src/HOL/Analysis/Polytope.thy
author paulson <lp15@cam.ac.uk>
Thu Jul 27 15:22:35 2017 +0100 (2017-07-27)
changeset 66297 d425bdf419f5
parent 66287 005a30862ed0
child 66515 85c505c98332
permissions -rw-r--r--
polytopes: simplical subdivisions, etc.
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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"  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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qed
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lp15@66287
   272
lemma subset_of_face_of_affine_hull:
lp15@66287
   273
    fixes S :: "'a::euclidean_space set"
lp15@66287
   274
  assumes T: "T face_of S" and "convex S" "U \<subseteq> S" and dis: "~disjnt (affine hull T) (rel_interior U)"
lp15@66287
   275
  shows "U \<subseteq> T"
lp15@66287
   276
  apply (rule subset_of_face_of [OF T \<open>U \<subseteq> S\<close>])
lp15@66287
   277
  using face_of_imp_eq_affine_Int [OF \<open>convex S\<close> T]
lp15@66287
   278
  using rel_interior_subset [of U] dis
lp15@66287
   279
  using \<open>U \<subseteq> S\<close> disjnt_def by fastforce
lp15@66287
   280
lp15@66287
   281
lemma affine_hull_face_of_disjoint_rel_interior:
lp15@66287
   282
    fixes S :: "'a::euclidean_space set"
lp15@66287
   283
  assumes "convex S" "F face_of S" "F \<noteq> S"
lp15@66287
   284
  shows "affine hull F \<inter> rel_interior S = {}"
lp15@66287
   285
  by (metis assms disjnt_def face_of_imp_subset order_refl subset_antisym subset_of_face_of_affine_hull)
lp15@63078
   286
lp15@63078
   287
lemma affine_diff_divide:
lp15@63078
   288
    assumes "affine S" "k \<noteq> 0" "k \<noteq> 1" and xy: "x \<in> S" "y /\<^sub>R (1 - k) \<in> S"
lp15@63078
   289
      shows "(x - y) /\<^sub>R k \<in> S"
lp15@63078
   290
proof -
lp15@63078
   291
  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
   292
    using assms
lp15@63078
   293
    by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] divide_simps)
lp15@63078
   294
  then show ?thesis
lp15@63078
   295
    using \<open>affine S\<close> xy by (auto simp: affine_alt)
lp15@63078
   296
qed
lp15@63078
   297
lp15@63078
   298
lemma face_of_convex_hulls:
lp15@63078
   299
      assumes S: "finite S" "T \<subseteq> S" and disj: "affine hull T \<inter> convex hull (S - T) = {}"
lp15@63078
   300
      shows  "(convex hull T) face_of (convex hull S)"
lp15@63078
   301
proof -
lp15@63078
   302
  have fin: "finite T" "finite (S - T)" using assms
lp15@63078
   303
    by (auto simp: finite_subset)
lp15@63078
   304
  have *: "x \<in> convex hull T"
lp15@63078
   305
          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
   306
          for x y w
lp15@63078
   307
  proof -
lp15@63078
   308
    have waff: "w \<in> affine hull T"
lp15@63078
   309
      using convex_hull_subset_affine_hull w by blast
nipkow@64267
   310
    obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "sum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
nipkow@64267
   311
                 and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "sum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
lp15@63078
   312
      using x y by (auto simp: assms convex_hull_finite)
lp15@63078
   313
    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
   314
               and u01: "0 < u" "u < 1"
lp15@63078
   315
      using w by (auto simp: open_segment_image_interval split: if_split_asm)
wenzelm@63148
   316
    define c where "c i = (1 - u) * a i + u * b i" for i
lp15@63078
   317
    have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
lp15@63078
   318
      using a b u01 by (simp add: c_def)
nipkow@64267
   319
    have sumc1: "sum c S = 1"
nipkow@64267
   320
      by (simp add: c_def sum.distrib sum_distrib_left [symmetric] asum bsum)
lp15@63078
   321
    have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
nipkow@64267
   322
      apply (simp add: c_def sum.distrib scaleR_left_distrib)
nipkow@64267
   323
      by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.sum [symmetric] aeqx beqy)
lp15@63078
   324
    show ?thesis
nipkow@64267
   325
    proof (cases "sum c (S - T) = 0")
lp15@63078
   326
      case True
lp15@63078
   327
      have ci0: "\<And>i. i \<in> (S - T) \<Longrightarrow> c i = 0"
lp15@65680
   328
        using True cge0 fin(2) sum_nonneg_eq_0_iff by auto
lp15@63078
   329
      have a0: "a i = 0" if "i \<in> (S - T)" for i
lp15@63078
   330
        using ci0 [OF that] u01 a [of i] b [of i] that
lp15@63078
   331
        by (simp add: c_def Groups.ordered_comm_monoid_add_class.add_nonneg_eq_0_iff)
nipkow@64267
   332
      have [simp]: "sum a T = 1"
nipkow@64267
   333
        using assms by (metis sum.mono_neutral_cong_right a0 asum)
lp15@63078
   334
      show ?thesis
lp15@63078
   335
        apply (simp add: convex_hull_finite \<open>finite T\<close>)
lp15@63078
   336
        apply (rule_tac x=a in exI)
lp15@63078
   337
        using a0 assms
nipkow@64267
   338
        apply (auto simp: cge0 a aeqx [symmetric] sum.mono_neutral_right)
lp15@63078
   339
        done
lp15@63078
   340
    next
lp15@63078
   341
      case False
nipkow@64267
   342
      define k where "k = sum c (S - T)"
lp15@63078
   343
      have "k > 0" using False
nipkow@64267
   344
        unfolding k_def by (metis DiffD1 antisym_conv cge0 sum_nonneg not_less)
nipkow@64267
   345
      have weq_sumsum: "w = sum (\<lambda>x. c x *\<^sub>R x) T + sum (\<lambda>x. c x *\<^sub>R x) (S - T)"
nipkow@64267
   346
        by (metis (no_types) add.commute S(1) S(2) sum.subset_diff sumci_xy weq)
lp15@63078
   347
      show ?thesis
lp15@63078
   348
      proof (cases "k = 1")
lp15@63078
   349
        case True
nipkow@64267
   350
        then have "sum c T = 0"
nipkow@64267
   351
          by (simp add: S k_def sum_diff sumc1)
nipkow@64267
   352
        then have [simp]: "sum c (S - T) = 1"
nipkow@64267
   353
          by (simp add: S sum_diff sumc1)
lp15@63078
   354
        have ci0: "\<And>i. i \<in> T \<Longrightarrow> c i = 0"
nipkow@64267
   355
          by (meson \<open>finite T\<close> \<open>sum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 sum_nonneg_eq_0_iff subsetCE)
lp15@63078
   356
        then have [simp]: "(\<Sum>i\<in>S-T. c i *\<^sub>R i) = w"
lp15@63078
   357
          by (simp add: weq_sumsum)
lp15@63078
   358
        have "w \<in> convex hull (S - T)"
lp15@63078
   359
          apply (simp add: convex_hull_finite fin)
lp15@63078
   360
          apply (rule_tac x=c in exI)
lp15@63078
   361
          apply (auto simp: cge0 weq True k_def)
lp15@63078
   362
          done
lp15@63078
   363
        then show ?thesis
lp15@63078
   364
          using disj waff by blast
lp15@63078
   365
      next
lp15@63078
   366
        case False
nipkow@64267
   367
        then have sumcf: "sum c T = 1 - k"
nipkow@64267
   368
          by (simp add: S k_def sum_diff sumc1)
lp15@63078
   369
        have "(\<Sum>i\<in>T. c i *\<^sub>R i) /\<^sub>R (1 - k) \<in> convex hull T"
lp15@63078
   370
          apply (simp add: convex_hull_finite fin)
lp15@63078
   371
          apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
lp15@63078
   372
          apply auto
nipkow@64267
   373
          apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) sum_nonneg subsetCE)
nipkow@64267
   374
          apply (metis False mult.commute right_inverse right_minus_eq sum_distrib_left sumcf)
nipkow@64267
   375
          by (metis (mono_tags, lifting) scaleR_right.sum scaleR_scaleR sum.cong)
nipkow@64267
   376
        with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - sum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
lp15@63078
   377
          by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
nipkow@64267
   378
        moreover have "inverse(k) *\<^sub>R (w - sum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
lp15@63078
   379
          apply (simp add: weq_sumsum convex_hull_finite fin)
lp15@63078
   380
          apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
lp15@63078
   381
          using \<open>k > 0\<close> cge0
nipkow@64267
   382
          apply (auto simp: scaleR_right.sum sum_distrib_left [symmetric] k_def [symmetric])
lp15@63078
   383
          done
lp15@63078
   384
        ultimately show ?thesis
lp15@63078
   385
          using disj by blast
lp15@63078
   386
      qed
lp15@63078
   387
    qed
lp15@63078
   388
  qed
lp15@63078
   389
  have [simp]: "convex hull T \<subseteq> convex hull S"
lp15@63078
   390
    by (simp add: \<open>T \<subseteq> S\<close> hull_mono)
lp15@63078
   391
  show ?thesis
lp15@63078
   392
    using open_segment_commute by (auto simp: face_of_def intro: *)
lp15@63078
   393
qed
lp15@63078
   394
lp15@63078
   395
proposition face_of_convex_hull_insert:
lp15@63078
   396
   "\<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
   397
  apply (rule face_of_trans, blast)
lp15@63078
   398
  apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
lp15@63078
   399
  done
lp15@63078
   400
lp15@63078
   401
proposition face_of_affine_trivial:
lp15@63078
   402
    assumes "affine S" "T face_of S"
lp15@63078
   403
    shows "T = {} \<or> T = S"
lp15@63078
   404
proof (rule ccontr, clarsimp)
lp15@63078
   405
  assume "T \<noteq> {}" "T \<noteq> S"
lp15@63078
   406
  then obtain a where "a \<in> T" by auto
lp15@63078
   407
  then have "a \<in> S"
lp15@63078
   408
    using \<open>T face_of S\<close> face_of_imp_subset by blast
lp15@63078
   409
  have "S \<subseteq> T"
lp15@63078
   410
  proof
lp15@63078
   411
    fix b  assume "b \<in> S"
lp15@63078
   412
    show "b \<in> T"
lp15@63078
   413
    proof (cases "a = b")
lp15@63078
   414
      case True with \<open>a \<in> T\<close> show ?thesis by auto
lp15@63078
   415
    next
lp15@63078
   416
      case False
lp15@63078
   417
      then have "a \<in> open_segment (2 *\<^sub>R a - b) b"
lp15@63078
   418
        apply (auto simp: open_segment_def closed_segment_def)
lp15@63078
   419
        apply (rule_tac x="1/2" in exI)
lp15@63078
   420
        apply (simp add: algebra_simps)
lp15@63078
   421
        by (simp add: scaleR_2)
lp15@63078
   422
      moreover have "2 *\<^sub>R a - b \<in> S"
lp15@63078
   423
        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
   424
      moreover note \<open>b \<in> S\<close> \<open>a \<in> T\<close>
lp15@63078
   425
      ultimately show ?thesis
lp15@63078
   426
        by (rule face_ofD [OF \<open>T face_of S\<close>, THEN conjunct2])
lp15@63078
   427
    qed
lp15@63078
   428
  qed
lp15@63078
   429
  then show False
wenzelm@63145
   430
    using \<open>T \<noteq> S\<close> \<open>T face_of S\<close> face_of_imp_subset by blast
lp15@63078
   431
qed
lp15@63078
   432
lp15@63078
   433
lp15@63078
   434
lemma face_of_affine_eq:
lp15@63078
   435
   "affine S \<Longrightarrow> (T face_of S \<longleftrightarrow> T = {} \<or> T = S)"
lp15@63078
   436
using affine_imp_convex face_of_affine_trivial face_of_refl by auto
lp15@63078
   437
lp15@63078
   438
lp15@63078
   439
lemma Inter_faces_finite_altbound:
lp15@63078
   440
    fixes T :: "'a::euclidean_space set set"
lp15@63078
   441
    assumes cfaI: "\<And>c. c \<in> T \<Longrightarrow> c face_of S"
lp15@63078
   442
    shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<and> \<Inter>F' = \<Inter>T"
lp15@63078
   443
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
   444
  case True
lp15@63078
   445
  then obtain c where c:
lp15@63078
   446
       "\<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
   447
    by metis
wenzelm@63148
   448
  define d where "d = rec_nat {c{}} (\<lambda>n r. insert (c r) r)"
lp15@63078
   449
  have [simp]: "d 0 = {c {}}"
lp15@63078
   450
    by (simp add: d_def)
lp15@63078
   451
  have dSuc [simp]: "\<And>n. d (Suc n) = insert (c (d n)) (d n)"
lp15@63078
   452
    by (simp add: d_def)
lp15@63078
   453
  have dn_notempty: "d n \<noteq> {}" for n
lp15@63078
   454
    by (induction n) auto
lp15@63078
   455
  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
   456
  using that
lp15@63078
   457
  proof (induction n)
lp15@63078
   458
    case 0
lp15@63078
   459
    then show ?case by (simp add: c)
lp15@63078
   460
  next
lp15@63078
   461
    case (Suc n)
lp15@63078
   462
    then show ?case by (auto simp: c card_insert_if)
lp15@63078
   463
  qed
lp15@63078
   464
  have aff_dim_le: "aff_dim(\<Inter>(d n)) \<le> DIM('a) - int n" if "n \<le> DIM('a) + 2" for n
lp15@63078
   465
  using that
lp15@63078
   466
  proof (induction n)
lp15@63078
   467
    case 0
lp15@63078
   468
    then show ?case
lp15@63078
   469
      by (simp add: aff_dim_le_DIM)
lp15@63078
   470
  next
lp15@63078
   471
    case (Suc n)
lp15@63078
   472
    have fs: "\<Inter>d (Suc n) face_of S"
lp15@63078
   473
      by (meson Suc.prems cfaI dn_le_Suc dn_notempty face_of_Inter subsetCE)
lp15@63078
   474
    have condn: "convex (\<Inter>d n)"
lp15@63078
   475
      using Suc.prems nat_le_linear not_less_eq_eq
lp15@63078
   476
      by (blast intro: face_of_imp_convex cfaI convex_Inter dest: dn_le_Suc)
lp15@63078
   477
    have fdn: "\<Inter>d (Suc n) face_of \<Inter>d n"
lp15@63078
   478
      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
   479
    have ne: "\<Inter>d (Suc n) \<noteq> \<Inter>d n"
lp15@63078
   480
      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
   481
    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
   482
      by arith
lp15@63078
   483
    have "aff_dim (\<Inter>d (Suc n)) < aff_dim (\<Inter>d n)"
lp15@63078
   484
      by (rule face_of_aff_dim_lt [OF condn fdn ne])
lp15@63078
   485
    moreover have "aff_dim (\<Inter>d n) \<le> int (DIM('a)) - int n"
lp15@63078
   486
      using Suc by auto
lp15@63078
   487
    ultimately
lp15@63078
   488
    have "aff_dim (\<Inter>d (Suc n)) \<le> int (DIM('a)) - (n+1)" by arith
lp15@63078
   489
    then show ?case by linarith
lp15@63078
   490
  qed
lp15@63078
   491
  have "aff_dim (\<Inter>d (DIM('a) + 2)) \<le> -2"
lp15@63078
   492
      using aff_dim_le [OF order_refl] by simp
lp15@63078
   493
  with aff_dim_geq [of "\<Inter>d (DIM('a) + 2)"] show ?thesis
lp15@63078
   494
    using order.trans by fastforce
lp15@63078
   495
next
lp15@63078
   496
  case False
lp15@63078
   497
  then show ?thesis
lp15@63078
   498
    apply simp
lp15@63078
   499
    apply (erule ex_forward)
lp15@63078
   500
    by blast
lp15@63078
   501
qed
lp15@63078
   502
lp15@63078
   503
lemma faces_of_translation:
lp15@63078
   504
   "{F. F face_of image (\<lambda>x. a + x) S} = image (image (\<lambda>x. a + x)) {F. F face_of S}"
lp15@63078
   505
apply (rule subset_antisym, clarify)
lp15@63078
   506
apply (auto simp: image_iff)
lp15@63078
   507
apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
lp15@63078
   508
done
lp15@63078
   509
lp15@63078
   510
proposition face_of_Times:
lp15@63078
   511
  assumes "F face_of S" and "F' face_of S'"
lp15@63078
   512
    shows "(F \<times> F') face_of (S \<times> S')"
lp15@63078
   513
proof -
lp15@63078
   514
  have "F \<times> F' \<subseteq> S \<times> S'"
lp15@63078
   515
    using assms [unfolded face_of_def] by blast
lp15@63078
   516
  moreover
lp15@63078
   517
  have "convex (F \<times> F')"
lp15@63078
   518
    using assms [unfolded face_of_def] by (blast intro: convex_Times)
lp15@63078
   519
  moreover
lp15@63078
   520
    have "a \<in> F \<and> a' \<in> F' \<and> b \<in> F \<and> b' \<in> F'"
lp15@63078
   521
       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
   522
       for a b a' b' x
lp15@63078
   523
  proof (cases "b=a \<or> b'=a'")
lp15@63078
   524
    case True with that show ?thesis
lp15@63078
   525
      using assms
lp15@63078
   526
      by (force simp: in_segment dest: face_ofD)
lp15@63078
   527
  next
lp15@63078
   528
    case False with assms [unfolded face_of_def] that show ?thesis
lp15@63078
   529
      by (blast dest!: open_segment_PairD)
lp15@63078
   530
  qed
lp15@63078
   531
  ultimately show ?thesis
lp15@63078
   532
    unfolding face_of_def by blast
lp15@63078
   533
qed
lp15@63078
   534
lp15@63078
   535
corollary face_of_Times_decomp:
lp15@63078
   536
    fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
lp15@63078
   537
    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
   538
     (is "?lhs = ?rhs")
lp15@63078
   539
proof
lp15@63078
   540
  assume c: ?lhs
lp15@63078
   541
  show ?rhs
lp15@63078
   542
  proof (cases "c = {}")
lp15@63078
   543
    case True then show ?thesis by auto
lp15@63078
   544
  next
lp15@63078
   545
    case False
lp15@63078
   546
    have 1: "fst ` c \<subseteq> S" "snd ` c \<subseteq> S'"
lp15@63078
   547
      using c face_of_imp_subset by fastforce+
lp15@63078
   548
    have "convex c"
lp15@63078
   549
      using c by (metis face_of_imp_convex)
lp15@63078
   550
    have conv: "convex (fst ` c)" "convex (snd ` c)"
lp15@63078
   551
      by (simp_all add: \<open>convex c\<close> convex_linear_image fst_linear snd_linear)
lp15@63078
   552
    have fstab: "a \<in> fst ` c \<and> b \<in> fst ` c"
lp15@63078
   553
            if "a \<in> S" "b \<in> S" "x \<in> open_segment a b" "(x,x') \<in> c" for a b x x'
lp15@63078
   554
    proof -
lp15@63078
   555
      have *: "(x,x') \<in> open_segment (a,x') (b,x')"
lp15@63078
   556
        using that by (auto simp: in_segment)
lp15@63078
   557
      show ?thesis
lp15@63078
   558
        using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
lp15@63078
   559
    qed
lp15@63078
   560
    have fst: "fst ` c face_of S"
lp15@63078
   561
      by (force simp: face_of_def 1 conv fstab)
lp15@63078
   562
    have sndab: "a' \<in> snd ` c \<and> b' \<in> snd ` c"
lp15@63078
   563
            if "a' \<in> S'" "b' \<in> S'" "x' \<in> open_segment a' b'" "(x,x') \<in> c" for a' b' x x'
lp15@63078
   564
    proof -
lp15@63078
   565
      have *: "(x,x') \<in> open_segment (x,a') (x,b')"
lp15@63078
   566
        using that by (auto simp: in_segment)
lp15@63078
   567
      show ?thesis
lp15@63078
   568
        using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
lp15@63078
   569
    qed
lp15@63078
   570
    have snd: "snd ` c face_of S'"
lp15@63078
   571
      by (force simp: face_of_def 1 conv sndab)
lp15@63078
   572
    have cc: "rel_interior c \<subseteq> rel_interior (fst ` c) \<times> rel_interior (snd ` c)"
lp15@63078
   573
      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
   574
    have "c = fst ` c \<times> snd ` c"
lp15@63078
   575
      apply (rule face_of_eq [OF c])
lp15@63078
   576
      apply (simp_all add: face_of_Times rel_interior_Times conv fst snd)
lp15@63078
   577
      using False rel_interior_eq_empty \<open>convex c\<close> cc
lp15@63078
   578
      apply blast
lp15@63078
   579
      done
lp15@63078
   580
    with fst snd show ?thesis by metis
lp15@63078
   581
  qed
lp15@63078
   582
next
lp15@63078
   583
  assume ?rhs with face_of_Times show ?lhs by auto
lp15@63078
   584
qed
lp15@63078
   585
lp15@63078
   586
lemma face_of_Times_eq:
lp15@63078
   587
    fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
lp15@63078
   588
    shows "(F \<times> F') face_of (S \<times> S') \<longleftrightarrow>
lp15@63078
   589
           F = {} \<or> F' = {} \<or> F face_of S \<and> F' face_of S'"
lp15@63078
   590
by (auto simp: face_of_Times_decomp times_eq_iff)
lp15@63078
   591
lp15@63078
   592
lemma hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
lp15@63078
   593
proof -
lp15@63078
   594
  have "{x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
lp15@63078
   595
    by auto
lp15@63078
   596
  with face_of_Int_supporting_hyperplane_le [OF convex_halfspace_le [of a b], of a b]
lp15@63078
   597
  show ?thesis by auto
lp15@63078
   598
qed
lp15@63078
   599
lp15@63078
   600
lemma hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
lp15@63078
   601
proof -
lp15@63078
   602
  have "{x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
lp15@63078
   603
    by auto
lp15@63078
   604
  with face_of_Int_supporting_hyperplane_ge [OF convex_halfspace_ge [of b a], of b a]
lp15@63078
   605
  show ?thesis by auto
lp15@63078
   606
qed
lp15@63078
   607
lp15@63078
   608
lemma face_of_halfspace_le:
lp15@63078
   609
  fixes a :: "'n::euclidean_space"
lp15@63078
   610
  shows "F face_of {x. a \<bullet> x \<le> b} \<longleftrightarrow>
lp15@63078
   611
         F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b}"
lp15@63078
   612
     (is "?lhs = ?rhs")
lp15@63078
   613
proof (cases "a = 0")
lp15@63078
   614
  case True then show ?thesis
lp15@63078
   615
    using face_of_affine_eq affine_UNIV by auto
lp15@63078
   616
next
lp15@63078
   617
  case False
lp15@63078
   618
  then have ine: "interior {x. a \<bullet> x \<le> b} \<noteq> {}"
lp15@63078
   619
    using halfspace_eq_empty_lt interior_halfspace_le by blast
lp15@63078
   620
  show ?thesis
lp15@63078
   621
  proof
lp15@63078
   622
    assume L: ?lhs
lp15@63078
   623
    have "F \<noteq> {x. a \<bullet> x \<le> b} \<Longrightarrow> F face_of {x. a \<bullet> x = b}"
lp15@63078
   624
      using False
lp15@63078
   625
      apply (simp add: frontier_halfspace_le [symmetric] rel_frontier_nonempty_interior [OF ine, symmetric])
lp15@63078
   626
      apply (rule face_of_subset [OF L])
lp15@63078
   627
      apply (simp add: face_of_subset_rel_frontier [OF L])
lp15@63078
   628
      apply (force simp: rel_frontier_def closed_halfspace_le)
lp15@63078
   629
      done
lp15@63078
   630
    with L show ?rhs
lp15@63078
   631
      using affine_hyperplane face_of_affine_eq by blast
lp15@63078
   632
  next
lp15@63078
   633
    assume ?rhs
lp15@63078
   634
    then show ?lhs
lp15@63078
   635
      by (metis convex_halfspace_le empty_face_of face_of_refl hyperplane_face_of_halfspace_le)
lp15@63078
   636
  qed
lp15@63078
   637
qed
lp15@63078
   638
lp15@63078
   639
lemma face_of_halfspace_ge:
lp15@63078
   640
  fixes a :: "'n::euclidean_space"
lp15@63078
   641
  shows "F face_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow>
lp15@63078
   642
         F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<ge> b}"
lp15@63078
   643
using face_of_halfspace_le [of F "-a" "-b"] by simp
lp15@63078
   644
lp15@63078
   645
subsection\<open>Exposed faces\<close>
lp15@63078
   646
lp15@63078
   647
text\<open>That is, faces that are intersection with supporting hyperplane\<close>
lp15@63078
   648
lp15@63078
   649
definition exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
lp15@63078
   650
                               (infixr "(exposed'_face'_of)" 50)
lp15@63078
   651
  where "T exposed_face_of S \<longleftrightarrow>
lp15@63078
   652
         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
   653
lp15@63078
   654
lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
lp15@63078
   655
  apply (simp add: exposed_face_of_def)
lp15@63078
   656
  apply (rule_tac x=0 in exI)
lp15@63078
   657
  apply (rule_tac x=1 in exI, force)
lp15@63078
   658
  done
lp15@63078
   659
lp15@63078
   660
lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
lp15@63078
   661
  apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
lp15@63078
   662
  apply (rule_tac x=0 in exI)+
lp15@63078
   663
  apply force
lp15@63078
   664
  done
lp15@63078
   665
lp15@63078
   666
lemma exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
lp15@63078
   667
  by simp
lp15@63078
   668
lp15@63078
   669
lemma exposed_face_of:
lp15@63078
   670
    "T exposed_face_of S \<longleftrightarrow>
lp15@63078
   671
     T face_of S \<and>
lp15@63078
   672
     (T = {} \<or> T = S \<or>
lp15@63078
   673
      (\<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
   674
proof (cases "T = {}")
lp15@63078
   675
  case True then show ?thesis
lp15@63078
   676
    by simp
lp15@63078
   677
next
lp15@63078
   678
  case False
lp15@63078
   679
  show ?thesis
lp15@63078
   680
  proof (cases "T = S")
lp15@63078
   681
    case True then show ?thesis
lp15@63078
   682
      by (simp add: face_of_refl_eq)
lp15@63078
   683
  next
lp15@63078
   684
    case False
lp15@63078
   685
    with \<open>T \<noteq> {}\<close> show ?thesis
lp15@63078
   686
      apply (auto simp: exposed_face_of_def)
lp15@63078
   687
      apply (metis inner_zero_left)
lp15@63078
   688
      done
lp15@63078
   689
  qed
lp15@63078
   690
qed
lp15@63078
   691
lp15@63078
   692
lemma exposed_face_of_Int_supporting_hyperplane_le:
lp15@63078
   693
   "\<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
   694
by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)
lp15@63078
   695
lp15@63078
   696
lemma exposed_face_of_Int_supporting_hyperplane_ge:
lp15@63078
   697
   "\<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
   698
using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp
lp15@63078
   699
lp15@63078
   700
proposition exposed_face_of_Int:
lp15@63078
   701
  assumes "T exposed_face_of S"
lp15@63078
   702
      and "u exposed_face_of S"
lp15@63078
   703
    shows "(T \<inter> u) exposed_face_of S"
lp15@63078
   704
proof -
lp15@63078
   705
  obtain a b where T: "S \<inter> {x. a \<bullet> x = b} face_of S"
lp15@63078
   706
               and S: "S \<subseteq> {x. a \<bullet> x \<le> b}"
lp15@63078
   707
               and teq: "T = S \<inter> {x. a \<bullet> x = b}"
lp15@63078
   708
    using assms by (auto simp: exposed_face_of_def)
lp15@63078
   709
  obtain a' b' where u: "S \<inter> {x. a' \<bullet> x = b'} face_of S"
lp15@63078
   710
                 and s': "S \<subseteq> {x. a' \<bullet> x \<le> b'}"
lp15@63078
   711
                 and ueq: "u = S \<inter> {x. a' \<bullet> x = b'}"
lp15@63078
   712
    using assms by (auto simp: exposed_face_of_def)
lp15@63078
   713
  have tu: "T \<inter> u face_of S"
lp15@63078
   714
    using T teq u ueq by (simp add: face_of_Int)
lp15@63078
   715
  have ss: "S \<subseteq> {x. (a + a') \<bullet> x \<le> b + b'}"
lp15@63078
   716
    using S s' by (force simp: inner_left_distrib)
lp15@63078
   717
  show ?thesis
lp15@63078
   718
    apply (simp add: exposed_face_of_def tu)
lp15@63078
   719
    apply (rule_tac x="a+a'" in exI)
lp15@63078
   720
    apply (rule_tac x="b+b'" in exI)
lp15@63078
   721
    using S s'
lp15@63078
   722
    apply (fastforce simp: ss inner_left_distrib teq ueq)
lp15@63078
   723
    done
lp15@63078
   724
qed
lp15@63078
   725
lp15@63078
   726
proposition exposed_face_of_Inter:
lp15@63078
   727
    fixes P :: "'a::euclidean_space set set"
lp15@63078
   728
  assumes "P \<noteq> {}"
lp15@63078
   729
      and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S"
lp15@63078
   730
    shows "\<Inter>P exposed_face_of S"
lp15@63078
   731
proof -
lp15@63078
   732
  obtain Q where "finite Q" and QsubP: "Q \<subseteq> P" "card Q \<le> DIM('a) + 2" and IntQ: "\<Inter>Q = \<Inter>P"
lp15@63078
   733
    using Inter_faces_finite_altbound [of P S] assms [unfolded exposed_face_of]
lp15@63078
   734
    by force
lp15@63078
   735
  show ?thesis
lp15@63078
   736
  proof (cases "Q = {}")
lp15@63078
   737
    case True then show ?thesis
lp15@63078
   738
      by (metis Inf_empty Inf_lower IntQ assms ex_in_conv subset_antisym top_greatest)
lp15@63078
   739
  next
lp15@63078
   740
    case False
lp15@63078
   741
    have "Q \<subseteq> {T. T exposed_face_of S}"
lp15@63078
   742
      using QsubP assms by blast
lp15@63078
   743
    moreover have "Q \<subseteq> {T. T exposed_face_of S} \<Longrightarrow> \<Inter>Q exposed_face_of S"
lp15@63078
   744
      using \<open>finite Q\<close> False
lp15@63078
   745
      apply (induction Q rule: finite_induct)
lp15@63078
   746
      using exposed_face_of_Int apply fastforce+
lp15@63078
   747
      done
lp15@63078
   748
    ultimately show ?thesis
lp15@63078
   749
      by (simp add: IntQ)
lp15@63078
   750
  qed
lp15@63078
   751
qed
lp15@63078
   752
lp15@63078
   753
proposition exposed_face_of_sums:
lp15@63078
   754
  assumes "convex S" and "convex T"
lp15@63078
   755
      and "F exposed_face_of {x + y | x y. x \<in> S \<and> y \<in> T}"
lp15@63078
   756
          (is "F exposed_face_of ?ST")
lp15@63078
   757
  obtains k l
lp15@63078
   758
    where "k exposed_face_of S" "l exposed_face_of T"
lp15@63078
   759
          "F = {x + y | x y. x \<in> k \<and> y \<in> l}"
lp15@63078
   760
proof (cases "F = {}")
lp15@63078
   761
  case True then show ?thesis
lp15@63078
   762
    using that by blast
lp15@63078
   763
next
lp15@63078
   764
  case False
lp15@63078
   765
  show ?thesis
lp15@63078
   766
  proof (cases "F = ?ST")
lp15@63078
   767
    case True then show ?thesis
lp15@63078
   768
      using assms exposed_face_of_refl_eq that by blast
lp15@63078
   769
  next
lp15@63078
   770
    case False
lp15@63078
   771
    obtain p where "p \<in> F" using \<open>F \<noteq> {}\<close> by blast
lp15@63078
   772
    moreover
lp15@63078
   773
    obtain u z where T: "?ST \<inter> {x. u \<bullet> x = z} face_of ?ST"
lp15@63078
   774
                 and S: "?ST \<subseteq> {x. u \<bullet> x \<le> z}"
lp15@63078
   775
                 and feq: "F = ?ST \<inter> {x. u \<bullet> x = z}"
lp15@63078
   776
      using assms by (auto simp: exposed_face_of_def)
lp15@63078
   777
    ultimately obtain a0 b0
lp15@63078
   778
            where p: "p = a0 + b0" and "a0 \<in> S" "b0 \<in> T" and z: "u \<bullet> p = z"
lp15@63078
   779
      by auto
lp15@63078
   780
    have lez: "u \<bullet> (x + y) \<le> z" if "x \<in> S" "y \<in> T" for x y
lp15@63078
   781
      using S that by auto
lp15@63078
   782
    have sef: "S \<inter> {x. u \<bullet> x = u \<bullet> a0} exposed_face_of S"
lp15@63078
   783
      apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex S\<close>])
lp15@63078
   784
      apply (metis p z add_le_cancel_right inner_right_distrib lez [OF _ \<open>b0 \<in> T\<close>])
lp15@63078
   785
      done
lp15@63078
   786
    have tef: "T \<inter> {x. u \<bullet> x = u \<bullet> b0} exposed_face_of T"
lp15@63078
   787
      apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex T\<close>])
lp15@63078
   788
      apply (metis p z add.commute add_le_cancel_right inner_right_distrib lez [OF \<open>a0 \<in> S\<close>])
lp15@63078
   789
      done
lp15@63078
   790
    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
   791
      by (auto simp: feq) (metis inner_right_distrib p z)
lp15@63078
   792
    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
   793
      apply (auto simp: feq)
lp15@63078
   794
      apply (rename_tac x y)
lp15@63078
   795
      apply (rule_tac x=x in exI)
lp15@63078
   796
      apply (rule_tac x=y in exI, simp)
lp15@63078
   797
      using z p \<open>a0 \<in> S\<close> \<open>b0 \<in> T\<close>
lp15@63078
   798
      apply clarify
lp15@63078
   799
      apply (simp add: inner_right_distrib)
lp15@63078
   800
      apply (metis add_le_cancel_right antisym lez [unfolded inner_right_distrib] add.commute)
lp15@63078
   801
      done
lp15@63078
   802
    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
   803
      by blast
lp15@63078
   804
    then show ?thesis
lp15@63078
   805
      by (rule that [OF sef tef])
lp15@63078
   806
  qed
lp15@63078
   807
qed
lp15@63078
   808
lp15@63078
   809
subsection\<open>Extreme points of a set: its singleton faces\<close>
lp15@63078
   810
lp15@63078
   811
definition extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
lp15@63078
   812
                               (infixr "(extreme'_point'_of)" 50)
lp15@63078
   813
  where "x extreme_point_of S \<longleftrightarrow>
lp15@63078
   814
         x \<in> S \<and> (\<forall>a \<in> S. \<forall>b \<in> S. x \<notin> open_segment a b)"
lp15@63078
   815
lp15@63078
   816
lemma extreme_point_of_stillconvex:
lp15@63078
   817
   "convex S \<Longrightarrow> (x extreme_point_of S \<longleftrightarrow> x \<in> S \<and> convex(S - {x}))"
lp15@63078
   818
  by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)
lp15@63078
   819
lp15@63078
   820
lemma face_of_singleton:
lp15@63078
   821
   "{x} face_of S \<longleftrightarrow> x extreme_point_of S"
lp15@63078
   822
by (fastforce simp add: extreme_point_of_def face_of_def)
lp15@63078
   823
lp15@63078
   824
lemma extreme_point_not_in_REL_INTERIOR:
lp15@63078
   825
    fixes S :: "'a::real_normed_vector set"
lp15@63078
   826
    shows "\<lbrakk>x extreme_point_of S; S \<noteq> {x}\<rbrakk> \<Longrightarrow> x \<notin> rel_interior S"
lp15@63078
   827
apply (simp add: face_of_singleton [symmetric])
lp15@63078
   828
apply (blast dest: face_of_disjoint_rel_interior)
lp15@63078
   829
done
lp15@63078
   830
lp15@63078
   831
lemma extreme_point_not_in_interior:
lp15@63078
   832
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
lp15@63078
   833
    shows "x extreme_point_of S \<Longrightarrow> x \<notin> interior S"
lp15@63078
   834
apply (case_tac "S = {x}")
lp15@63078
   835
apply (simp add: empty_interior_finite)
lp15@63078
   836
by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)
lp15@63078
   837
lp15@63078
   838
lemma extreme_point_of_face:
lp15@63078
   839
     "F face_of S \<Longrightarrow> v extreme_point_of F \<longleftrightarrow> v extreme_point_of S \<and> v \<in> F"
lp15@63078
   840
  by (meson empty_subsetI face_of_face face_of_singleton insert_subset)
lp15@63078
   841
lp15@63078
   842
lemma extreme_point_of_convex_hull:
lp15@63078
   843
   "x extreme_point_of (convex hull S) \<Longrightarrow> x \<in> S"
lp15@63078
   844
apply (simp add: extreme_point_of_stillconvex)
lp15@63078
   845
using hull_minimal [of S "(convex hull S) - {x}" convex]
lp15@63078
   846
using hull_subset [of S convex]
lp15@63078
   847
apply blast
lp15@63078
   848
done
lp15@63078
   849
lp15@63078
   850
lemma extreme_points_of_convex_hull:
lp15@63078
   851
   "{x. x extreme_point_of (convex hull S)} \<subseteq> S"
lp15@63078
   852
using extreme_point_of_convex_hull by auto
lp15@63078
   853
lp15@63078
   854
lemma extreme_point_of_empty [simp]: "~ (x extreme_point_of {})"
lp15@63078
   855
  by (simp add: extreme_point_of_def)
lp15@63078
   856
lp15@63078
   857
lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
lp15@63078
   858
  using extreme_point_of_stillconvex by auto
lp15@63078
   859
lp15@63078
   860
lemma extreme_point_of_translation_eq:
lp15@63078
   861
   "(a + x) extreme_point_of (image (\<lambda>x. a + x) S) \<longleftrightarrow> x extreme_point_of S"
lp15@63078
   862
by (auto simp: extreme_point_of_def)
lp15@63078
   863
lp15@63078
   864
lemma extreme_points_of_translation:
lp15@63078
   865
   "{x. x extreme_point_of (image (\<lambda>x. a + x) S)} =
lp15@63078
   866
    (\<lambda>x. a + x) ` {x. x extreme_point_of S}"
lp15@63078
   867
using extreme_point_of_translation_eq
lp15@63078
   868
by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
lp15@63078
   869
lp15@63078
   870
lemma extreme_point_of_Int:
lp15@63078
   871
   "\<lbrakk>x extreme_point_of S; x extreme_point_of T\<rbrakk> \<Longrightarrow> x extreme_point_of (S \<inter> T)"
lp15@63078
   872
by (simp add: extreme_point_of_def)
lp15@63078
   873
lp15@63078
   874
lemma extreme_point_of_Int_supporting_hyperplane_le:
lp15@63078
   875
   "\<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
   876
apply (simp add: face_of_singleton [symmetric])
lp15@63078
   877
by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)
lp15@63078
   878
lp15@63078
   879
lemma extreme_point_of_Int_supporting_hyperplane_ge:
lp15@63078
   880
   "\<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
   881
apply (simp add: face_of_singleton [symmetric])
lp15@63078
   882
by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)
lp15@63078
   883
lp15@63078
   884
lemma exposed_point_of_Int_supporting_hyperplane_le:
lp15@63078
   885
   "\<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
   886
apply (simp add: exposed_face_of_def face_of_singleton)
lp15@63078
   887
apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
lp15@63078
   888
done
lp15@63078
   889
lp15@63078
   890
lemma exposed_point_of_Int_supporting_hyperplane_ge:
lp15@63078
   891
    "\<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
   892
using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
lp15@63078
   893
by simp
lp15@63078
   894
lp15@63078
   895
lemma extreme_point_of_convex_hull_insert:
lp15@63078
   896
   "\<lbrakk>finite S; a \<notin> convex hull S\<rbrakk> \<Longrightarrow> a extreme_point_of (convex hull (insert a S))"
lp15@63078
   897
apply (case_tac "a \<in> S")
lp15@63078
   898
apply (simp add: hull_inc)
lp15@63078
   899
using face_of_convex_hulls [of "insert a S" "{a}"]
lp15@63078
   900
apply (auto simp: face_of_singleton hull_same)
lp15@63078
   901
done
lp15@63078
   902
lp15@63078
   903
subsection\<open>Facets\<close>
lp15@63078
   904
lp15@63078
   905
definition facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
lp15@63078
   906
                    (infixr "(facet'_of)" 50)
lp15@63078
   907
  where "F facet_of S \<longleftrightarrow> F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1"
lp15@63078
   908
lp15@63078
   909
lemma facet_of_empty [simp]: "~ S facet_of {}"
lp15@63078
   910
  by (simp add: facet_of_def)
lp15@63078
   911
lp15@63078
   912
lemma facet_of_irrefl [simp]: "~ S facet_of S "
lp15@63078
   913
  by (simp add: facet_of_def)
lp15@63078
   914
lp15@63078
   915
lemma facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
lp15@63078
   916
  by (simp add: facet_of_def)
lp15@63078
   917
lp15@63078
   918
lemma facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
lp15@63078
   919
  by (simp add: face_of_imp_subset facet_of_def)
lp15@63078
   920
lp15@63078
   921
lemma hyperplane_facet_of_halfspace_le:
lp15@63078
   922
   "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
lp15@63078
   923
unfolding facet_of_def hyperplane_eq_empty
lp15@63078
   924
by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
lp15@63078
   925
           DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)
lp15@63078
   926
lp15@63078
   927
lemma hyperplane_facet_of_halfspace_ge:
lp15@63078
   928
    "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<ge> b}"
lp15@63078
   929
unfolding facet_of_def hyperplane_eq_empty
lp15@63078
   930
by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
lp15@63078
   931
           DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)
lp15@63078
   932
lp15@63078
   933
lemma facet_of_halfspace_le:
lp15@63078
   934
    "F facet_of {x. a \<bullet> x \<le> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
lp15@63078
   935
    (is "?lhs = ?rhs")
lp15@63078
   936
proof
lp15@63078
   937
  assume c: ?lhs
lp15@63078
   938
  with c facet_of_irrefl show ?rhs
lp15@63078
   939
    by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
lp15@63078
   940
next
lp15@63078
   941
  assume ?rhs then show ?lhs
lp15@63078
   942
    by (simp add: hyperplane_facet_of_halfspace_le)
lp15@63078
   943
qed
lp15@63078
   944
lp15@63078
   945
lemma facet_of_halfspace_ge:
lp15@63078
   946
    "F facet_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
lp15@63078
   947
using facet_of_halfspace_le [of F "-a" "-b"] by simp
lp15@63078
   948
lp15@63078
   949
subsection \<open>Edges: faces of affine dimension 1\<close>
lp15@63078
   950
lp15@63078
   951
definition edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
lp15@63078
   952
  where "e edge_of S \<longleftrightarrow> e face_of S \<and> aff_dim e = 1"
lp15@63078
   953
lp15@63078
   954
lemma edge_of_imp_subset:
lp15@63078
   955
   "S edge_of T \<Longrightarrow> S \<subseteq> T"
lp15@63078
   956
by (simp add: edge_of_def face_of_imp_subset)
lp15@63078
   957
lp15@63078
   958
subsection\<open>Existence of extreme points\<close>
lp15@63078
   959
lp15@63078
   960
lemma different_norm_3_collinear_points:
lp15@63078
   961
  fixes a :: "'a::euclidean_space"
lp15@63078
   962
  assumes "x \<in> open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
lp15@63078
   963
  shows False
lp15@63078
   964
proof -
lp15@63078
   965
  obtain u where "norm ((1 - u) *\<^sub>R a + u *\<^sub>R b) = norm b"
lp15@63078
   966
             and "a \<noteq> b"
lp15@63078
   967
             and u01: "0 < u" "u < 1"
lp15@63078
   968
    using assms by (auto simp: open_segment_image_interval if_splits)
lp15@63078
   969
  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
   970
             (1 - u * u) *\<^sub>R (a \<bullet> a)"
lp15@63078
   971
    using assms by (simp add: norm_eq algebra_simps inner_commute)
lp15@63078
   972
  then have "(1 - u) *\<^sub>R ((1 - u) *\<^sub>R a \<bullet> a + (2 * u) *\<^sub>R  a \<bullet> b) =
lp15@63078
   973
             (1 - u) *\<^sub>R ((1 + u) *\<^sub>R (a \<bullet> a))"
lp15@63078
   974
    by (simp add: algebra_simps)
lp15@63078
   975
  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
   976
    using u01 by auto
lp15@63078
   977
  then have "a \<bullet> b = a \<bullet> a"
lp15@63078
   978
    using u01 by (simp add: algebra_simps)
lp15@63078
   979
  then have "a = b"
lp15@63078
   980
    using \<open>norm(a) = norm(b)\<close> norm_eq vector_eq by fastforce
lp15@63078
   981
  then show ?thesis
lp15@63078
   982
    using \<open>a \<noteq> b\<close> by force
lp15@63078
   983
qed
lp15@63078
   984
lp15@63078
   985
proposition extreme_point_exists_convex:
lp15@63078
   986
  fixes S :: "'a::euclidean_space set"
lp15@63078
   987
  assumes "compact S" "convex S" "S \<noteq> {}"
lp15@63078
   988
  obtains x where "x extreme_point_of S"
lp15@63078
   989
proof -
lp15@63078
   990
  obtain x where "x \<in> S" and xsup: "\<And>y. y \<in> S \<Longrightarrow> norm y \<le> norm x"
lp15@63078
   991
    using distance_attains_sup [of S 0] assms by auto
lp15@63078
   992
  have False if "a \<in> S" "b \<in> S" and x: "x \<in> open_segment a b" for a b
lp15@63078
   993
  proof -
lp15@63078
   994
    have noax: "norm a \<le> norm x" and nobx: "norm b \<le> norm x" using xsup that by auto
lp15@63078
   995
    have "a \<noteq> b"
lp15@63078
   996
      using empty_iff open_segment_idem x by auto
lp15@63078
   997
    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
   998
    proof -
lp15@63078
   999
      have "(1 - u) * na + u * nb < (1 - u) * norm x + u * nb"
lp15@63078
  1000
        by (simp add: that)
lp15@63078
  1001
      also have "... \<le> (1 - u) * norm x + u * norm x"
lp15@63078
  1002
        by (simp add: that)
lp15@63078
  1003
      finally have "(1 - u) * na + u * nb < (1 - u) * norm x + u * norm x" .
lp15@63078
  1004
      then show ?thesis
lp15@63078
  1005
      using scaleR_collapse [symmetric, of "norm x" u] by auto
lp15@63078
  1006
    qed
lp15@63078
  1007
    have "norm x < norm x" if "norm a < norm x"
lp15@63078
  1008
      using x
lp15@63078
  1009
      apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
lp15@63078
  1010
      apply (rule norm_triangle_lt)
lp15@63078
  1011
      apply (simp add: norm_mult)
lp15@63078
  1012
      using * [of "norm a" "norm b"] nobx that
lp15@63078
  1013
        apply blast
lp15@63078
  1014
      done
lp15@63078
  1015
    moreover have "norm x < norm x" if "norm b < norm x"
lp15@63078
  1016
      using x
lp15@63078
  1017
      apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
lp15@63078
  1018
      apply (rule norm_triangle_lt)
lp15@63078
  1019
      apply (simp add: norm_mult)
lp15@63078
  1020
      using * [of "norm b" "norm a" "1-u" for u] noax that
lp15@63078
  1021
        apply (simp add: add.commute)
lp15@63078
  1022
      done
lp15@63078
  1023
    ultimately have "~ (norm a < norm x) \<and> ~ (norm b < norm x)"
lp15@63078
  1024
      by auto
lp15@63078
  1025
    then show ?thesis
lp15@63078
  1026
      using different_norm_3_collinear_points noax nobx that(3) by fastforce
lp15@63078
  1027
  qed
lp15@63078
  1028
  then show ?thesis
lp15@63078
  1029
    apply (rule_tac x=x in that)
lp15@63078
  1030
    apply (force simp: extreme_point_of_def \<open>x \<in> S\<close>)
lp15@63078
  1031
    done
lp15@63078
  1032
qed
lp15@63078
  1033
lp15@63078
  1034
subsection\<open>Krein-Milman, the weaker form\<close>
lp15@63078
  1035
lp15@63078
  1036
proposition Krein_Milman:
lp15@63078
  1037
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1038
  assumes "compact S" "convex S"
lp15@63078
  1039
    shows "S = closure(convex hull {x. x extreme_point_of S})"
lp15@63078
  1040
proof (cases "S = {}")
lp15@63078
  1041
  case True then show ?thesis   by simp
lp15@63078
  1042
next
lp15@63078
  1043
  case False
lp15@63078
  1044
  have "closed S"
lp15@63078
  1045
    by (simp add: \<open>compact S\<close> compact_imp_closed)
lp15@63078
  1046
  have "closure (convex hull {x. x extreme_point_of S}) \<subseteq> S"
lp15@63078
  1047
    apply (rule closure_minimal [OF hull_minimal \<open>closed S\<close>])
lp15@63078
  1048
    using assms
lp15@63078
  1049
    apply (auto simp: extreme_point_of_def)
lp15@63078
  1050
    done
lp15@63078
  1051
  moreover have "u \<in> closure (convex hull {x. x extreme_point_of S})"
lp15@63078
  1052
                if "u \<in> S" for u
lp15@63078
  1053
  proof (rule ccontr)
lp15@63078
  1054
    assume unot: "u \<notin> closure(convex hull {x. x extreme_point_of S})"
lp15@63078
  1055
    then obtain a b where "a \<bullet> u < b"
lp15@63078
  1056
          and ab: "\<And>x. x \<in> closure(convex hull {x. x extreme_point_of S}) \<Longrightarrow> b < a \<bullet> x"
lp15@63078
  1057
      using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
lp15@63078
  1058
      by blast
lp15@63078
  1059
    have "continuous_on S (op \<bullet> a)"
lp15@63078
  1060
      by (rule continuous_intros)+
lp15@63078
  1061
    then obtain m where "m \<in> S" and m: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> m \<le> a \<bullet> y"
lp15@63078
  1062
      using continuous_attains_inf [of S "\<lambda>x. a \<bullet> x"] \<open>compact S\<close> \<open>u \<in> S\<close>
lp15@63078
  1063
      by auto
wenzelm@63148
  1064
    define T where "T = S \<inter> {x. a \<bullet> x = a \<bullet> m}"
lp15@63078
  1065
    have "m \<in> T"
lp15@63078
  1066
      by (simp add: T_def \<open>m \<in> S\<close>)
lp15@63078
  1067
    moreover have "compact T"
lp15@63078
  1068
      by (simp add: T_def compact_Int_closed [OF \<open>compact S\<close> closed_hyperplane])
lp15@63078
  1069
    moreover have "convex T"
lp15@63078
  1070
      by (simp add: T_def convex_Int [OF \<open>convex S\<close> convex_hyperplane])
lp15@63078
  1071
    ultimately obtain v where v: "v extreme_point_of T"
lp15@63078
  1072
      using extreme_point_exists_convex [of T] by auto
lp15@63078
  1073
    then have "{v} face_of T"
lp15@63078
  1074
      by (simp add: face_of_singleton)
lp15@63078
  1075
    also have "T face_of S"
lp15@63078
  1076
      by (simp add: T_def m face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
lp15@63078
  1077
    finally have "v extreme_point_of S"
lp15@63078
  1078
      by (simp add: face_of_singleton)
lp15@63078
  1079
    then have "b < a \<bullet> v"
lp15@63078
  1080
      using closure_subset by (simp add: closure_hull hull_inc ab)
lp15@63078
  1081
    then show False
lp15@63078
  1082
      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
  1083
  qed
lp15@63078
  1084
  ultimately show ?thesis
lp15@63078
  1085
    by blast
lp15@63078
  1086
qed
lp15@63078
  1087
lp15@63078
  1088
text\<open>Now the sharper form.\<close>
lp15@63078
  1089
lp15@63078
  1090
lemma Krein_Milman_Minkowski_aux:
lp15@63078
  1091
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1092
  assumes n: "dim S = n" and S: "compact S" "convex S" "0 \<in> S"
lp15@63078
  1093
    shows "0 \<in> convex hull {x. x extreme_point_of S}"
lp15@63078
  1094
using n S
lp15@63078
  1095
proof (induction n arbitrary: S rule: less_induct)
lp15@63078
  1096
  case (less n S) show ?case
lp15@63078
  1097
  proof (cases "0 \<in> rel_interior S")
lp15@63078
  1098
    case True with Krein_Milman show ?thesis
lp15@63078
  1099
      by (metis subsetD convex_convex_hull convex_rel_interior_closure less.prems(2) less.prems(3) rel_interior_subset)
lp15@63078
  1100
  next
lp15@63078
  1101
    case False
lp15@63078
  1102
    have "rel_interior S \<noteq> {}"
lp15@63078
  1103
      by (simp add: rel_interior_convex_nonempty_aux less)
lp15@63078
  1104
    then obtain c where c: "c \<in> rel_interior S" by blast
lp15@63078
  1105
    obtain a where "a \<noteq> 0"
lp15@63078
  1106
              and le_ay: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> 0 \<le> a \<bullet> y"
lp15@63078
  1107
              and less_ay: "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> 0 < a \<bullet> y"
lp15@63078
  1108
      by (blast intro: supporting_hyperplane_rel_boundary intro!: less False)
lp15@63078
  1109
    have face: "S \<inter> {x. a \<bullet> x = 0} face_of S"
lp15@63078
  1110
      apply (rule face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
lp15@63078
  1111
      using le_ay by auto
lp15@63078
  1112
    then have co: "compact (S \<inter> {x. a \<bullet> x = 0})" "convex (S \<inter> {x. a \<bullet> x = 0})"
lp15@63078
  1113
      using less.prems by (blast intro: face_of_imp_compact face_of_imp_convex)+
lp15@63078
  1114
    have "a \<bullet> y = 0" if "y \<in> span (S \<inter> {x. a \<bullet> x = 0})" for y
lp15@63078
  1115
    proof -
lp15@63078
  1116
      have "y \<in> span {x. a \<bullet> x = 0}"
lp15@63078
  1117
        by (metis inf.cobounded2 span_mono subsetCE that)
lp15@63469
  1118
      then show ?thesis
lp15@63469
  1119
        by (blast intro: span_induct [OF _ subspace_hyperplane])
lp15@63078
  1120
    qed
lp15@63078
  1121
    then have "dim (S \<inter> {x. a \<bullet> x = 0}) < n"
lp15@63078
  1122
      by (metis (no_types) less_ay c subsetD dim_eq_span inf.strict_order_iff
lp15@63078
  1123
           inf_le1 \<open>dim S = n\<close> not_le rel_interior_subset span_0 span_clauses(1))
lp15@63078
  1124
    then have "0 \<in> convex hull {x. x extreme_point_of (S \<inter> {x. a \<bullet> x = 0})}"
lp15@63078
  1125
      by (rule less.IH) (auto simp: co less.prems)
lp15@63078
  1126
    then show ?thesis
lp15@63078
  1127
      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
  1128
  qed
lp15@63078
  1129
qed
lp15@63078
  1130
lp15@63078
  1131
lp15@63078
  1132
theorem Krein_Milman_Minkowski:
lp15@63078
  1133
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1134
  assumes "compact S" "convex S"
lp15@63078
  1135
    shows "S = convex hull {x. x extreme_point_of S}"
lp15@63078
  1136
proof
lp15@63078
  1137
  show "S \<subseteq> convex hull {x. x extreme_point_of S}"
lp15@63078
  1138
  proof
lp15@63078
  1139
    fix a assume [simp]: "a \<in> S"
lp15@63078
  1140
    have 1: "compact (op + (- a) ` S)"
lp15@63078
  1141
      by (simp add: \<open>compact S\<close> compact_translation)
lp15@63078
  1142
    have 2: "convex (op + (- a) ` S)"
lp15@63078
  1143
      by (simp add: \<open>convex S\<close> convex_translation)
lp15@63078
  1144
    show a_invex: "a \<in> convex hull {x. x extreme_point_of S}"
lp15@63078
  1145
      using Krein_Milman_Minkowski_aux [OF refl 1 2]
lp15@63078
  1146
            convex_hull_translation [of "-a"]
lp15@63078
  1147
      by (auto simp: extreme_points_of_translation translation_assoc)
lp15@63078
  1148
    qed
lp15@63078
  1149
next
lp15@63078
  1150
  show "convex hull {x. x extreme_point_of S} \<subseteq> S"
lp15@63078
  1151
  proof -
lp15@63078
  1152
    have "{a. a extreme_point_of S} \<subseteq> S"
lp15@63078
  1153
      using extreme_point_of_def by blast
lp15@63078
  1154
    then show ?thesis
lp15@63078
  1155
      by (simp add: \<open>convex S\<close> hull_minimal)
lp15@63078
  1156
  qed
lp15@63078
  1157
qed
lp15@63078
  1158
lp15@63078
  1159
lp15@63078
  1160
subsection\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
lp15@63078
  1161
lp15@63078
  1162
lemma Krein_Milman_polytope:
lp15@63078
  1163
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1164
  shows
lp15@63078
  1165
   "finite S
lp15@63078
  1166
       \<Longrightarrow> convex hull S =
lp15@63078
  1167
           convex hull {x. x extreme_point_of (convex hull S)}"
lp15@63078
  1168
by (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
lp15@63078
  1169
lp15@63078
  1170
lemma extreme_points_of_convex_hull_eq:
lp15@63078
  1171
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1172
  shows
lp15@63078
  1173
   "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
lp15@63078
  1174
        \<Longrightarrow> {x. x extreme_point_of (convex hull S)} = S"
lp15@63078
  1175
by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)
lp15@63078
  1176
lp15@63078
  1177
lp15@63078
  1178
lemma extreme_point_of_convex_hull_eq:
lp15@63078
  1179
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1180
  shows
lp15@63078
  1181
   "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
lp15@63078
  1182
    \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
lp15@63078
  1183
using extreme_points_of_convex_hull_eq by auto
lp15@63078
  1184
lp15@63078
  1185
lemma extreme_point_of_convex_hull_convex_independent:
lp15@63078
  1186
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1187
  assumes "compact S" and S: "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})"
lp15@63078
  1188
  shows "(x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
lp15@63078
  1189
proof -
lp15@63078
  1190
  have "convex hull T \<noteq> convex hull S" if "T \<subset> S" for T
lp15@63078
  1191
  proof -
lp15@63078
  1192
    obtain a where  "T \<subseteq> S" "a \<in> S" "a \<notin> T" using \<open>T \<subset> S\<close> by blast
lp15@63078
  1193
    then show ?thesis
lp15@63078
  1194
      by (metis (full_types) Diff_eq_empty_iff Diff_insert0 S hull_mono hull_subset insert_Diff_single subsetCE)
lp15@63078
  1195
  qed
lp15@63078
  1196
  then show ?thesis
lp15@63078
  1197
    by (rule extreme_point_of_convex_hull_eq [OF \<open>compact S\<close>])
lp15@63078
  1198
qed
lp15@63078
  1199
lp15@63078
  1200
lemma extreme_point_of_convex_hull_affine_independent:
lp15@63078
  1201
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1202
  shows
lp15@63078
  1203
   "~ affine_dependent S
lp15@63078
  1204
         \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
lp15@63078
  1205
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
  1206
lp15@63078
  1207
text\<open>Elementary proofs exist, not requiring Euclidean spaces and all this development\<close>
lp15@63078
  1208
lemma extreme_point_of_convex_hull_2:
lp15@63078
  1209
  fixes x :: "'a::euclidean_space"
lp15@63078
  1210
  shows "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x = a \<or> x = b"
lp15@63078
  1211
proof -
lp15@63078
  1212
  have "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x \<in> {a,b}"
lp15@63078
  1213
    by (intro extreme_point_of_convex_hull_affine_independent affine_independent_2)
lp15@63078
  1214
  then show ?thesis
lp15@63078
  1215
    by simp
lp15@63078
  1216
qed
lp15@63078
  1217
lp15@63078
  1218
lemma extreme_point_of_segment:
lp15@63078
  1219
  fixes x :: "'a::euclidean_space"
lp15@63078
  1220
  shows
lp15@63078
  1221
   "x extreme_point_of closed_segment a b \<longleftrightarrow> x = a \<or> x = b"
lp15@63078
  1222
by (simp add: extreme_point_of_convex_hull_2 segment_convex_hull)
lp15@63078
  1223
lp15@63078
  1224
lemma face_of_convex_hull_subset:
lp15@63078
  1225
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1226
  assumes "compact S" and T: "T face_of (convex hull S)"
lp15@63078
  1227
  obtains s' where "s' \<subseteq> S" "T = convex hull s'"
lp15@63078
  1228
apply (rule_tac s' = "{x. x extreme_point_of T}" in that)
lp15@63078
  1229
using T extreme_point_of_convex_hull extreme_point_of_face apply blast
lp15@63078
  1230
by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)
lp15@63078
  1231
lp15@63078
  1232
lp15@66297
  1233
lemma face_of_convex_hull_aux:
lp15@66297
  1234
  assumes eq: "x *\<^sub>R p = u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c"
lp15@66297
  1235
    and x: "u + v + w = x" "x \<noteq> 0" and S: "affine S" "a \<in> S" "b \<in> S" "c \<in> S"
lp15@66297
  1236
  shows "p \<in> S"
lp15@66297
  1237
proof -
lp15@66297
  1238
  have "p = (u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x"
lp15@66297
  1239
    by (metis \<open>x \<noteq> 0\<close> eq mult.commute right_inverse scaleR_one scaleR_scaleR)
lp15@66297
  1240
  moreover have "affine hull {a,b,c} \<subseteq> S"
lp15@66297
  1241
    by (simp add: S hull_minimal)
lp15@66297
  1242
  moreover have "(u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x \<in> affine hull {a,b,c}"
lp15@66297
  1243
    apply (simp add: affine_hull_3)
lp15@66297
  1244
    apply (rule_tac x="u/x" in exI)
lp15@66297
  1245
    apply (rule_tac x="v/x" in exI)
lp15@66297
  1246
    apply (rule_tac x="w/x" in exI)
lp15@66297
  1247
    using x apply (auto simp: algebra_simps divide_simps)
lp15@66297
  1248
    done
lp15@66297
  1249
  ultimately show ?thesis by force
lp15@66297
  1250
qed
lp15@66297
  1251
lp15@66297
  1252
proposition face_of_convex_hull_insert_eq:
lp15@66297
  1253
  fixes a :: "'a :: euclidean_space"
lp15@66297
  1254
  assumes "finite S" and a: "a \<notin> affine hull S"
lp15@66297
  1255
  shows "(F face_of (convex hull (insert a S)) \<longleftrightarrow>
lp15@66297
  1256
          F face_of (convex hull S) \<or>
lp15@66297
  1257
          (\<exists>F'. F' face_of (convex hull S) \<and> F = convex hull (insert a F')))"
lp15@66297
  1258
         (is "F face_of ?CAS \<longleftrightarrow> _")
lp15@66297
  1259
proof safe
lp15@66297
  1260
  assume F: "F face_of ?CAS"
lp15@66297
  1261
    and *: "\<nexists>F'. F' face_of convex hull S \<and> F = convex hull insert a F'"
lp15@66297
  1262
  obtain T where T: "T \<subseteq> insert a S" and FeqT: "F = convex hull T"
lp15@66297
  1263
    by (metis F \<open>finite S\<close> compact_insert finite_imp_compact face_of_convex_hull_subset)
lp15@66297
  1264
  show "F face_of convex hull S"
lp15@66297
  1265
  proof (cases "a \<in> T")
lp15@66297
  1266
    case True
lp15@66297
  1267
    have "F = convex hull insert a (convex hull T \<inter> convex hull S)"
lp15@66297
  1268
    proof
lp15@66297
  1269
      have "T \<subseteq> insert a (convex hull T \<inter> convex hull S)"
lp15@66297
  1270
        using T hull_subset by fastforce
lp15@66297
  1271
      then show "F \<subseteq> convex hull insert a (convex hull T \<inter> convex hull S)"
lp15@66297
  1272
        by (simp add: FeqT hull_mono)
lp15@66297
  1273
      show "convex hull insert a (convex hull T \<inter> convex hull S) \<subseteq> F"
lp15@66297
  1274
        apply (rule hull_minimal)
lp15@66297
  1275
        using True by (auto simp: \<open>F = convex hull T\<close> hull_inc)
lp15@66297
  1276
    qed
lp15@66297
  1277
    moreover have "convex hull T \<inter> convex hull S face_of convex hull S"
lp15@66297
  1278
      by (metis F FeqT convex_convex_hull face_of_slice hull_mono inf.absorb_iff2 subset_insertI)
lp15@66297
  1279
    ultimately show ?thesis
lp15@66297
  1280
      using * by force
lp15@66297
  1281
  next
lp15@66297
  1282
    case False
lp15@66297
  1283
    then show ?thesis
lp15@66297
  1284
      by (metis FeqT F T face_of_subset hull_mono subset_insert subset_insertI)
lp15@66297
  1285
  qed
lp15@66297
  1286
next
lp15@66297
  1287
  assume "F face_of convex hull S"
lp15@66297
  1288
  show "F face_of ?CAS"
lp15@66297
  1289
    by (simp add: \<open>F face_of convex hull S\<close> a face_of_convex_hull_insert \<open>finite S\<close>)
lp15@66297
  1290
next
lp15@66297
  1291
  fix F
lp15@66297
  1292
  assume F: "F face_of convex hull S"
lp15@66297
  1293
  show "convex hull insert a F face_of ?CAS"
lp15@66297
  1294
  proof (cases "S = {}")
lp15@66297
  1295
    case True
lp15@66297
  1296
    then show ?thesis
lp15@66297
  1297
      using F face_of_affine_eq by auto
lp15@66297
  1298
  next
lp15@66297
  1299
    case False
lp15@66297
  1300
    have anotc: "a \<notin> convex hull S"
lp15@66297
  1301
      by (metis (no_types) a affine_hull_convex_hull hull_inc)
lp15@66297
  1302
    show ?thesis
lp15@66297
  1303
    proof (cases "F = {}")
lp15@66297
  1304
      case True show ?thesis
lp15@66297
  1305
        using anotc by (simp add: \<open>F = {}\<close> \<open>finite S\<close> extreme_point_of_convex_hull_insert face_of_singleton)
lp15@66297
  1306
    next
lp15@66297
  1307
      case False
lp15@66297
  1308
      have "convex hull insert a F \<subseteq> ?CAS"
lp15@66297
  1309
        by (simp add: F a \<open>finite S\<close> convex_hull_subset face_of_convex_hull_insert face_of_imp_subset hull_inc)
lp15@66297
  1310
      moreover
lp15@66297
  1311
      have "(\<exists>y v. (1 - ub) *\<^sub>R a + ub *\<^sub>R b = (1 - v) *\<^sub>R a + v *\<^sub>R y \<and>
lp15@66297
  1312
                   0 \<le> v \<and> v \<le> 1 \<and> y \<in> F) \<and>
lp15@66297
  1313
            (\<exists>x u. (1 - uc) *\<^sub>R a + uc *\<^sub>R c = (1 - u) *\<^sub>R a + u *\<^sub>R x \<and>
lp15@66297
  1314
                   0 \<le> u \<and> u \<le> 1 \<and> x \<in> F)"
lp15@66297
  1315
        if *: "(1 - ux) *\<^sub>R a + ux *\<^sub>R x
lp15@66297
  1316
               \<in> open_segment ((1 - ub) *\<^sub>R a + ub *\<^sub>R b) ((1 - uc) *\<^sub>R a + uc *\<^sub>R c)"
lp15@66297
  1317
          and "0 \<le> ub" "ub \<le> 1" "0 \<le> uc" "uc \<le> 1" "0 \<le> ux" "ux \<le> 1"
lp15@66297
  1318
          and b: "b \<in> convex hull S" and c: "c \<in> convex hull S" and "x \<in> F"
lp15@66297
  1319
        for b c ub uc ux x
lp15@66297
  1320
      proof -
lp15@66297
  1321
        obtain v where ne: "(1 - ub) *\<^sub>R a + ub *\<^sub>R b \<noteq> (1 - uc) *\<^sub>R a + uc *\<^sub>R c"
lp15@66297
  1322
          and eq: "(1 - ux) *\<^sub>R a + ux *\<^sub>R x =
lp15@66297
  1323
                    (1 - v) *\<^sub>R ((1 - ub) *\<^sub>R a + ub *\<^sub>R b) + v *\<^sub>R ((1 - uc) *\<^sub>R a + uc *\<^sub>R c)"
lp15@66297
  1324
          and "0 < v" "v < 1"
lp15@66297
  1325
          using * by (auto simp: in_segment)
lp15@66297
  1326
        then have 0: "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *\<^sub>R a +
lp15@66297
  1327
                      (ux *\<^sub>R x - (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c)) = 0"
lp15@66297
  1328
          by (auto simp: algebra_simps)
lp15@66297
  1329
        then have "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *\<^sub>R a =
lp15@66297
  1330
                   ((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c + (-ux) *\<^sub>R x"
lp15@66297
  1331
          by (auto simp: algebra_simps)
lp15@66297
  1332
        then have "a \<in> affine hull S" if "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) \<noteq> 0"
lp15@66297
  1333
          apply (rule face_of_convex_hull_aux)
lp15@66297
  1334
          using b c that apply (auto simp: algebra_simps)
lp15@66297
  1335
          using F convex_hull_subset_affine_hull face_of_imp_subset \<open>x \<in> F\<close> apply blast+
lp15@66297
  1336
          done
lp15@66297
  1337
        then have "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) = 0"
lp15@66297
  1338
          using a by blast
lp15@66297
  1339
        with 0 have equx: "(1 - v) * ub + v * uc = ux"
lp15@66297
  1340
          and uxx: "ux *\<^sub>R x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c)"
lp15@66297
  1341
          by auto (auto simp: algebra_simps)
lp15@66297
  1342
        show ?thesis
lp15@66297
  1343
        proof (cases "uc = 0")
lp15@66297
  1344
          case True
lp15@66297
  1345
          then show ?thesis
lp15@66297
  1346
            using equx 0 \<open>0 \<le> ub\<close> \<open>ub \<le> 1\<close> \<open>v < 1\<close> \<open>x \<in> F\<close>
lp15@66297
  1347
            apply (auto simp: algebra_simps)
lp15@66297
  1348
             apply (rule_tac x=x in exI, simp)
lp15@66297
  1349
             apply (rule_tac x=ub in exI, auto)
lp15@66297
  1350
             apply (metis add.left_neutral diff_eq_eq less_irrefl mult.commute mult_cancel_right1 real_vector.scale_cancel_left real_vector.scale_left_diff_distrib)
lp15@66297
  1351
            using \<open>x \<in> F\<close> \<open>uc \<le> 1\<close> apply blast
lp15@66297
  1352
            done
lp15@66297
  1353
        next
lp15@66297
  1354
          case False
lp15@66297
  1355
          show ?thesis
lp15@66297
  1356
          proof (cases "ub = 0")
lp15@66297
  1357
            case True
lp15@66297
  1358
            then show ?thesis
lp15@66297
  1359
              using equx 0 \<open>0 \<le> uc\<close> \<open>uc \<le> 1\<close> \<open>0 < v\<close> \<open>x \<in> F\<close> \<open>uc \<noteq> 0\<close> by (force simp: algebra_simps)
lp15@66297
  1360
          next
lp15@66297
  1361
            case False
lp15@66297
  1362
            then have "0 < ub" "0 < uc"
lp15@66297
  1363
              using \<open>uc \<noteq> 0\<close> \<open>0 \<le> ub\<close> \<open>0 \<le> uc\<close> by auto
lp15@66297
  1364
            then have "ux \<noteq> 0"
lp15@66297
  1365
              by (metis \<open>0 < v\<close> \<open>v < 1\<close> diff_ge_0_iff_ge dual_order.strict_implies_order equx leD le_add_same_cancel2 zero_le_mult_iff zero_less_mult_iff)
lp15@66297
  1366
            have "b \<in> F \<and> c \<in> F"
lp15@66297
  1367
            proof (cases "b = c")
lp15@66297
  1368
              case True
lp15@66297
  1369
              then show ?thesis
lp15@66297
  1370
                by (metis \<open>ux \<noteq> 0\<close> equx real_vector.scale_cancel_left scaleR_add_left uxx \<open>x \<in> F\<close>)
lp15@66297
  1371
            next
lp15@66297
  1372
              case False
lp15@66297
  1373
              have "x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c) /\<^sub>R ux"
lp15@66297
  1374
                by (metis \<open>ux \<noteq> 0\<close> uxx mult.commute right_inverse scaleR_one scaleR_scaleR)
lp15@66297
  1375
              also have "... = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c"
lp15@66297
  1376
                using \<open>ux \<noteq> 0\<close> equx apply (auto simp: algebra_simps divide_simps)
lp15@66297
  1377
                by (metis add.commute add_diff_eq add_divide_distrib diff_add_cancel scaleR_add_left)
lp15@66297
  1378
              finally have "x = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c" .
lp15@66297
  1379
              then have "x \<in> open_segment b c"
lp15@66297
  1380
                apply (simp add: in_segment \<open>b \<noteq> c\<close>)
lp15@66297
  1381
                apply (rule_tac x="(v * uc) / ux" in exI)
lp15@66297
  1382
                using \<open>0 \<le> ux\<close> \<open>ux \<noteq> 0\<close> \<open>0 < uc\<close> \<open>0 < v\<close> \<open>0 < ub\<close> \<open>v < 1\<close> equx
lp15@66297
  1383
                apply (force simp: algebra_simps divide_simps)
lp15@66297
  1384
                done
lp15@66297
  1385
              then show ?thesis
lp15@66297
  1386
                by (rule face_ofD [OF F _ b c \<open>x \<in> F\<close>])
lp15@66297
  1387
            qed
lp15@66297
  1388
            with \<open>0 \<le> ub\<close> \<open>ub \<le> 1\<close> \<open>0 \<le> uc\<close> \<open>uc \<le> 1\<close> show ?thesis by blast
lp15@66297
  1389
          qed
lp15@66297
  1390
        qed
lp15@66297
  1391
      qed
lp15@66297
  1392
      moreover have "convex hull F = F"
lp15@66297
  1393
        by (meson F convex_hull_eq face_of_imp_convex)
lp15@66297
  1394
      ultimately show ?thesis
lp15@66297
  1395
        unfolding face_of_def by (fastforce simp: convex_hull_insert_alt \<open>S \<noteq> {}\<close> \<open>F \<noteq> {}\<close>)
lp15@66297
  1396
    qed
lp15@66297
  1397
  qed
lp15@66297
  1398
qed
lp15@66297
  1399
lp15@66297
  1400
lemma face_of_convex_hull_insert2:
lp15@66297
  1401
  fixes a :: "'a :: euclidean_space"
lp15@66297
  1402
  assumes S: "finite S" and a: "a \<notin> affine hull S" and F: "F face_of convex hull S"
lp15@66297
  1403
  shows "convex hull (insert a F) face_of convex hull (insert a S)"
lp15@66297
  1404
  by (metis F face_of_convex_hull_insert_eq [OF S a])
lp15@66297
  1405
lp15@63078
  1406
proposition face_of_convex_hull_affine_independent:
lp15@63078
  1407
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1408
  assumes "~ affine_dependent S"
lp15@63078
  1409
    shows "(T face_of (convex hull S) \<longleftrightarrow> (\<exists>c. c \<subseteq> S \<and> T = convex hull c))"
lp15@63078
  1410
          (is "?lhs = ?rhs")
lp15@63078
  1411
proof
lp15@63078
  1412
  assume ?lhs
lp15@63078
  1413
  then show ?rhs
lp15@63078
  1414
    by (meson \<open>T face_of convex hull S\<close> aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
lp15@63078
  1415
next
lp15@63078
  1416
  assume ?rhs
lp15@63078
  1417
  then obtain c where "c \<subseteq> S" and T: "T = convex hull c"
lp15@63078
  1418
    by blast
lp15@63078
  1419
  have "affine hull c \<inter> affine hull (S - c) = {}"
lp15@63078
  1420
    apply (rule disjoint_affine_hull [OF assms \<open>c \<subseteq> S\<close>], auto)
lp15@63078
  1421
    done
lp15@63078
  1422
  then have "affine hull c \<inter> convex hull (S - c) = {}"
lp15@63078
  1423
    using convex_hull_subset_affine_hull by fastforce
lp15@63078
  1424
  then show ?lhs
lp15@63078
  1425
    by (metis face_of_convex_hulls \<open>c \<subseteq> S\<close> aff_independent_finite assms T)
lp15@63078
  1426
qed
lp15@63078
  1427
lp15@63078
  1428
lemma facet_of_convex_hull_affine_independent:
lp15@63078
  1429
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1430
  assumes "~ affine_dependent S"
lp15@63078
  1431
    shows "T facet_of (convex hull S) \<longleftrightarrow>
lp15@63078
  1432
           T \<noteq> {} \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u}))"
lp15@63078
  1433
          (is "?lhs = ?rhs")
lp15@63078
  1434
proof
lp15@63078
  1435
  assume ?lhs
lp15@63078
  1436
  then have "T face_of (convex hull S)" "T \<noteq> {}"
lp15@63078
  1437
        and afft: "aff_dim T = aff_dim (convex hull S) - 1"
lp15@63078
  1438
    by (auto simp: facet_of_def)
lp15@63078
  1439
  then obtain c where "c \<subseteq> S" and c: "T = convex hull c"
lp15@63078
  1440
    by (auto simp: face_of_convex_hull_affine_independent [OF assms])
lp15@63078
  1441
  then have affs: "aff_dim S = aff_dim c + 1"
lp15@63078
  1442
    by (metis aff_dim_convex_hull afft eq_diff_eq)
lp15@63078
  1443
  have "~ affine_dependent c"
lp15@63078
  1444
    using \<open>c \<subseteq> S\<close> affine_dependent_subset assms by blast
lp15@63078
  1445
  with affs have "card (S - c) = 1"
lp15@63078
  1446
    apply (simp add: aff_dim_affine_independent [symmetric] aff_dim_convex_hull)
lp15@63078
  1447
    by (metis aff_dim_affine_independent aff_independent_finite One_nat_def \<open>c \<subseteq> S\<close> add.commute
lp15@63078
  1448
                add_diff_cancel_right' assms card_Diff_subset card_mono of_nat_1 of_nat_diff of_nat_eq_iff)
lp15@63078
  1449
  then obtain u where u: "u \<in> S - c"
lp15@63078
  1450
    by (metis DiffI \<open>c \<subseteq> S\<close> aff_independent_finite assms cancel_comm_monoid_add_class.diff_cancel
lp15@63078
  1451
                card_Diff_subset subsetI subset_antisym zero_neq_one)
lp15@63078
  1452
  then have u: "S = insert u c"
lp15@63078
  1453
    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
  1454
  have "T = convex hull (c - {u})"
lp15@63078
  1455
    by (metis Diff_empty Diff_insert0 \<open>T facet_of convex hull S\<close> c facet_of_irrefl insert_absorb u)
lp15@63078
  1456
  with \<open>T \<noteq> {}\<close> show ?rhs
lp15@63078
  1457
    using c u by auto
lp15@63078
  1458
next
lp15@63078
  1459
  assume ?rhs
lp15@63078
  1460
  then obtain u where "T \<noteq> {}" "u \<in> S" and u: "T = convex hull (S - {u})"
lp15@63078
  1461
    by (force simp: facet_of_def)
lp15@63078
  1462
  then have "\<not> S \<subseteq> {u}"
lp15@63078
  1463
    using \<open>T \<noteq> {}\<close> u by auto
lp15@63078
  1464
  have [simp]: "aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1"
lp15@63078
  1465
    using assms \<open>u \<in> S\<close>
lp15@63078
  1466
    apply (simp add: aff_dim_convex_hull affine_dependent_def)
lp15@63078
  1467
    apply (drule bspec, assumption)
lp15@63078
  1468
    by (metis add_diff_cancel_right' aff_dim_insert insert_Diff [of u S])
lp15@63078
  1469
  show ?lhs
lp15@63078
  1470
    apply (subst u)
lp15@63078
  1471
    apply (simp add: \<open>\<not> S \<subseteq> {u}\<close> facet_of_def face_of_convex_hull_affine_independent [OF assms], blast)
lp15@63078
  1472
    done
lp15@63078
  1473
qed
lp15@63078
  1474
lp15@63078
  1475
lemma facet_of_convex_hull_affine_independent_alt:
lp15@63078
  1476
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1477
  shows
lp15@63078
  1478
   "~affine_dependent S
lp15@63078
  1479
        \<Longrightarrow> (T facet_of (convex hull S) \<longleftrightarrow>
lp15@63078
  1480
             2 \<le> card S \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
lp15@63078
  1481
apply (simp add: facet_of_convex_hull_affine_independent)
lp15@63078
  1482
apply (auto simp: Set.subset_singleton_iff)
lp15@63078
  1483
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
  1484
done
lp15@63078
  1485
lp15@63078
  1486
lemma segment_face_of:
lp15@63078
  1487
  assumes "(closed_segment a b) face_of S"
lp15@63078
  1488
  shows "a extreme_point_of S" "b extreme_point_of S"
lp15@63078
  1489
proof -
lp15@63078
  1490
  have as: "{a} face_of S"
lp15@63078
  1491
    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
  1492
  moreover have "{b} face_of S"
lp15@63078
  1493
  proof -
lp15@63078
  1494
    have "b \<in> convex hull {a} \<or> b extreme_point_of convex hull {b, a}"
lp15@63078
  1495
      by (meson extreme_point_of_convex_hull_insert finite.emptyI finite.insertI)
lp15@63078
  1496
    moreover have "closed_segment a b = convex hull {b, a}"
lp15@63078
  1497
      using closed_segment_commute segment_convex_hull by blast
lp15@63078
  1498
    ultimately show ?thesis
lp15@63078
  1499
      by (metis as assms face_of_face convex_hull_singleton empty_iff face_of_singleton insertE)
lp15@63078
  1500
    qed
lp15@63078
  1501
  ultimately show "a extreme_point_of S" "b extreme_point_of S"
lp15@63078
  1502
    using face_of_singleton by blast+
lp15@63078
  1503
qed
lp15@63078
  1504
lp15@63078
  1505
lp15@63078
  1506
lemma Krein_Milman_frontier:
lp15@63078
  1507
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1508
  assumes "convex S" "compact S"
lp15@63078
  1509
    shows "S = convex hull (frontier S)"
lp15@63078
  1510
          (is "?lhs = ?rhs")
lp15@63078
  1511
proof
lp15@63078
  1512
  have "?lhs \<subseteq> convex hull {x. x extreme_point_of S}"
lp15@63078
  1513
    using Krein_Milman_Minkowski assms by blast
lp15@63078
  1514
  also have "... \<subseteq> ?rhs"
lp15@63078
  1515
    apply (rule hull_mono)
lp15@63078
  1516
    apply (auto simp: frontier_def extreme_point_not_in_interior)
lp15@63078
  1517
    using closure_subset apply (force simp: extreme_point_of_def)
lp15@63078
  1518
    done
lp15@63078
  1519
  finally show "?lhs \<subseteq> ?rhs" .
lp15@63078
  1520
next
lp15@63078
  1521
  have "?rhs \<subseteq> convex hull S"
lp15@63078
  1522
    by (metis Diff_subset \<open>compact S\<close> closure_closed compact_eq_bounded_closed frontier_def hull_mono)
lp15@63078
  1523
  also have "... \<subseteq> ?lhs"
lp15@63078
  1524
    by (simp add: \<open>convex S\<close> hull_same)
lp15@63078
  1525
  finally show "?rhs \<subseteq> ?lhs" .
lp15@63078
  1526
qed
lp15@63078
  1527
lp15@63078
  1528
subsection\<open>Polytopes\<close>
lp15@63078
  1529
lp15@63078
  1530
definition polytope where
lp15@63078
  1531
 "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
lp15@63078
  1532
lp15@63078
  1533
lemma polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
lp15@63078
  1534
apply (simp add: polytope_def, safe)
lp15@63078
  1535
apply (metis convex_hull_translation finite_imageI translation_galois)
lp15@63078
  1536
by (metis convex_hull_translation finite_imageI)
lp15@63078
  1537
lp15@63078
  1538
lemma polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
lp15@63078
  1539
  unfolding polytope_def using convex_hull_linear_image by blast
lp15@63078
  1540
lp15@63078
  1541
lemma polytope_empty: "polytope {}"
lp15@63078
  1542
  using convex_hull_empty polytope_def by blast
lp15@63078
  1543
lp15@63078
  1544
lemma polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
lp15@63078
  1545
  using polytope_def by auto
lp15@63078
  1546
lp15@63078
  1547
lemma polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
lp15@63078
  1548
  unfolding polytope_def
lp15@63078
  1549
  by (metis finite_cartesian_product convex_hull_Times)
lp15@63078
  1550
lp15@63078
  1551
lemma face_of_polytope_polytope:
lp15@63078
  1552
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1553
  shows "\<lbrakk>polytope S; F face_of S\<rbrakk> \<Longrightarrow> polytope F"
lp15@63078
  1554
unfolding polytope_def
lp15@63078
  1555
by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)
lp15@63078
  1556
lp15@63078
  1557
lemma finite_polytope_faces:
lp15@63078
  1558
  fixes S :: "'a::euclidean_space set"
lp15@63078
  1559
  assumes "polytope S"
lp15@63078
  1560
  shows "finite {F. F face_of S}"
lp15@63078
  1561
proof -
lp15@63078
  1562
  obtain v where "finite v" "S = convex hull v"
lp15@63078
  1563
    using assms polytope_def by auto
lp15@63078
  1564
  have "finite (op hull convex ` {T. T \<subseteq> v})"
lp15@63078
  1565
    by (simp add: \<open>finite v\<close>)
lp15@63078
  1566
  moreover have "{F. F face_of S} \<subseteq> (op hull convex ` {T. T \<subseteq> v})"
lp15@63078
  1567
    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
  1568
  ultimately show ?thesis
lp15@63078
  1569
    by (blast intro: finite_subset)
lp15@63078
  1570
qed
lp15@63078
  1571
lp15@63078
  1572
lemma finite_polytope_facets:
lp15@63078
  1573
  assumes "polytope S"
lp15@63078
  1574
  shows "finite {T. T facet_of S}"
lp15@63078
  1575
by (simp add: assms facet_of_def finite_polytope_faces)
lp15@63078
  1576
lp15@63078
  1577
lemma polytope_scaling:
lp15@63078
  1578
  assumes "polytope S"  shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
lp15@63078
  1579
by (simp add: assms polytope_linear_image)
lp15@63078
  1580
lp15@63078
  1581
lemma polytope_imp_compact:
lp15@63078
  1582
  fixes S :: "'a::real_normed_vector set"
lp15@63078
  1583
  shows "polytope S \<Longrightarrow> compact S"
lp15@63078
  1584
by (metis finite_imp_compact_convex_hull polytope_def)
lp15@63078
  1585
lp15@63078
  1586
lemma polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
lp15@63078
  1587
  by (metis convex_convex_hull polytope_def)
lp15@63078
  1588
lp15@63078
  1589
lemma polytope_imp_closed:
lp15@63078
  1590
  fixes S :: "'a::real_normed_vector set"
lp15@63078
  1591
  shows "polytope S \<Longrightarrow> closed S"
lp15@63078
  1592
by (simp add: compact_imp_closed polytope_imp_compact)
lp15@63078
  1593
lp15@63078
  1594
lemma polytope_imp_bounded:
lp15@63078
  1595
  fixes S :: "'a::real_normed_vector set"
lp15@63078
  1596
  shows "polytope S \<Longrightarrow> bounded S"
lp15@63078
  1597
by (simp add: compact_imp_bounded polytope_imp_compact)
lp15@63078
  1598
lp15@63078
  1599
lemma polytope_interval: "polytope(cbox a b)"
lp15@63078
  1600
  unfolding polytope_def by (meson closed_interval_as_convex_hull)
lp15@63078
  1601
lp15@63078
  1602
lemma polytope_sing: "polytope {a}"
lp15@63078
  1603
  using polytope_def by force
lp15@63078
  1604
lp15@66297
  1605
lemma face_of_polytope_insert:
lp15@66297
  1606
     "\<lbrakk>polytope S; a \<notin> affine hull S; F face_of S\<rbrakk> \<Longrightarrow> F face_of convex hull (insert a S)"
lp15@66297
  1607
  by (metis (no_types, lifting) affine_hull_convex_hull face_of_convex_hull_insert hull_insert polytope_def)
lp15@66297
  1608
lp15@66297
  1609
lemma face_of_polytope_insert2:
lp15@66297
  1610
  fixes a :: "'a :: euclidean_space"
lp15@66297
  1611
  assumes "polytope S" "a \<notin> affine hull S" "F face_of S"
lp15@66297
  1612
  shows "convex hull (insert a F) face_of convex hull (insert a S)"
lp15@66297
  1613
proof -
lp15@66297
  1614
  obtain V where "finite V" "S = convex hull V"
lp15@66297
  1615
    using assms by (auto simp: polytope_def)
lp15@66297
  1616
  then have "convex hull (insert a F) face_of convex hull (insert a V)"
lp15@66297
  1617
    using affine_hull_convex_hull assms face_of_convex_hull_insert2 by blast
lp15@66297
  1618
  then show ?thesis
lp15@66297
  1619
    by (metis \<open>S = convex hull V\<close> hull_insert)
lp15@66297
  1620
qed
lp15@66297
  1621
lp15@63078
  1622
lp15@63078
  1623
subsection\<open>Polyhedra\<close>
lp15@63078
  1624
lp15@63078
  1625
definition polyhedron where
lp15@63078
  1626
 "polyhedron S \<equiv>
lp15@63078
  1627
        \<exists>F. finite F \<and>
lp15@63078
  1628
            S = \<Inter> F \<and>
lp15@63078
  1629
            (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b})"
lp15@63078
  1630
lp15@63078
  1631
lemma polyhedron_Int [intro,simp]:
lp15@63078
  1632
   "\<lbrakk>polyhedron S; polyhedron T\<rbrakk> \<Longrightarrow> polyhedron (S \<inter> T)"
lp15@63078
  1633
  apply (simp add: polyhedron_def, clarify)
lp15@63078
  1634
  apply (rename_tac F G)
lp15@63078
  1635
  apply (rule_tac x="F \<union> G" in exI, auto)
lp15@63078
  1636
  done
lp15@63078
  1637
lp15@63078
  1638
lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
lp15@63078
  1639
  unfolding polyhedron_def
lp15@63078
  1640
  by (rule_tac x="{}" in exI) auto
lp15@63078
  1641
lp15@63078
  1642
lemma polyhedron_Inter [intro,simp]:
lp15@63078
  1643
   "\<lbrakk>finite F; \<And>S. S \<in> F \<Longrightarrow> polyhedron S\<rbrakk> \<Longrightarrow> polyhedron(\<Inter>F)"
lp15@63078
  1644
by (induction F rule: finite_induct) auto
lp15@63078
  1645
lp15@63078
  1646
lp15@63078
  1647
lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
lp15@63078
  1648
proof -
lp15@63078
  1649
  have "\<exists>a. a \<noteq> 0 \<and>
lp15@63078
  1650
             (\<exists>b. {x. (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b})"
lp15@63078
  1651
    by (rule_tac x="(SOME i. i \<in> Basis)" in exI) (force simp: SOME_Basis nonzero_Basis)
lp15@63078
  1652
  moreover have "\<exists>a b. a \<noteq> 0 \<and>
lp15@63078
  1653
                       {x. - (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b}"
lp15@63078
  1654
      apply (rule_tac x="-(SOME i. i \<in> Basis)" in exI)
lp15@63078
  1655
      apply (rule_tac x="-1" in exI)
lp15@63078
  1656
      apply (simp add: SOME_Basis nonzero_Basis)
lp15@63078
  1657
      done
lp15@63078
  1658
  ultimately show ?thesis
lp15@63078
  1659
    unfolding polyhedron_def
lp15@63078
  1660
    apply (rule_tac x="{{x. (SOME i. i \<in> Basis) \<bullet> x \<le> -1},
lp15@63078
  1661
                        {x. -(SOME i. i \<in> Basis) \<bullet> x \<le> -1}}" in exI)
lp15@63078
  1662
    apply force
lp15@63078
  1663
    done
lp15@63078
  1664
qed
lp15@63078
  1665
lp15@63078
  1666
lemma polyhedron_halfspace_le:
lp15@63078
  1667
  fixes a :: "'a :: euclidean_space"
lp15@63078
  1668
  shows "polyhedron {x. a \<bullet> x \<le> b}"
lp15@63078
  1669
proof (cases "a = 0")
lp15@63078
  1670
  case True then show ?thesis by auto
lp15@63078
  1671
next
lp15@63078
  1672
  case False
lp15@63078
  1673
  then show ?thesis
lp15@63078
  1674
    unfolding polyhedron_def
lp15@63078
  1675
    by (rule_tac x="{{x. a \<bullet> x \<le> b}}" in exI) auto
lp15@63078
  1676
qed
lp15@63078
  1677
lp15@63078
  1678
lemma polyhedron_halfspace_ge:
lp15@63078
  1679
  fixes a :: "'a :: euclidean_space"
lp15@63078
  1680
  shows "polyhedron {x. a \<bullet> x \<ge> b}"
lp15@63078
  1681
using polyhedron_halfspace_le [of "-a" "-b"] by simp
lp15@63078
  1682
lp15@63078
  1683
lemma polyhedron_hyperplane:
lp15@63078
  1684
  fixes a :: "'a :: euclidean_space"
lp15@63078
  1685
  shows "polyhedron {x. a \<bullet> x = b}"
lp15@63078
  1686
proof -
lp15@63078
  1687
  have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
lp15@63078
  1688
    by force
lp15@63078
  1689
  then show ?thesis
lp15@63078
  1690
    by (simp add: polyhedron_halfspace_ge polyhedron_halfspace_le)
lp15@63078
  1691
qed
lp15@63078
  1692
lp15@63078
  1693
lemma affine_imp_polyhedron:
lp15@63078
  1694
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1695
  shows "affine S \<Longrightarrow> polyhedron S"
lp15@63078
  1696
by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])
lp15@63078
  1697
lp15@63078
  1698
lemma polyhedron_imp_closed:
lp15@63078
  1699
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1700
  shows "polyhedron S \<Longrightarrow> closed S"
lp15@63078
  1701
apply (simp add: polyhedron_def)
lp15@63078
  1702
using closed_halfspace_le by fastforce
lp15@63078
  1703
lp15@63078
  1704
lemma polyhedron_imp_convex:
lp15@63078
  1705
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1706
  shows "polyhedron S \<Longrightarrow> convex S"
lp15@63078
  1707
apply (simp add: polyhedron_def)
lp15@63078
  1708
using convex_Inter convex_halfspace_le by fastforce
lp15@63078
  1709
lp15@63078
  1710
lemma polyhedron_affine_hull:
lp15@63078
  1711
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1712
  shows "polyhedron(affine hull S)"
lp15@63078
  1713
by (simp add: affine_imp_polyhedron)
lp15@63078
  1714
lp15@63078
  1715
lp15@63078
  1716
subsection\<open>Canonical polyhedron representation making facial structure explicit\<close>
lp15@63078
  1717
lp15@63078
  1718
lemma polyhedron_Int_affine:
lp15@63078
  1719
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1720
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1721
           (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
lp15@63078
  1722
                (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}))"
lp15@63078
  1723
        (is "?lhs = ?rhs")
lp15@63078
  1724
proof
lp15@63078
  1725
  assume ?lhs then show ?rhs
lp15@63078
  1726
    apply (simp add: polyhedron_def)
lp15@63078
  1727
    apply (erule ex_forward)
lp15@63078
  1728
    using hull_subset apply force
lp15@63078
  1729
    done
lp15@63078
  1730
next
lp15@63078
  1731
  assume ?rhs then show ?lhs
lp15@63078
  1732
    apply clarify
lp15@63078
  1733
    apply (erule ssubst)
lp15@63078
  1734
    apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le)
lp15@63078
  1735
    done
lp15@63078
  1736
qed
lp15@63078
  1737
lp15@63078
  1738
proposition rel_interior_polyhedron_explicit:
lp15@63078
  1739
  assumes "finite F"
lp15@63078
  1740
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  1741
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1742
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1743
    shows "rel_interior S = {x \<in> S. \<forall>h \<in> F. a h \<bullet> x < b h}"
lp15@63078
  1744
proof -
lp15@63078
  1745
  have rels: "\<And>x. x \<in> rel_interior S \<Longrightarrow> x \<in> S"
lp15@63078
  1746
    by (meson IntE mem_rel_interior)
lp15@63078
  1747
  moreover have "a i \<bullet> x < b i" if x: "x \<in> rel_interior S" and "i \<in> F" for x i
lp15@63078
  1748
  proof -
lp15@63078
  1749
    have fif: "F - {i} \<subset> F"
lp15@63078
  1750
      using \<open>i \<in> F\<close> Diff_insert_absorb Diff_subset set_insert psubsetI by blast
lp15@63078
  1751
    then have "S \<subset> affine hull S \<inter> \<Inter>(F - {i})"
lp15@63078
  1752
      by (rule psub)
lp15@63078
  1753
    then obtain z where ssub: "S \<subseteq> \<Inter>(F - {i})" and zint: "z \<in> \<Inter>(F - {i})"
lp15@63078
  1754
                    and "z \<notin> S" and zaff: "z \<in> affine hull S"
lp15@63078
  1755
      by auto
lp15@63078
  1756
    have "z \<noteq> x"
lp15@63078
  1757
      using \<open>z \<notin> S\<close> rels x by blast
lp15@63078
  1758
    have "z \<notin> affine hull S \<inter> \<Inter>F"
lp15@63078
  1759
      using \<open>z \<notin> S\<close> seq by auto
lp15@63078
  1760
    then have aiz: "a i \<bullet> z > b i"
lp15@63078
  1761
      using faceq zint zaff by fastforce
lp15@63078
  1762
    obtain e where "e > 0" "x \<in> S" and e: "ball x e \<inter> affine hull S \<subseteq> S"
lp15@63078
  1763
      using x by (auto simp: mem_rel_interior_ball)
lp15@63078
  1764
    then have ins: "\<And>y. \<lbrakk>norm (x - y) < e; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
lp15@63078
  1765
      by (metis IntI subsetD dist_norm mem_ball)
wenzelm@63148
  1766
    define \<xi> where "\<xi> = min (1/2) (e / 2 / norm(z - x))"
lp15@63078
  1767
    have "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) = norm (\<xi> *\<^sub>R (x - z))"
lp15@63078
  1768
      by (simp add: \<xi>_def algebra_simps norm_mult)
lp15@63078
  1769
    also have "... = \<xi> * norm (x - z)"
lp15@63078
  1770
      using \<open>e > 0\<close> by (simp add: \<xi>_def)
lp15@63078
  1771
    also have "... < e"
lp15@63078
  1772
      using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def divide_simps norm_minus_commute)
lp15@63078
  1773
    finally have lte: "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) < e" .
lp15@63078
  1774
    have \<xi>_aff: "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> affine hull S"
lp15@63078
  1775
      by (metis \<open>x \<in> S\<close> add.commute affine_affine_hull diff_add_cancel hull_inc mem_affine zaff)
lp15@63078
  1776
    have "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> S"
lp15@63078
  1777
      apply (rule ins [OF _ \<xi>_aff])
lp15@63078
  1778
      apply (simp add: algebra_simps lte)
lp15@63078
  1779
      done
lp15@63078
  1780
    then obtain l where l: "0 < l" "l < 1" and ls: "(l *\<^sub>R z + (1 - l) *\<^sub>R x) \<in> S"
lp15@63078
  1781
      apply (rule_tac l = \<xi> in that)
lp15@63078
  1782
      using \<open>e > 0\<close> \<open>z \<noteq> x\<close>  apply (auto simp: \<xi>_def)
lp15@63078
  1783
      done
lp15@63078
  1784
    then have i: "l *\<^sub>R z + (1 - l) *\<^sub>R x \<in> i"
lp15@63078
  1785
      using seq \<open>i \<in> F\<close> by auto
lp15@63078
  1786
    have "b i * l + (a i \<bullet> x) * (1 - l) < a i \<bullet> (l *\<^sub>R z + (1 - l) *\<^sub>R x)"
lp15@63078
  1787
      using l by (simp add: algebra_simps aiz)
lp15@63078
  1788
    also have "\<dots> \<le> b i" using i l
lp15@63078
  1789
      using faceq mem_Collect_eq \<open>i \<in> F\<close> by blast
lp15@63078
  1790
    finally have "(a i \<bullet> x) * (1 - l) < b i * (1 - l)"
lp15@63078
  1791
      by (simp add: algebra_simps)
lp15@63078
  1792
    with l show ?thesis
lp15@63078
  1793
      by simp
lp15@63078
  1794
  qed
lp15@63078
  1795
  moreover have "x \<in> rel_interior S"
lp15@63078
  1796
           if "x \<in> S" and less: "\<And>h. h \<in> F \<Longrightarrow> a h \<bullet> x < b h" for x
lp15@63078
  1797
  proof -
lp15@63078
  1798
    have 1: "\<And>h. h \<in> F \<Longrightarrow> x \<in> interior h"
lp15@63078
  1799
      by (metis interior_halfspace_le mem_Collect_eq less faceq)
lp15@63078
  1800
    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
  1801
      by (metis IntI Inter_iff contra_subsetD interior_subset seq)
lp15@63078
  1802
    show ?thesis
lp15@63078
  1803
      apply (simp add: rel_interior \<open>x \<in> S\<close>)
lp15@63078
  1804
      apply (rule_tac x="\<Inter>h\<in>F. interior h" in exI)
lp15@63078
  1805
      apply (auto simp: \<open>finite F\<close> open_INT 1 2)
lp15@63078
  1806
      done
lp15@63078
  1807
  qed
lp15@63078
  1808
  ultimately show ?thesis by blast
lp15@63078
  1809
qed
lp15@63078
  1810
lp15@63078
  1811
lp15@63078
  1812
lemma polyhedron_Int_affine_parallel:
lp15@63078
  1813
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1814
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1815
         (\<exists>F. finite F \<and>
lp15@63078
  1816
              S = (affine hull S) \<inter> (\<Inter>F) \<and>
lp15@63078
  1817
              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
lp15@63078
  1818
                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)))"
lp15@63078
  1819
    (is "?lhs = ?rhs")
lp15@63078
  1820
proof
lp15@63078
  1821
  assume ?lhs
lp15@63078
  1822
  then obtain F where "finite F" and seq: "S = (affine hull S) \<inter> \<Inter>F"
lp15@63078
  1823
                  and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
lp15@63078
  1824
    by (fastforce simp add: polyhedron_Int_affine)
lp15@63078
  1825
  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
  1826
    by metis
lp15@63078
  1827
  show ?rhs
lp15@63078
  1828
  proof -
lp15@63078
  1829
    have "\<exists>a' b'. a' \<noteq> 0 \<and>
lp15@63078
  1830
                  affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> h \<and>
lp15@63078
  1831
                  (\<forall>w \<in> affine hull S. (w + a') \<in> affine hull S)"
lp15@63078
  1832
        if "h \<in> F" "~(affine hull S \<subseteq> h)" for h
lp15@63078
  1833
    proof -
lp15@63078
  1834
      have "a h \<noteq> 0" and "h = {x. a h \<bullet> x \<le> b h}" "h \<inter> \<Inter>F = \<Inter>F"
lp15@63078
  1835
        using \<open>h \<in> F\<close> ab by auto
lp15@63078
  1836
      then have "(affine hull S) \<inter> {x. a h \<bullet> x \<le> b h} \<noteq> {}"
lp15@63078
  1837
        by (metis (no_types) affine_hull_eq_empty inf.absorb_iff2 inf_assoc inf_bot_right inf_commute seq that(2))
lp15@63078
  1838
      moreover have "~ (affine hull S \<subseteq> {x. a h \<bullet> x \<le> b h})"
lp15@63078
  1839
        using \<open>h = {x. a h \<bullet> x \<le> b h}\<close> that(2) by blast
lp15@63078
  1840
      ultimately show ?thesis
lp15@63078
  1841
        using affine_parallel_slice [of "affine hull S"]
lp15@63078
  1842
        by (metis \<open>h = {x. a h \<bullet> x \<le> b h}\<close> affine_affine_hull)
lp15@63078
  1843
    qed
lp15@63078
  1844
    then obtain a b
lp15@63078
  1845
         where ab: "\<And>h. \<lbrakk>h \<in> F; ~ (affine hull S \<subseteq> h)\<rbrakk>
lp15@63078
  1846
             \<Longrightarrow> a h \<noteq> 0 \<and>
lp15@63078
  1847
                  affine hull S \<inter> {x. a h \<bullet> x \<le> b h} = affine hull S \<inter> h \<and>
lp15@63078
  1848
                  (\<forall>w \<in> affine hull S. (w + a h) \<in> affine hull S)"
lp15@63078
  1849
      by metis
lp15@63078
  1850
    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
  1851
      by (subst seq) (auto simp: ab INT_extend_simps)
lp15@63078
  1852
    show ?thesis
lp15@63078
  1853
      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
  1854
      apply (intro conjI seq2)
lp15@63078
  1855
        using \<open>finite F\<close> apply force
lp15@63078
  1856
       using ab apply blast
lp15@63078
  1857
       done
lp15@63078
  1858
  qed
lp15@63078
  1859
next
lp15@63078
  1860
  assume ?rhs then show ?lhs
lp15@63078
  1861
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  1862
    by metis
lp15@63078
  1863
qed
lp15@63078
  1864
lp15@63078
  1865
lp15@63078
  1866
proposition polyhedron_Int_affine_parallel_minimal:
lp15@63078
  1867
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1868
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1869
         (\<exists>F. finite F \<and>
lp15@63078
  1870
              S = (affine hull S) \<inter> (\<Inter>F) \<and>
lp15@63078
  1871
              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
lp15@63078
  1872
                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)) \<and>
lp15@63078
  1873
              (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> (\<Inter>F')))"
lp15@63078
  1874
    (is "?lhs = ?rhs")
lp15@63078
  1875
proof
lp15@63078
  1876
  assume ?lhs
lp15@63078
  1877
  then obtain f0
lp15@63078
  1878
           where f0: "finite f0"
lp15@63078
  1879
                 "S = (affine hull S) \<inter> (\<Inter>f0)"
lp15@63078
  1880
                   (is "?P f0")
lp15@63078
  1881
                 "\<forall>h \<in> f0. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
lp15@63078
  1882
                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)"
lp15@63078
  1883
                   (is "?Q f0")
lp15@63078
  1884
    by (force simp: polyhedron_Int_affine_parallel)
wenzelm@63148
  1885
  define n where "n = (LEAST n. \<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F)"
lp15@63078
  1886
  have nf: "\<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F"
lp15@63078
  1887
    apply (simp add: n_def)
lp15@63078
  1888
    apply (rule LeastI [where k = "card f0"])
lp15@63078
  1889
    using f0 apply auto
lp15@63078
  1890
    done
lp15@63078
  1891
  then obtain F where F: "card F = n" "finite F" and seq: "?P F" and aff: "?Q F"
lp15@63078
  1892
    by blast
lp15@63078
  1893
  then have "~ (finite g \<and> ?P g \<and> ?Q g)" if "card g < n" for g
lp15@63078
  1894
    using that by (auto simp: n_def dest!: not_less_Least)
lp15@63078
  1895
  then have *: "~ (?P g \<and> ?Q g)" if "g \<subset> F" for g
lp15@63078
  1896
    using that \<open>finite F\<close> psubset_card_mono \<open>card F = n\<close>
lp15@63078
  1897
    by (metis finite_Int inf.strict_order_iff)
lp15@63078
  1898
  have 1: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subseteq> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1899
    by (subst seq) blast
lp15@63078
  1900
  have 2: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<noteq> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1901
    apply (frule *)
lp15@63078
  1902
    by (metis aff subsetCE subset_iff_psubset_eq)
lp15@63078
  1903
  show ?rhs
lp15@63078
  1904
    by (metis \<open>finite F\<close> seq aff psubsetI 1 2)
lp15@63078
  1905
next
lp15@63078
  1906
  assume ?rhs then show ?lhs
lp15@63078
  1907
    by (auto simp: polyhedron_Int_affine_parallel)
lp15@63078
  1908
qed
lp15@63078
  1909
lp15@63078
  1910
lp15@63078
  1911
lemma polyhedron_Int_affine_minimal:
lp15@63078
  1912
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  1913
  shows "polyhedron S \<longleftrightarrow>
lp15@63078
  1914
         (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
lp15@63078
  1915
              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}) \<and>
lp15@63078
  1916
              (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'))"
lp15@63078
  1917
apply (rule iffI)
lp15@63078
  1918
 apply (force simp: polyhedron_Int_affine_parallel_minimal elim!: ex_forward)
lp15@63078
  1919
apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
lp15@63078
  1920
done
lp15@63078
  1921
lp15@63078
  1922
proposition facet_of_polyhedron_explicit:
lp15@63078
  1923
  assumes "finite F"
lp15@63078
  1924
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  1925
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  1926
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  1927
    shows "c facet_of S \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
lp15@63078
  1928
proof (cases "S = {}")
lp15@63078
  1929
  case True with psub show ?thesis by force
lp15@63078
  1930
next
lp15@63078
  1931
  case False
lp15@63078
  1932
  have "polyhedron S"
lp15@63078
  1933
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  1934
    apply (rule_tac x=F in exI)
lp15@63078
  1935
    using assms  apply force
lp15@63078
  1936
    done
lp15@63078
  1937
  then have "convex S"
lp15@63078
  1938
    by (rule polyhedron_imp_convex)
lp15@63078
  1939
  with False rel_interior_eq_empty have "rel_interior S \<noteq> {}" by blast
lp15@63078
  1940
  then obtain x where "x \<in> rel_interior S" by auto
lp15@63078
  1941
  then obtain T where "open T" "x \<in> T" "x \<in> S" "T \<inter> affine hull S \<subseteq> S"
lp15@63078
  1942
    by (force simp: mem_rel_interior)
lp15@63078
  1943
  then have xaff: "x \<in> affine hull S" and xint: "x \<in> \<Inter>F"
lp15@63078
  1944
    using seq hull_inc by auto
lp15@63078
  1945
  have "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
lp15@63078
  1946
    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
lp15@63078
  1947
  with \<open>x \<in> rel_interior S\<close>
lp15@63078
  1948
  have [simp]: "\<And>h. h\<in>F \<Longrightarrow> a h \<bullet> x < b h" by blast
lp15@63078
  1949
  have *: "(S \<inter> {x. a h \<bullet> x = b h}) facet_of S" if "h \<in> F" for h
lp15@63078
  1950
  proof -
lp15@63078
  1951
    have "S \<subset> affine hull S \<inter> \<Inter>(F - {h})"
lp15@63078
  1952
      using psub that by (metis Diff_disjoint Diff_subset insert_disjoint(2) psubsetI)
lp15@63078
  1953
    then obtain z where zaff: "z \<in> affine hull S" and zint: "z \<in> \<Inter>(F - {h})" and "z \<notin> S"
lp15@63078
  1954
      by force
lp15@63078
  1955
    then have "z \<noteq> x" "z \<notin> h" using seq \<open>x \<in> S\<close> by auto
lp15@63078
  1956
    have "x \<in> h" using that xint by auto
lp15@63078
  1957
    then have able: "a h \<bullet> x \<le> b h"
lp15@63078
  1958
      using faceq that by blast
lp15@63078
  1959
    also have "... < a h \<bullet> z" using \<open>z \<notin> h\<close> faceq [OF that] xint by auto
lp15@63078
  1960
    finally have xltz: "a h \<bullet> x < a h \<bullet> z" .
wenzelm@63148
  1961
    define l where "l = (b h - a h \<bullet> x) / (a h \<bullet> z - a h \<bullet> x)"
wenzelm@63148
  1962
    define w where "w = (1 - l) *\<^sub>R x + l *\<^sub>R z"
lp15@63078
  1963
    have "0 < l" "l < 1"
lp15@63078
  1964
      using able xltz \<open>b h < a h \<bullet> z\<close> \<open>h \<in> F\<close>
lp15@63078
  1965
      by (auto simp: l_def divide_simps)
lp15@63078
  1966
    have awlt: "a i \<bullet> w < b i" if "i \<in> F" "i \<noteq> h" for i
lp15@63078
  1967
    proof -
lp15@63078
  1968
      have "(1 - l) * (a i \<bullet> x) < (1 - l) * b i"
lp15@63078
  1969
        by (simp add: \<open>l < 1\<close> \<open>i \<in> F\<close>)
lp15@63078
  1970
      moreover have "l * (a i \<bullet> z) \<le> l * b i"
lp15@63078
  1971
        apply (rule mult_left_mono)
lp15@63078
  1972
        apply (metis Diff_insert_absorb Inter_iff Set.set_insert \<open>h \<in> F\<close> faceq insertE mem_Collect_eq that zint)
lp15@63078
  1973
        using \<open>0 < l\<close>
lp15@63078
  1974
        apply simp
lp15@63078
  1975
        done
lp15@63078
  1976
      ultimately show ?thesis by (simp add: w_def algebra_simps)
lp15@63078
  1977
    qed
lp15@63078
  1978
    have weq: "a h \<bullet> w = b h"
lp15@63078
  1979
      using xltz unfolding w_def l_def
lp15@63078
  1980
      by (simp add: algebra_simps) (simp add: field_simps)
lp15@63078
  1981
    have "w \<in> affine hull S"
lp15@63078
  1982
      by (simp add: w_def mem_affine xaff zaff)
lp15@63078
  1983
    moreover have "w \<in> \<Inter>F"
lp15@63078
  1984
      using \<open>a h \<bullet> w = b h\<close> awlt faceq less_eq_real_def by blast
lp15@63078
  1985
    ultimately have "w \<in> S"
lp15@63078
  1986
      using seq by blast
lp15@63078
  1987
    with weq have "S \<inter> {x. a h \<bullet> x = b h} \<noteq> {}" by blast
lp15@63078
  1988
    moreover have "S \<inter> {x. a h \<bullet> x = b h} face_of S"
lp15@63078
  1989
      apply (rule face_of_Int_supporting_hyperplane_le)
lp15@63078
  1990
      apply (rule \<open>convex S\<close>)
lp15@63078
  1991
      apply (subst (asm) seq)
lp15@63078
  1992
      using faceq that apply fastforce
lp15@63078
  1993
      done
lp15@63078
  1994
    moreover have "affine hull (S \<inter> {x. a h \<bullet> x = b h}) =
lp15@63078
  1995
                   (affine hull S) \<inter> {x. a h \<bullet> x = b h}"
lp15@63078
  1996
    proof
lp15@63078
  1997
      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
  1998
        apply (intro Int_greatest hull_mono Int_lower1)
lp15@63078
  1999
        apply (metis affine_hull_eq affine_hyperplane hull_mono inf_le2)
lp15@63078
  2000
        done
lp15@63078
  2001
    next
lp15@63078
  2002
      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
  2003
      proof
lp15@63078
  2004
        fix y
lp15@63078
  2005
        assume yaff: "y \<in> affine hull S \<inter> {y. a h \<bullet> y = b h}"
lp15@63078
  2006
        obtain T where "0 < T"
lp15@63078
  2007
                 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
  2008
        proof (cases "F - {h} = {}")
lp15@63078
  2009
          case True then show ?thesis
lp15@63078
  2010
            by (rule_tac T=1 in that) auto
lp15@63078
  2011
        next
lp15@63078
  2012
          case False
lp15@63078
  2013
          then obtain h' where h': "h' \<in> F - {h}" by auto
wenzelm@63148
  2014
          define inff where "inff =
wenzelm@63148
  2015
            (INF j:F - {h}.
wenzelm@63148
  2016
              if 0 < a j \<bullet> y - a j \<bullet> w
wenzelm@63148
  2017
              then (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)
wenzelm@63148
  2018
              else 1)"
lp15@63078
  2019
          have "0 < inff"
lp15@63078
  2020
            apply (simp add: inff_def)
lp15@63078
  2021
            apply (rule finite_imp_less_Inf)
lp15@63078
  2022
              using \<open>finite F\<close> apply blast
lp15@63078
  2023
             using h' apply blast
lp15@63078
  2024
            apply simp
lp15@63078
  2025
            using awlt apply (force simp: divide_simps)
lp15@63078
  2026
            done
lp15@63078
  2027
          moreover have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
lp15@63078
  2028
                        if "j \<in> F" "j \<noteq> h" for j
lp15@63078
  2029
          proof (cases "a j \<bullet> w < a j \<bullet> y")
lp15@63078
  2030
            case True
lp15@63078
  2031
            then have "inff \<le> (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)"
lp15@63078
  2032
              apply (simp add: inff_def)
lp15@63078
  2033
              apply (rule cInf_le_finite)
lp15@63078
  2034
              using \<open>finite F\<close> apply blast
lp15@63078
  2035
              apply (simp add: that split: if_split_asm)
lp15@63078
  2036
              done
lp15@63078
  2037
            then show ?thesis
lp15@63078
  2038
              using \<open>0 < inff\<close> awlt [OF that] mult_strict_left_mono
lp15@63078
  2039
              by (fastforce simp add: algebra_simps divide_simps split: if_split_asm)
lp15@63078
  2040
          next
lp15@63078
  2041
            case False
lp15@63078
  2042
            with \<open>0 < inff\<close> have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> 0"
lp15@63078
  2043
              by (simp add: mult_le_0_iff)
lp15@63078
  2044
            also have "... < b j - a j \<bullet> w"
lp15@63078
  2045
              by (simp add: awlt that)
lp15@63078
  2046
            finally show ?thesis by simp
lp15@63078
  2047
          qed
lp15@63078
  2048
          ultimately show ?thesis
lp15@63078
  2049
            by (blast intro: that)
lp15@63078
  2050
        qed
wenzelm@63148
  2051
        define c where "c = (1 - T) *\<^sub>R w + T *\<^sub>R y"
lp15@63078
  2052
        have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> j" if "j \<in> F" for j
lp15@63078
  2053
        proof (cases "j = h")
lp15@63078
  2054
          case True
lp15@63078
  2055
          have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a h \<bullet> x \<le> b h}"
lp15@63078
  2056
            using weq yaff by (auto simp: algebra_simps)
lp15@63078
  2057
          with True faceq [OF that] show ?thesis by metis
lp15@63078
  2058
        next
lp15@63078
  2059
          case False
lp15@63078
  2060
          with T that have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a j \<bullet> x \<le> b j}"
lp15@63078
  2061
            by (simp add: algebra_simps)
lp15@63078
  2062
          with faceq [OF that] show ?thesis by simp
lp15@63078
  2063
        qed
lp15@63078
  2064
        moreover have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> affine hull S"
wenzelm@63170
  2065
          apply (rule affine_affine_hull [simplified affine_alt, rule_format])
lp15@63078
  2066
          apply (simp add: \<open>w \<in> affine hull S\<close>)
lp15@63078
  2067
          using yaff apply blast
lp15@63078
  2068
          done
lp15@63078
  2069
        ultimately have "c \<in> S"
lp15@63078
  2070
          using seq by (force simp: c_def)
lp15@63078
  2071
        moreover have "a h \<bullet> c = b h"
lp15@63078
  2072
          using yaff by (force simp: c_def algebra_simps weq)
lp15@63078
  2073
        ultimately have caff: "c \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
lp15@63078
  2074
          by (simp add: hull_inc)
lp15@63078
  2075
        have waff: "w \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
lp15@63078
  2076
          using \<open>w \<in> S\<close> weq by (blast intro: hull_inc)
lp15@63078
  2077
        have yeq: "y = (1 - inverse T) *\<^sub>R w + c /\<^sub>R T"
lp15@63078
  2078
          using \<open>0 < T\<close> by (simp add: c_def algebra_simps)
lp15@63078
  2079
        show "y \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
wenzelm@63170
  2080
          by (metis yeq affine_affine_hull [simplified affine_alt, rule_format, OF waff caff])
lp15@63078
  2081
      qed
lp15@63078
  2082
    qed
lp15@63078
  2083
    ultimately show ?thesis
lp15@63078
  2084
      apply (simp add: facet_of_def)
lp15@63078
  2085
      apply (subst aff_dim_affine_hull [symmetric])
lp15@63078
  2086
      using  \<open>b h < a h \<bullet> z\<close> zaff
lp15@63078
  2087
      apply (force simp: aff_dim_affine_Int_hyperplane)
lp15@63078
  2088
      done
lp15@63078
  2089
  qed
lp15@63078
  2090
  show ?thesis
lp15@63078
  2091
  proof
lp15@63078
  2092
    show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h} \<Longrightarrow> c facet_of S"
lp15@63078
  2093
      using * by blast
lp15@63078
  2094
  next
lp15@63078
  2095
    assume "c facet_of S"
lp15@63078
  2096
    then have "c face_of S" "convex c" "c \<noteq> {}" and affc: "aff_dim c = aff_dim S - 1"
lp15@63078
  2097
      by (auto simp: facet_of_def face_of_imp_convex)
lp15@63078
  2098
    then obtain x where x: "x \<in> rel_interior c"
lp15@63078
  2099
      by (force simp: rel_interior_eq_empty)
lp15@63078
  2100
    then have "x \<in> c"
lp15@63078
  2101
      by (meson subsetD rel_interior_subset)
lp15@63078
  2102
    then have "x \<in> S"
lp15@63078
  2103
      using \<open>c facet_of S\<close> facet_of_imp_subset by blast
lp15@63078
  2104
    have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
lp15@63078
  2105
      by (rule rel_interior_polyhedron_explicit [OF assms])
lp15@63078
  2106
    have "c \<noteq> S"
lp15@63078
  2107
      using \<open>c facet_of S\<close> facet_of_irrefl by blast
lp15@63078
  2108
    then have "x \<notin> rel_interior S"
lp15@63078
  2109
      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
  2110
    with rels \<open>x \<in> S\<close> obtain i where "i \<in> F" and i: "a i \<bullet> x \<ge> b i"
lp15@63078
  2111
      by force
lp15@63078
  2112
    have "x \<in> {u. a i \<bullet> u \<le> b i}"
lp15@63078
  2113
      by (metis IntD2 InterE \<open>i \<in> F\<close> \<open>x \<in> S\<close> faceq seq)
lp15@63078
  2114
    then have "a i \<bullet> x \<le> b i" by simp
lp15@63078
  2115
    then have "a i \<bullet> x = b i" using i by auto
lp15@63078
  2116
    have "c \<subseteq> S \<inter> {x. a i \<bullet> x = b i}"
lp15@63078
  2117
      apply (rule subset_of_face_of [of _ S])
lp15@63078
  2118
        apply (simp add: "*" \<open>i \<in> F\<close> facet_of_imp_face_of)
lp15@63078
  2119
       apply (simp add: \<open>c face_of S\<close> face_of_imp_subset)
lp15@63078
  2120
      using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> S\<close> x by blast
lp15@63078
  2121
    then have cface: "c face_of (S \<inter> {x. a i \<bullet> x = b i})"
lp15@63078
  2122
      by (meson \<open>c face_of S\<close> face_of_subset inf_le1)
lp15@63078
  2123
    have con: "convex (S \<inter> {x. a i \<bullet> x = b i})"
lp15@63078
  2124
      by (simp add: \<open>convex S\<close> convex_Int convex_hyperplane)
lp15@63078
  2125
    show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h}"
lp15@63078
  2126
      apply (rule_tac x=i in exI)
lp15@63078
  2127
      apply (simp add: \<open>i \<in> F\<close>)
lp15@63078
  2128
      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
  2129
  qed
lp15@63078
  2130
qed
lp15@63078
  2131
lp15@63078
  2132
lp15@63078
  2133
lemma face_of_polyhedron_subset_explicit:
lp15@63078
  2134
  fixes S :: "'a :: euclidean_space set"
lp15@63078
  2135
  assumes "finite F"
lp15@63078
  2136
      and seq: "S = affine hull S \<inter> \<Inter>F"
lp15@63078
  2137
      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
lp15@63078
  2138
      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
lp15@63078
  2139
      and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
lp15@63078
  2140
   obtains h where "h \<in> F" "c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
lp15@63078
  2141
proof -
lp15@63078
  2142
  have "c \<subseteq> S" using \<open>c face_of S\<close>
lp15@63078
  2143
    by (simp add: face_of_imp_subset)
lp15@63078
  2144
  have "polyhedron S"
lp15@63078
  2145
    apply (simp add: polyhedron_Int_affine)
lp15@63078
  2146
    by (metis \<open>finite F\<close> faceq seq)
lp15@63078
  2147
  then have "convex S"
lp15@63078
  2148
    by (simp add: polyhedron_imp_convex)
lp15@63078
  2149
  then have *: "(S \<inter> {x. a h \<bullet> x = b h}) face_of S" if "h \<in> F" for h
lp15@63078
  2150
    apply (rule face_of_Int_supporting_hyperplane_le)
lp15@63078
  2151
    using faceq seq that by fastforce
lp15@63078
  2152
  have "rel_interior c \<noteq> {}"
lp15@63078
  2153
    using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
lp15@63078
  2154
  then obtain x where "x \<in> rel_interior c" by auto
lp15@63078
  2155
  have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
lp15@63078
  2156
    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
lp15@63078
  2157
  then have xnot: "x \<notin> rel_interior S"
lp15@63078
  2158
    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
  2159
  then have "x \<in> S"
lp15@63078
  2160
    using \<open>c \<subseteq> S\<close> \<open>x \<in> rel_interior c\<close> rel_interior_subset by auto
lp15@63078
  2161
  then have xint: "x \<in> \<Inter>F"
lp15@63078
  2162
    using seq by blast
lp15@63078
  2163
  have "F \<noteq> {}" using assms
lp15@63078
  2164
    by (metis affine_Int affine_Inter affine_affine_hull ex_in_conv face_of_affine_trivial)
lp15@63078
  2165
  then obtain i where "i \<in> F" "~ (a i \<bullet> x < b i)"
lp15@63078
  2166
    using \<open>x \<in> S\<close> rels xnot by auto
lp15@63078
  2167
  with xint have "a i \<bullet> x = b i"
lp15@63078
  2168
    by (metis eq_iff mem_Collect_eq not_le Inter_iff faceq)