull_same)
lp15@63078
   887
done
lp15@63078
   888
lp15@63078
   889
subsection\<open>Facets\<close>
lp15@63078
   890
lp15@63078
   891
definition facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
lp15@63078
   892
                    (infixr "(facet'_of)" 50)
lp15@63078
   893
  where "F facet_of S \<longleftrightarrow> F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1"
lp15@63078
   894
lp15@63078
   895
lemma facet_of_empty [simp]: "~ S facet_of {}"
lp15@63078
   896
  by (simp add: facet_of_def)
lp15@63078
   897
lp15@63078
   898
lemma facet_of_irrefl [simp]: "~ S facet_of S "
lp15@63078
   899
  by (simp add: facet_of_def)
lp15@63078
   900
lp15@63078
   901
lemma facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
lp15@63078
   902
  by (simp add: facet_of_def)
lp15@63078
   903
lp15@63078
   904
lemma facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
lp15@63078
   905
  by (simp add: face_of_imp_subset facet_of_def)
lp15@63078
   906
lp15@63078
   907
lemma hyperplane_facet_of_halfspace_le:
lp15@63078
   908
   "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
lp15@63078
   909
unfolding facet_of_def hyperplane_eq_empty
lp15@63078
   910
by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
lp15@63078
   911
           DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)
lp15@63078
   912
lp15@63078
   913
lemma hyperplane_facet_of_halfspace_ge:
lp15@63078
   914
    "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<ge> b}"
lp15@63078
   915
unfolding facet_of_def hyperplane_eq_empty
lp15@63078
   916
by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
lp15@63078
   917
           DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)
lp15@63078
   918
lp15@63078
   919
lemma facet_of_halfspace_le:
lp15@63078
   920
    "F facet_of {x. a \<bullet> x \<le> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
lp15@63078
   921
    (is "?lhs = ?rhs")
lp15@63078
   922
proof
lp15@63078
   923
  assume c: ?lhs
lp15@63078
   924
  with c facet_of_irrefl show ?rhs
lp15@63078
   925
    by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
lp15@63078
   926
next
lp15@63078
   927
  assume ?rhs then show ?lhs
lp15@63078
   928
    by (simp add: hyperplane_facet_of_halfspace_le)
lp15@63078
   929
qed
lp15@63078
   930
lp15@63078
   931
lemma facet_of_halfspace_ge:
lp15@63078
   932
    "F facet_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
lp15@63078
   933
using facet_of_halfspace_le [of F "-a" "-b"] by simp
lp15@63078
   934
lp15@63078
   935
subsection \<open>Edges: faces of affine dimension 1\<close>
lp15@63078
   936
lp15@63078
   937
definition edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
lp15@63078
   938
  where "e edge_of S \<longleftrightarrow> e face_of S \<and> aff_dim e = 1"
lp15@63078
   939
lp15@63078
   940
lemma edge_of_imp_subset:
lp15@63078
   941
   "S edge_of T \<Longrightarrow> S \<subseteq> T"
lp15@63078
   942
by (simp add: edge_of_def face_of_imp_subset)
lp15@63078
   943
lp15@63078
   944
subsection\<open>Existence of extreme points\<close>
lp15@63078
   945
lp15@63078
   946
lemma different_norm_3_collinear_points:
lp15@63078
   947
  fixes a :: "'a::euclidean_space"
lp15@63078
   948
  assumes "x \<in> open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
lp15@63078
   949
  shows False
lp15@63078
   950
proof -
lp15@63078
   951
  obtain u where "norm ((1 - u) *\<^sub>R a + u *\<^sub>R b) = norm b"
lp15@63078
   952
             and "a \<noteq> b"
lp15@63078
   953
             and u01: "0 < u" "u < 1"
lp15@63078
   954
    using assms by (auto simp: open_segment_image_interval if_splits)
lp15@63078
   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 =
lp15@63078
   956
             (1 - u * u) *\<^sub>R (a \<bullet> a)"
lp15@63078
   957
    using assms by (simp add: norm_eq algebra_simps inner_commute)
lp15@63078
   958
  then have "(1 - u) *\<^sub>R ((1 - u) *\<^sub>R a \<bullet> a + (2 * u) *\<^sub>R  a \<bullet> b) =
lp15@63078
   959
             (1 - u) *\<^sub>R ((1 + u) *\<^sub>R (a \<bullet> a))"
lp15@63078
   960
    by (simp add: algebra_simps)
lp15@63078
   961
  then have "(1 - u) *\<^sub>R (a \<bullet> a) + (2 * u) *\<^sub>R (a \<bullet> b) = (1 + u) *\<^sub>R (a \<bullet> a)"
lp15@63078
   962
    using u01 by auto
lp15@63078
   963
  then have "a \<bullet> b = a \<bullet> a"
lp15@63078
   964
    using u01 by (simp add: algebra_simps)
lp15@63078
   965
  then have "a = b"
lp15@63078
   966
    using \<open>norm(a) = norm(b)\<close> norm_eq vector_eq by fastforce
lp15@63078
   967
  then show ?thesis
lp15@63078
   968
    using \<open>a \<noteq> b\<close> by force
lp15@63078
   969
qed
lp15@63078
   970
lp15@63078
   971
proposition extreme_point_exists_convex:
lp15@63078
   972
  fixes S :: "'a::euclidean_space set"
lp15@63078
   973
  assumes "compact S" "convex S" "S \<noteq> {}"
lp15@63078
   974
  obtains x where "x extreme_point_of S"
lp15@63078
   975
proof -
lp15@63078
   976
  obtain x where "x \<in> S" and xsup: "\<And>y. y \<in> S \<Longrightarrow> norm y \<le> norm x"
lp15@63078
   977
    using distance_attains_sup [of S 0] assms by auto
lp15@63078
   978
  have False if "a \<in> S" "b \<in> S" and x: "x \<in> open_segment a b" for a b
lp15@63078
   979
  proof -
lp15@63078
   980
    have noax: "norm a \<le> norm x" and nobx: "norm b \<le> norm x" using xsup that by auto
lp15@63078
   981
    have "a \<noteq> b"
lp15@63078
   982
      using empty_iff open_segment_idem x by auto
lp15@63078
   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
lp15@63078
   984
    proof -
lp15@63078
   985
      have "(1 - u) * na + u * nb < (1 - u) * norm x + u * nb"
lp15@63078
   986
        by (simp add: that)
lp15@63078
   987
      also have "... \<le> (1 - u) * norm x + u * norm x"
lp15@63078
   988
        by (simp add: that)
lp15@63078
   989
      finally have "(1 - u) * na + u * nb < (1 - u) * norm x + u * norm x" .
lp15@63078
   990
      then show ?thesis
lp15@63078
   991
      using scaleR_collapse [symmetric, of "norm x" u] by auto
lp15@63078
   992
    qed
lp15@63078
   993
    have "norm x < norm x" if "norm a < norm x"
lp15@63078
   994
      using x
lp15@63078
   995
      apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
lp15@63078
   996
      apply (rule norm_triangle_lt)
lp15@63078
   997
      apply (simp add: norm_mult)
lp15@63078
   998
      using * [of "norm a" "norm b"] nobx that
lp15@63078
   999
        apply blast
lp15@63078
  1000
      done
lp15@63078
  1001
    moreover have "norm x < norm x" if "norm b < norm x"
lp15@63078
  1002
      using x
lp15@63078
  1003
      apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
lp15@63078
  1004
      apply (rule norm_triangle_lt)
lp15@63078
  1005
      apply (simp add: norm_mult)
lp15@63078
  1006
      using * [of "norm b" "norm a" "1-u" for u] noax that
lp15@63078
  1007
        apply (simp add: add.commute)
lp15@63078
  1008
      done
lp15@63078
  1009
    ultimately have "~ (norm a < norm x) \<and> ~ (norm b < norm x)"
lp15@63078
  1010
      by auto
lp15@63078
  1011
    then show ?thesis
lp15@63078
  1012
      using different_norm_3_collinear_points noax nobx that(3) by fastforce
lp15@63078
  1013
  qed
lp15@63078
  1014
  then show ?thesis
lp15@63078
  1015
    apply (rule_tac x=x in that)
lp15@63078
  1016
    apply (force simp: extreme_point_of_def \<open>x \<in> S\<close>)
lp15@63078
  1017
    done
lp15@63078
  1018
qed
lp15@63078
  1019
lp15@63078
  1020
subsection\<open>Krein-Milman, the weaker form\<close>
lp15@63078
  1021
lp15@63078
  1022
proposition Krein_Milman:
lp15@63078
  1023
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1024
  assumes "compact S" "convex S"
lp15@63078
  1025
    shows "S = closure(convex hull {x. x extreme_point_of S})"
lp15@63078
  1026
proof (cases "S = {}")
lp15@63078
  1027
  case True then show ?thesis   by simp
lp15@63078
  1028
next
lp15@63078
  1029
  case False
lp15@63078
  1030
  have "closed S"
lp15@63078
  1031
    by (simp add: \<open>compact S\<close> compact_imp_closed)
lp15@63078
  1032
  have "closure (convex hull {x. x extreme_point_of S}) \<subseteq> S"
lp15@63078
  1033
    apply (rule closure_minimal [OF hull_minimal \<open>closed S\<close>])
lp15@63078
  1034
    using assms
lp15@63078
  1035
    apply (auto simp: extreme_point_of_def)
lp15@63078
  1036
    done
lp15@63078
  1037
  moreover have "u \<in> closure (convex hull {x. x extreme_point_of S})"
lp15@63078
  1038
                if "u \<in> S" for u
lp15@63078
  1039
  proof (rule ccontr)
lp15@63078
  1040
    assume unot: "u \<notin> closure(convex hull {x. x extreme_point_of S})"
lp15@63078
  1041
    then obtain a b where "a \<bullet> u < b"
lp15@63078
  1042
          and ab: "\<And>x. x \<in> closure(convex hull {x. x extreme_point_of S}) \<Longrightarrow> b < a \<bullet> x"
lp15@63078
  1043
      using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
lp15@63078
  1044
      by blast
lp15@63078
  1045
    have "continuous_on S (op \<bullet> a)"
lp15@63078
  1046
      by (rule continuous_intros)+
lp15@63078
  1047
    then obtain m where "m \<in> S" and m: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> m \<le> a \<bullet> y"
lp15@63078
  1048
      using continuous_attains_inf [of S "\<lambda>x. a \<bullet> x"] \<open>compact S\<close> \<open>u \<in> S\<close>
lp15@63078
  1049
      by auto
wenzelm@63148
  1050
    define T where "T = S \<inter> {x. a \<bullet> x = a \<bullet> m}"
lp15@63078
  1051
    have "m \<in> T"
lp15@63078
  1052
      by (simp add: T_def \<open>m \<in> S\<close>)
lp15@63078
  1053
    moreover have "compact T"
lp15@63078
  1054
      by (simp add: T_def compact_Int_closed [OF \<open>compact S\<close> closed_hyperplane])
lp15@63078
  1055
    moreover have "convex T"
lp15@63078
  1056
      by (simp add: T_def convex_Int [OF \<open>convex S\<close> convex_hyperplane])
lp15@63078
  1057
    ultimately obtain v where v: "v extreme_point_of T"
lp15@63078
  1058
      using extreme_point_exists_convex [of T] by auto
lp15@63078
  1059
    then have "{v} face_of T"
lp15@63078
  1060
      by (simp add: face_of_singleton)
lp15@63078
  1061
    also have "T face_of S"
lp15@63078
  1062
      by (simp add: T_def m face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
lp15@63078
  1063
    finally have "v extreme_point_of S"
lp15@63078
  1064
      by (simp add: face_of_singleton)
lp15@63078
  1065
    then have "b < a \<bullet> v"
lp15@63078
  1066
      using closure_subset by (simp add: closure_hull hull_inc ab)
lp15@63078
  1067
    then show False
lp15@63078
  1068
      using \<open>a \<bullet> u < b\<close> \<open>{v} face_of T\<close> face_of_imp_subset m T_def that by fastforce
lp15@63078
  1069
  qed
lp15@63078
  1070
  ultimately show ?thesis
lp15@63078
  1071
    by blast
lp15@63078
  1072
qed
lp15@63078
  1073
lp15@63078
  1074
text\<open>Now the sharper form.\<close>
lp15@63078
  1075
lp15@63078
  1076
lemma Krein_Milman_Minkowski_aux:
lp15@63078
  1077
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1078
  assumes n: "dim S = n" and S: "compact S" "convex S" "0 \<in> S"
lp15@63078
  1079
    shows "0 \<in> convex hull {x. x extreme_point_of S}"
lp15@63078
  1080
using n S
lp15@63078
  1081
proof (induction n arbitrary: S rule: less_induct)
lp15@63078
  1082
  case (less n S) show ?case
lp15@63078
  1083
  proof (cases "0 \<in> rel_interior S")
lp15@63078
  1084
    case True with Krein_Milman show ?thesis
lp15@63078
  1085
      by (metis subsetD convex_convex_hull convex_rel_interior_closure less.prems(2) less.prems(3) rel_interior_subset)
lp15@63078
  1086
  next
lp15@63078
  1087
    case False
lp15@63078
  1088
    have "rel_interior S \<noteq> {}"
lp15@63078
  1089
      by (simp add: rel_interior_convex_nonempty_aux less)
lp15@63078
  1090
    then obtain c where c: "c \<in> rel_interior S" by blast
lp15@63078
  1091
    obtain a where "a \<noteq> 0"
lp15@63078
  1092
              and le_ay: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> 0 \<le> a \<bullet> y"
lp15@63078
  1093
              and less_ay: "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> 0 < a \<bullet> y"
lp15@63078
  1094
      by (blast intro: supporting_hyperplane_rel_boundary intro!: less False)
lp15@63078
  1095
    have face: "S \<inter> {x. a \<bullet> x = 0} face_of S"
lp15@63078
  1096
      apply (rule face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
lp15@63078
  1097
      using le_ay by auto
lp15@63078
  1098
    then have co: "compact (S \<inter> {x. a \<bullet> x = 0})" "convex (S \<inter> {x. a \<bullet> x = 0})"
lp15@63078
  1099
      using less.prems by (blast intro: face_of_imp_compact face_of_imp_convex)+
lp15@63078
  1100
    have "a \<bullet> y = 0" if "y \<in> span (S \<inter> {x. a \<bullet> x = 0})" for y
lp15@63078
  1101
    proof -
lp15@63078
  1102
      have "y \<in> span {x. a \<bullet> x = 0}"
lp15@63078
  1103
        by (metis inf.cobounded2 span_mono subsetCE that)
lp15@63469
  1104
      then show ?thesis
lp15@63469
  1105
        by (blast intro: span_induct [OF _ subspace_hyperplane])
lp15@63078
  1106
    qed
lp15@63078
  1107
    then have "dim (S \<inter> {x. a \<bullet> x = 0}) < n"
lp15@63078
  1108
      by (metis (no_types) less_ay c subsetD dim_eq_span inf.strict_order_iff
lp15@63078
  1109
           inf_le1 \<open>dim S = n\<close> not_le rel_interior_subset span_0 span_clauses(1))
lp15@63078
  1110
    then have "0 \<in> convex hull {x. x extreme_point_of (S \<inter> {x. a \<bullet> x = 0})}"
lp15@63078
  1111
      by (rule less.IH) (auto simp: co less.prems)
lp15@63078
  1112
    then show ?thesis
lp15@63078
  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)
lp15@63078
  1114
  qed
lp15@63078
  1115
qed
lp15@63078
  1116
lp15@63078
  1117
lp15@63078
  1118
theorem Krein_Milman_Minkowski:
lp15@63078
  1119
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1120
  assumes "compact S" "convex S"
lp15@63078
  1121
    shows "S = convex hull {x. x extreme_point_of S}"
lp15@63078
  1122
proof
lp15@63078
  1123
  show "S \<subseteq> convex hull {x. x extreme_point_of S}"
lp15@63078
  1124
  proof
lp15@63078
  1125
    fix a assume [simp]: "a \<in> S"
lp15@63078
  1126
    have 1: "compact (op + (- a) ` S)"
lp15@63078
  1127
      by (simp add: \<open>compact S\<close> compact_translation)
lp15@63078
  1128
    have 2: "convex (op + (- a) ` S)"
lp15@63078
  1129
      by (simp add: \<open>convex S\<close> convex_translation)
lp15@63078
  1130
    show a_invex: "a \<in> convex hull {x. x extreme_point_of S}"
lp15@63078
  1131
      using Krein_Milman_Minkowski_aux [OF refl 1 2]
lp15@63078
  1132
            convex_hull_translation [of "-a"]
lp15@63078
  1133
      by (auto simp: extreme_points_of_translation translation_assoc)
lp15@63078
  1134
    qed
lp15@63078
  1135
next
lp15@63078
  1136
  show "convex hull {x. x extreme_point_of S} \<subseteq> S"
lp15@63078
  1137
  proof -
lp15@63078
  1138
    have "{a. a extreme_point_of S} \<subseteq> S"
lp15@63078
  1139
      using extreme_point_of_def by blast
lp15@63078
  1140
    then show ?thesis
lp15@63078
  1141
      by (simp add: \<open>convex S\<close> hull_minimal)
lp15@63078
  1142
  qed
lp15@63078
  1143
qed
lp15@63078
  1144
lp15@63078
  1145
lp15@63078
  1146
subsection\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
lp15@63078
  1147
lp15@63078
  1148
lemma Krein_Milman_polytope:
lp15@63078
  1149
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1150
  shows
lp15@63078
  1151
   "finite S
lp15@63078
  1152
       \<Longrightarrow> convex hull S =
lp15@63078
  1153
           convex hull {x. x extreme_point_of (convex hull S)}"
lp15@63078
  1154
by (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
lp15@63078
  1155
lp15@63078
  1156
lemma extreme_points_of_convex_hull_eq:
lp15@63078
  1157
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1158
  shows
lp15@63078
  1159
   "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
lp15@63078
  1160
        \<Longrightarrow> {x. x extreme_point_of (convex hull S)} = S"
lp15@63078
  1161
by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)
lp15@63078
  1162
lp15@63078
  1163
lp15@63078
  1164
lemma extreme_point_of_convex_hull_eq:
lp15@63078
  1165
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1166
  shows
lp15@63078
  1167
   "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
lp15@63078
  1168
    \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
lp15@63078
  1169
using extreme_points_of_convex_hull_eq by auto
lp15@63078
  1170
lp15@63078
  1171
lemma extreme_point_of_convex_hull_convex_independent:
lp15@63078
  1172
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1173
  assumes "compact S" and S: "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})"
lp15@63078
  1174
  shows "(x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
lp15@63078
  1175
proof -
lp15@63078
  1176
  have "convex hull T \<noteq> convex hull S" if "T \<subset> S" for T
lp15@63078
  1177
  proof -
lp15@63078
  1178
    obtain a where  "T \<subseteq> S" "a \<in> S" "a \<notin> T" using \<open>T \<subset> S\<close> by blast
lp15@63078
  1179
    then show ?thesis
lp15@63078
  1180
      by (metis (full_types) Diff_eq_empty_iff Diff_insert0 S hull_mono hull_subset insert_Diff_single subsetCE)
lp15@63078
  1181
  qed
lp15@63078
  1182
  then show ?thesis
lp15@63078
  1183
    by (rule extreme_point_of_convex_hull_eq [OF \<open>compact S\<close>])
lp15@63078
  1184
qed
lp15@63078
  1185
lp15@63078
  1186
lemma extreme_point_of_convex_hull_affine_independent:
lp15@63078
  1187
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1188
  shows
lp15@63078
  1189
   "~ affine_dependent S
lp15@63078
  1190
         \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
lp15@63078
  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)
lp15@63078
  1192
lp15@63078
  1193
text\<open>Elementary proofs exist, not requiring Euclidean spaces and all this development\<close>
lp15@63078
  1194
lemma extreme_point_of_convex_hull_2:
lp15@63078
  1195
  fixes x :: "'a::euclidean_space"
lp15@63078
  1196
  shows "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x = a \<or> x = b"
lp15@63078
  1197
proof -
lp15@63078
  1198
  have "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x \<in> {a,b}"
lp15@63078
  1199
    by (intro extreme_point_of_convex_hull_affine_independent affine_independent_2)
lp15@63078
  1200
  then show ?thesis
lp15@63078
  1201
    by simp
lp15@63078
  1202
qed
lp15@63078
  1203
lp15@63078
  1204
lemma extreme_point_of_segment:
lp15@63078
  1205
  fixes x :: "'a::euclidean_space"
lp15@63078
  1206
  shows
lp15@63078
  1207
   "x extreme_point_of closed_segment a b \<longleftrightarrow> x = a \<or> x = b"
lp15@63078
  1208
by (simp add: extreme_point_of_convex_hull_2 segment_convex_hull)
lp15@63078
  1209
lp15@63078
  1210
lemma face_of_convex_hull_subset:
lp15@63078
  1211
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1212
  assumes "compact S" and T: "T face_of (convex hull S)"
lp15@63078
  1213
  obtains s' where "s' \<subseteq> S" "T = convex hull s'"
lp15@63078
  1214
apply (rule_tac s' = "{x. x extreme_point_of T}" in that)
lp15@63078
  1215
using T extreme_point_of_convex_hull extreme_point_of_face apply blast
lp15@63078
  1216
by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)
lp15@63078
  1217
lp15@63078
  1218
lp15@63078
  1219
proposition face_of_convex_hull_affine_independent:
lp15@63078
  1220
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1221
  assumes "~ affine_dependent S"
lp15@63078
  1222
    shows "(T face_of (convex hull S) \<longleftrightarrow> (\<exists>c. c \<subseteq> S \<and> T = convex hull c))"
lp15@63078
  1223
          (is "?lhs = ?rhs")
lp15@63078
  1224
proof
lp15@63078
  1225
  assume ?lhs
lp15@63078
  1226
  then show ?rhs
lp15@63078
  1227
    by (meson \<open>T face_of convex hull S\<close> aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
lp15@63078
  1228
next
lp15@63078
  1229
  assume ?rhs
lp15@63078
  1230
  then obtain c where "c \<subseteq> S" and T: "T = convex hull c"
lp15@63078
  1231
    by blast
lp15@63078
  1232
  have "affine hull c \<inter> affine hull (S - c) = {}"
lp15@63078
  1233
    apply (rule disjoint_affine_hull [OF assms \<open>c \<subseteq> S\<close>], auto)
lp15@63078
  1234
    done
lp15@63078
  1235
  then have "affine hull c \<inter> convex hull (S - c) = {}"
lp15@63078
  1236
    using convex_hull_subset_affine_hull by fastforce
lp15@63078
  1237
  then show ?lhs
lp15@63078
  1238
    by (metis face_of_convex_hulls \<open>c \<subseteq> S\<close> aff_independent_finite assms T)
lp15@63078
  1239
qed
lp15@63078
  1240
lp15@63078
  1241
lemma facet_of_convex_hull_affine_independent:
lp15@63078
  1242
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1243
  assumes "~ affine_dependent S"
lp15@63078
  1244
    shows "T facet_of (convex hull S) \<longleftrightarrow>
lp15@63078
  1245
           T \<noteq> {} \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u}))"
lp15@63078
  1246
          (is "?lhs = ?rhs")
lp15@63078
  1247
proof
lp15@63078
  1248
  assume ?lhs
lp15@63078
  1249
  then have "T face_of (convex hull S)" "T \<noteq> {}"
lp15@63078
  1250
        and afft: "aff_dim T = aff_dim (convex hull S) - 1"
lp15@63078
  1251
    by (auto simp: facet_of_def)
lp15@63078
  1252
  then obtain c where "c \<subseteq> S" and c: "T = convex hull c"
lp15@63078
  1253
    by (auto simp: face_of_convex_hull_affine_independent [OF assms])
lp15@63078
  1254
  then have affs: "aff_dim S = aff_dim c + 1"
lp15@63078
  1255
    by (metis aff_dim_convex_hull afft eq_diff_eq)
lp15@63078
  1256
  have "~ affine_dependent c"
lp15@63078
  1257
    using \<open>c \<subseteq> S\<close> affine_dependent_subset assms by blast
lp15@63078
  1258
  with affs have "card (S - c) = 1"
lp15@63078
  1259
    apply (simp add: aff_dim_affine_independent [symmetric] aff_dim_convex_hull)
lp15@63078
  1260
    by (metis aff_dim_affine_independent aff_independent_finite One_nat_def \<open>c \<subseteq> S\<close> add.commute
lp15@63078
  1261
                add_diff_cancel_right' assms card_Diff_subset card_mono of_nat_1 of_nat_diff of_nat_eq_iff)
lp15@63078
  1262
  then obtain u where u: "u \<in> S - c"
lp15@63078
  1263
    by (metis DiffI \<open>c \<subseteq> S\<close> aff_independent_finite assms cancel_comm_monoid_add_class.diff_cancel
lp15@63078
  1264
                card_Diff_subset subsetI subset_antisym zero_neq_one)
lp15@63078
  1265
  then have u: "S = insert u c"
lp15@63078
  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)
lp15@63078
  1267
  have "T = convex hull (c - {u})"
lp15@63078
  1268
    by (metis Diff_empty Diff_insert0 \<open>T facet_of convex hull S\<close> c facet_of_irrefl insert_absorb u)
lp15@63078
  1269
  with \<open>T \<noteq> {}\<close> show ?rhs
lp15@63078
  1270
    using c u by auto
lp15@63078
  1271
next
lp15@63078
  1272
  assume ?rhs
lp15@63078
  1273
  then obtain u where "T \<noteq> {}" "u \<in> S" and u: "T = convex hull (S - {u})"
lp15@63078
  1274
    by (force simp: facet_of_def)
lp15@63078
  1275
  then have "\<not> S \<subseteq> {u}"
lp15@63078
  1276
    using \<open>T \<noteq> {}\<close> u by auto
lp15@63078
  1277
  have [simp]: "aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1"
lp15@63078
  1278
    using assms \<open>u \<in> S\<close>
lp15@63078
  1279
    apply (simp add: aff_dim_convex_hull affine_dependent_def)
lp15@63078
  1280
    apply (drule bspec, assumption)
lp15@63078
  1281
    by (metis add_diff_cancel_right' aff_dim_insert insert_Diff [of u S])
lp15@63078
  1282
  show ?lhs
lp15@63078
  1283
    apply (subst u)
lp15@63078
  1284
    apply (simp add: \<open>\<not> S \<subseteq> {u}\<close> facet_of_def face_of_convex_hull_affine_independent [OF assms], blast)
lp15@63078
  1285
    done
lp15@63078
  1286
qed
lp15@63078
  1287
lp15@63078
  1288
lemma facet_of_convex_hull_affine_independent_alt:
lp15@63078
  1289
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1290
  shows
lp15@63078
  1291
   "~affine_dependent S
lp15@63078
  1292
        \<Longrightarrow> (T facet_of (convex hull S) \<longleftrightarrow>
lp15@63078
  1293
             2 \<le> card S \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
lp15@63078
  1294
apply (simp add: facet_of_convex_hull_affine_independent)
lp15@63078
  1295
apply (auto simp: Set.subset_singleton_iff)
lp15@63078
  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)
lp15@63078
  1297
done
lp15@63078
  1298
lp15@63078
  1299
lemma segment_face_of:
lp15@63078
  1300
  assumes "(closed_segment a b) face_of S"
lp15@63078
  1301
  shows "a extreme_point_of S" "b extreme_point_of S"
lp15@63078
  1302
proof -
lp15@63078
  1303
  have as: "{a} face_of S"
lp15@63078
  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)
lp15@63078
  1305
  moreover have "{b} face_of S"
lp15@63078
  1306
  proof -
lp15@63078
  1307
    have "b \<in> convex hull {a} \<or> b extreme_point_of convex hull {b, a}"
lp15@63078
  1308
      by (meson extreme_point_of_convex_hull_insert finite.emptyI finite.insertI)
lp15@63078
  1309
    moreover have "closed_segment a b = convex hull {b, a}"
lp15@63078
  1310
      using closed_segment_commute segment_convex_hull by blast
lp15@63078
  1311
    ultimately show ?thesis
lp15@63078
  1312
      by (metis as assms face_of_face convex_hull_singleton empty_iff face_of_singleton insertE)
lp15@63078
  1313
    qed
lp15@63078
  1314
  ultimately show "a extreme_point_of S" "b extreme_point_of S"
lp15@63078
  1315
    using face_of_singleton by blast+
lp15@63078
  1316
qed
lp15@63078
  1317
lp15@63078
  1318
lp15@63078
  1319
lemma Krein_Milman_frontier:
lp15@63078
  1320
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1321
  assumes "convex S" "compact S"
lp15@63078
  1322
    shows "S = convex hull (frontier S)"
lp15@63078
  1323
          (is "?lhs = ?rhs")
lp15@63078
  1324
proof
lp15@63078
  1325
  have "?lhs \<subseteq> convex hull {x. x extreme_point_of S}"
lp15@63078
  1326
    using Krein_Milman_Minkowski assms by blast
lp15@63078
  1327
  also have "... \<subseteq> ?rhs"
lp15@63078
  1328
    apply (rule hull_mono)
lp15@63078
  1329
    apply (auto simp: frontier_def extreme_point_not_in_interior)
lp15@63078
  1330
    using closure_subset apply (force simp: extreme_point_of_def)
lp15@63078
  1331
    done
lp15@63078
  1332
  finally show "?lhs \<subseteq> ?rhs" .
lp15@63078
  1333
next
lp15@63078
  1334
  have "?rhs \<subseteq> convex hull S"
lp15@63078
  1335
    by (metis Diff_subset \<open>compact S\<close> closure_closed compact_eq_bounded_closed frontier_def hull_mono)
lp15@63078
  1336
  also have "... \<subseteq> ?lhs"
lp15@63078
  1337
    by (simp add: \<open>convex S\<close> hull_same)
lp15@63078
  1338
  finally show "?rhs \<subseteq> ?lhs" .
lp15@63078
  1339
qed
lp15@63078
  1340
lp15@63078
  1341
subsection\<open>Polytopes\<close>
lp15@63078
  1342
lp15@63078
  1343
definition polytope where
lp15@63078
  1344
 "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
lp15@63078
  1345
lp15@63078
  1346
lemma polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
lp15@63078
  1347
apply (simp add: polytope_def, safe)
lp15@63078
  1348
apply (metis convex_hull_translation finite_imageI translation_galois)
lp15@63078
  1349
by (metis convex_hull_translation finite_imageI)
lp15@63078
  1350
lp15@63078
  1351
lemma polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
lp15@63078
  1352
  unfolding polytope_def using convex_hull_linear_image by blast
lp15@63078
  1353
lp15@63078
  1354
lemma polytope_empty: "polytope {}"
lp15@63078
  1355
  using convex_hull_empty polytope_def by blast
lp15@63078
  1356
lp15@63078
  1357
lemma polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
lp15@63078
  1358
  using polytope_def by auto
lp15@63078
  1359
lp15@63078
  1360
lemma polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
lp15@63078
  1361
  unfolding polytope_def
lp15@63078
  1362
  by (metis finite_cartesian_product convex_hull_Times)
lp15@63078
  1363
lp15@63078
  1364
lemma face_of_polytope_polytope:
lp15@63078
  1365
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1366
  shows "\<lbrakk>polytope S; F face_of S\<rbrakk> \<Longrightarrow> polytope F"
lp15@63078
  1367
unfolding polytope_def
lp15@63078
  1368
by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)
lp15@63078
  1369
lp15@63078
  1370
lemma finite_polytope_faces:
lp15@63078
  1371
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1372
  assumes "polytope S"
lp15@63078
  1373
  shows "finite {F. F face_of S}"
lp15@63078
  1374
proof -
lp15@63078
  1375
  obtain v where "finite v" "S = convex hull v"
lp15@63078
  1376
    using assms polytope_def by auto
lp15@63078
  1377
  have "finite (op hull convex ` {T. T \<subseteq> v})"
lp15@63078
  1378
    by (simp add: \<open>finite v\<close>)
lp15@63078
  1379
  moreover have "{F. F face_of S} \<subseteq> (op hull convex ` {T. T \<subseteq> v})"
lp15@63078
  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)
lp15@63078
  1381
  ultimately show ?thesis
lp15@63078
  1382
    by (blast intro: finite_subset)
lp15@63078
  1383
qed
lp15@63078
  1384
lp15@63078
  1385
lemma finite_polytope_facets:
lp15@63078
  1386
  assumes "polytope S"
lp15@63078
  1387
  shows "finite {T. T facet_of S}"
lp15@63078
  1388
by (simp add: assms facet_of_def finite_polytope_faces)
lp15@63078
  1389
lp15@63078
  1390
lemma polytope_scaling:
lp15@63078
  1391
  assumes "polytope S"  shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
lp15@63078
  1392
by (simp add: assms polytope_linear_image)
lp15@63078
  1393
lp15@63078
  1394
lemma polytope_imp_compact:
lp15@63078
  1395
  fixes S :: "'a::real_normed_vector set"
lp15@63078
  1396
  shows "polytope S \<Longrightarrow> compact S"
lp15@63078
  1397
by (metis finite_imp_compact_convex_hull polytope_def)
lp15@63078
  1398
lp15@63078
  1399
lemma polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
lp15@63078
  1400
  by (metis convex_convex_hull polytope_def)
lp15@63078
  1401
lp15@63078
  1402
lemma polytope_imp_closed:
lp15@63078
  1403
  fixes S :: "'a::real_normed_vector set"
lp15@63078
  1404
  shows "polytope S \<Longrightarrow> closed S"
lp15@63078
  1405
by (simp add: compact_imp_closed polytope_imp_compact)
lp15@63078
  1406
lp15@63078
  1407
lemma polytope_imp_bounded:
lp15@63078
  1408
  fixes S :: "'a::real_normed_vector set"
lp15@63078
  1409
  shows "polytope S \<Longrightarrow> bounded S"
lp15@63078
  1410
by (simp add: compact_imp_bounded polytope_imp_compact)
lp15@63078
  1411
lp15@63078
  1412
lemma polytope_interval: "polytope(cbox a b)"
lp15@63078
  1413
  unfolding polytope_def by (meson closed_interval_as_convex_hull)
lp15@63078
  1414
lp15@63078
  1415
lemma polytope_sing: "polytope {a}"
lp15@63078
  1416
  using polytope_def by force
lp15@63078
  1417
lp15@63078
  1418
lp15@63078
  1419
subsection\<open>Polyhedra\<close>
lp15@63078
  1420
lp15@63078
  1421
definition polyhedron where
lp15@63078
  1422
 "polyhedron S \<equiv>
lp15@63078
  1423
        \<exists>F. finite F \<and>
lp15@63078
  1424
            S = \<Inter> F \<and>
lp15@63078
  1425
            (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b})"
lp15@63078
  1426
lp15@63078
  1427
lemma polyhedron_Int [intro,simp]:
lp15@63078
  1428
   "\<lbrakk>polyhedron S; polyhedron T\<rbrakk> \<Longrightarrow> polyhedron (S \<inter> T)"
lp15@63078
  1429
  apply (simp add: polyhedron_def, clarify)
lp15@63078
  1430
  apply (rename_tac F G)
lp15@63078
  1431
  apply (rule_tac x="F \<union> G" in exI, auto)
lp15@63078
  1432
  done
lp15@63078
  1433
lp15@63078
  1434
lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
lp15@63078
  1435
  unfolding polyhedron_def
lp15@63078
  1436
  by (rule_tac x="{}" in exI) auto
lp15@63078
  1437
lp15@63078
  1438
lemma polyhedron_Inter [intro,simp]:
lp15@63078
  1439
   "\<lbrakk>finite F; \<And>S. S \<in> F \<Longrightarrow> polyhedron S\<rbrakk> \<Longrightarrow> polyhedron(\<Inter>F)"
lp15@63078
  1440
by (induction F rule: finite_induct) auto
lp15@63078
  1441
lp15@63078
  1442
lp15@63078
  1443
lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
lp15@63078
  1444
proof -
lp15@63078
  1445
  have "\<exists>a. a \<noteq> 0 \<and>
lp15@63078
  1446
             (\<exists>b. {x. (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b})"
lp15@63078
  1447
    by (rule_tac x="(SOME i. i \<in> Basis)" in exI) (force simp: SOME_Basis nonzero_Basis)
lp15@63078
  1448
  moreover have "\<exists>a b. a \<noteq> 0 \<and>
lp15@63078
  1449
                       {x. - (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b}"
lp15@63078
  1450
      apply (rule_tac x="-(SOME i. i \<in> Basis)" in exI)
lp15@63078
  1451
      apply (rule_tac x="-1" in exI)
lp15@63078
  1452
      apply (simp add: SOME_Basis nonzero_Basis)
lp15@63078
  1453
      done
lp15@63078
  1454
  ultimately show ?thesis
lp15@63078
  1455
    unfolding polyhedron_def
lp15@63078
  1456
    apply (rule_tac x="{{x. (SOME i. i \<in> Basis) \<bullet> x \<le> -1},
lp15@63078
  1457
                        {x. -(SOME i. i \<in> Basis) \<bullet> x \<le> -1}}" in exI)
lp15@63078
  1458
    apply force
lp15@63078
  1459
    done
lp15@63078
  1460
qed
lp15@63078
  1461
lp15@63078
  1462
lemma polyhedron_halfspace_le:
lp15@63078
  1463
  fixes a :: "'a :: euclidean_space"
lp15@63078
  1464
  shows "polyhedron {x. a \<bullet> x \<le> b}"
lp15@63078
  1465
proof (cases "a = 0")
lp15@63078
  1466
  case True then show ?thesis by auto
lp15@63078
  1467
next
lp15@63078
  1468
  case False
lp15@63078
  1469
  then show ?thesis
lp15@63078
  1470
    unfolding polyhedron_def
lp15@63078
  1471
    by (rule_tac x="{{x. a \<bullet> x \<le> b}}" in exI) auto
lp15@63078
  1472
qed
lp15@63078
  1473
lp15@63078
  1474
lemma polyhedron_halfspace_ge:
lp15@63078
  1475
  fixes a :: "'a :: euclidean_space"
lp15@63078
  1476
  shows "polyhedron {x. a \<bullet> x \<ge> b}"
lp15@63078
  1477
using polyhedron_halfspace_le [of "-a" "-b"] by simp
lp15@63078
  1478
lp15@63078
  1479
lemma polyhedron_hyperplane:
lp15@63078
  1480
  fixes a :: "'a :: euclidean_space"
lp15@63078
  1481
  shows "polyhedron {x. a \<bullet> x = b}"
lp15@63078
  1482
proof -
lp15@63078
  1483
  have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
lp15@63078
  1484
    by force
lp15@63078
  1485
  then show ?thesis
lp15@63078
  1486
    by (simp add: polyhedron_halfspace_ge polyhedron_halfspace_le)
lp15@63078
  1487
qed
lp15@63078
  1488
lp15@63078
  1489
lemma affine_imp_polyhedron:
lp15@63078
  1490
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1491
  shows "affine S \<Longrightarrow> polyhedron S"
lp15@63078
  1492
by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])
lp15@63078
  1493
lp15@63078
  1494
lemma polyhedron_imp_closed:
lp15@63078
  1495
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1496
  shows "polyhedron S \<Longrightarrow> closed S"
lp15@63078
  1497
apply (simp add: polyhedron_def)
lp15@63078
  1498
using closed_halfspace_le by fastforce
lp15@63078
  1499
lp15@63078
  1500
lemma polyhedron_imp_convex:
lp15@63078
  1501
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1502
  shows "polyhedron S \<Longrightarrow> convex S"
lp15@63078
  1503
apply (simp add: polyhedron_def)
lp15@63078
  1504
using convex_Inter convex_halfspace_le by fastforce
lp15@63078
  1505
lp15@63078
  1506
lemma polyhedron_affine_hull:
lp15@63078
  1507
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1508
  shows "polyhedron(affine hull S)"
lp15@63078
  1509
by (simp add: affine_imp_polyhedron)
lp15@63078
  1510
lp15@63078
  1511
lp15@63078
  1512
subsection\<open>Canonical polyhedron representation making facial structure explicit\<close>
lp15@63078
  1513
lp15@63078
  1514
lemma polyhedron_Int_affine:
lp15@63078
  1515
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1516
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1517
           (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
lp15@63078
  1518
                (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}))"
lp15@63078
  1519
        (is "?lhs = ?rhs")
lp15@63078
  1520
proof
lp15@63078
  1521
  assume ?lhs then show ?rhs
lp15@63078
  1522
    apply (simp add: polyhedron_def)
lp15@63078
  1523
    apply (erule ex_forward)
lp15@63078
  1524
    using hull_subset apply force
lp15@63078
  1525
    done
lp15@63078
  1526
next
lp15@63078
  1527
  assume ?rhs then show ?lhs
lp15@63078
  1528
    apply clarify
lp15@63078
  1529
    apply (erule ssubst)
lp15@63078
  1530
    apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le)
lp15@63078
  1531
    done
lp15@63078
  1532
qed
lp15@63078
  1533
lp15@63078
  1534
proposition rel_interior_polyhedron_explicit:
lp15@63078
  1535
  assumes "finite F"
lp15@63078
  1536
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  1537
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1538
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1539
    shows "rel_interior S = {x \<in> S. \<forall>h \<in> F. a h \<bullet> x < b h}"
lp15@63078
  1540
proof -
lp15@63078
  1541
  have rels: "\<And>x. x \<in> rel_interior S \<Longrightarrow> x \<in> S"
lp15@63078
  1542
    by (meson IntE mem_rel_interior)
lp15@63078
  1543
  moreover have "a i \<bullet> x < b i" if x: "x \<in> rel_interior S" and "i \<in> F" for x i
lp15@63078
  1544
  proof -
lp15@63078
  1545
    have fif: "F - {i} \<subset> F"
lp15@63078
  1546
      using \<open>i \<in> F\<close> Diff_insert_absorb Diff_subset set_insert psubsetI by blast
lp15@63078
  1547
    then have "S \<subset> affine hull S \<inter> \<Inter>(F - {i})"
lp15@63078
  1548
      by (rule psub)
lp15@63078
  1549
    then obtain z where ssub: "S \<subseteq> \<Inter>(F - {i})" and zint: "z \<in> \<Inter>(F - {i})"
lp15@63078
  1550
                    and "z \<notin> S" and zaff: "z \<in> affine hull S"
lp15@63078
  1551
      by auto
lp15@63078
  1552
    have "z \<noteq> x"
lp15@63078
  1553
      using \<open>z \<notin> S\<close> rels x by blast
lp15@63078
  1554
    have "z \<notin> affine hull S \<inter> \<Inter>F"
lp15@63078
  1555
      using \<open>z \<notin> S\<close> seq by auto
lp15@63078
  1556
    then have aiz: "a i \<bullet> z > b i"
lp15@63078
  1557
      using faceq zint zaff by fastforce
lp15@63078
  1558
    obtain e where "e > 0" "x \<in> S" and e: "ball x e \<inter> affine hull S \<subseteq> S"
lp15@63078
  1559
      using x by (auto simp: mem_rel_interior_ball)
lp15@63078
  1560
    then have ins: "\<And>y. \<lbrakk>norm (x - y) < e; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
lp15@63078
  1561
      by (metis IntI subsetD dist_norm mem_ball)
wenzelm@63148
  1562
    define \<xi> where "\<xi> = min (1/2) (e / 2 / norm(z - x))"
lp15@63078
  1563
    have "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) = norm (\<xi> *\<^sub>R (x - z))"
lp15@63078
  1564
      by (simp add: \<xi>_def algebra_simps norm_mult)
lp15@63078
  1565
    also have "... = \<xi> * norm (x - z)"
lp15@63078
  1566
      using \<open>e > 0\<close> by (simp add: \<xi>_def)
lp15@63078
  1567
    also have "... < e"
lp15@63078
  1568
      using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def divide_simps norm_minus_commute)
lp15@63078
  1569
    finally have lte: "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) < e" .
lp15@63078
  1570
    have \<xi>_aff: "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> affine hull S"
lp15@63078
  1571
      by (metis \<open>x \<in> S\<close> add.commute affine_affine_hull diff_add_cancel hull_inc mem_affine zaff)
lp15@63078
  1572
    have "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> S"
lp15@63078
  1573
      apply (rule ins [OF _ \<xi>_aff])
lp15@63078
  1574
      apply (simp add: algebra_simps lte)
lp15@63078
  1575
      done
lp15@63078
  1576
    then obtain l where l: "0 < l" "l < 1" and ls: "(l *\<^sub>R z + (1 - l) *\<^sub>R x) \<in> S"
lp15@63078
  1577
      apply (rule_tac l = \<xi> in that)
lp15@63078
  1578
      using \<open>e > 0\<close> \<open>z \<noteq> x\<close>  apply (auto simp: \<xi>_def)
lp15@63078
  1579
      done
lp15@63078
  1580
    then have i: "l *\<^sub>R z + (1 - l) *\<^sub>R x \<in> i"
lp15@63078
  1581
      using seq \<open>i \<in> F\<close> by auto
lp15@63078
  1582
    have "b i * l + (a i \<bullet> x) * (1 - l) < a i \<bullet> (l *\<^sub>R z + (1 - l) *\<^sub>R x)"
lp15@63078
  1583
      using l by (simp add: algebra_simps aiz)
lp15@63078
  1584
    also have "\<dots> \<le> b i" using i l
lp15@63078
  1585
      using faceq mem_Collect_eq \<open>i \<in> F\<close> by blast
lp15@63078
  1586
    finally have "(a i \<bullet> x) * (1 - l) < b i * (1 - l)"
lp15@63078
  1587
      by (simp add: algebra_simps)
lp15@63078
  1588
    with l show ?thesis
lp15@63078
  1589
      by simp
lp15@63078
  1590
  qed
lp15@63078
  1591
  moreover have "x \<in> rel_interior S"
lp15@63078
  1592
           if "x \<in> S" and less: "\<And>h. h \<in> F \<Longrightarrow> a h \<bullet> x < b h" for x
lp15@63078
  1593
  proof -
lp15@63078
  1594
    have 1: "\<And>h. h \<in> F \<Longrightarrow> x \<in> interior h"
lp15@63078
  1595
      by (metis interior_halfspace_le mem_Collect_eq less faceq)
lp15@63078
  1596
    have 2: "\<And>y. \<lbrakk>\<forall>h\<in>F. y \<in> interior h; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
lp15@63078
  1597
      by (metis IntI Inter_iff contra_subsetD interior_subset seq)
lp15@63078
  1598
    show ?thesis
lp15@63078
  1599
      apply (simp add: rel_interior \<open>x \<in> S\<close>)
lp15@63078
  1600
      apply (rule_tac x="\<Inter>h\<in>F. interior h" in exI)
lp15@63078
  1601
      apply (auto simp: \<open>finite F\<close> open_INT 1 2)
lp15@63078
  1602
      done
lp15@63078
  1603
  qed
lp15@63078
  1604
  ultimately show ?thesis by blast
lp15@63078
  1605
qed
lp15@63078
  1606
lp15@63078
  1607
lp15@63078
  1608
lemma polyhedron_Int_affine_parallel:
lp15@63078
  1609
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1610
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1611
         (\<exists>F. finite F \<and>
lp15@63078
  1612
              S = (affine hull S) \<inter> (\<Inter>F) \<and>
lp15@63078
  1613
              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
lp15@63078
  1614
                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)))"
lp15@63078
  1615
    (is "?lhs = ?rhs")
lp15@63078
  1616
proof
lp15@63078
  1617
  assume ?lhs
lp15@63078
  1618
  then obtain F where "finite F" and seq: "S = (affine hull S) \<inter> \<Inter>F"
lp15@63078
  1619
                  and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
lp15@63078
  1620
    by (fastforce simp add: polyhedron_Int_affine)
lp15@63078
  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}"
lp15@63078
  1622
    by metis
lp15@63078
  1623
  show ?rhs
lp15@63078
  1624
  proof -
lp15@63078
  1625
    have "\<exists>a' b'. a' \<noteq> 0 \<and>
lp15@63078
  1626
                  affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> h \<and>
lp15@63078
  1627
                  (\<forall>w \<in> affine hull S. (w + a') \<in> affine hull S)"
lp15@63078
  1628
        if "h \<in> F" "~(affine hull S \<subseteq> h)" for h
lp15@63078
  1629
    proof -
lp15@63078
  1630
      have "a h \<noteq> 0" and "h = {x. a h \<bullet> x \<le> b h}" "h \<inter> \<Inter>F = \<Inter>F"
lp15@63078
  1631
        using \<open>h \<in> F\<close> ab by auto
lp15@63078
  1632
      then have "(affine hull S) \<inter> {x. a h \<bullet> x \<le> b h} \<noteq> {}"
lp15@63078
  1633
        by (metis (no_types) affine_hull_eq_empty inf.absorb_iff2 inf_assoc inf_bot_right inf_commute seq that(2))
lp15@63078
  1634
      moreover have "~ (affine hull S \<subseteq> {x. a h \<bullet> x \<le> b h})"
lp15@63078
  1635
        using \<open>h = {x. a h \<bullet> x \<le> b h}\<close> that(2) by blast
lp15@63078
  1636
      ultimately show ?thesis
lp15@63078
  1637
        using affine_parallel_slice [of "affine hull S"]
lp15@63078
  1638
        by (metis \<open>h = {x. a h \<bullet> x \<le> b h}\<close> affine_affine_hull)
lp15@63078
  1639
    qed
lp15@63078
  1640
    then obtain a b
lp15@63078
  1641
         where ab: "\<And>h. \<lbrakk>h \<in> F; ~ (affine hull S \<subseteq> h)\<rbrakk>
lp15@63078
  1642
             \<Longrightarrow> a h \<noteq> 0 \<and>
lp15@63078
  1643
                  affine hull S \<inter> {x. a h \<bullet> x \<le> b h} = affine hull S \<inter> h \<and>
lp15@63078
  1644
                  (\<forall>w \<in> affine hull S. (w + a h) \<in> affine hull S)"
lp15@63078
  1645
      by metis
lp15@63078
  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})"
lp15@63078
  1647
      by (subst seq) (auto simp: ab INT_extend_simps)
lp15@63078
  1648
    show ?thesis
lp15@63078
  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)
lp15@63078
  1650
      apply (intro conjI seq2)
lp15@63078
  1651
        using \<open>finite F\<close> apply force
lp15@63078
  1652
       using ab apply blast
lp15@63078
  1653
       done
lp15@63078
  1654
  qed
lp15@63078
  1655
next
lp15@63078
  1656
  assume ?rhs then show ?lhs
lp15@63078
  1657
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  1658
    by metis
lp15@63078
  1659
qed
lp15@63078
  1660
lp15@63078
  1661
lp15@63078
  1662
proposition polyhedron_Int_affine_parallel_minimal:
lp15@63078
  1663
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1664
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1665
         (\<exists>F. finite F \<and>
lp15@63078
  1666
              S = (affine hull S) \<inter> (\<Inter>F) \<and>
lp15@63078
  1667
              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
lp15@63078
  1668
                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)) \<and>
lp15@63078
  1669
              (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> (\<Inter>F')))"
lp15@63078
  1670
    (is "?lhs = ?rhs")
lp15@63078
  1671
proof
lp15@63078
  1672
  assume ?lhs
lp15@63078
  1673
  then obtain f0
lp15@63078
  1674
           where f0: "finite f0"
lp15@63078
  1675
                 "S = (affine hull S) \<inter> (\<Inter>f0)"
lp15@63078
  1676
                   (is "?P f0")
lp15@63078
  1677
                 "\<forall>h \<in> f0. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
lp15@63078
  1678
                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)"
lp15@63078
  1679
                   (is "?Q f0")
lp15@63078
  1680
    by (force simp: polyhedron_Int_affine_parallel)
wenzelm@63148
  1681
  define n where "n = (LEAST n. \<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F)"
lp15@63078
  1682
  have nf: "\<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F"
lp15@63078
  1683
    apply (simp add: n_def)
lp15@63078
  1684
    apply (rule LeastI [where k = "card f0"])
lp15@63078
  1685
    using f0 apply auto
lp15@63078
  1686
    done
lp15@63078
  1687
  then obtain F where F: "card F = n" "finite F" and seq: "?P F" and aff: "?Q F"
lp15@63078
  1688
    by blast
lp15@63078
  1689
  then have "~ (finite g \<and> ?P g \<and> ?Q g)" if "card g < n" for g
lp15@63078
  1690
    using that by (auto simp: n_def dest!: not_less_Least)
lp15@63078
  1691
  then have *: "~ (?P g \<and> ?Q g)" if "g \<subset> F" for g
lp15@63078
  1692
    using that \<open>finite F\<close> psubset_card_mono \<open>card F = n\<close>
lp15@63078
  1693
    by (metis finite_Int inf.strict_order_iff)
lp15@63078
  1694
  have 1: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subseteq> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1695
    by (subst seq) blast
lp15@63078
  1696
  have 2: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<noteq> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1697
    apply (frule *)
lp15@63078
  1698
    by (metis aff subsetCE subset_iff_psubset_eq)
lp15@63078
  1699
  show ?rhs
lp15@63078
  1700
    by (metis \<open>finite F\<close> seq aff psubsetI 1 2)
lp15@63078
  1701
next
lp15@63078
  1702
  assume ?rhs then show ?lhs
lp15@63078
  1703
    by (auto simp: polyhedron_Int_affine_parallel)
lp15@63078
  1704
qed
lp15@63078
  1705
lp15@63078
  1706
lp15@63078
  1707
lemma polyhedron_Int_affine_minimal:
lp15@63078
  1708
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1709
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1710
         (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
lp15@63078
  1711
              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}) \<and>
lp15@63078
  1712
              (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'))"
lp15@63078
  1713
apply (rule iffI)
lp15@63078
  1714
 apply (force simp: polyhedron_Int_affine_parallel_minimal elim!: ex_forward)
lp15@63078
  1715
apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
lp15@63078
  1716
done
lp15@63078
  1717
lp15@63078
  1718
proposition facet_of_polyhedron_explicit:
lp15@63078
  1719
  assumes "finite F"
lp15@63078
  1720
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  1721
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1722
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1723
    shows "c facet_of S \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
lp15@63078
  1724
proof (cases "S = {}")
lp15@63078
  1725
  case True with psub show ?thesis by force
lp15@63078
  1726
next
lp15@63078
  1727
  case False
lp15@63078
  1728
  have "polyhedron S"
lp15@63078
  1729
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  1730
    apply (rule_tac x=F in exI)
lp15@63078
  1731
    using assms  apply force
lp15@63078
  1732
    done
lp15@63078
  1733
  then have "convex S"
lp15@63078
  1734
    by (rule polyhedron_imp_convex)
lp15@63078
  1735
  with False rel_interior_eq_empty have "rel_interior S \<noteq> {}" by blast
lp15@63078
  1736
  then obtain x where "x \<in> rel_interior S" by auto
lp15@63078
  1737
  then obtain T where "open T" "x \<in> T" "x \<in> S" "T \<inter> affine hull S \<subseteq> S"
lp15@63078
  1738
    by (force simp: mem_rel_interior)
lp15@63078
  1739
  then have xaff: "x \<in> affine hull S" and xint: "x \<in> \<Inter>F"
lp15@63078
  1740
    using seq hull_inc by auto
lp15@63078
  1741
  have "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
lp15@63078
  1742
    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
lp15@63078
  1743
  with \<open>x \<in> rel_interior S\<close>
lp15@63078
  1744
  have [simp]: "\<And>h. h\<in>F \<Longrightarrow> a h \<bullet> x < b h" by blast
lp15@63078
  1745
  have *: "(S \<inter> {x. a h \<bullet> x = b h}) facet_of S" if "h \<in> F" for h
lp15@63078
  1746
  proof -
lp15@63078
  1747
    have "S \<subset> affine hull S \<inter> \<Inter>(F - {h})"
lp15@63078
  1748
      using psub that by (metis Diff_disjoint Diff_subset insert_disjoint(2) psubsetI)
lp15@63078
  1749
    then obtain z where zaff: "z \<in> affine hull S" and zint: "z \<in> \<Inter>(F - {h})" and "z \<notin> S"
lp15@63078
  1750
      by force
lp15@63078
  1751
    then have "z \<noteq> x" "z \<notin> h" using seq \<open>x \<in> S\<close> by auto
lp15@63078
  1752
    have "x \<in> h" using that xint by auto
lp15@63078
  1753
    then have able: "a h \<bullet> x \<le> b h"
lp15@63078
  1754
      using faceq that by blast
lp15@63078
  1755
    also have "... < a h \<bullet> z" using \<open>z \<notin> h\<close> faceq [OF that] xint by auto
lp15@63078
  1756
    finally have xltz: "a h \<bullet> x < a h \<bullet> z" .
wenzelm@63148
  1757
    define l where "l = (b h - a h \<bullet> x) / (a h \<bullet> z - a h \<bullet> x)"
wenzelm@63148
  1758
    define w where "w = (1 - l) *\<^sub>R x + l *\<^sub>R z"
lp15@63078
  1759
    have "0 < l" "l < 1"
lp15@63078
  1760
      using able xltz \<open>b h < a h \<bullet> z\<close> \<open>h \<in> F\<close>
lp15@63078
  1761
      by (auto simp: l_def divide_simps)
lp15@63078
  1762
    have awlt: "a i \<bullet> w < b i" if "i \<in> F" "i \<noteq> h" for i
lp15@63078
  1763
    proof -
lp15@63078
  1764
      have "(1 - l) * (a i \<bullet> x) < (1 - l) * b i"
lp15@63078
  1765
        by (simp add: \<open>l < 1\<close> \<open>i \<in> F\<close>)
lp15@63078
  1766
      moreover have "l * (a i \<bullet> z) \<le> l * b i"
lp15@63078
  1767
        apply (rule mult_left_mono)
lp15@63078
  1768
        apply (metis Diff_insert_absorb Inter_iff Set.set_insert \<open>h \<in> F\<close> faceq insertE mem_Collect_eq that zint)
lp15@63078
  1769
        using \<open>0 < l\<close>
lp15@63078
  1770
        apply simp
lp15@63078
  1771
        done
lp15@63078
  1772
      ultimately show ?thesis by (simp add: w_def algebra_simps)
lp15@63078
  1773
    qed
lp15@63078
  1774
    have weq: "a h \<bullet> w = b h"
lp15@63078
  1775
      using xltz unfolding w_def l_def
lp15@63078
  1776
      by (simp add: algebra_simps) (simp add: field_simps)
lp15@63078
  1777
    have "w \<in> affine hull S"
lp15@63078
  1778
      by (simp add: w_def mem_affine xaff zaff)
lp15@63078
  1779
    moreover have "w \<in> \<Inter>F"
lp15@63078
  1780
      using \<open>a h \<bullet> w = b h\<close> awlt faceq less_eq_real_def by blast
lp15@63078
  1781
    ultimately have "w \<in> S"
lp15@63078
  1782
      using seq by blast
lp15@63078
  1783
    with weq have "S \<inter> {x. a h \<bullet> x = b h} \<noteq> {}" by blast
lp15@63078
  1784
    moreover have "S \<inter> {x. a h \<bullet> x = b h} face_of S"
lp15@63078
  1785
      apply (rule face_of_Int_supporting_hyperplane_le)
lp15@63078
  1786
      apply (rule \<open>convex S\<close>)
lp15@63078
  1787
      apply (subst (asm) seq)
lp15@63078
  1788
      using faceq that apply fastforce
lp15@63078
  1789
      done
lp15@63078
  1790
    moreover have "affine hull (S \<inter> {x. a h \<bullet> x = b h}) =
lp15@63078
  1791
                   (affine hull S) \<inter> {x. a h \<bullet> x = b h}"
lp15@63078
  1792
    proof
lp15@63078
  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}"
lp15@63078
  1794
        apply (intro Int_greatest hull_mono Int_lower1)
lp15@63078
  1795
        apply (metis affine_hull_eq affine_hyperplane hull_mono inf_le2)
lp15@63078
  1796
        done
lp15@63078
  1797
    next
lp15@63078
  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})"
lp15@63078
  1799
      proof
lp15@63078
  1800
        fix y
lp15@63078
  1801
        assume yaff: "y \<in> affine hull S \<inter> {y. a h \<bullet> y = b h}"
lp15@63078
  1802
        obtain T where "0 < T"
lp15@63078
  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"
lp15@63078
  1804
        proof (cases "F - {h} = {}")
lp15@63078
  1805
          case True then show ?thesis
lp15@63078
  1806
            by (rule_tac T=1 in that) auto
lp15@63078
  1807
        next
lp15@63078
  1808
          case False
lp15@63078
  1809
          then obtain h' where h': "h' \<in> F - {h}" by auto
wenzelm@63148
  1810
          define inff where "inff =
wenzelm@63148
  1811
            (INF j:F - {h}.
wenzelm@63148
  1812
              if 0 < a j \<bullet> y - a j \<bullet> w
wenzelm@63148
  1813
              then (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)
wenzelm@63148
  1814
              else 1)"
lp15@63078
  1815
          have "0 < inff"
lp15@63078
  1816
            apply (simp add: inff_def)
lp15@63078
  1817
            apply (rule finite_imp_less_Inf)
lp15@63078
  1818
              using \<open>finite F\<close> apply blast
lp15@63078
  1819
             using h' apply blast
lp15@63078
  1820
            apply simp
lp15@63078
  1821
            using awlt apply (force simp: divide_simps)
lp15@63078
  1822
            done
lp15@63078
  1823
          moreover have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
lp15@63078
  1824
                        if "j \<in> F" "j \<noteq> h" for j
lp15@63078
  1825
          proof (cases "a j \<bullet> w < a j \<bullet> y")
lp15@63078
  1826
            case True
lp15@63078
  1827
            then have "inff \<le> (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)"
lp15@63078
  1828
              apply (simp add: inff_def)
lp15@63078
  1829
              apply (rule cInf_le_finite)
lp15@63078
  1830
              using \<open>finite F\<close> apply blast
lp15@63078
  1831
              apply (simp add: that split: if_split_asm)
lp15@63078
  1832
              done
lp15@63078
  1833
            then show ?thesis
lp15@63078
  1834
              using \<open>0 < inff\<close> awlt [OF that] mult_strict_left_mono
lp15@63078
  1835
              by (fastforce simp add: algebra_simps divide_simps split: if_split_asm)
lp15@63078
  1836
          next
lp15@63078
  1837
            case False
lp15@63078
  1838
            with \<open>0 < inff\<close> have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> 0"
lp15@63078
  1839
              by (simp add: mult_le_0_iff)
lp15@63078
  1840
            also have "... < b j - a j \<bullet> w"
lp15@63078
  1841
              by (simp add: awlt that)
lp15@63078
  1842
            finally show ?thesis by simp
lp15@63078
  1843
          qed
lp15@63078
  1844
          ultimately show ?thesis
lp15@63078
  1845
            by (blast intro: that)
lp15@63078
  1846
        qed
wenzelm@63148
  1847
        define c where "c = (1 - T) *\<^sub>R w + T *\<^sub>R y"
lp15@63078
  1848
        have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> j" if "j \<in> F" for j
lp15@63078
  1849
        proof (cases "j = h")
lp15@63078
  1850
          case True
lp15@63078
  1851
          have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1852
            using weq yaff by (auto simp: algebra_simps)
lp15@63078
  1853
          with True faceq [OF that] show ?thesis by metis
lp15@63078
  1854
        next
lp15@63078
  1855
          case False
lp15@63078
  1856
          with T that have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a j \<bullet> x \<le> b j}"
lp15@63078
  1857
            by (simp add: algebra_simps)
lp15@63078
  1858
          with faceq [OF that] show ?thesis by simp
lp15@63078
  1859
        qed
lp15@63078
  1860
        moreover have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> affine hull S"
wenzelm@63170
  1861
          apply (rule affine_affine_hull [simplified affine_alt, rule_format])
lp15@63078
  1862
          apply (simp add: \<open>w \<in> affine hull S\<close>)
lp15@63078
  1863
          using yaff apply blast
lp15@63078
  1864
          done
lp15@63078
  1865
        ultimately have "c \<in> S"
lp15@63078
  1866
          using seq by (force simp: c_def)
lp15@63078
  1867
        moreover have "a h \<bullet> c = b h"
lp15@63078
  1868
          using yaff by (force simp: c_def algebra_simps weq)
lp15@63078
  1869
        ultimately have caff: "c \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
lp15@63078
  1870
          by (simp add: hull_inc)
lp15@63078
  1871
        have waff: "w \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
lp15@63078
  1872
          using \<open>w \<in> S\<close> weq by (blast intro: hull_inc)
lp15@63078
  1873
        have yeq: "y = (1 - inverse T) *\<^sub>R w + c /\<^sub>R T"
lp15@63078
  1874
          using \<open>0 < T\<close> by (simp add: c_def algebra_simps)
lp15@63078
  1875
        show "y \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
wenzelm@63170
  1876
          by (metis yeq affine_affine_hull [simplified affine_alt, rule_format, OF waff caff])
lp15@63078
  1877
      qed
lp15@63078
  1878
    qed
lp15@63078
  1879
    ultimately show ?thesis
lp15@63078
  1880
      apply (simp add: facet_of_def)
lp15@63078
  1881
      apply (subst aff_dim_affine_hull [symmetric])
lp15@63078
  1882
      using  \<open>b h < a h \<bullet> z\<close> zaff
lp15@63078
  1883
      apply (force simp: aff_dim_affine_Int_hyperplane)
lp15@63078
  1884
      done
lp15@63078
  1885
  qed
lp15@63078
  1886
  show ?thesis
lp15@63078
  1887
  proof
lp15@63078
  1888
    show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h} \<Longrightarrow> c facet_of S"
lp15@63078
  1889
      using * by blast
lp15@63078
  1890
  next
lp15@63078
  1891
    assume "c facet_of S"
lp15@63078
  1892
    then have "c face_of S" "convex c" "c \<noteq> {}" and affc: "aff_dim c = aff_dim S - 1"
lp15@63078
  1893
      by (auto simp: facet_of_def face_of_imp_convex)
lp15@63078
  1894
    then obtain x where x: "x \<in> rel_interior c"
lp15@63078
  1895
      by (force simp: rel_interior_eq_empty)
lp15@63078
  1896
    then have "x \<in> c"
lp15@63078
  1897
      by (meson subsetD rel_interior_subset)
lp15@63078
  1898
    then have "x \<in> S"
lp15@63078
  1899
      using \<open>c facet_of S\<close> facet_of_imp_subset by blast
lp15@63078
  1900
    have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
lp15@63078
  1901
      by (rule rel_interior_polyhedron_explicit [OF assms])
lp15@63078
  1902
    have "c \<noteq> S"
lp15@63078
  1903
      using \<open>c facet_of S\<close> facet_of_irrefl by blast
lp15@63078
  1904
    then have "x \<notin> rel_interior S"
lp15@63078
  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)
lp15@63078
  1906
    with rels \<open>x \<in> S\<close> obtain i where "i \<in> F" and i: "a i \<bullet> x \<ge> b i"
lp15@63078
  1907
      by force
lp15@63078
  1908
    have "x \<in> {u. a i \<bullet> u \<le> b i}"
lp15@63078
  1909
      by (metis IntD2 InterE \<open>i \<in> F\<close> \<open>x \<in> S\<close> faceq seq)
lp15@63078
  1910
    then have "a i \<bullet> x \<le> b i" by simp
lp15@63078
  1911
    then have "a i \<bullet> x = b i" using i by auto
lp15@63078
  1912
    have "c \<subseteq> S \<inter> {x. a i \<bullet> x = b i}"
lp15@63078
  1913
      apply (rule subset_of_face_of [of _ S])
lp15@63078
  1914
        apply (simp add: "*" \<open>i \<in> F\<close> facet_of_imp_face_of)
lp15@63078
  1915
       apply (simp add: \<open>c face_of S\<close> face_of_imp_subset)
lp15@63078
  1916
      using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> S\<close> x by blast
lp15@63078
  1917
    then have cface: "c face_of (S \<inter> {x. a i \<bullet> x = b i})"
lp15@63078
  1918
      by (meson \<open>c face_of S\<close> face_of_subset inf_le1)
lp15@63078
  1919
    have con: "convex (S \<inter> {x. a i \<bullet> x = b i})"
lp15@63078
  1920
      by (simp add: \<open>convex S\<close> convex_Int convex_hyperplane)
lp15@63078
  1921
    show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h}"
lp15@63078
  1922
      apply (rule_tac x=i in exI)
lp15@63078
  1923
      apply (simp add: \<open>i \<in> F\<close>)
lp15@63078
  1924
      by (metis (no_types) * \<open>i \<in> F\<close> affc facet_of_def less_irrefl face_of_aff_dim_lt [OF con cface])
lp15@63078
  1925
  qed
lp15@63078
  1926
qed
lp15@63078
  1927
lp15@63078
  1928
lp15@63078
  1929
lemma face_of_polyhedron_subset_explicit:
lp15@63078
  1930
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1931
  assumes "finite F"
lp15@63078
  1932
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  1933
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1934
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1935
      and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
lp15@63078
  1936
   obtains h where "h \<in> F" "c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
lp15@63078
  1937
proof -
lp15@63078
  1938
  have "c \<subseteq> S" using \<open>c face_of S\<close>
lp15@63078
  1939
    by (simp add: face_of_imp_subset)
lp15@63078
  1940
  have "polyhedron S"
lp15@63078
  1941
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  1942
    by (metis \<open>finite F\<close> faceq seq)
lp15@63078
  1943
  then have "convex S"
lp15@63078
  1944
    by (simp add: polyhedron_imp_convex)
lp15@63078
  1945
  then have *: "(S \<inter> {x. a h \<bullet> x = b h}) face_of S" if "h \<in> F" for h
lp15@63078
  1946
    apply (rule face_of_Int_supporting_hyperplane_le)
lp15@63078
  1947
    using faceq seq that by fastforce
lp15@63078
  1948
  have "rel_interior c \<noteq> {}"
lp15@63078
  1949
    using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
lp15@63078
  1950
  then obtain x where "x \<in> rel_interior c" by auto
lp15@63078
  1951
  have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
lp15@63078
  1952
    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
lp15@63078
  1953
  then have xnot: "x \<notin> rel_interior S"
lp15@63078
  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)
lp15@63078
  1955
  then have "x \<in> S"
lp15@63078
  1956
    using \<open>c \<subseteq> S\<close> \<open>x \<in> rel_interior c\<close> rel_interior_subset by auto
lp15@63078
  1957
  then have xint: "x \<in> \<Inter>F"
lp15@63078
  1958
    using seq by blast
lp15@63078
  1959
  have "F \<noteq> {}" using assms
lp15@63078
  1960
    by (metis affine_Int affine_Inter affine_affine_hull ex_in_conv face_of_affine_trivial)
lp15@63078
  1961
  then obtain i where "i \<in> F" "~ (a i \<bullet> x < b i)"
lp15@63078
  1962
    using \<open>x \<in> S\<close> rels xnot by auto
lp15@63078
  1963
  with xint have "a i \<bullet> x = b i"
lp15@63078
  1964
    by (metis eq_iff mem_Collect_eq not_le Inter_iff faceq)
lp15@63078
  1965
  have face: "S \<inter> {x. a i \<bullet> x = b i} face_of S"
lp15@63078
  1966
    by (simp add: "*" \<open>i \<in> F\<close>)
lp15@63078
  1967
  show ?thesis
lp15@63078
  1968
    apply (rule_tac h = i in that)
lp15@63078
  1969
     apply (rule \<open>i \<in> F\<close>)
lp15@63078
  1970
    apply (rule subset_of_face_of [OF face \<open>c \<subseteq> S\<close>])
lp15@63078
  1971
    using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> rel_interior c\<close> \<open>x \<in> S\<close> apply blast
lp15@63078
  1972
    done
lp15@63078
  1973
qed
lp15@63078
  1974
lp15@63078
  1975
text\<open>Initial part of proof duplicates that above\<close>
lp15@63078
  1976
proposition face_of_polyhedron_explicit:
lp15@63078
  1977
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1978
  assumes "finite F"
lp15@63078
  1979
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  1980
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1981
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1982
      and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
lp15@63078
  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}}"
lp15@63078
  1984
proof -
lp15@63078
  1985
  let ?ab = "\<lambda>h. {x. a h \<bullet> x = b h}"
lp15@63078
  1986
  have "c \<subseteq> S" using \<open>c face_of S\<close>
lp15@63078
  1987
    by (simp add: face_of_imp_subset)
lp15@63078
  1988
  have "polyhedron S"
lp15@63078
  1989
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  1990
    by (metis \<open>finite F\<close> faceq seq)
lp15@63078
  1991
  then have "convex S"
lp15@63078
  1992
    by (simp add: polyhedron_imp_convex)
lp15@63078
  1993
  then have *: "(S \<inter> ?ab h) face_of S" if "h \<in> F" for h
lp15@63078
  1994
    apply (rule face_of_Int_supporting_hyperplane_le)
lp15@63078
  1995
    using faceq seq that by fastforce
lp15@63078
  1996
  have "rel_interior c \<noteq> {}"
lp15@63078
  1997
    using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
lp15@63078
  1998
  then obtain z where z: "z \<in> rel_interior c" by auto
lp15@63078
  1999
  have rels: "rel_interior S = {z \<in> S. \<forall>h\<in>F. a h \<bullet> z < b h}"
lp15@63078
  2000
    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
lp15@63078
  2001
  then have xnot: "z \<notin> rel_interior S"
lp15@63078
  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)
lp15@63078
  2003
  then have "z \<in> S"
lp15@63078
  2004
    using \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close> rel_interior_subset by auto
lp15@63078
  2005
  with seq have xint: "z \<in> \<Inter>F" by blast
lp15@63078
  2006
  have "open (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
lp15@63078
  2007
    by (auto simp: \<open>finite F\<close> open_halfspace_lt open_INT)
lp15@63078
  2008
  then obtain e where "0 < e"
lp15@63078
  2009
                 "ball z e \<subseteq> (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
lp15@63078
  2010
    by (auto intro: openE [of _ z])
lp15@63078
  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}"
lp15@63078
  2012
    by blast
lp15@63078
  2013
  have "c \<subseteq> (S \<inter> ?ab h) \<longleftrightarrow> z \<in> S \<inter> ?ab h" if "h \<in> F" for h
lp15@63078
  2014
  proof
lp15@63078
  2015
    show "z \<in> S \<inter> ?ab h \<Longrightarrow> c \<subseteq> S \<inter> ?ab h"
lp15@63078
  2016
      apply (rule subset_of_face_of [of _ S])
lp15@63078
  2017
      using that \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close>
lp15@63078
  2018
      using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub]
lp15@63078
  2019
            unfolding facet_of_def
lp15@63078
  2020
      apply auto
lp15@63078
  2021
      done
lp15@63078
  2022
  next
lp15@63078
  2023
    show "c \<subseteq> S \<inter> ?ab h \<Longrightarrow> z \<in> S \<inter> ?ab h"
lp15@63078
  2024
      using \<open>z \<in> rel_interior c\<close> rel_interior_subset by force
lp15@63078
  2025
  qed
lp15@63078
  2026
  then have **: "{S \<inter> ?ab h | h. h \<in> F \<and> c \<subseteq> S \<and> c \<subseteq> ?ab h} =
lp15@63078
  2027
                 {S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<inter> ?ab h}"
lp15@63078
  2028
    by blast
lp15@63078
  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}
lp15@63078
  2030
             \<subseteq> affine hull S \<inter> \<Inter>F \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
lp15@63078
  2031
            if "i \<in> F" and i: "a i \<bullet> z = b i" for i
lp15@63078
  2032
  proof -
lp15@63078
  2033
    have sub: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> j"
lp15@63078
  2034
             if "j \<in> F" for j
lp15@63078
  2035
    proof -
lp15@63078
  2036
      have "a j \<bullet> z \<le> b j" using faceq that xint by auto
lp15@63078
  2037
      then consider "a j \<bullet> z < b j" | "a j \<bullet> z = b j" by linarith
lp15@63078
  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"
lp15@63078
  2039
      proof cases
lp15@63078
  2040
        assume "a j \<bullet> z < b j"
lp15@63078
  2041
        then have "ball z e \<inter> {x. a i \<bullet> x = b i} \<subseteq> j"
lp15@63078
  2042
          using e [OF \<open>j \<in> F\<close>] faceq that
lp15@63078
  2043
          by (fastforce simp: ball_def)
lp15@63078
  2044
        then show ?thesis
lp15@63078
  2045
          by (rule_tac x="{x. a i \<bullet> x = b i}" in exI) (force simp: \<open>i \<in> F\<close> i)
lp15@63078
  2046
      next
lp15@63078
  2047
        assume eq: "a j \<bullet> z = b j"
lp15@63078
  2048
        with faceq that show ?thesis
lp15@63078
  2049
          by (rule_tac x="{x. a j \<bullet> x = b j}" in exI) (fastforce simp add: \<open>j \<in> F\<close>)
lp15@63078
  2050
      qed
lp15@63078
  2051
      then show ?thesis  by blast
lp15@63078
  2052
    qed
lp15@63078
  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"
lp15@63078
  2054
      apply (rule hull_mono)
lp15@63078
  2055
      using that \<open>z \<in> S\<close> by auto
lp15@63078
  2056
    have 2: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
lp15@63078
  2057
          \<subseteq> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
lp15@63078
  2058
      by (rule hull_minimal) (auto intro: affine_hyperplane)
lp15@63078
  2059
    have 3: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> \<Inter>F"
lp15@63078
  2060
      by (iprover intro: sub Inter_greatest)
lp15@63078
  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"
lp15@63078
  2062
             for A B C D E  by blast
lp15@63078
  2063
    show ?thesis by (intro * 1 2 3)
lp15@63078
  2064
  qed
lp15@63078
  2065
  have "\<exists>h. h \<in> F \<and> c \<subseteq> ?ab h"
lp15@63078
  2066
    apply (rule face_of_polyhedron_subset_explicit [OF \<open>finite F\<close> seq faceq psub])
lp15@63078
  2067
    using assms by auto
lp15@63078
  2068
  then have fac: "\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> c \<subseteq> S \<inter> ?ab h} face_of S"
lp15@63078
  2069
    using * by (force simp: \<open>c \<subseteq> S\<close> intro: face_of_Inter)
lp15@63078
  2070
  have red:
lp15@63078
  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}"
lp15@63078
  2072
     for P T F   by blast
lp15@63078
  2073
  have "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
lp15@63078
  2074
        \<subseteq> \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
lp15@63078
  2075
    apply (rule red)
lp15@63078
  2076
    apply (metis seq bsub)
lp15@63078
  2077
    done
lp15@63078
  2078
  with \<open>0 < e\<close> have zinrel: "z \<in> rel_interior
lp15@63078
  2079
                    (\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<and> a h \<bullet> z = b h})"
lp15@63078
  2080
    by (auto simp: mem_rel_interior_ball \<open>z \<in> S\<close>)
lp15@63078
  2081
  show ?thesis
lp15@63078
  2082
    apply (rule face_of_eq [OF c fac])
lp15@63078
  2083
    using z zinrel apply (force simp: **)
lp15@63078
  2084
    done
lp15@63078
  2085
qed
lp15@63078
  2086
lp15@63078
  2087
lp15@63078
  2088
subsection\<open>More general corollaries from the explicit representation\<close>
lp15@63078
  2089
lp15@63078
  2090
corollary facet_of_polyhedron:
lp15@63078
  2091
  assumes "polyhedron S" and "c facet_of S"
lp15@63078
  2092
  obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x \<le> b}" "c = S \<inter> {x. a \<bullet> x = b}"
lp15@63078
  2093
proof -
lp15@63078
  2094
  obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  2095
             and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
lp15@63078
  2096
             and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
lp15@63078
  2097
    using assms by (simp add: polyhedron_Int_affine_minimal) meson
lp15@63078
  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}"
lp15@63078
  2099
    by metis
lp15@63078
  2100
  obtain i where "i \<in> F" and c: "c = S \<inter> {x. a i \<bullet> x = b i}"
lp15@63078
  2101
    using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] assms
lp15@63078
  2102
    by force
lp15@63078
  2103
  moreover have ssub: "S \<subseteq> {x. a i \<bullet> x \<le> b i}"
lp15@63078
  2104
     apply (subst seq)
lp15@63078
  2105
     using \<open>i \<in> F\<close> ab by auto
lp15@63078
  2106
  ultimately show ?thesis
lp15@63078
  2107
    by (rule_tac a = "a i" and b = "b i" in that) (simp_all add: ab)
lp15@63078
  2108
qed
lp15@63078
  2109
lp15@63078
  2110
corollary face_of_polyhedron:
lp15@63078
  2111
  assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
lp15@63078
  2112
    shows "c = \<Inter>{F. F facet_of S \<and> c \<subseteq> F}"
lp15@63078
  2113
proof -
lp15@63078
  2114
  obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  2115
             and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
lp15@63078
  2116
             and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
lp15@63078
  2117
    using assms by (simp add: polyhedron_Int_affine_minimal) meson
lp15@63078
  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}"
lp15@63078
  2119
    by metis
lp15@63078
  2120
  show ?thesis
lp15@63078
  2121
    apply (subst face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
lp15@63078
  2122
    apply (auto simp: assms facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] cong: Collect_cong)
lp15@63078
  2123
    done
lp15@63078
  2124
qed
lp15@63078
  2125
lp15@63078
  2126
lemma face_of_polyhedron_subset_facet:
lp15@63078
  2127
  assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
lp15@63078
  2128
  obtains F where "F facet_of S" "c \<subseteq> F"
lp15@63078
  2129
using face_of_polyhedron assms
lp15@63078
  2130
by (metis (no_types, lifting) Inf_greatest antisym_conv face_of_imp_subset mem_Collect_eq)
lp15@63078
  2131
lp15@63078
  2132