--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Affine.thy Thu Dec 05 21:03:06 2019 +0100
@@ -0,0 +1,1653 @@
+theory Affine
+imports Linear_Algebra
+begin
+
+lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
+ by (fact if_distrib)
+
+lemma sum_delta_notmem:
+ assumes "x \<notin> s"
+ shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
+ and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
+ and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
+ and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
+ apply (rule_tac [!] sum.cong)
+ using assms
+ apply auto
+ done
+
+lemmas independent_finite = independent_imp_finite
+
+lemma span_substd_basis:
+ assumes d: "d \<subseteq> Basis"
+ shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
+ (is "_ = ?B")
+proof -
+ have "d \<subseteq> ?B"
+ using d by (auto simp: inner_Basis)
+ moreover have s: "subspace ?B"
+ using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
+ ultimately have "span d \<subseteq> ?B"
+ using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
+ moreover have *: "card d \<le> dim (span d)"
+ using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
+ span_superset[of d]
+ by auto
+ moreover from * have "dim ?B \<le> dim (span d)"
+ using dim_substandard[OF assms] by auto
+ ultimately show ?thesis
+ using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
+qed
+
+lemma basis_to_substdbasis_subspace_isomorphism:
+ fixes B :: "'a::euclidean_space set"
+ assumes "independent B"
+ shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
+ f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
+proof -
+ have B: "card B = dim B"
+ using dim_unique[of B B "card B"] assms span_superset[of B] by auto
+ have "dim B \<le> card (Basis :: 'a set)"
+ using dim_subset_UNIV[of B] by simp
+ from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
+ by auto
+ let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
+ have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
+ proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
+ show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
+ using d inner_not_same_Basis by blast
+ qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
+ with t \<open>card B = dim B\<close> d show ?thesis by auto
+qed
+
+subsection \<open>Affine set and affine hull\<close>
+
+definition\<^marker>\<open>tag important\<close> affine :: "'a::real_vector set \<Rightarrow> bool"
+ where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
+
+lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
+ unfolding affine_def by (metis eq_diff_eq')
+
+lemma affine_empty [iff]: "affine {}"
+ unfolding affine_def by auto
+
+lemma affine_sing [iff]: "affine {x}"
+ unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
+
+lemma affine_UNIV [iff]: "affine UNIV"
+ unfolding affine_def by auto
+
+lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
+ unfolding affine_def by auto
+
+lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
+ unfolding affine_def by auto
+
+lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
+ apply (clarsimp simp add: affine_def)
+ apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
+ apply (auto simp: algebra_simps)
+ done
+
+lemma affine_affine_hull [simp]: "affine(affine hull s)"
+ unfolding hull_def
+ using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
+
+lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
+ by (metis affine_affine_hull hull_same)
+
+lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
+ by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some explicit formulations\<close>
+
+text "Formalized by Lars Schewe."
+
+lemma affine:
+ fixes V::"'a::real_vector set"
+ shows "affine V \<longleftrightarrow>
+ (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
+proof -
+ have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
+ and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
+ proof (cases "x = y")
+ case True
+ then show ?thesis
+ using that by (metis scaleR_add_left scaleR_one)
+ next
+ case False
+ then show ?thesis
+ using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
+ qed
+ moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
+ if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
+ and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
+ proof -
+ define n where "n = card S"
+ consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
+ then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
+ proof cases
+ assume "card S = 1"
+ then obtain a where "S={a}"
+ by (auto simp: card_Suc_eq)
+ then show ?thesis
+ using that by simp
+ next
+ assume "card S = 2"
+ then obtain a b where "S = {a, b}"
+ by (metis Suc_1 card_1_singletonE card_Suc_eq)
+ then show ?thesis
+ using *[of a b] that
+ by (auto simp: sum_clauses(2))
+ next
+ assume "card S > 2"
+ then show ?thesis using that n_def
+ proof (induct n arbitrary: u S)
+ case 0
+ then show ?case by auto
+ next
+ case (Suc n u S)
+ have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
+ using that unfolding card_eq_sum by auto
+ with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
+ have c: "card (S - {x}) = card S - 1"
+ by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
+ have "sum u (S - {x}) = 1 - u x"
+ by (simp add: Suc.prems sum_diff1 \<open>x \<in> S\<close>)
+ with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
+ by auto
+ have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
+ proof (cases "card (S - {x}) > 2")
+ case True
+ then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
+ using Suc.prems c by force+
+ show ?thesis
+ proof (rule Suc.hyps)
+ show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
+ by (auto simp: eq1 sum_distrib_left[symmetric])
+ qed (use S Suc.prems True in auto)
+ next
+ case False
+ then have "card (S - {x}) = Suc (Suc 0)"
+ using Suc.prems c by auto
+ then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
+ unfolding card_Suc_eq by auto
+ then show ?thesis
+ using eq1 \<open>S \<subseteq> V\<close>
+ by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
+ qed
+ have "u x + (1 - u x) = 1 \<Longrightarrow>
+ u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
+ by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
+ moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
+ by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
+ ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
+ by (simp add: x)
+ qed
+ qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
+ qed
+ ultimately show ?thesis
+ unfolding affine_def by meson
+qed
+
+
+lemma affine_hull_explicit:
+ "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
+ (is "_ = ?rhs")
+proof (rule hull_unique)
+ show "p \<subseteq> ?rhs"
+ proof (intro subsetI CollectI exI conjI)
+ show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
+ by auto
+ qed auto
+ show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
+ using that unfolding affine by blast
+ show "affine ?rhs"
+ unfolding affine_def
+ proof clarify
+ fix u v :: real and sx ux sy uy
+ assume uv: "u + v = 1"
+ and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
+ and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
+ have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
+ by auto
+ show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
+ sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
+ proof (intro exI conjI)
+ show "finite (sx \<union> sy)"
+ using x y by auto
+ show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
+ using x y uv
+ by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
+ have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
+ = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
+ using x y
+ unfolding scaleR_left_distrib scaleR_zero_left if_smult
+ by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] **)
+ also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
+ unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
+ finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
+ = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
+ qed (use x y in auto)
+ qed
+qed
+
+lemma affine_hull_finite:
+ assumes "finite S"
+ shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
+proof -
+ have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
+ if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
+ proof -
+ have "S \<inter> F = F"
+ using that by auto
+ show ?thesis
+ proof (intro exI conjI)
+ show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
+ by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
+ show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
+ by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
+ qed
+ qed
+ show ?thesis
+ unfolding affine_hull_explicit using assms
+ by (fastforce dest: *)
+qed
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems and hence small special cases\<close>
+
+lemma affine_hull_empty[simp]: "affine hull {} = {}"
+ by simp
+
+lemma affine_hull_finite_step:
+ fixes y :: "'a::real_vector"
+ shows "finite S \<Longrightarrow>
+ (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
+ (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
+proof -
+ assume fin: "finite S"
+ show "?lhs = ?rhs"
+ proof
+ assume ?lhs
+ then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
+ by auto
+ show ?rhs
+ proof (cases "a \<in> S")
+ case True
+ then show ?thesis
+ using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
+ next
+ case False
+ show ?thesis
+ by (rule exI [where x="u a"]) (use u fin False in auto)
+ qed
+ next
+ assume ?rhs
+ then obtain v u where vu: "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
+ by auto
+ have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
+ by auto
+ show ?lhs
+ proof (cases "a \<in> S")
+ case True
+ show ?thesis
+ by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
+ (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
+ next
+ case False
+ then show ?thesis
+ apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
+ apply (simp add: vu sum_clauses(2)[OF fin] *)
+ by (simp add: sum_delta_notmem(3) vu)
+ qed
+ qed
+qed
+
+lemma affine_hull_2:
+ fixes a b :: "'a::real_vector"
+ shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
+ (is "?lhs = ?rhs")
+proof -
+ have *:
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
+ have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
+ using affine_hull_finite[of "{a,b}"] by auto
+ also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
+ by (simp add: affine_hull_finite_step[of "{b}" a])
+ also have "\<dots> = ?rhs" unfolding * by auto
+ finally show ?thesis by auto
+qed
+
+lemma affine_hull_3:
+ fixes a b c :: "'a::real_vector"
+ shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
+proof -
+ have *:
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
+ show ?thesis
+ apply (simp add: affine_hull_finite affine_hull_finite_step)
+ unfolding *
+ apply safe
+ apply (metis add.assoc)
+ apply (rule_tac x=u in exI, force)
+ done
+qed
+
+lemma mem_affine:
+ assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
+ shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
+ using assms affine_def[of S] by auto
+
+lemma mem_affine_3:
+ assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
+ shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
+proof -
+ have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
+ using affine_hull_3[of x y z] assms by auto
+ moreover
+ have "affine hull {x, y, z} \<subseteq> affine hull S"
+ using hull_mono[of "{x, y, z}" "S"] assms by auto
+ moreover
+ have "affine hull S = S"
+ using assms affine_hull_eq[of S] by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma mem_affine_3_minus:
+ assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
+ shows "x + v *\<^sub>R (y-z) \<in> S"
+ using mem_affine_3[of S x y z 1 v "-v"] assms
+ by (simp add: algebra_simps)
+
+corollary%unimportant mem_affine_3_minus2:
+ "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
+ by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some relations between affine hull and subspaces\<close>
+
+lemma affine_hull_insert_subset_span:
+ "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
+proof -
+ have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
+ if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
+ for x F u
+ proof -
+ have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
+ using that by auto
+ show ?thesis
+ proof (intro exI conjI)
+ show "finite ((\<lambda>x. x - a) ` (F - {a}))"
+ by (simp add: that(1))
+ show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
+ by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
+ sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
+ qed (use \<open>F \<subseteq> insert a S\<close> in auto)
+ qed
+ then show ?thesis
+ unfolding affine_hull_explicit span_explicit by blast
+qed
+
+lemma affine_hull_insert_span:
+ assumes "a \<notin> S"
+ shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x. x \<in> S}}"
+proof -
+ have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
+ if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
+ proof -
+ from that
+ obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
+ unfolding span_explicit by auto
+ define F where "F = (\<lambda>x. x + a) ` T"
+ have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
+ unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
+ have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
+ using F assms by auto
+ show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
+ apply (rule_tac x = "insert a F" in exI)
+ apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
+ using assms F
+ apply (auto simp: sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
+ done
+ qed
+ show ?thesis
+ by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
+qed
+
+lemma affine_hull_span:
+ assumes "a \<in> S"
+ shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
+ using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Parallel affine sets\<close>
+
+definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
+ where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
+
+lemma affine_parallel_expl_aux:
+ fixes S T :: "'a::real_vector set"
+ assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
+ shows "T = (\<lambda>x. a + x) ` S"
+proof -
+ have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
+ using that
+ by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
+ moreover have "T \<ge> (\<lambda>x. a + x) ` S"
+ using assms by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
+ by (auto simp add: affine_parallel_def)
+ (use affine_parallel_expl_aux [of S _ T] in blast)
+
+lemma affine_parallel_reflex: "affine_parallel S S"
+ unfolding affine_parallel_def
+ using image_add_0 by blast
+
+lemma affine_parallel_commut:
+ assumes "affine_parallel A B"
+ shows "affine_parallel B A"
+proof -
+ from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
+ unfolding affine_parallel_def by auto
+ have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
+ from B show ?thesis
+ using translation_galois [of B a A]
+ unfolding affine_parallel_def by blast
+qed
+
+lemma affine_parallel_assoc:
+ assumes "affine_parallel A B"
+ and "affine_parallel B C"
+ shows "affine_parallel A C"
+proof -
+ from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
+ unfolding affine_parallel_def by auto
+ moreover
+ from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
+ unfolding affine_parallel_def by auto
+ ultimately show ?thesis
+ using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
+qed
+
+lemma affine_translation_aux:
+ fixes a :: "'a::real_vector"
+ assumes "affine ((\<lambda>x. a + x) ` S)"
+ shows "affine S"
+proof -
+ {
+ fix x y u v
+ assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
+ then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
+ by auto
+ then have h1: "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
+ using xy assms unfolding affine_def by auto
+ have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
+ by (simp add: algebra_simps)
+ also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
+ using \<open>u + v = 1\<close> by auto
+ ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
+ using h1 by auto
+ then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
+ }
+ then show ?thesis unfolding affine_def by auto
+qed
+
+lemma affine_translation:
+ "affine S \<longleftrightarrow> affine ((+) a ` S)" for a :: "'a::real_vector"
+proof
+ show "affine ((+) a ` S)" if "affine S"
+ using that translation_assoc [of "- a" a S]
+ by (auto intro: affine_translation_aux [of "- a" "((+) a ` S)"])
+ show "affine S" if "affine ((+) a ` S)"
+ using that by (rule affine_translation_aux)
+qed
+
+lemma parallel_is_affine:
+ fixes S T :: "'a::real_vector set"
+ assumes "affine S" "affine_parallel S T"
+ shows "affine T"
+proof -
+ from assms obtain a where "T = (\<lambda>x. a + x) ` S"
+ unfolding affine_parallel_def by auto
+ then show ?thesis
+ using affine_translation assms by auto
+qed
+
+lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
+ unfolding subspace_def affine_def by auto
+
+lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
+ by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Subspace parallel to an affine set\<close>
+
+lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
+proof -
+ have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
+ using subspace_imp_affine[of S] subspace_0 by auto
+ {
+ assume assm: "affine S \<and> 0 \<in> S"
+ {
+ fix c :: real
+ fix x
+ assume x: "x \<in> S"
+ have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
+ moreover
+ have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
+ using affine_alt[of S] assm x by auto
+ ultimately have "c *\<^sub>R x \<in> S" by auto
+ }
+ then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
+
+ {
+ fix x y
+ assume xy: "x \<in> S" "y \<in> S"
+ define u where "u = (1 :: real)/2"
+ have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
+ by auto
+ moreover
+ have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
+ by (simp add: algebra_simps)
+ moreover
+ have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
+ using affine_alt[of S] assm xy by auto
+ ultimately
+ have "(1/2) *\<^sub>R (x+y) \<in> S"
+ using u_def by auto
+ moreover
+ have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
+ by auto
+ ultimately
+ have "x + y \<in> S"
+ using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
+ }
+ then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
+ by auto
+ then have "subspace S"
+ using h1 assm unfolding subspace_def by auto
+ }
+ then show ?thesis using h0 by metis
+qed
+
+lemma affine_diffs_subspace:
+ assumes "affine S" "a \<in> S"
+ shows "subspace ((\<lambda>x. (-a)+x) ` S)"
+proof -
+ have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
+ have "affine ((\<lambda>x. (-a)+x) ` S)"
+ using affine_translation assms by blast
+ moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
+ using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
+ ultimately show ?thesis using subspace_affine by auto
+qed
+
+lemma affine_diffs_subspace_subtract:
+ "subspace ((\<lambda>x. x - a) ` S)" if "affine S" "a \<in> S"
+ using that affine_diffs_subspace [of _ a] by simp
+
+lemma parallel_subspace_explicit:
+ assumes "affine S"
+ and "a \<in> S"
+ assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
+ shows "subspace L \<and> affine_parallel S L"
+proof -
+ from assms have "L = plus (- a) ` S" by auto
+ then have par: "affine_parallel S L"
+ unfolding affine_parallel_def ..
+ then have "affine L" using assms parallel_is_affine by auto
+ moreover have "0 \<in> L"
+ using assms by auto
+ ultimately show ?thesis
+ using subspace_affine par by auto
+qed
+
+lemma parallel_subspace_aux:
+ assumes "subspace A"
+ and "subspace B"
+ and "affine_parallel A B"
+ shows "A \<supseteq> B"
+proof -
+ from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
+ using affine_parallel_expl[of A B] by auto
+ then have "-a \<in> A"
+ using assms subspace_0[of B] by auto
+ then have "a \<in> A"
+ using assms subspace_neg[of A "-a"] by auto
+ then show ?thesis
+ using assms a unfolding subspace_def by auto
+qed
+
+lemma parallel_subspace:
+ assumes "subspace A"
+ and "subspace B"
+ and "affine_parallel A B"
+ shows "A = B"
+proof
+ show "A \<supseteq> B"
+ using assms parallel_subspace_aux by auto
+ show "A \<subseteq> B"
+ using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
+qed
+
+lemma affine_parallel_subspace:
+ assumes "affine S" "S \<noteq> {}"
+ shows "\<exists>!L. subspace L \<and> affine_parallel S L"
+proof -
+ have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
+ using assms parallel_subspace_explicit by auto
+ {
+ fix L1 L2
+ assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
+ then have "affine_parallel L1 L2"
+ using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
+ then have "L1 = L2"
+ using ass parallel_subspace by auto
+ }
+ then show ?thesis using ex by auto
+qed
+
+
+subsection \<open>Affine Dependence\<close>
+
+text "Formalized by Lars Schewe."
+
+definition\<^marker>\<open>tag important\<close> affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
+ where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
+
+lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
+ unfolding affine_dependent_def dependent_def
+ using affine_hull_subset_span by auto
+
+lemma affine_dependent_subset:
+ "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
+apply (simp add: affine_dependent_def Bex_def)
+apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
+done
+
+lemma affine_independent_subset:
+ shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
+by (metis affine_dependent_subset)
+
+lemma affine_independent_Diff:
+ "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
+by (meson Diff_subset affine_dependent_subset)
+
+proposition affine_dependent_explicit:
+ "affine_dependent p \<longleftrightarrow>
+ (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
+proof -
+ have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
+ if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
+ proof (intro exI conjI)
+ have "x \<notin> S"
+ using that by auto
+ then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
+ using that by (simp add: sum_delta_notmem)
+ show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
+ using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
+ qed (use that in auto)
+ moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
+ if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
+ proof (intro bexI exI conjI)
+ have "S \<noteq> {v}"
+ using that by auto
+ then show "S - {v} \<noteq> {}"
+ using that by auto
+ show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
+ unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
+ show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
+ unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
+ scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
+ using that by auto
+ show "S - {v} \<subseteq> p - {v}"
+ using that by auto
+ qed (use that in auto)
+ ultimately show ?thesis
+ unfolding affine_dependent_def affine_hull_explicit by auto
+qed
+
+lemma affine_dependent_explicit_finite:
+ fixes S :: "'a::real_vector set"
+ assumes "finite S"
+ shows "affine_dependent S \<longleftrightarrow>
+ (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
+ (is "?lhs = ?rhs")
+proof
+ have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
+ by auto
+ assume ?lhs
+ then obtain t u v where
+ "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
+ unfolding affine_dependent_explicit by auto
+ then show ?rhs
+ apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
+ apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
+ done
+next
+ assume ?rhs
+ then obtain u v where "sum u S = 0" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
+ by auto
+ then show ?lhs unfolding affine_dependent_explicit
+ using assms by auto
+qed
+
+lemma dependent_imp_affine_dependent:
+ assumes "dependent {x - a| x . x \<in> s}"
+ and "a \<notin> s"
+ shows "affine_dependent (insert a s)"
+proof -
+ from assms(1)[unfolded dependent_explicit] obtain S u v
+ where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
+ by auto
+ define t where "t = (\<lambda>x. x + a) ` S"
+
+ have inj: "inj_on (\<lambda>x. x + a) S"
+ unfolding inj_on_def by auto
+ have "0 \<notin> S"
+ using obt(2) assms(2) unfolding subset_eq by auto
+ have fin: "finite t" and "t \<subseteq> s"
+ unfolding t_def using obt(1,2) by auto
+ then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
+ by auto
+ moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
+ apply (rule sum.cong)
+ using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
+ apply auto
+ done
+ have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
+ unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
+ moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
+ using obt(3,4) \<open>0\<notin>S\<close>
+ by (rule_tac x="v + a" in bexI) (auto simp: t_def)
+ moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
+ using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
+ have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
+ unfolding scaleR_left.sum
+ unfolding t_def and sum.reindex[OF inj] and o_def
+ using obt(5)
+ by (auto simp: sum.distrib scaleR_right_distrib)
+ then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
+ unfolding sum_clauses(2)[OF fin]
+ using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
+ by (auto simp: *)
+ ultimately show ?thesis
+ unfolding affine_dependent_explicit
+ apply (rule_tac x="insert a t" in exI, auto)
+ done
+qed
+
+lemma affine_dependent_biggerset:
+ fixes s :: "'a::euclidean_space set"
+ assumes "finite s" "card s \<ge> DIM('a) + 2"
+ shows "affine_dependent s"
+proof -
+ have "s \<noteq> {}" using assms by auto
+ then obtain a where "a\<in>s" by auto
+ have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
+ by auto
+ have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
+ unfolding * by (simp add: card_image inj_on_def)
+ also have "\<dots> > DIM('a)" using assms(2)
+ unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
+ finally show ?thesis
+ apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
+ apply (rule dependent_imp_affine_dependent)
+ apply (rule dependent_biggerset, auto)
+ done
+qed
+
+lemma affine_dependent_biggerset_general:
+ assumes "finite (S :: 'a::euclidean_space set)"
+ and "card S \<ge> dim S + 2"
+ shows "affine_dependent S"
+proof -
+ from assms(2) have "S \<noteq> {}" by auto
+ then obtain a where "a\<in>S" by auto
+ have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
+ by auto
+ have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
+ by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
+ have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
+ using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
+ also have "\<dots> < dim S + 1" by auto
+ also have "\<dots> \<le> card (S - {a})"
+ using assms
+ using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
+ by auto
+ finally show ?thesis
+ apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
+ apply (rule dependent_imp_affine_dependent)
+ apply (rule dependent_biggerset_general)
+ unfolding **
+ apply auto
+ done
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some Properties of Affine Dependent Sets\<close>
+
+lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
+ by (simp add: affine_dependent_def)
+
+lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
+ by (simp add: affine_dependent_def)
+
+lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
+ by (simp add: affine_dependent_def insert_Diff_if hull_same)
+
+lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (affine hull S)"
+proof -
+ have "affine ((\<lambda>x. a + x) ` (affine hull S))"
+ using affine_translation affine_affine_hull by blast
+ moreover have "(\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
+ using hull_subset[of S] by auto
+ ultimately have h1: "affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
+ by (metis hull_minimal)
+ have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)))"
+ using affine_translation affine_affine_hull by blast
+ moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S))"
+ using hull_subset[of "(\<lambda>x. a + x) ` S"] by auto
+ moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S"
+ using translation_assoc[of "-a" a] by auto
+ ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)) >= (affine hull S)"
+ by (metis hull_minimal)
+ then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
+ by auto
+ then show ?thesis using h1 by auto
+qed
+
+lemma affine_dependent_translation:
+ assumes "affine_dependent S"
+ shows "affine_dependent ((\<lambda>x. a + x) ` S)"
+proof -
+ obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
+ using assms affine_dependent_def by auto
+ have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
+ by auto
+ then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
+ using affine_hull_translation[of a "S - {x}"] x by auto
+ moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
+ using x by auto
+ ultimately show ?thesis
+ unfolding affine_dependent_def by auto
+qed
+
+lemma affine_dependent_translation_eq:
+ "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
+proof -
+ {
+ assume "affine_dependent ((\<lambda>x. a + x) ` S)"
+ then have "affine_dependent S"
+ using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
+ by auto
+ }
+ then show ?thesis
+ using affine_dependent_translation by auto
+qed
+
+lemma affine_hull_0_dependent:
+ assumes "0 \<in> affine hull S"
+ shows "dependent S"
+proof -
+ obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+ using assms affine_hull_explicit[of S] by auto
+ then have "\<exists>v\<in>s. u v \<noteq> 0" by auto
+ then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
+ using s_u by auto
+ then show ?thesis
+ unfolding dependent_explicit[of S] by auto
+qed
+
+lemma affine_dependent_imp_dependent2:
+ assumes "affine_dependent (insert 0 S)"
+ shows "dependent S"
+proof -
+ obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
+ using affine_dependent_def[of "(insert 0 S)"] assms by blast
+ then have "x \<in> span (insert 0 S - {x})"
+ using affine_hull_subset_span by auto
+ moreover have "span (insert 0 S - {x}) = span (S - {x})"
+ using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
+ ultimately have "x \<in> span (S - {x})" by auto
+ then have "x \<noteq> 0 \<Longrightarrow> dependent S"
+ using x dependent_def by auto
+ moreover
+ {
+ assume "x = 0"
+ then have "0 \<in> affine hull S"
+ using x hull_mono[of "S - {0}" S] by auto
+ then have "dependent S"
+ using affine_hull_0_dependent by auto
+ }
+ ultimately show ?thesis by auto
+qed
+
+lemma affine_dependent_iff_dependent:
+ assumes "a \<notin> S"
+ shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
+proof -
+ have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
+ then show ?thesis
+ using affine_dependent_translation_eq[of "(insert a S)" "-a"]
+ affine_dependent_imp_dependent2 assms
+ dependent_imp_affine_dependent[of a S]
+ by (auto simp del: uminus_add_conv_diff)
+qed
+
+lemma affine_dependent_iff_dependent2:
+ assumes "a \<in> S"
+ shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
+proof -
+ have "insert a (S - {a}) = S"
+ using assms by auto
+ then show ?thesis
+ using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
+qed
+
+lemma affine_hull_insert_span_gen:
+ "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
+proof -
+ have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
+ by auto
+ {
+ assume "a \<notin> s"
+ then have ?thesis
+ using affine_hull_insert_span[of a s] h1 by auto
+ }
+ moreover
+ {
+ assume a1: "a \<in> s"
+ have "\<exists>x. x \<in> s \<and> -a+x=0"
+ apply (rule exI[of _ a])
+ using a1
+ apply auto
+ done
+ then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
+ by auto
+ then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
+ using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
+ moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
+ by auto
+ moreover have "insert a (s - {a}) = insert a s"
+ by auto
+ ultimately have ?thesis
+ using affine_hull_insert_span[of "a" "s-{a}"] by auto
+ }
+ ultimately show ?thesis by auto
+qed
+
+lemma affine_hull_span2:
+ assumes "a \<in> s"
+ shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
+ using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
+ by auto
+
+lemma affine_hull_span_gen:
+ assumes "a \<in> affine hull s"
+ shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
+proof -
+ have "affine hull (insert a s) = affine hull s"
+ using hull_redundant[of a affine s] assms by auto
+ then show ?thesis
+ using affine_hull_insert_span_gen[of a "s"] by auto
+qed
+
+lemma affine_hull_span_0:
+ assumes "0 \<in> affine hull S"
+ shows "affine hull S = span S"
+ using affine_hull_span_gen[of "0" S] assms by auto
+
+lemma extend_to_affine_basis_nonempty:
+ fixes S V :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
+ shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
+proof -
+ obtain a where a: "a \<in> S"
+ using assms by auto
+ then have h0: "independent ((\<lambda>x. -a + x) ` (S-{a}))"
+ using affine_dependent_iff_dependent2 assms by auto
+ obtain B where B:
+ "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
+ using assms
+ by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
+ define T where "T = (\<lambda>x. a+x) ` insert 0 B"
+ then have "T = insert a ((\<lambda>x. a+x) ` B)"
+ by auto
+ then have "affine hull T = (\<lambda>x. a+x) ` span B"
+ using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
+ by auto
+ then have "V \<subseteq> affine hull T"
+ using B assms translation_inverse_subset[of a V "span B"]
+ by auto
+ moreover have "T \<subseteq> V"
+ using T_def B a assms by auto
+ ultimately have "affine hull T = affine hull V"
+ by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
+ moreover have "S \<subseteq> T"
+ using T_def B translation_inverse_subset[of a "S-{a}" B]
+ by auto
+ moreover have "\<not> affine_dependent T"
+ using T_def affine_dependent_translation_eq[of "insert 0 B"]
+ affine_dependent_imp_dependent2 B
+ by auto
+ ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
+qed
+
+lemma affine_basis_exists:
+ fixes V :: "'n::euclidean_space set"
+ shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
+proof (cases "V = {}")
+ case True
+ then show ?thesis
+ using affine_independent_0 by auto
+next
+ case False
+ then obtain x where "x \<in> V" by auto
+ then show ?thesis
+ using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
+ by auto
+qed
+
+proposition extend_to_affine_basis:
+ fixes S V :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent S" "S \<subseteq> V"
+ obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
+proof (cases "S = {}")
+ case True then show ?thesis
+ using affine_basis_exists by (metis empty_subsetI that)
+next
+ case False
+ then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
+qed
+
+
+subsection \<open>Affine Dimension of a Set\<close>
+
+definition\<^marker>\<open>tag important\<close> aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
+ where "aff_dim V =
+ (SOME d :: int.
+ \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
+
+lemma aff_dim_basis_exists:
+ fixes V :: "('n::euclidean_space) set"
+ shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
+proof -
+ obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
+ using affine_basis_exists[of V] by auto
+ then show ?thesis
+ unfolding aff_dim_def
+ some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
+ apply auto
+ apply (rule exI[of _ "int (card B) - (1 :: int)"])
+ apply (rule exI[of _ "B"], auto)
+ done
+qed
+
+lemma affine_hull_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
+by (metis affine_empty subset_empty subset_hull)
+
+lemma empty_eq_affine_hull[simp]: "{} = affine hull S \<longleftrightarrow> S = {}"
+by (metis affine_hull_eq_empty)
+
+lemma aff_dim_parallel_subspace_aux:
+ fixes B :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent B" "a \<in> B"
+ shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
+proof -
+ have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
+ using affine_dependent_iff_dependent2 assms by auto
+ then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
+ "finite ((\<lambda>x. -a + x) ` (B - {a}))"
+ using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
+ show ?thesis
+ proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
+ case True
+ have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
+ using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
+ then have "B = {a}" using True by auto
+ then show ?thesis using assms fin by auto
+ next
+ case False
+ then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
+ using fin by auto
+ moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
+ by (rule card_image) (use translate_inj_on in blast)
+ ultimately have "card (B-{a}) > 0" by auto
+ then have *: "finite (B - {a})"
+ using card_gt_0_iff[of "(B - {a})"] by auto
+ then have "card (B - {a}) = card B - 1"
+ using card_Diff_singleton assms by auto
+ with * show ?thesis using fin h1 by auto
+ qed
+qed
+
+lemma aff_dim_parallel_subspace:
+ fixes V L :: "'n::euclidean_space set"
+ assumes "V \<noteq> {}"
+ and "subspace L"
+ and "affine_parallel (affine hull V) L"
+ shows "aff_dim V = int (dim L)"
+proof -
+ obtain B where
+ B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
+ using aff_dim_basis_exists by auto
+ then have "B \<noteq> {}"
+ using assms B
+ by auto
+ then obtain a where a: "a \<in> B" by auto
+ define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
+ moreover have "affine_parallel (affine hull B) Lb"
+ using Lb_def B assms affine_hull_span2[of a B] a
+ affine_parallel_commut[of "Lb" "(affine hull B)"]
+ unfolding affine_parallel_def
+ by auto
+ moreover have "subspace Lb"
+ using Lb_def subspace_span by auto
+ moreover have "affine hull B \<noteq> {}"
+ using assms B by auto
+ ultimately have "L = Lb"
+ using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
+ by auto
+ then have "dim L = dim Lb"
+ by auto
+ moreover have "card B - 1 = dim Lb" and "finite B"
+ using Lb_def aff_dim_parallel_subspace_aux a B by auto
+ ultimately show ?thesis
+ using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
+qed
+
+lemma aff_independent_finite:
+ fixes B :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent B"
+ shows "finite B"
+proof -
+ {
+ assume "B \<noteq> {}"
+ then obtain a where "a \<in> B" by auto
+ then have ?thesis
+ using aff_dim_parallel_subspace_aux assms by auto
+ }
+ then show ?thesis by auto
+qed
+
+
+lemma aff_dim_empty:
+ fixes S :: "'n::euclidean_space set"
+ shows "S = {} \<longleftrightarrow> aff_dim S = -1"
+proof -
+ obtain B where *: "affine hull B = affine hull S"
+ and "\<not> affine_dependent B"
+ and "int (card B) = aff_dim S + 1"
+ using aff_dim_basis_exists by auto
+ moreover
+ from * have "S = {} \<longleftrightarrow> B = {}"
+ by auto
+ ultimately show ?thesis
+ using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
+qed
+
+lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
+ by (simp add: aff_dim_empty [symmetric])
+
+lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
+ unfolding aff_dim_def using hull_hull[of _ S] by auto
+
+lemma aff_dim_affine_hull2:
+ assumes "affine hull S = affine hull T"
+ shows "aff_dim S = aff_dim T"
+ unfolding aff_dim_def using assms by auto
+
+lemma aff_dim_unique:
+ fixes B V :: "'n::euclidean_space set"
+ assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
+ shows "of_nat (card B) = aff_dim V + 1"
+proof (cases "B = {}")
+ case True
+ then have "V = {}"
+ using assms
+ by auto
+ then have "aff_dim V = (-1::int)"
+ using aff_dim_empty by auto
+ then show ?thesis
+ using \<open>B = {}\<close> by auto
+next
+ case False
+ then obtain a where a: "a \<in> B" by auto
+ define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
+ have "affine_parallel (affine hull B) Lb"
+ using Lb_def affine_hull_span2[of a B] a
+ affine_parallel_commut[of "Lb" "(affine hull B)"]
+ unfolding affine_parallel_def by auto
+ moreover have "subspace Lb"
+ using Lb_def subspace_span by auto
+ ultimately have "aff_dim B = int(dim Lb)"
+ using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
+ moreover have "(card B) - 1 = dim Lb" "finite B"
+ using Lb_def aff_dim_parallel_subspace_aux a assms by auto
+ ultimately have "of_nat (card B) = aff_dim B + 1"
+ using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
+ then show ?thesis
+ using aff_dim_affine_hull2 assms by auto
+qed
+
+lemma aff_dim_affine_independent:
+ fixes B :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent B"
+ shows "of_nat (card B) = aff_dim B + 1"
+ using aff_dim_unique[of B B] assms by auto
+
+lemma affine_independent_iff_card:
+ fixes s :: "'a::euclidean_space set"
+ shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
+ apply (rule iffI)
+ apply (simp add: aff_dim_affine_independent aff_independent_finite)
+ by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
+
+lemma aff_dim_sing [simp]:
+ fixes a :: "'n::euclidean_space"
+ shows "aff_dim {a} = 0"
+ using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
+
+lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
+proof (clarsimp)
+ assume "a \<noteq> b"
+ then have "aff_dim{a,b} = card{a,b} - 1"
+ using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
+ also have "\<dots> = 1"
+ using \<open>a \<noteq> b\<close> by simp
+ finally show "aff_dim {a, b} = 1" .
+qed
+
+lemma aff_dim_inner_basis_exists:
+ fixes V :: "('n::euclidean_space) set"
+ shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
+ \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
+proof -
+ obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
+ using affine_basis_exists[of V] by auto
+ then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
+ with B show ?thesis by auto
+qed
+
+lemma aff_dim_le_card:
+ fixes V :: "'n::euclidean_space set"
+ assumes "finite V"
+ shows "aff_dim V \<le> of_nat (card V) - 1"
+proof -
+ obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
+ using aff_dim_inner_basis_exists[of V] by auto
+ then have "card B \<le> card V"
+ using assms card_mono by auto
+ with B show ?thesis by auto
+qed
+
+lemma aff_dim_parallel_eq:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "affine_parallel (affine hull S) (affine hull T)"
+ shows "aff_dim S = aff_dim T"
+proof -
+ {
+ assume "T \<noteq> {}" "S \<noteq> {}"
+ then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
+ using affine_parallel_subspace[of "affine hull T"]
+ affine_affine_hull[of T]
+ by auto
+ then have "aff_dim T = int (dim L)"
+ using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
+ moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
+ using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
+ moreover from * have "aff_dim S = int (dim L)"
+ using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
+ ultimately have ?thesis by auto
+ }
+ moreover
+ {
+ assume "S = {}"
+ then have "S = {}" and "T = {}"
+ using assms
+ unfolding affine_parallel_def
+ by auto
+ then have ?thesis using aff_dim_empty by auto
+ }
+ moreover
+ {
+ assume "T = {}"
+ then have "S = {}" and "T = {}"
+ using assms
+ unfolding affine_parallel_def
+ by auto
+ then have ?thesis
+ using aff_dim_empty by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma aff_dim_translation_eq:
+ "aff_dim ((+) a ` S) = aff_dim S" for a :: "'n::euclidean_space"
+proof -
+ have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
+ unfolding affine_parallel_def
+ apply (rule exI[of _ "a"])
+ using affine_hull_translation[of a S]
+ apply auto
+ done
+ then show ?thesis
+ using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
+qed
+
+lemma aff_dim_translation_eq_subtract:
+ "aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S" for a :: "'n::euclidean_space"
+ using aff_dim_translation_eq [of "- a"] by (simp cong: image_cong_simp)
+
+lemma aff_dim_affine:
+ fixes S L :: "'n::euclidean_space set"
+ assumes "S \<noteq> {}"
+ and "affine S"
+ and "subspace L"
+ and "affine_parallel S L"
+ shows "aff_dim S = int (dim L)"
+proof -
+ have *: "affine hull S = S"
+ using assms affine_hull_eq[of S] by auto
+ then have "affine_parallel (affine hull S) L"
+ using assms by (simp add: *)
+ then show ?thesis
+ using assms aff_dim_parallel_subspace[of S L] by blast
+qed
+
+lemma dim_affine_hull:
+ fixes S :: "'n::euclidean_space set"
+ shows "dim (affine hull S) = dim S"
+proof -
+ have "dim (affine hull S) \<ge> dim S"
+ using dim_subset by auto
+ moreover have "dim (span S) \<ge> dim (affine hull S)"
+ using dim_subset affine_hull_subset_span by blast
+ moreover have "dim (span S) = dim S"
+ using dim_span by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma aff_dim_subspace:
+ fixes S :: "'n::euclidean_space set"
+ assumes "subspace S"
+ shows "aff_dim S = int (dim S)"
+proof (cases "S={}")
+ case True with assms show ?thesis
+ by (simp add: subspace_affine)
+next
+ case False
+ with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
+ show ?thesis by auto
+qed
+
+lemma aff_dim_zero:
+ fixes S :: "'n::euclidean_space set"
+ assumes "0 \<in> affine hull S"
+ shows "aff_dim S = int (dim S)"
+proof -
+ have "subspace (affine hull S)"
+ using subspace_affine[of "affine hull S"] affine_affine_hull assms
+ by auto
+ then have "aff_dim (affine hull S) = int (dim (affine hull S))"
+ using assms aff_dim_subspace[of "affine hull S"] by auto
+ then show ?thesis
+ using aff_dim_affine_hull[of S] dim_affine_hull[of S]
+ by auto
+qed
+
+lemma aff_dim_eq_dim:
+ "aff_dim S = int (dim ((+) (- a) ` S))" if "a \<in> affine hull S"
+ for S :: "'n::euclidean_space set"
+proof -
+ have "0 \<in> affine hull (+) (- a) ` S"
+ unfolding affine_hull_translation
+ using that by (simp add: ac_simps)
+ with aff_dim_zero show ?thesis
+ by (metis aff_dim_translation_eq)
+qed
+
+lemma aff_dim_eq_dim_subtract:
+ "aff_dim S = int (dim ((\<lambda>x. x - a) ` S))" if "a \<in> affine hull S"
+ for S :: "'n::euclidean_space set"
+ using aff_dim_eq_dim [of a] that by (simp cong: image_cong_simp)
+
+lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
+ using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
+ dim_UNIV[where 'a="'n::euclidean_space"]
+ by auto
+
+lemma aff_dim_geq:
+ fixes V :: "'n::euclidean_space set"
+ shows "aff_dim V \<ge> -1"
+proof -
+ obtain B where "affine hull B = affine hull V"
+ and "\<not> affine_dependent B"
+ and "int (card B) = aff_dim V + 1"
+ using aff_dim_basis_exists by auto
+ then show ?thesis by auto
+qed
+
+lemma aff_dim_negative_iff [simp]:
+ fixes S :: "'n::euclidean_space set"
+ shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
+by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
+
+lemma aff_lowdim_subset_hyperplane:
+ fixes S :: "'a::euclidean_space set"
+ assumes "aff_dim S < DIM('a)"
+ obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
+proof (cases "S={}")
+ case True
+ moreover
+ have "(SOME b. b \<in> Basis) \<noteq> 0"
+ by (metis norm_some_Basis norm_zero zero_neq_one)
+ ultimately show ?thesis
+ using that by blast
+next
+ case False
+ then obtain c S' where "c \<notin> S'" "S = insert c S'"
+ by (meson equals0I mk_disjoint_insert)
+ have "dim ((+) (-c) ` S) < DIM('a)"
+ by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
+ then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
+ using lowdim_subset_hyperplane by blast
+ moreover
+ have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
+ proof -
+ have "w-c \<in> span ((+) (- c) ` S)"
+ by (simp add: span_base \<open>w \<in> S\<close>)
+ with that have "w-c \<in> {x. a \<bullet> x = 0}"
+ by blast
+ then show ?thesis
+ by (auto simp: algebra_simps)
+ qed
+ ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
+ by blast
+ then show ?thesis
+ by (rule that[OF \<open>a \<noteq> 0\<close>])
+qed
+
+lemma affine_independent_card_dim_diffs:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "\<not> affine_dependent S" "a \<in> S"
+ shows "card S = dim {x - a|x. x \<in> S} + 1"
+proof -
+ have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
+ have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
+ proof (cases "x = a")
+ case True then show ?thesis by (simp add: span_clauses)
+ next
+ case False then show ?thesis
+ using assms by (blast intro: span_base that)
+ qed
+ have "\<not> affine_dependent (insert a S)"
+ by (simp add: assms insert_absorb)
+ then have 3: "independent {b - a |b. b \<in> S - {a}}"
+ using dependent_imp_affine_dependent by fastforce
+ have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
+ by blast
+ then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
+ by simp
+ also have "\<dots> = card (S - {a})"
+ by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
+ also have "\<dots> = card S - 1"
+ by (simp add: aff_independent_finite assms)
+ finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
+ have "finite S"
+ by (meson assms aff_independent_finite)
+ with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
+ moreover have "dim {x - a |x. x \<in> S} = card S - 1"
+ using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
+ ultimately show ?thesis
+ by auto
+qed
+
+lemma independent_card_le_aff_dim:
+ fixes B :: "'n::euclidean_space set"
+ assumes "B \<subseteq> V"
+ assumes "\<not> affine_dependent B"
+ shows "int (card B) \<le> aff_dim V + 1"
+proof -
+ obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
+ by (metis assms extend_to_affine_basis[of B V])
+ then have "of_nat (card T) = aff_dim V + 1"
+ using aff_dim_unique by auto
+ then show ?thesis
+ using T card_mono[of T B] aff_independent_finite[of T] by auto
+qed
+
+lemma aff_dim_subset:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "S \<subseteq> T"
+ shows "aff_dim S \<le> aff_dim T"
+proof -
+ obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
+ "of_nat (card B) = aff_dim S + 1"
+ using aff_dim_inner_basis_exists[of S] by auto
+ then have "int (card B) \<le> aff_dim T + 1"
+ using assms independent_card_le_aff_dim[of B T] by auto
+ with B show ?thesis by auto
+qed
+
+lemma aff_dim_le_DIM:
+ fixes S :: "'n::euclidean_space set"
+ shows "aff_dim S \<le> int (DIM('n))"
+proof -
+ have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
+ using aff_dim_UNIV by auto
+ then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
+ using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
+qed
+
+lemma affine_dim_equal:
+ fixes S :: "'n::euclidean_space set"
+ assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
+ shows "S = T"
+proof -
+ obtain a where "a \<in> S" using assms by auto
+ then have "a \<in> T" using assms by auto
+ define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
+ then have ls: "subspace LS" "affine_parallel S LS"
+ using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
+ then have h1: "int(dim LS) = aff_dim S"
+ using assms aff_dim_affine[of S LS] by auto
+ have "T \<noteq> {}" using assms by auto
+ define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
+ then have lt: "subspace LT \<and> affine_parallel T LT"
+ using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
+ then have "int(dim LT) = aff_dim T"
+ using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
+ then have "dim LS = dim LT"
+ using h1 assms by auto
+ moreover have "LS \<le> LT"
+ using LS_def LT_def assms by auto
+ ultimately have "LS = LT"
+ using subspace_dim_equal[of LS LT] ls lt by auto
+ moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
+ using LS_def by auto
+ moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
+ using LT_def by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma aff_dim_eq_0:
+ fixes S :: "'a::euclidean_space set"
+ shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by auto
+next
+ case False
+ then obtain a where "a \<in> S" by auto
+ show ?thesis
+ proof safe
+ assume 0: "aff_dim S = 0"
+ have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
+ by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
+ then show "\<exists>a. S = {a}"
+ using \<open>a \<in> S\<close> by blast
+ qed auto
+qed
+
+lemma affine_hull_UNIV:
+ fixes S :: "'n::euclidean_space set"
+ assumes "aff_dim S = int(DIM('n))"
+ shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
+proof -
+ have "S \<noteq> {}"
+ using assms aff_dim_empty[of S] by auto
+ have h0: "S \<subseteq> affine hull S"
+ using hull_subset[of S _] by auto
+ have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
+ using aff_dim_UNIV assms by auto
+ then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
+ using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
+ have h3: "aff_dim S \<le> aff_dim (affine hull S)"
+ using h0 aff_dim_subset[of S "affine hull S"] assms by auto
+ then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
+ using h0 h1 h2 by auto
+ then show ?thesis
+ using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
+ affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
+ by auto
+qed
+
+lemma disjoint_affine_hull:
+ fixes s :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
+ shows "(affine hull t) \<inter> (affine hull u) = {}"
+proof -
+ have "finite s" using assms by (simp add: aff_independent_finite)
+ then have "finite t" "finite u" using assms finite_subset by blast+
+ { fix y
+ assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
+ then obtain a b
+ where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
+ and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
+ by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
+ define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
+ have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
+ have "sum c s = 0"
+ by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
+ moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
+ by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum.neutral zero_neq_one)
+ moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
+ by (simp add: c_def if_smult sum_negf
+ comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
+ ultimately have False
+ using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
+ }
+ then show ?thesis by blast
+qed
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Convex.thy Thu Dec 05 19:53:41 2019 +0100
+++ b/src/HOL/Analysis/Convex.thy Thu Dec 05 21:03:06 2019 +0100
@@ -10,7 +10,7 @@
theory Convex
imports
- Linear_Algebra
+ Affine
"HOL-Library.Set_Algebras"
begin
@@ -1073,17 +1073,6 @@
obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
-lemma sum_delta_notmem:
- assumes "x \<notin> s"
- shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
- and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
- and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
- and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
- apply (rule_tac [!] sum.cong)
- using assms
- apply auto
- done
-
lemma sum_delta'':
fixes s::"'a::real_vector set"
assumes "finite s"
@@ -1095,9 +1084,6 @@
unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
qed
-lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
- by (fact if_distrib)
-
lemma dist_triangle_eq:
fixes x y z :: "'a::real_inner"
shows "dist x z = dist x y + dist y z \<longleftrightarrow>
@@ -1109,596 +1095,6 @@
qed
-subsection \<open>Affine set and affine hull\<close>
-
-definition\<^marker>\<open>tag important\<close> affine :: "'a::real_vector set \<Rightarrow> bool"
- where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
-
-lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
- unfolding affine_def by (metis eq_diff_eq')
-
-lemma affine_empty [iff]: "affine {}"
- unfolding affine_def by auto
-
-lemma affine_sing [iff]: "affine {x}"
- unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
-
-lemma affine_UNIV [iff]: "affine UNIV"
- unfolding affine_def by auto
-
-lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
- unfolding affine_def by auto
-
-lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
- unfolding affine_def by auto
-
-lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
- apply (clarsimp simp add: affine_def)
- apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
- apply (auto simp: algebra_simps)
- done
-
-lemma affine_affine_hull [simp]: "affine(affine hull s)"
- unfolding hull_def
- using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
-
-lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
- by (metis affine_affine_hull hull_same)
-
-lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
- by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some explicit formulations\<close>
-
-text "Formalized by Lars Schewe."
-
-lemma affine:
- fixes V::"'a::real_vector set"
- shows "affine V \<longleftrightarrow>
- (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
-proof -
- have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
- and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
- proof (cases "x = y")
- case True
- then show ?thesis
- using that by (metis scaleR_add_left scaleR_one)
- next
- case False
- then show ?thesis
- using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
- qed
- moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
- if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
- and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
- proof -
- define n where "n = card S"
- consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
- then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
- proof cases
- assume "card S = 1"
- then obtain a where "S={a}"
- by (auto simp: card_Suc_eq)
- then show ?thesis
- using that by simp
- next
- assume "card S = 2"
- then obtain a b where "S = {a, b}"
- by (metis Suc_1 card_1_singletonE card_Suc_eq)
- then show ?thesis
- using *[of a b] that
- by (auto simp: sum_clauses(2))
- next
- assume "card S > 2"
- then show ?thesis using that n_def
- proof (induct n arbitrary: u S)
- case 0
- then show ?case by auto
- next
- case (Suc n u S)
- have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
- using that unfolding card_eq_sum by auto
- with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
- have c: "card (S - {x}) = card S - 1"
- by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
- have "sum u (S - {x}) = 1 - u x"
- by (simp add: Suc.prems sum_diff1 \<open>x \<in> S\<close>)
- with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
- by auto
- have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
- proof (cases "card (S - {x}) > 2")
- case True
- then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
- using Suc.prems c by force+
- show ?thesis
- proof (rule Suc.hyps)
- show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
- by (auto simp: eq1 sum_distrib_left[symmetric])
- qed (use S Suc.prems True in auto)
- next
- case False
- then have "card (S - {x}) = Suc (Suc 0)"
- using Suc.prems c by auto
- then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
- unfolding card_Suc_eq by auto
- then show ?thesis
- using eq1 \<open>S \<subseteq> V\<close>
- by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
- qed
- have "u x + (1 - u x) = 1 \<Longrightarrow>
- u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
- by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
- moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
- by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
- ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
- by (simp add: x)
- qed
- qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
- qed
- ultimately show ?thesis
- unfolding affine_def by meson
-qed
-
-
-lemma affine_hull_explicit:
- "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
- (is "_ = ?rhs")
-proof (rule hull_unique)
- show "p \<subseteq> ?rhs"
- proof (intro subsetI CollectI exI conjI)
- show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
- by auto
- qed auto
- show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
- using that unfolding affine by blast
- show "affine ?rhs"
- unfolding affine_def
- proof clarify
- fix u v :: real and sx ux sy uy
- assume uv: "u + v = 1"
- and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
- and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
- have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
- by auto
- show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
- sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
- proof (intro exI conjI)
- show "finite (sx \<union> sy)"
- using x y by auto
- show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
- using x y uv
- by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
- have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
- = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
- using x y
- unfolding scaleR_left_distrib scaleR_zero_left if_smult
- by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] **)
- also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
- unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
- finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
- = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
- qed (use x y in auto)
- qed
-qed
-
-lemma affine_hull_finite:
- assumes "finite S"
- shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
-proof -
- have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
- if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
- proof -
- have "S \<inter> F = F"
- using that by auto
- show ?thesis
- proof (intro exI conjI)
- show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
- by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
- show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
- by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
- qed
- qed
- show ?thesis
- unfolding affine_hull_explicit using assms
- by (fastforce dest: *)
-qed
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems and hence small special cases\<close>
-
-lemma affine_hull_empty[simp]: "affine hull {} = {}"
- by simp
-
-lemma affine_hull_finite_step:
- fixes y :: "'a::real_vector"
- shows "finite S \<Longrightarrow>
- (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
- (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
-proof -
- assume fin: "finite S"
- show "?lhs = ?rhs"
- proof
- assume ?lhs
- then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
- by auto
- show ?rhs
- proof (cases "a \<in> S")
- case True
- then show ?thesis
- using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
- next
- case False
- show ?thesis
- by (rule exI [where x="u a"]) (use u fin False in auto)
- qed
- next
- assume ?rhs
- then obtain v u where vu: "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
- by auto
- have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
- by auto
- show ?lhs
- proof (cases "a \<in> S")
- case True
- show ?thesis
- by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
- (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
- next
- case False
- then show ?thesis
- apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
- apply (simp add: vu sum_clauses(2)[OF fin] *)
- by (simp add: sum_delta_notmem(3) vu)
- qed
- qed
-qed
-
-lemma affine_hull_2:
- fixes a b :: "'a::real_vector"
- shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
- (is "?lhs = ?rhs")
-proof -
- have *:
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
- have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
- using affine_hull_finite[of "{a,b}"] by auto
- also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
- by (simp add: affine_hull_finite_step[of "{b}" a])
- also have "\<dots> = ?rhs" unfolding * by auto
- finally show ?thesis by auto
-qed
-
-lemma affine_hull_3:
- fixes a b c :: "'a::real_vector"
- shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
-proof -
- have *:
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
- show ?thesis
- apply (simp add: affine_hull_finite affine_hull_finite_step)
- unfolding *
- apply safe
- apply (metis add.assoc)
- apply (rule_tac x=u in exI, force)
- done
-qed
-
-lemma mem_affine:
- assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
- shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
- using assms affine_def[of S] by auto
-
-lemma mem_affine_3:
- assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
- shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
-proof -
- have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
- using affine_hull_3[of x y z] assms by auto
- moreover
- have "affine hull {x, y, z} \<subseteq> affine hull S"
- using hull_mono[of "{x, y, z}" "S"] assms by auto
- moreover
- have "affine hull S = S"
- using assms affine_hull_eq[of S] by auto
- ultimately show ?thesis by auto
-qed
-
-lemma mem_affine_3_minus:
- assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
- shows "x + v *\<^sub>R (y-z) \<in> S"
- using mem_affine_3[of S x y z 1 v "-v"] assms
- by (simp add: algebra_simps)
-
-corollary%unimportant mem_affine_3_minus2:
- "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
- by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some relations between affine hull and subspaces\<close>
-
-lemma affine_hull_insert_subset_span:
- "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
-proof -
- have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
- if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
- for x F u
- proof -
- have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
- using that by auto
- show ?thesis
- proof (intro exI conjI)
- show "finite ((\<lambda>x. x - a) ` (F - {a}))"
- by (simp add: that(1))
- show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
- by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
- sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
- qed (use \<open>F \<subseteq> insert a S\<close> in auto)
- qed
- then show ?thesis
- unfolding affine_hull_explicit span_explicit by blast
-qed
-
-lemma affine_hull_insert_span:
- assumes "a \<notin> S"
- shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x. x \<in> S}}"
-proof -
- have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
- if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
- proof -
- from that
- obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
- unfolding span_explicit by auto
- define F where "F = (\<lambda>x. x + a) ` T"
- have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
- unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
- have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
- using F assms by auto
- show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
- apply (rule_tac x = "insert a F" in exI)
- apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
- using assms F
- apply (auto simp: sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
- done
- qed
- show ?thesis
- by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
-qed
-
-lemma affine_hull_span:
- assumes "a \<in> S"
- shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
- using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Parallel affine sets\<close>
-
-definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
- where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
-
-lemma affine_parallel_expl_aux:
- fixes S T :: "'a::real_vector set"
- assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
- shows "T = (\<lambda>x. a + x) ` S"
-proof -
- have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
- using that
- by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
- moreover have "T \<ge> (\<lambda>x. a + x) ` S"
- using assms by auto
- ultimately show ?thesis by auto
-qed
-
-lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
- by (auto simp add: affine_parallel_def)
- (use affine_parallel_expl_aux [of S _ T] in blast)
-
-lemma affine_parallel_reflex: "affine_parallel S S"
- unfolding affine_parallel_def
- using image_add_0 by blast
-
-lemma affine_parallel_commut:
- assumes "affine_parallel A B"
- shows "affine_parallel B A"
-proof -
- from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
- unfolding affine_parallel_def by auto
- have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
- from B show ?thesis
- using translation_galois [of B a A]
- unfolding affine_parallel_def by blast
-qed
-
-lemma affine_parallel_assoc:
- assumes "affine_parallel A B"
- and "affine_parallel B C"
- shows "affine_parallel A C"
-proof -
- from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
- unfolding affine_parallel_def by auto
- moreover
- from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
- unfolding affine_parallel_def by auto
- ultimately show ?thesis
- using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
-qed
-
-lemma affine_translation_aux:
- fixes a :: "'a::real_vector"
- assumes "affine ((\<lambda>x. a + x) ` S)"
- shows "affine S"
-proof -
- {
- fix x y u v
- assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
- then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
- by auto
- then have h1: "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
- using xy assms unfolding affine_def by auto
- have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
- by (simp add: algebra_simps)
- also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
- using \<open>u + v = 1\<close> by auto
- ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
- using h1 by auto
- then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
- }
- then show ?thesis unfolding affine_def by auto
-qed
-
-lemma affine_translation:
- "affine S \<longleftrightarrow> affine ((+) a ` S)" for a :: "'a::real_vector"
-proof
- show "affine ((+) a ` S)" if "affine S"
- using that translation_assoc [of "- a" a S]
- by (auto intro: affine_translation_aux [of "- a" "((+) a ` S)"])
- show "affine S" if "affine ((+) a ` S)"
- using that by (rule affine_translation_aux)
-qed
-
-lemma parallel_is_affine:
- fixes S T :: "'a::real_vector set"
- assumes "affine S" "affine_parallel S T"
- shows "affine T"
-proof -
- from assms obtain a where "T = (\<lambda>x. a + x) ` S"
- unfolding affine_parallel_def by auto
- then show ?thesis
- using affine_translation assms by auto
-qed
-
-lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
- unfolding subspace_def affine_def by auto
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Subspace parallel to an affine set\<close>
-
-lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
-proof -
- have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
- using subspace_imp_affine[of S] subspace_0 by auto
- {
- assume assm: "affine S \<and> 0 \<in> S"
- {
- fix c :: real
- fix x
- assume x: "x \<in> S"
- have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
- moreover
- have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
- using affine_alt[of S] assm x by auto
- ultimately have "c *\<^sub>R x \<in> S" by auto
- }
- then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
-
- {
- fix x y
- assume xy: "x \<in> S" "y \<in> S"
- define u where "u = (1 :: real)/2"
- have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
- by auto
- moreover
- have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
- by (simp add: algebra_simps)
- moreover
- have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
- using affine_alt[of S] assm xy by auto
- ultimately
- have "(1/2) *\<^sub>R (x+y) \<in> S"
- using u_def by auto
- moreover
- have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
- by auto
- ultimately
- have "x + y \<in> S"
- using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
- }
- then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
- by auto
- then have "subspace S"
- using h1 assm unfolding subspace_def by auto
- }
- then show ?thesis using h0 by metis
-qed
-
-lemma affine_diffs_subspace:
- assumes "affine S" "a \<in> S"
- shows "subspace ((\<lambda>x. (-a)+x) ` S)"
-proof -
- have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
- have "affine ((\<lambda>x. (-a)+x) ` S)"
- using affine_translation assms by blast
- moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
- using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
- ultimately show ?thesis using subspace_affine by auto
-qed
-
-lemma affine_diffs_subspace_subtract:
- "subspace ((\<lambda>x. x - a) ` S)" if "affine S" "a \<in> S"
- using that affine_diffs_subspace [of _ a] by simp
-
-lemma parallel_subspace_explicit:
- assumes "affine S"
- and "a \<in> S"
- assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
- shows "subspace L \<and> affine_parallel S L"
-proof -
- from assms have "L = plus (- a) ` S" by auto
- then have par: "affine_parallel S L"
- unfolding affine_parallel_def ..
- then have "affine L" using assms parallel_is_affine by auto
- moreover have "0 \<in> L"
- using assms by auto
- ultimately show ?thesis
- using subspace_affine par by auto
-qed
-
-lemma parallel_subspace_aux:
- assumes "subspace A"
- and "subspace B"
- and "affine_parallel A B"
- shows "A \<supseteq> B"
-proof -
- from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
- using affine_parallel_expl[of A B] by auto
- then have "-a \<in> A"
- using assms subspace_0[of B] by auto
- then have "a \<in> A"
- using assms subspace_neg[of A "-a"] by auto
- then show ?thesis
- using assms a unfolding subspace_def by auto
-qed
-
-lemma parallel_subspace:
- assumes "subspace A"
- and "subspace B"
- and "affine_parallel A B"
- shows "A = B"
-proof
- show "A \<supseteq> B"
- using assms parallel_subspace_aux by auto
- show "A \<subseteq> B"
- using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
-qed
-
-lemma affine_parallel_subspace:
- assumes "affine S" "S \<noteq> {}"
- shows "\<exists>!L. subspace L \<and> affine_parallel S L"
-proof -
- have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
- using assms parallel_subspace_explicit by auto
- {
- fix L1 L2
- assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
- then have "affine_parallel L1 L2"
- using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
- then have "L1 = L2"
- using ass parallel_subspace by auto
- }
- then show ?thesis using ex by auto
-qed
subsection \<open>Cones\<close>
@@ -1864,85 +1260,25 @@
ultimately show ?thesis by auto
qed
-
-subsection \<open>Affine Dependence\<close>
-
-text "Formalized by Lars Schewe."
-
-definition\<^marker>\<open>tag important\<close> affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
- where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
-
-lemma affine_dependent_subset:
- "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
-apply (simp add: affine_dependent_def Bex_def)
-apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
-done
-
-lemma affine_independent_subset:
- shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
-by (metis affine_dependent_subset)
-
-lemma affine_independent_Diff:
- "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
-by (meson Diff_subset affine_dependent_subset)
-
-proposition affine_dependent_explicit:
- "affine_dependent p \<longleftrightarrow>
- (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
+lemma convex_cone:
+ "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
+ (is "?lhs = ?rhs")
proof -
- have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
- if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
- proof (intro exI conjI)
- have "x \<notin> S"
- using that by auto
- then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
- using that by (simp add: sum_delta_notmem)
- show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
- using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
- qed (use that in auto)
- moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
- if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
- proof (intro bexI exI conjI)
- have "S \<noteq> {v}"
- using that by auto
- then show "S - {v} \<noteq> {}"
- using that by auto
- show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
- unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
- show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
- unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
- scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
- using that by auto
- show "S - {v} \<subseteq> p - {v}"
- using that by auto
- qed (use that in auto)
- ultimately show ?thesis
- unfolding affine_dependent_def affine_hull_explicit by auto
-qed
-
-lemma affine_dependent_explicit_finite:
- fixes S :: "'a::real_vector set"
- assumes "finite S"
- shows "affine_dependent S \<longleftrightarrow>
- (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
- (is "?lhs = ?rhs")
-proof
- have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
- by auto
- assume ?lhs
- then obtain t u v where
- "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
- unfolding affine_dependent_explicit by auto
- then show ?rhs
- apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
- apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
- done
-next
- assume ?rhs
- then obtain u v where "sum u S = 0" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
- by auto
- then show ?lhs unfolding affine_dependent_explicit
- using assms by auto
+ {
+ fix x y
+ assume "x\<in>s" "y\<in>s" and ?lhs
+ then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
+ unfolding cone_def by auto
+ then have "x + y \<in> s"
+ using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
+ apply (erule_tac x="2*\<^sub>R x" in ballE)
+ apply (erule_tac x="2*\<^sub>R y" in ballE)
+ apply (erule_tac x="1/2" in allE, simp)
+ apply (erule_tac x="1/2" in allE, auto)
+ done
+ }
+ then show ?thesis
+ unfolding convex_def cone_def by blast
qed
@@ -2502,993 +1838,12 @@
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
using subspace_imp_affine affine_imp_convex by auto
-lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
- by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
-
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
-lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
- unfolding affine_dependent_def dependent_def
- using affine_hull_subset_span by auto
-
-lemma dependent_imp_affine_dependent:
- assumes "dependent {x - a| x . x \<in> s}"
- and "a \<notin> s"
- shows "affine_dependent (insert a s)"
-proof -
- from assms(1)[unfolded dependent_explicit] obtain S u v
- where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
- by auto
- define t where "t = (\<lambda>x. x + a) ` S"
-
- have inj: "inj_on (\<lambda>x. x + a) S"
- unfolding inj_on_def by auto
- have "0 \<notin> S"
- using obt(2) assms(2) unfolding subset_eq by auto
- have fin: "finite t" and "t \<subseteq> s"
- unfolding t_def using obt(1,2) by auto
- then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
- by auto
- moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
- apply (rule sum.cong)
- using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
- apply auto
- done
- have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
- unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
- moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
- using obt(3,4) \<open>0\<notin>S\<close>
- by (rule_tac x="v + a" in bexI) (auto simp: t_def)
- moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
- using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
- have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
- unfolding scaleR_left.sum
- unfolding t_def and sum.reindex[OF inj] and o_def
- using obt(5)
- by (auto simp: sum.distrib scaleR_right_distrib)
- then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
- unfolding sum_clauses(2)[OF fin]
- using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
- by (auto simp: *)
- ultimately show ?thesis
- unfolding affine_dependent_explicit
- apply (rule_tac x="insert a t" in exI, auto)
- done
-qed
-
-lemma convex_cone:
- "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
- (is "?lhs = ?rhs")
-proof -
- {
- fix x y
- assume "x\<in>s" "y\<in>s" and ?lhs
- then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
- unfolding cone_def by auto
- then have "x + y \<in> s"
- using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
- apply (erule_tac x="2*\<^sub>R x" in ballE)
- apply (erule_tac x="2*\<^sub>R y" in ballE)
- apply (erule_tac x="1/2" in allE, simp)
- apply (erule_tac x="1/2" in allE, auto)
- done
- }
- then show ?thesis
- unfolding convex_def cone_def by blast
-qed
-
-lemma affine_dependent_biggerset:
- fixes s :: "'a::euclidean_space set"
- assumes "finite s" "card s \<ge> DIM('a) + 2"
- shows "affine_dependent s"
-proof -
- have "s \<noteq> {}" using assms by auto
- then obtain a where "a\<in>s" by auto
- have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
- by auto
- have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
- unfolding * by (simp add: card_image inj_on_def)
- also have "\<dots> > DIM('a)" using assms(2)
- unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
- finally show ?thesis
- apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
- apply (rule dependent_imp_affine_dependent)
- apply (rule dependent_biggerset, auto)
- done
-qed
-
-lemma affine_dependent_biggerset_general:
- assumes "finite (S :: 'a::euclidean_space set)"
- and "card S \<ge> dim S + 2"
- shows "affine_dependent S"
-proof -
- from assms(2) have "S \<noteq> {}" by auto
- then obtain a where "a\<in>S" by auto
- have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
- by auto
- have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
- by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
- have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
- using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
- also have "\<dots> < dim S + 1" by auto
- also have "\<dots> \<le> card (S - {a})"
- using assms
- using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
- by auto
- finally show ?thesis
- apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
- apply (rule dependent_imp_affine_dependent)
- apply (rule dependent_biggerset_general)
- unfolding **
- apply auto
- done
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some Properties of Affine Dependent Sets\<close>
-
-lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
- by (simp add: affine_dependent_def)
-
-lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
- by (simp add: affine_dependent_def)
-
-lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
- by (simp add: affine_dependent_def insert_Diff_if hull_same)
-
-lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (affine hull S)"
-proof -
- have "affine ((\<lambda>x. a + x) ` (affine hull S))"
- using affine_translation affine_affine_hull by blast
- moreover have "(\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
- using hull_subset[of S] by auto
- ultimately have h1: "affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
- by (metis hull_minimal)
- have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)))"
- using affine_translation affine_affine_hull by blast
- moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S))"
- using hull_subset[of "(\<lambda>x. a + x) ` S"] by auto
- moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S"
- using translation_assoc[of "-a" a] by auto
- ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)) >= (affine hull S)"
- by (metis hull_minimal)
- then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
- by auto
- then show ?thesis using h1 by auto
-qed
-
-lemma affine_dependent_translation:
- assumes "affine_dependent S"
- shows "affine_dependent ((\<lambda>x. a + x) ` S)"
-proof -
- obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
- using assms affine_dependent_def by auto
- have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
- by auto
- then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
- using affine_hull_translation[of a "S - {x}"] x by auto
- moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
- using x by auto
- ultimately show ?thesis
- unfolding affine_dependent_def by auto
-qed
-
-lemma affine_dependent_translation_eq:
- "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
-proof -
- {
- assume "affine_dependent ((\<lambda>x. a + x) ` S)"
- then have "affine_dependent S"
- using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
- by auto
- }
- then show ?thesis
- using affine_dependent_translation by auto
-qed
-
-lemma affine_hull_0_dependent:
- assumes "0 \<in> affine hull S"
- shows "dependent S"
-proof -
- obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
- using assms affine_hull_explicit[of S] by auto
- then have "\<exists>v\<in>s. u v \<noteq> 0" by auto
- then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
- using s_u by auto
- then show ?thesis
- unfolding dependent_explicit[of S] by auto
-qed
-
-lemma affine_dependent_imp_dependent2:
- assumes "affine_dependent (insert 0 S)"
- shows "dependent S"
-proof -
- obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
- using affine_dependent_def[of "(insert 0 S)"] assms by blast
- then have "x \<in> span (insert 0 S - {x})"
- using affine_hull_subset_span by auto
- moreover have "span (insert 0 S - {x}) = span (S - {x})"
- using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
- ultimately have "x \<in> span (S - {x})" by auto
- then have "x \<noteq> 0 \<Longrightarrow> dependent S"
- using x dependent_def by auto
- moreover
- {
- assume "x = 0"
- then have "0 \<in> affine hull S"
- using x hull_mono[of "S - {0}" S] by auto
- then have "dependent S"
- using affine_hull_0_dependent by auto
- }
- ultimately show ?thesis by auto
-qed
-
-lemma affine_dependent_iff_dependent:
- assumes "a \<notin> S"
- shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
-proof -
- have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
- then show ?thesis
- using affine_dependent_translation_eq[of "(insert a S)" "-a"]
- affine_dependent_imp_dependent2 assms
- dependent_imp_affine_dependent[of a S]
- by (auto simp del: uminus_add_conv_diff)
-qed
-
-lemma affine_dependent_iff_dependent2:
- assumes "a \<in> S"
- shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
-proof -
- have "insert a (S - {a}) = S"
- using assms by auto
- then show ?thesis
- using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
-qed
-
-lemma affine_hull_insert_span_gen:
- "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
-proof -
- have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
- by auto
- {
- assume "a \<notin> s"
- then have ?thesis
- using affine_hull_insert_span[of a s] h1 by auto
- }
- moreover
- {
- assume a1: "a \<in> s"
- have "\<exists>x. x \<in> s \<and> -a+x=0"
- apply (rule exI[of _ a])
- using a1
- apply auto
- done
- then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
- by auto
- then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
- using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
- moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
- by auto
- moreover have "insert a (s - {a}) = insert a s"
- by auto
- ultimately have ?thesis
- using affine_hull_insert_span[of "a" "s-{a}"] by auto
- }
- ultimately show ?thesis by auto
-qed
-
-lemma affine_hull_span2:
- assumes "a \<in> s"
- shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
- using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
- by auto
-
-lemma affine_hull_span_gen:
- assumes "a \<in> affine hull s"
- shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
-proof -
- have "affine hull (insert a s) = affine hull s"
- using hull_redundant[of a affine s] assms by auto
- then show ?thesis
- using affine_hull_insert_span_gen[of a "s"] by auto
-qed
-
-lemma affine_hull_span_0:
- assumes "0 \<in> affine hull S"
- shows "affine hull S = span S"
- using affine_hull_span_gen[of "0" S] assms by auto
-
-lemma extend_to_affine_basis_nonempty:
- fixes S V :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
- shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
-proof -
- obtain a where a: "a \<in> S"
- using assms by auto
- then have h0: "independent ((\<lambda>x. -a + x) ` (S-{a}))"
- using affine_dependent_iff_dependent2 assms by auto
- obtain B where B:
- "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
- using assms
- by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
- define T where "T = (\<lambda>x. a+x) ` insert 0 B"
- then have "T = insert a ((\<lambda>x. a+x) ` B)"
- by auto
- then have "affine hull T = (\<lambda>x. a+x) ` span B"
- using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
- by auto
- then have "V \<subseteq> affine hull T"
- using B assms translation_inverse_subset[of a V "span B"]
- by auto
- moreover have "T \<subseteq> V"
- using T_def B a assms by auto
- ultimately have "affine hull T = affine hull V"
- by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
- moreover have "S \<subseteq> T"
- using T_def B translation_inverse_subset[of a "S-{a}" B]
- by auto
- moreover have "\<not> affine_dependent T"
- using T_def affine_dependent_translation_eq[of "insert 0 B"]
- affine_dependent_imp_dependent2 B
- by auto
- ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
-qed
-
-lemma affine_basis_exists:
- fixes V :: "'n::euclidean_space set"
- shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
-proof (cases "V = {}")
- case True
- then show ?thesis
- using affine_independent_0 by auto
-next
- case False
- then obtain x where "x \<in> V" by auto
- then show ?thesis
- using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
- by auto
-qed
-
-proposition extend_to_affine_basis:
- fixes S V :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent S" "S \<subseteq> V"
- obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
-proof (cases "S = {}")
- case True then show ?thesis
- using affine_basis_exists by (metis empty_subsetI that)
-next
- case False
- then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
-qed
-
-subsection \<open>Affine Dimension of a Set\<close>
-
-definition\<^marker>\<open>tag important\<close> aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
- where "aff_dim V =
- (SOME d :: int.
- \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
-
-lemma aff_dim_basis_exists:
- fixes V :: "('n::euclidean_space) set"
- shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
-proof -
- obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
- using affine_basis_exists[of V] by auto
- then show ?thesis
- unfolding aff_dim_def
- some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
- apply auto
- apply (rule exI[of _ "int (card B) - (1 :: int)"])
- apply (rule exI[of _ "B"], auto)
- done
-qed
-
-lemma affine_hull_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
-by (metis affine_empty subset_empty subset_hull)
-
-lemma empty_eq_affine_hull[simp]: "{} = affine hull S \<longleftrightarrow> S = {}"
-by (metis affine_hull_eq_empty)
-
-lemma aff_dim_parallel_subspace_aux:
- fixes B :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent B" "a \<in> B"
- shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
-proof -
- have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
- using affine_dependent_iff_dependent2 assms by auto
- then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
- "finite ((\<lambda>x. -a + x) ` (B - {a}))"
- using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
- show ?thesis
- proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
- case True
- have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
- using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
- then have "B = {a}" using True by auto
- then show ?thesis using assms fin by auto
- next
- case False
- then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
- using fin by auto
- moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
- by (rule card_image) (use translate_inj_on in blast)
- ultimately have "card (B-{a}) > 0" by auto
- then have *: "finite (B - {a})"
- using card_gt_0_iff[of "(B - {a})"] by auto
- then have "card (B - {a}) = card B - 1"
- using card_Diff_singleton assms by auto
- with * show ?thesis using fin h1 by auto
- qed
-qed
-
-lemma aff_dim_parallel_subspace:
- fixes V L :: "'n::euclidean_space set"
- assumes "V \<noteq> {}"
- and "subspace L"
- and "affine_parallel (affine hull V) L"
- shows "aff_dim V = int (dim L)"
-proof -
- obtain B where
- B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
- using aff_dim_basis_exists by auto
- then have "B \<noteq> {}"
- using assms B
- by auto
- then obtain a where a: "a \<in> B" by auto
- define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
- moreover have "affine_parallel (affine hull B) Lb"
- using Lb_def B assms affine_hull_span2[of a B] a
- affine_parallel_commut[of "Lb" "(affine hull B)"]
- unfolding affine_parallel_def
- by auto
- moreover have "subspace Lb"
- using Lb_def subspace_span by auto
- moreover have "affine hull B \<noteq> {}"
- using assms B by auto
- ultimately have "L = Lb"
- using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
- by auto
- then have "dim L = dim Lb"
- by auto
- moreover have "card B - 1 = dim Lb" and "finite B"
- using Lb_def aff_dim_parallel_subspace_aux a B by auto
- ultimately show ?thesis
- using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
-qed
-
-lemma aff_independent_finite:
- fixes B :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent B"
- shows "finite B"
-proof -
- {
- assume "B \<noteq> {}"
- then obtain a where "a \<in> B" by auto
- then have ?thesis
- using aff_dim_parallel_subspace_aux assms by auto
- }
- then show ?thesis by auto
-qed
-
-lemmas independent_finite = independent_imp_finite
-
-lemma span_substd_basis:
- assumes d: "d \<subseteq> Basis"
- shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- (is "_ = ?B")
-proof -
- have "d \<subseteq> ?B"
- using d by (auto simp: inner_Basis)
- moreover have s: "subspace ?B"
- using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
- ultimately have "span d \<subseteq> ?B"
- using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
- moreover have *: "card d \<le> dim (span d)"
- using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
- span_superset[of d]
- by auto
- moreover from * have "dim ?B \<le> dim (span d)"
- using dim_substandard[OF assms] by auto
- ultimately show ?thesis
- using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
-qed
-
-lemma basis_to_substdbasis_subspace_isomorphism:
- fixes B :: "'a::euclidean_space set"
- assumes "independent B"
- shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
- f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
-proof -
- have B: "card B = dim B"
- using dim_unique[of B B "card B"] assms span_superset[of B] by auto
- have "dim B \<le> card (Basis :: 'a set)"
- using dim_subset_UNIV[of B] by simp
- from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
- by auto
- let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
- proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
- show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
- using d inner_not_same_Basis by blast
- qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
- with t \<open>card B = dim B\<close> d show ?thesis by auto
-qed
-
-lemma aff_dim_empty:
- fixes S :: "'n::euclidean_space set"
- shows "S = {} \<longleftrightarrow> aff_dim S = -1"
-proof -
- obtain B where *: "affine hull B = affine hull S"
- and "\<not> affine_dependent B"
- and "int (card B) = aff_dim S + 1"
- using aff_dim_basis_exists by auto
- moreover
- from * have "S = {} \<longleftrightarrow> B = {}"
- by auto
- ultimately show ?thesis
- using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
-qed
-
-lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
- by (simp add: aff_dim_empty [symmetric])
-
-lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
- unfolding aff_dim_def using hull_hull[of _ S] by auto
-
-lemma aff_dim_affine_hull2:
- assumes "affine hull S = affine hull T"
- shows "aff_dim S = aff_dim T"
- unfolding aff_dim_def using assms by auto
-
-lemma aff_dim_unique:
- fixes B V :: "'n::euclidean_space set"
- assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
- shows "of_nat (card B) = aff_dim V + 1"
-proof (cases "B = {}")
- case True
- then have "V = {}"
- using assms
- by auto
- then have "aff_dim V = (-1::int)"
- using aff_dim_empty by auto
- then show ?thesis
- using \<open>B = {}\<close> by auto
-next
- case False
- then obtain a where a: "a \<in> B" by auto
- define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
- have "affine_parallel (affine hull B) Lb"
- using Lb_def affine_hull_span2[of a B] a
- affine_parallel_commut[of "Lb" "(affine hull B)"]
- unfolding affine_parallel_def by auto
- moreover have "subspace Lb"
- using Lb_def subspace_span by auto
- ultimately have "aff_dim B = int(dim Lb)"
- using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
- moreover have "(card B) - 1 = dim Lb" "finite B"
- using Lb_def aff_dim_parallel_subspace_aux a assms by auto
- ultimately have "of_nat (card B) = aff_dim B + 1"
- using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
- then show ?thesis
- using aff_dim_affine_hull2 assms by auto
-qed
-
-lemma aff_dim_affine_independent:
- fixes B :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent B"
- shows "of_nat (card B) = aff_dim B + 1"
- using aff_dim_unique[of B B] assms by auto
-
-lemma affine_independent_iff_card:
- fixes s :: "'a::euclidean_space set"
- shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
- apply (rule iffI)
- apply (simp add: aff_dim_affine_independent aff_independent_finite)
- by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
-
-lemma aff_dim_sing [simp]:
- fixes a :: "'n::euclidean_space"
- shows "aff_dim {a} = 0"
- using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
-
-lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
-proof (clarsimp)
- assume "a \<noteq> b"
- then have "aff_dim{a,b} = card{a,b} - 1"
- using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
- also have "\<dots> = 1"
- using \<open>a \<noteq> b\<close> by simp
- finally show "aff_dim {a, b} = 1" .
-qed
-
-lemma aff_dim_inner_basis_exists:
- fixes V :: "('n::euclidean_space) set"
- shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
- \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
-proof -
- obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
- using affine_basis_exists[of V] by auto
- then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
- with B show ?thesis by auto
-qed
-
-lemma aff_dim_le_card:
- fixes V :: "'n::euclidean_space set"
- assumes "finite V"
- shows "aff_dim V \<le> of_nat (card V) - 1"
-proof -
- obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
- using aff_dim_inner_basis_exists[of V] by auto
- then have "card B \<le> card V"
- using assms card_mono by auto
- with B show ?thesis by auto
-qed
-
-lemma aff_dim_parallel_eq:
- fixes S T :: "'n::euclidean_space set"
- assumes "affine_parallel (affine hull S) (affine hull T)"
- shows "aff_dim S = aff_dim T"
-proof -
- {
- assume "T \<noteq> {}" "S \<noteq> {}"
- then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
- using affine_parallel_subspace[of "affine hull T"]
- affine_affine_hull[of T]
- by auto
- then have "aff_dim T = int (dim L)"
- using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
- moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
- using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
- moreover from * have "aff_dim S = int (dim L)"
- using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
- ultimately have ?thesis by auto
- }
- moreover
- {
- assume "S = {}"
- then have "S = {}" and "T = {}"
- using assms
- unfolding affine_parallel_def
- by auto
- then have ?thesis using aff_dim_empty by auto
- }
- moreover
- {
- assume "T = {}"
- then have "S = {}" and "T = {}"
- using assms
- unfolding affine_parallel_def
- by auto
- then have ?thesis
- using aff_dim_empty by auto
- }
- ultimately show ?thesis by blast
-qed
-
-lemma aff_dim_translation_eq:
- "aff_dim ((+) a ` S) = aff_dim S" for a :: "'n::euclidean_space"
-proof -
- have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
- unfolding affine_parallel_def
- apply (rule exI[of _ "a"])
- using affine_hull_translation[of a S]
- apply auto
- done
- then show ?thesis
- using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
-qed
-
-lemma aff_dim_translation_eq_subtract:
- "aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S" for a :: "'n::euclidean_space"
- using aff_dim_translation_eq [of "- a"] by (simp cong: image_cong_simp)
-
-lemma aff_dim_affine:
- fixes S L :: "'n::euclidean_space set"
- assumes "S \<noteq> {}"
- and "affine S"
- and "subspace L"
- and "affine_parallel S L"
- shows "aff_dim S = int (dim L)"
-proof -
- have *: "affine hull S = S"
- using assms affine_hull_eq[of S] by auto
- then have "affine_parallel (affine hull S) L"
- using assms by (simp add: *)
- then show ?thesis
- using assms aff_dim_parallel_subspace[of S L] by blast
-qed
-
-lemma dim_affine_hull:
- fixes S :: "'n::euclidean_space set"
- shows "dim (affine hull S) = dim S"
-proof -
- have "dim (affine hull S) \<ge> dim S"
- using dim_subset by auto
- moreover have "dim (span S) \<ge> dim (affine hull S)"
- using dim_subset affine_hull_subset_span by blast
- moreover have "dim (span S) = dim S"
- using dim_span by auto
- ultimately show ?thesis by auto
-qed
-
-lemma aff_dim_subspace:
- fixes S :: "'n::euclidean_space set"
- assumes "subspace S"
- shows "aff_dim S = int (dim S)"
-proof (cases "S={}")
- case True with assms show ?thesis
- by (simp add: subspace_affine)
-next
- case False
- with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
- show ?thesis by auto
-qed
-
-lemma aff_dim_zero:
- fixes S :: "'n::euclidean_space set"
- assumes "0 \<in> affine hull S"
- shows "aff_dim S = int (dim S)"
-proof -
- have "subspace (affine hull S)"
- using subspace_affine[of "affine hull S"] affine_affine_hull assms
- by auto
- then have "aff_dim (affine hull S) = int (dim (affine hull S))"
- using assms aff_dim_subspace[of "affine hull S"] by auto
- then show ?thesis
- using aff_dim_affine_hull[of S] dim_affine_hull[of S]
- by auto
-qed
-
-lemma aff_dim_eq_dim:
- "aff_dim S = int (dim ((+) (- a) ` S))" if "a \<in> affine hull S"
- for S :: "'n::euclidean_space set"
-proof -
- have "0 \<in> affine hull (+) (- a) ` S"
- unfolding affine_hull_translation
- using that by (simp add: ac_simps)
- with aff_dim_zero show ?thesis
- by (metis aff_dim_translation_eq)
-qed
-
-lemma aff_dim_eq_dim_subtract:
- "aff_dim S = int (dim ((\<lambda>x. x - a) ` S))" if "a \<in> affine hull S"
- for S :: "'n::euclidean_space set"
- using aff_dim_eq_dim [of a] that by (simp cong: image_cong_simp)
-
-lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
- using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
- dim_UNIV[where 'a="'n::euclidean_space"]
- by auto
-
-lemma aff_dim_geq:
- fixes V :: "'n::euclidean_space set"
- shows "aff_dim V \<ge> -1"
-proof -
- obtain B where "affine hull B = affine hull V"
- and "\<not> affine_dependent B"
- and "int (card B) = aff_dim V + 1"
- using aff_dim_basis_exists by auto
- then show ?thesis by auto
-qed
-
-lemma aff_dim_negative_iff [simp]:
- fixes S :: "'n::euclidean_space set"
- shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
-by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
-
-lemma aff_lowdim_subset_hyperplane:
- fixes S :: "'a::euclidean_space set"
- assumes "aff_dim S < DIM('a)"
- obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
-proof (cases "S={}")
- case True
- moreover
- have "(SOME b. b \<in> Basis) \<noteq> 0"
- by (metis norm_some_Basis norm_zero zero_neq_one)
- ultimately show ?thesis
- using that by blast
-next
- case False
- then obtain c S' where "c \<notin> S'" "S = insert c S'"
- by (meson equals0I mk_disjoint_insert)
- have "dim ((+) (-c) ` S) < DIM('a)"
- by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
- then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
- using lowdim_subset_hyperplane by blast
- moreover
- have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
- proof -
- have "w-c \<in> span ((+) (- c) ` S)"
- by (simp add: span_base \<open>w \<in> S\<close>)
- with that have "w-c \<in> {x. a \<bullet> x = 0}"
- by blast
- then show ?thesis
- by (auto simp: algebra_simps)
- qed
- ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
- by blast
- then show ?thesis
- by (rule that[OF \<open>a \<noteq> 0\<close>])
-qed
-
-lemma affine_independent_card_dim_diffs:
- fixes S :: "'a :: euclidean_space set"
- assumes "\<not> affine_dependent S" "a \<in> S"
- shows "card S = dim {x - a|x. x \<in> S} + 1"
-proof -
- have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
- have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
- proof (cases "x = a")
- case True then show ?thesis by (simp add: span_clauses)
- next
- case False then show ?thesis
- using assms by (blast intro: span_base that)
- qed
- have "\<not> affine_dependent (insert a S)"
- by (simp add: assms insert_absorb)
- then have 3: "independent {b - a |b. b \<in> S - {a}}"
- using dependent_imp_affine_dependent by fastforce
- have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
- by blast
- then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
- by simp
- also have "\<dots> = card (S - {a})"
- by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
- also have "\<dots> = card S - 1"
- by (simp add: aff_independent_finite assms)
- finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
- have "finite S"
- by (meson assms aff_independent_finite)
- with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
- moreover have "dim {x - a |x. x \<in> S} = card S - 1"
- using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
- ultimately show ?thesis
- by auto
-qed
-
-lemma independent_card_le_aff_dim:
- fixes B :: "'n::euclidean_space set"
- assumes "B \<subseteq> V"
- assumes "\<not> affine_dependent B"
- shows "int (card B) \<le> aff_dim V + 1"
-proof -
- obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
- by (metis assms extend_to_affine_basis[of B V])
- then have "of_nat (card T) = aff_dim V + 1"
- using aff_dim_unique by auto
- then show ?thesis
- using T card_mono[of T B] aff_independent_finite[of T] by auto
-qed
-
-lemma aff_dim_subset:
- fixes S T :: "'n::euclidean_space set"
- assumes "S \<subseteq> T"
- shows "aff_dim S \<le> aff_dim T"
-proof -
- obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
- "of_nat (card B) = aff_dim S + 1"
- using aff_dim_inner_basis_exists[of S] by auto
- then have "int (card B) \<le> aff_dim T + 1"
- using assms independent_card_le_aff_dim[of B T] by auto
- with B show ?thesis by auto
-qed
-
-lemma aff_dim_le_DIM:
- fixes S :: "'n::euclidean_space set"
- shows "aff_dim S \<le> int (DIM('n))"
-proof -
- have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
- using aff_dim_UNIV by auto
- then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
- using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
-qed
-
-lemma affine_dim_equal:
- fixes S :: "'n::euclidean_space set"
- assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
- shows "S = T"
-proof -
- obtain a where "a \<in> S" using assms by auto
- then have "a \<in> T" using assms by auto
- define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
- then have ls: "subspace LS" "affine_parallel S LS"
- using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
- then have h1: "int(dim LS) = aff_dim S"
- using assms aff_dim_affine[of S LS] by auto
- have "T \<noteq> {}" using assms by auto
- define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
- then have lt: "subspace LT \<and> affine_parallel T LT"
- using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
- then have "int(dim LT) = aff_dim T"
- using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
- then have "dim LS = dim LT"
- using h1 assms by auto
- moreover have "LS \<le> LT"
- using LS_def LT_def assms by auto
- ultimately have "LS = LT"
- using subspace_dim_equal[of LS LT] ls lt by auto
- moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
- using LS_def by auto
- moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
- using LT_def by auto
- ultimately show ?thesis by auto
-qed
-
-lemma aff_dim_eq_0:
- fixes S :: "'a::euclidean_space set"
- shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by auto
-next
- case False
- then obtain a where "a \<in> S" by auto
- show ?thesis
- proof safe
- assume 0: "aff_dim S = 0"
- have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
- by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
- then show "\<exists>a. S = {a}"
- using \<open>a \<in> S\<close> by blast
- qed auto
-qed
-
-lemma affine_hull_UNIV:
- fixes S :: "'n::euclidean_space set"
- assumes "aff_dim S = int(DIM('n))"
- shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
-proof -
- have "S \<noteq> {}"
- using assms aff_dim_empty[of S] by auto
- have h0: "S \<subseteq> affine hull S"
- using hull_subset[of S _] by auto
- have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
- using aff_dim_UNIV assms by auto
- then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
- using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
- have h3: "aff_dim S \<le> aff_dim (affine hull S)"
- using h0 aff_dim_subset[of S "affine hull S"] assms by auto
- then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
- using h0 h1 h2 by auto
- then show ?thesis
- using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
- affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
- by auto
-qed
-
-lemma disjoint_affine_hull:
- fixes s :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
- shows "(affine hull t) \<inter> (affine hull u) = {}"
-proof -
- have "finite s" using assms by (simp add: aff_independent_finite)
- then have "finite t" "finite u" using assms finite_subset by blast+
- { fix y
- assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
- then obtain a b
- where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
- and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
- by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
- define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
- have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
- have "sum c s = 0"
- by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
- moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
- by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum.neutral zero_neq_one)
- moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
- by (simp add: c_def if_smult sum_negf
- comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
- ultimately have False
- using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
- }
- then show ?thesis by blast
-qed
-
lemma aff_dim_convex_hull:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (convex hull S) = aff_dim S"
@@ -3497,6 +1852,7 @@
aff_dim_subset[of "convex hull S" "affine hull S"]
by auto
+
subsection \<open>Caratheodory's theorem\<close>
lemma convex_hull_caratheodory_aff_dim: