isabelle: comparison src/HOL/Multivariate_Analysis/Convex_Euclidean

equal deleted inserted replaced

-:7faf67b411b4
+:8d68162b7826
 using assms convex_def[of S] by auto
 lemma mem_convex_alt:
 assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
 shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
 apply (subst mem_convex_2)
 using assms apply (auto simp add: algebra_simps zero_le_divide_iff)
 using add_divide_distrib[of u v "u+v"] apply auto
 done
 lemma inj_on_image_mem_iff: "inj_on f B ==> (A <= B) ==> (f a : f`A) ==> (a : B) ==> (a : A)"
 by (blast dest: inj_onD)
 lemma independent_injective_on_span_image:
 assumes iS: "independent S"
 and lf: "linear f" and fi: "inj_on f (span S)"
 shows "independent (f ` S)"
 proof -
 {
 fix a
 then show ?thesis unfolding dependent_def by blast
 qed
 lemma dim_image_eq:
 fixes f :: "'n::euclidean_space => 'm::euclidean_space"
 assumes lf: "linear f" and fi: "inj_on f (span S)"
 shows "dim (f ` S) = dim (S:: ('n::euclidean_space) set)"
 proof -
 obtain B where B_def: "B<=S & independent B & S <= span B & card B = dim S"
 using basis_exists[of S] by auto
 then have "span S = span B"
 using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
 then have "independent (f ` B)"
 using independent_injective_on_span_image[of B f] B_def assms by auto
 proof -
 have "inj_on f S <-> (!x : S. !y : S. f x = f y --> x = y)" by (simp add: inj_on_def)
 also have "... <-> (!x : S. !y : S. f x - f y = 0 --> x - y = 0)" by simp
 also have "... <-> (!x : S. !y : S. f (x - y) = 0 --> x - y = 0)"
 by (simp add: linear_sub[OF lf])
 also have "... <-> (! x : S. f x = 0 --> x = 0)"
 using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
 finally show ?thesis .
 qed
 lemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"
 unfolding subspace_def by auto
 lemma span_eq[simp]: "(span s = s) <-> subspace s"
 unfolding span_def by (rule hull_eq, rule subspace_Inter)
 lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
 by (auto simp add: inj_on_def euclidean_eq[where 'a='n])
 lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
 proof -
 have eq: "?S = basis ` d" by blast
 show ?thesis
 unfolding eq
 apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
 using independent_basis[where 'a='a] assms apply (auto simp: *)
 done
 qed
 lemma dim_cball:
 assumes "0<e"
 shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
 proof -
 { fix x :: "'n::euclidean_space"
 def y == "(e/norm x) *\<^sub>R x"
 then have "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto
 moreover have "x = (norm x/e) *\<^sub>R y" using y_def assms by simp
 moreover hence "x = (norm x/e) *\<^sub>R y" by auto
 ultimately have "x : span (cball 0 e)"
 using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
 } then have "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto
 then show ?thesis
 using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
 qed
 lemma indep_card_eq_dim_span:
 fixes B :: "('n::euclidean_space) set"
 assumes "independent B"
 shows "finite B & card B = dim (span B)"
 using assms basis_card_eq_dim[of B "span B"] span_inc by auto
 lemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"
 by (rule ccontr) auto
 lemma translate_inj_on:
 fixes A :: "('a::ab_group_add) set"
 shows "inj_on (%x. a+x) A"
 unfolding inj_on_def by auto
 lemma translation_assoc:
 using translation_assoc[of "-a" a S] apply auto
 using translation_assoc[of a "-a" T] apply auto
 done
 lemma translation_inverse_subset:
 assumes "((%x. -a+x) ` V) <= (S :: 'n::ab_group_add set)"
 shows "V <= ((%x. a+x) ` S)"
 proof -
 { fix x
 assume "x:V"
 then have "x-a : S" using assms by auto
 lemma affine_UNIV[intro]: "affine UNIV"
 unfolding affine_def by auto
 lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
 unfolding affine_def by auto
 lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
 unfolding affine_def by auto
 lemma affine_affine_hull: "affine(affine hull s)"
 shows "affine V \<longleftrightarrow>
 (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
 unfolding affine_def
 apply rule
 apply(rule, rule, rule)
 apply(erule conjE)+
 defer
 apply (rule, rule, rule, rule, rule)
 proof -
 fix x y u v
 assume as: "x \<in> V" "y \<in> V" "u + v = (1::real)"
 have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V"
 proof (cases "card (s - {x}) > 2")
 case True
 then have "s - {x} \<noteq> {}" "card (s - {x}) = n"
 unfolding c and as(1)[symmetric]
 proof (rule_tac ccontr)
 assume "\<not> s - {x} \<noteq> {}"
 then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
 then show False using True by auto
 qed auto
 then show ?thesis
 apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
 unfolding setsum_right_distrib[symmetric] using as and *** and True
 proof (cases "a \<in> s")
 case True
 then show ?thesis
 apply (rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
 unfolding setsum_clauses(2)[OF `?as`] apply simp
 unfolding scaleR_left_distrib and setsum_addf
 unfolding vu and * and scaleR_zero_left
 apply (auto simp add: setsum_delta[OF `?as`])
 done
 next
 case False
 then have **:
 "\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
 "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
 from False show ?thesis
 apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
 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. setsum 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(2)[of "{b}" a])
 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}" (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
 show ?thesis
 apply (simp add: affine_hull_finite affine_hull_finite_step)
 unfolding *
 apply auto
 moreover
 have "affine hull {x, y, z} <= 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 : S" "y : S" "z : S"
 shows "x + v *\<^sub>R (y-z) : S"
 shows "affine hull (insert a s) = {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
 apply (rule, rule affine_hull_insert_subset_span)
 unfolding subset_eq Ball_def
 unfolding affine_hull_explicit and mem_Collect_eq
 proof (rule, rule, erule exE, erule conjE)
 fix y v
 assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
 then obtain t u where obt:"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
 def f \<equiv> "(\<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"
 definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
 where "affine_parallel S T = (? a. T = ((%x. a + x) ` S))"
 lemma affine_parallel_expl_aux:
 fixes S T :: "'a::real_vector set"
 assumes "!x. (x : S <-> (a+x) : T)"
 shows "T = ((%x. a + x) ` S)"
 proof -
 { fix x
 assume "x : T"
 then have "(-a)+x : S" using assms by auto
 then have "x : ((%x. a + x) ` S)"
 using imageI[of "-a+x" S "(%x. a+x)"] by auto }
 moreover have "T >= ((%x. a + x) ` S)" using assms by auto
 ultimately show ?thesis by auto
 qed
 lemma affine_parallel_expl: "affine_parallel S T = (? a. !x. (x : S <-> (a+x) : T))"
 unfolding affine_parallel_def
 using translation_galois[of B a A] unfolding affine_parallel_def by auto
 qed
 lemma affine_parallel_assoc:
 assumes "affine_parallel A B" "affine_parallel B C"
 shows "affine_parallel A C"
 proof -
 from assms obtain ab where "B=((%x. ab + x) ` A)"
 unfolding affine_parallel_def by auto
 moreover
 from assms obtain bc where "C=((%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
 fixes S T :: "'a::real_vector set"
 assumes "affine S" "affine_parallel S T"
 shows "affine T"
 proof -
 from assms obtain a where "T=((%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 subspace_affine: "subspace S <-> (affine S & 0 : S)"
 proof -
 have h0: "subspace S ==> (affine S & 0 : S)"
 using subspace_imp_affine[of S] subspace_0 by auto
 { assume assm: "affine S & 0 : S"
 { fix c :: real
 fix x assume x_def: "x : 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 : S" using affine_alt[of S] assm x_def by auto
 ultimately have "c *\<^sub>R x : S" by auto
 moreover
 have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" by auto
 ultimately
 have "(x+y) : S" using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
 }
 then have "!x : S. !y : S. (x+y) : 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 : S"
 shows "subspace ((%x. (-a)+x) ` S)"
 proof -
 have "affine ((%x. (-a)+x) ` S)"
 using  affine_translation assms by auto
 moreover have "0 : ((%x. (-a)+x) ` S)"
 using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
 ultimately show ?thesis using subspace_affine by auto
 qed
 lemma parallel_subspace_explicit:
 assumes "affine S" "a : S"
 assumes "L == {y. ? x : S. (-a)+x=y}"
 shows "subspace L & affine_parallel S L"
 proof -
 have par: "affine_parallel S L"
 unfolding affine_parallel_def using assms by auto
 then have "affine L" using assms parallel_is_affine by auto
 moreover have "0 : L"
 using assms apply auto
 using exI[of "(%x. x:S & -a+x=0)" a] apply auto
 done
 ultimately show ?thesis using subspace_affine par by auto
 qed
 lemma parallel_subspace_aux:
 assumes "subspace A" "subspace B" "affine_parallel A B"
 shows "A>=B"
 using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
 qed
 lemma affine_parallel_subspace:
 assumes "affine S" "S ~= {}"
 shows "?!L. subspace L & affine_parallel S L"
 proof -
 have ex: "? L. subspace L & affine_parallel S L"
 using assms parallel_subspace_explicit by auto
 { fix L1 L2
 assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & 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"
 } then have "cone ?rhs" unfolding cone_def by auto
 then have "?rhs : Collect cone" unfolding mem_Collect_eq by auto
 { fix x
 assume "x : S"
 then have "1 *\<^sub>R x : ?rhs"
 apply auto
 apply (rule_tac x="1" in exI)
 apply auto
 done
 then have "x : ?rhs" by auto
 } then have "S <= ?rhs" by auto
 lemma convex_connected:
 fixes s :: "'a::real_normed_vector set"
 assumes "convex s" shows "connected s"
 proof-
 { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2"
 assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
 then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
 hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
 { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
 using as(1,2)[unfolded open_dist] apply simp
 using as(1,2)[unfolded open_dist] apply simp
 using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
 using as(3) by auto
 then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
 hence False using as(4)
 using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
 using x1(2) x2(2) by auto  }
 thus ?thesis unfolding connected_def by auto
 qed
 ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
 qed
 lemma convex_ball:
 fixes x :: "'a::real_normed_vector"
 shows "convex (ball x e)"
 proof(auto simp add: convex_def)
 fix y z assume yz:"dist x y < e" "dist x z < e"
 fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
 have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
 using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
 proof(auto simp add: convex_def Ball_def)
 fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
 fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
 have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
 using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
 thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto
 qed
 lemma connected_ball:
 fixes x :: "'a::real_normed_vector"
 shows "connected (ball x e)"
 subsubsection {* Convex hull is "preserved" by a linear function *}
 lemma convex_hull_linear_image:
 assumes "bounded_linear f"
 shows "f ` (convex hull s) = convex hull (f ` s)"
 apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3
 apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
 apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
 proof-
 interpret f: bounded_linear f by fact
 show "convex {x. f x \<in> convex hull f ` s}"
 unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
 interpret f: bounded_linear f by fact
 show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s]
 unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
 qed auto
 lemma in_convex_hull_linear_image:
 assumes "bounded_linear f" "x \<in> convex hull s"
 assumes "s \<noteq> {}"
 shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
 b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
 apply(rule,rule hull_minimal,rule) unfolding insert_iff prefer 3 apply rule proof-
 fix x assume x:"x = a \<or> x \<in> s"
 thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer
 apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
 next
 fix x assume "x\<in>?hull"
 then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
 have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
 using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by arith+
 hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
 thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
 next
 have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
 also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
 also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
 case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
 apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
 using as(1,2) obt1(1,2) obt2(1,2) by auto
 thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
 apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
 apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
 unfolding add_divide_distrib[symmetric] and zero_le_divide_iff
 by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
 have u1:"u1 \<le> 1" unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
 have u2:"u2 \<le> 1" unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
 have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
 apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
 also have "\<dots> \<le> 1" unfolding right_distrib[symmetric] and as(3) using u1 u2 by auto
 finally
 show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
 apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
 using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
 qed
 qed
 from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" by auto
 from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" by auto
 have *:"\<And>P (x1::'a) x2 s1 s2 i.(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
 "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
 prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
 have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto
 show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
 apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
 apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
 unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
 unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric] proof-
 fixes s :: "'a::real_vector set"
 assumes "finite s"
 shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
 setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
 proof(rule hull_unique, auto simp add: convex_def[of ?set])
 fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x"
 apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
 unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto
 next
 fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
 fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
 fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
 { fix x assume "x\<in>s"
 moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
 unfolding setsum_addf and setsum_right_distrib[symmetric] and ux(2) uy(2) using uv(3) by auto
 moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
 unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric] by auto
 ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
 apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto
 next
 fix t assume t:"s \<subseteq> t" "convex t"
 fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
 thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
 using assms and t(1) by auto
 qed
 have fin:"finite {1..k}" by auto
 have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
 { fix j assume "j\<in>{1..k}"
 hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
 using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
 apply(rule setsum_nonneg) using obt(1) by auto }
 moreover
 have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
 unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
 moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
 using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
 unfolding scaleR_left.setsum using obt(3) by auto
 ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
 moreover
 { fix y assume "y\<in>?rhs"
 then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
 obtain f where f:"inj_on f {1..card s}" "f ` {1..card s} = s" using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
 { fix i::nat assume "i\<in>{1..card s}"
 hence "f i \<in> s"  apply(subst f(2)[symmetric]) by auto
 hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
 moreover have *:"finite {1..card s}" by auto
 { fix y assume "y\<in>s"
 hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
 "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
 by (auto simp add: setsum_constant_scaleR)   }
 hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
 unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
 unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
 using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
 ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
 apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastforce
 hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
 ultimately show ?thesis unfolding set_eq_iff by blast
 qed
 by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
 lemma affine_dependent_imp_dependent:
 shows "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}" "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
 def t \<equiv> "(\<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
 hence "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 setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
 have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
 unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * 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"
 apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
 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)"
 apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
 have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
 unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
 using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
 hence "(\<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 setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *)
 ultimately show ?thesis unfolding affine_dependent_explicit
 apply(rule_tac x="insert a t" in exI) by auto
 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-
 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 *
 apply(rule card_image) unfolding inj_on_def by auto
 also have "\<dots> > DIM('a)" using assms(2)
 unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
 finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, symmetric])
 apply(rule dependent_imp_affine_dependent)
 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})" unfolding *
 apply(rule card_image) unfolding inj_on_def by auto
 have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
 apply(rule subset_le_dim) unfolding subset_eq
 using `a\<in>s` by (auto simp add:span_superset span_sub)
 also have "\<dots> < dim s + 1" by auto
 using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
 thus False using wv(1) by auto
 qed hence "i\<noteq>{}" unfolding i_def by auto
 hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
 using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto
 have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
 fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
 show"0 \<le> u v + t * w v" proof(cases "w v < 0")
 case False thus ?thesis apply(rule_tac add_nonneg_nonneg)
 using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
 case True hence "t \<le> u v / (- w v)" using `v\<in>s`
 unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto
 thus ?thesis unfolding real_0_le_add_iff
 using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]] by auto
 qed qed
 obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
 using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
 hence a:"a\<in>s" "u a + t * w a = 0" by auto
 have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
 unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto
 have "(\<Sum>v\<in>s. u v + t * w v) = 1"
 unfolding setsum_addf wv(1) setsum_right_distrib[symmetric] obt(5) by auto
 moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
 unfolding setsum_addf obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
 using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
 ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
 apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a
 by (auto simp add: * scaleR_left_distrib)
 proof-
 have "affine ((%x. a + x) ` (affine hull S))" using affine_translation affine_affine_hull by auto
 moreover have "(%x. a + x) `  S <= (%x. a + x) ` (affine hull S)" using hull_subset[of S] by auto
 ultimately have h1: "affine hull ((%x. a + x) `  S) <= (%x. a + x) ` (affine hull S)" by (metis hull_minimal)
 have "affine((%x. -a + x) ` (affine hull ((%x. a + x) `  S)))"  using affine_translation affine_affine_hull by auto
 moreover have "(%x. -a + x) ` (%x. a + x) `  S <= (%x. -a + x) ` (affine hull ((%x. a + x) `  S))" using hull_subset[of "(%x. a + x) `  S"] by auto
 moreover have "S=(%x. -a + x) ` (%x. a + x) `  S" using  translation_assoc[of "-a" a] by auto
 ultimately have "(%x. -a + x) ` (affine hull ((%x. a + x) `  S)) >= (affine hull S)" by (metis hull_minimal)
 hence "affine hull ((%x. a + x) `  S) >= (%x. a + x) ` (affine hull S)" by auto
 from this show ?thesis using h1 by auto
 qed
 shows "affine_dependent ((%x. a + x) ` S)"
 proof-
 obtain x where x_def: "x : S & x : affine hull (S - {x})" using assms affine_dependent_def by auto
 have "op + a ` (S - {x}) = op + a ` S - {a + x}" by auto
 hence "a+x : affine hull ((%x. a + x) ` S - {a+x})" using  affine_hull_translation[of a "S-{x}"] x_def by auto
 moreover have "a+x : (%x. a + x) ` S" using x_def by auto
 ultimately show ?thesis unfolding affine_dependent_def by auto
 qed
 lemma affine_dependent_translation_eq:
 "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"
 proof-
 { assume "affine_dependent ((%x. a + x) ` S)"
 hence "affine_dependent S" using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] by auto
 } from this show ?thesis using affine_dependent_translation by auto
 qed
 lemma affine_hull_0_dependent:
 assumes "0 : affine hull S"
 shows "dependent S"
 proof-
 obtain s u where s_u_def: "finite s & s ~= {} & s <= S & setsum u s = 1 & (SUM v:s. u v *\<^sub>R v) = 0" using assms affine_hull_explicit[of S] by auto
 hence "EX v:s. u v ~= 0" using setsum_not_0[of "u" "s"] by auto
 hence "finite s & s <= S & (EX v:s. u v ~= 0 & (SUM v:s. u v *\<^sub>R v) = 0)" using s_u_def by auto
 from this show ?thesis unfolding dependent_explicit[of S] by auto
 qed
 lemma affine_dependent_imp_dependent2:
 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 : span (S - {x})" by auto
 hence "(x~=0) ==> dependent S" using x_def dependent_def by auto
 moreover
 { assume "x=0" hence "0 : affine hull S" using x_def hull_mono[of "S - {0}" S] by auto
 hence "dependent S" using affine_hull_0_dependent by auto
 } ultimately show ?thesis by auto
 qed
 lemma affine_dependent_iff_dependent:
 assumes "a ~: S"
 shows "affine_dependent (insert a S) <-> dependent ((%x. -a + x) ` S)"
 proof-
 have "(op + (- a) ` S)={x - a| x . x : S}" by auto
 from this 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
 qed
 lemma affine_dependent_iff_dependent2:
 assumes "a : S"
 shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"
 proof-
 have "insert a (S - {a})=S" using assms by auto
 from this show ?thesis using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
 qed
 lemma affine_hull_insert_span_gen:
 shows "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)"
 proof-
 have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)" by auto
 { assume "a ~: s" hence ?thesis using affine_hull_insert_span[of a s] h1 by auto}
 moreover
 { assume a1: "a : s"
 have "EX x. x:s & -a+x=0" apply (rule exI[of _ a]) using a1 by auto
 hence "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s" by auto
 hence "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)"
 using span_insert_0[of "op + (- a) ` (s - {a})"] by auto
 moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))" by auto
 moreover have "insert a (s - {a})=(insert a s)" using assms by auto
 ultimately have ?thesis using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
 }
 ultimately show ?thesis by auto
 qed
 lemma affine_hull_span2:
 assumes "a : s"
 shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` (s-{a}))"
 assumes "~(affine_dependent S)" "S <= V" "S~={}"
 shows "? T. ~(affine_dependent T) & S<=T & T<=V & affine hull T = affine hull V"
 proof-
 obtain a where a_def: "a : S" using assms by auto
 hence h0: "independent  ((%x. -a + x) ` (S-{a}))" using affine_dependent_iff_dependent2 assms by auto
 from this obtain B
 where B_def: "(%x. -a+x) ` (S - {a}) <= B & B <= (%x. -a+x) ` V & independent B & (%x. -a+x) ` V <= span B"
 using maximal_independent_subset_extend[of "(%x. -a+x) ` (S-{a})" "(%x. -a + x) ` V"] assms by blast
 def T == "(%x. a+x) ` (insert 0 B)" hence "T=insert a ((%x. a+x) ` B)" by auto
 hence "affine hull T = (%x. a+x) ` span B" using affine_hull_insert_span_gen[of a "((%x. a+x) ` B)"] translation_assoc[of "-a" a B] by auto
 hence "V <= affine hull T" using B_def assms translation_inverse_subset[of a V "span B"] by auto
 moreover have "T<=V" using T_def B_def a_def assms by auto
 ultimately have "affine hull T = affine hull V"
 by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
 moreover have "S<=T" using T_def B_def translation_inverse_subset[of a "S-{a}" B] by auto
 moreover have "~(affine_dependent T)" using T_def affine_dependent_translation_eq[of "insert 0 B"] affine_dependent_imp_dependent2 B_def by auto
 ultimately show ?thesis using `T<=V` by auto
 qed
 lemma affine_basis_exists:
 fixes V :: "('n::euclidean_space) set"
 shows "? B. B <= V & ~(affine_dependent B) & affine hull V = affine hull B"
 proof-
 { assume empt: "V={}" have "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)" using empt affine_independent_empty by auto
 }
 subsection {* Affine Dimension of a Set *}
 definition "aff_dim V = (SOME d :: int. ? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1))"
 lemma aff_dim_basis_exists:
 fixes V :: "('n::euclidean_space) set"
 shows "? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
 proof-
 obtain B where B_def: "~(affine_dependent B) & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
 from this show ?thesis unfolding aff_dim_def some_eq_ex[of "%d. ? (B :: ('n::euclidean_space) set). (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1)"] apply auto apply (rule exI[of _ "int (card B)-(1 :: int)"]) apply (rule exI[of _ "B"]) by auto
 qed
 lemma affine_hull_nonempty: "(S ~= {}) <-> affine hull S ~= {}"
 proof-
 have "(S = {}) ==> affine hull S = {}"using affine_hull_empty by auto
 moreover have "affine hull S = {} ==> S = {}" unfolding hull_def by auto
 ultimately show "(S ~= {}) <-> affine hull S ~= {}" by blast
 qed
 lemma aff_dim_parallel_subspace_aux:
 fixes B :: "('n::euclidean_space) set"
 assumes "~(affine_dependent B)" "a:B"
 shows "finite B & ((card B) - 1 = dim (span ((%x. -a+x) ` (B-{a}))))"
 proof-
 have "independent ((%x. -a + x) ` (B-{a}))" using affine_dependent_iff_dependent2 assms by auto
 hence fin: "dim (span ((%x. -a+x) ` (B-{a}))) = card ((%x. -a + x) ` (B-{a}))" "finite ((%x. -a + x) ` (B - {a}))"  using indep_card_eq_dim_span[of "(%x. -a+x) ` (B-{a})"] by auto
 { assume emp: "(%x. -a + x) ` (B - {a}) = {}"
 have "B=insert a ((%x. a + x) ` (%x. -a + x) ` (B - {a}))" using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
 hence "B={a}" using emp by auto
 hence ?thesis using assms fin by auto
 }
 moreover
 { assume "(%x. -a + x) ` (B - {a}) ~= {}"
 hence "card ((%x. -a + x) ` (B - {a}))>0" using fin by auto
 moreover have h1: "card ((%x. -a + x) ` (B-{a})) = card (B-{a})"
 apply (rule card_image) using translate_inj_on by auto
 ultimately have "card (B-{a})>0" by auto
 hence "finite(B-{a})" using card_gt_0_iff[of "(B - {a})"] by auto
 moreover hence "(card (B-{a})= (card B) - 1)" using card_Diff_singleton assms by auto
 ultimately have ?thesis using fin h1 by auto
 assumes "V ~= {}"
 assumes "subspace L" "affine_parallel (affine hull V) L"
 shows "aff_dim V=int(dim L)"
 proof-
 obtain B where B_def: "affine hull B = affine hull V & ~ affine_dependent B & int (card B) = aff_dim V + 1" using aff_dim_basis_exists by auto
 hence "B~={}" using assms B_def  affine_hull_nonempty[of V] affine_hull_nonempty[of B] by auto
 from this obtain a where a_def: "a : B" by auto
 def Lb == "span ((%x. -a+x) ` (B-{a}))"
 moreover have "affine_parallel (affine hull B) Lb"
 using Lb_def B_def assms affine_hull_span2[of a B] a_def  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 ~= {}" using assms B_def affine_hull_nonempty[of V] by auto
 ultimately have "L=Lb" using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B_def by auto
 hence "dim L=dim Lb" by auto
 moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def B_def by auto
 (*  hence "card B=dim Lb+1" using `B~={}` card_gt_0_iff[of B] by auto *)
 ultimately show ?thesis using B_def `B~={}` card_gt_0_iff[of B] by auto
 qed
 lemma aff_independent_finite:
 fixes B :: "('n::euclidean_space) set"
 assumes "~(affine_dependent B)"
 shows "finite B"
 proof-
 { assume "B~={}" from this obtain a where "a:B" by auto
 hence ?thesis using aff_dim_parallel_subspace_aux assms by auto
 } from this show ?thesis by auto
 qed
 lemma independent_finite:
 fixes B :: "('n::euclidean_space) set"
 assumes "independent B"
 shows "finite B"
 using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms by auto
 lemma subspace_dim_equal:
 assumes "subspace (S :: ('n::euclidean_space) set)" "subspace T" "S <= T" "dim S >= dim T"
 shows "S=T"
 proof-
 obtain B where B_def: "B <= S & independent B & S <= span B & (card B = dim S)" using basis_exists[of S] by auto
 hence "span B <= S" using span_mono[of B S] span_eq[of S] assms by metis
 hence "span B = S" using B_def by auto
 have "dim S = dim T" using assms dim_subset[of S T] by auto
 hence "T <= span B" using card_eq_dim[of B T] B_def independent_finite assms by auto
 from this show ?thesis using assms `span B=S` by auto
 qed
 (is "span ?A = ?B")
 proof-
 have "?A <= ?B" by auto
 moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .
 ultimately have "span ?A <= ?B" using span_mono[of "?A" "?B"] span_eq[of "?B"] by blast
 moreover have "card d <= dim (span ?A)" using independent_card_le_dim[of "?A" "span ?A"]
 independent_substdbasis[OF assms] card_substdbasis[OF assms] span_inc[of "?A"] by auto
 moreover hence "dim ?B <= dim (span ?A)" using dim_substandard[OF assms] by auto
 ultimately show ?thesis using s subspace_dim_equal[of "span ?A" "?B"]
 subspace_span[of "?A"] by auto
 qed
 lemma basis_to_substdbasis_subspace_isomorphism:
 fixes B :: "('a::euclidean_space) set"
 assumes "independent B"
 shows "EX f d. card d = card B & linear f & f ` B = {basis i::'a |i. i : (d :: nat set)} &
 f ` span B = {x. ALL i<DIM('a). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B) \<and> d\<subseteq>{..<DIM('a)}"
 proof-
 have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto
 def d \<equiv> "{..<dim B}" have t:"card d = dim B" unfolding d_def by auto
 have "dim B <= DIM('a)" using dim_subset_UNIV[of B] by auto
 hence d:"d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
 apply(rule subspace_span) apply(rule subspace_substandard) defer
 apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)
 unfolding span_substd_basis[OF d,symmetric] card_substdbasis[OF d] apply(rule span_inc)
 apply(rule independent_substdbasis[OF d]) apply(rule,assumption)
 unfolding t[symmetric] span_substd_basis[OF d] dim_substandard[OF d] by auto
 from this t `card B=dim B` show ?thesis using d by auto
 qed
 lemma aff_dim_empty:
 fixes S :: "('n::euclidean_space) set"
 shows "S = {} <-> aff_dim S = -1"
 proof-
 obtain B where "affine hull B = affine hull S & ~ affine_dependent B & int (card B) = aff_dim S + 1" using aff_dim_basis_exists by auto
 moreover hence "S={} <-> B={}" using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
 ultimately show ?thesis using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
 qed
 lemma aff_dim_affine_hull:
 shows "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) & ~(affine_dependent B)"
 shows "of_nat(card B) = aff_dim V+1"
 proof-
 { assume "B={}" hence "V={}" using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms by auto
 hence "aff_dim V = (-1::int)"  using aff_dim_empty by auto
 hence ?thesis using `B={}` by auto
 }
 moreover
 { assume "B~={}" from this obtain a where a_def: "a:B" by auto
 def Lb == "span ((%x. -a+x) ` (B-{a}))"
 have "affine_parallel (affine hull B) Lb"
 using Lb_def affine_hull_span2[of a B] a_def  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] `B~={}` by auto
 moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def assms by auto
 ultimately have "(of_nat(card B) = aff_dim B+1)" using  `B~={}` card_gt_0_iff[of B] by auto
 hence ?thesis using aff_dim_affine_hull2 assms by auto
 } ultimately show ?thesis by blast
 qed
 lemma aff_dim_affine_independent:
 fixes B :: "('n::euclidean_space) set"
 assumes "~(affine_dependent B)"
 shows "of_nat(card B) = aff_dim B+1"
 using aff_dim_unique[of B B] assms by auto
 lemma aff_dim_sing:
 fixes a :: "'n::euclidean_space"
 shows "aff_dim {a}=0"
 using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
 lemma aff_dim_inner_basis_exists:
 fixes V :: "('n::euclidean_space) set"
 shows "? B. B<=V & (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
 proof-
 obtain B where B_def: "~(affine_dependent B) & B<=V & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
 moreover hence "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
 ultimately show ?thesis by auto
 qed
 lemma aff_dim_le_card:
 fixes V :: "('n::euclidean_space) set"
 assumes "finite V"
 shows "aff_dim V <= of_nat(card V) - 1"
 proof-
 obtain B where B_def: "B<=V & (of_nat(card B) = aff_dim V+1)" using aff_dim_inner_basis_exists[of V] by auto
 moreover hence "card B <= card V" using assms card_mono by auto
 ultimately 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~={}" "S~={}"
 from this obtain L where L_def: "subspace L & affine_parallel (affine hull T) L"
 using affine_parallel_subspace[of "affine hull T"] affine_affine_hull[of T] affine_hull_nonempty by auto
 hence "aff_dim T = int(dim L)" using aff_dim_parallel_subspace `T~={}` by auto
 moreover have "subspace L & affine_parallel (affine hull S) L"
 using L_def affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
 moreover hence "aff_dim S = int(dim L)" using aff_dim_parallel_subspace `S~={}` by auto
 ultimately have ?thesis by auto
 }
 moreover
 { assume "S={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
 hence ?thesis using aff_dim_empty by auto
 ultimately show ?thesis by blast
 qed
 lemma aff_dim_translation_eq:
 fixes a :: "'n::euclidean_space"
 shows "aff_dim ((%x. a + x) ` S)=aff_dim S"
 proof-
 have "affine_parallel (affine hull S) (affine hull ((%x. a + x) ` S))" unfolding affine_parallel_def apply (rule exI[of _ "a"]) using affine_hull_translation[of a S] by auto
 from this show ?thesis using  aff_dim_parallel_eq[of S "(%x. a + x) ` S"] by auto
 qed
 lemma aff_dim_affine:
 fixes S L :: "('n::euclidean_space) set"
 assumes "S ~= {}" "affine S"
 assumes "subspace L" "affine_parallel S L"
 shows "aff_dim S=int(dim L)"
 proof-
 have 1: "(affine hull S) = S" using assms affine_hull_eq[of S] by auto
 hence "affine_parallel (affine hull S) L" using assms by (simp add:1)
 from this 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"
 qed
 lemma aff_dim_subspace:
 fixes S :: "('n::euclidean_space) set"
 assumes "S ~= {}" "subspace S"
 shows "aff_dim S=int(dim S)" using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] by auto
 lemma aff_dim_zero:
 fixes S :: "('n::euclidean_space) set"
 assumes "0 : 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
 hence "aff_dim (affine hull S) =int(dim (affine hull S))" using assms aff_dim_subspace[of "affine hull S"] by auto
 from this show ?thesis using aff_dim_affine_hull[of S] dim_affine_hull[of S] by auto
 qed
 lemma aff_dim_univ: "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))"
 using aff_dim_subspace[of "(UNIV :: ('n::euclidean_space) set)"]
 proof-
 obtain B where B_def: "affine hull B = affine hull V & ~ affine_dependent B & int (card B) = aff_dim V + 1" using aff_dim_basis_exists by auto
 from this show ?thesis by auto
 qed
 lemma independent_card_le_aff_dim:
 assumes "(B::('n::euclidean_space) set) <= V"
 assumes "~(affine_dependent B)"
 shows "int(card B) <= aff_dim V+1"
 proof-
 { assume "B~={}"
 from this obtain T where T_def: "~(affine_dependent T) & B<=T & T<=V & affine hull T = affine hull V"
 using assms extend_to_affine_basis[of B V] by auto
 hence "of_nat(card T) = aff_dim V+1" using aff_dim_unique by auto
 hence ?thesis using T_def card_mono[of T B] aff_independent_finite[of T] by auto
 }
 moreover
 qed
 lemma aff_dim_subset_univ:
 fixes S :: "('n::euclidean_space) set"
 shows "aff_dim S <= int(DIM('n))"
 proof -
 have "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))" using aff_dim_univ by auto
 from this show "aff_dim (S:: ('n::euclidean_space) set) <= int(DIM('n))" using assms aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
 qed
 lemma affine_dim_equal:
 assumes "affine (S :: ('n::euclidean_space) set)" "affine T" "S ~= {}" "S <= T" "aff_dim S = aff_dim T"
 shows "S=T"
 proof-
 obtain a where "a : S" using assms by auto
 hence "a : T" using assms by auto
 def LS == "{y. ? x : S. (-a)+x=y}"
 hence ls: "subspace LS & affine_parallel S LS" using assms parallel_subspace_explicit[of S a LS] `a : S` by auto
 hence h1: "int(dim LS) = aff_dim S" using assms aff_dim_affine[of S LS] by auto
 have "T ~= {}" using assms by auto
 def LT == "{y. ? x : T. (-a)+x=y}"
 hence lt: "subspace LT & affine_parallel T LT" using assms parallel_subspace_explicit[of T a LT] `a : T` by auto
 hence "int(dim LT) = aff_dim T" using assms aff_dim_affine[of T LT] `T ~= {}` by auto
 hence "dim LS = dim LT" using h1 assms by auto
 moreover have "LS <= 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. ? y : LS. a+y=x}" using LS_def by auto
 moreover have "T = {x. ? y : LT. a+y=x}" using LT_def by auto
 ultimately show ?thesis by 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 ~= {}" using assms aff_dim_empty[of S] by auto
 have h0: "S <= 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
 hence h2: "aff_dim (affine hull S) <= aff_dim (UNIV :: ('n::euclidean_space) set)" using aff_dim_subset_univ[of "affine hull S"] assms h0 by auto
 have h3: "aff_dim S <= aff_dim (affine hull S)" using h0 aff_dim_subset[of S "affine hull S"] assms by auto
 hence h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" using h0 h1 h2 by auto
 from this 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 `S ~= {}` by auto
 qed
 lemma aff_dim_convex_hull:
 fixes S :: "('n::euclidean_space) set"
 shows "aff_dim (convex hull S)=aff_dim S"
 using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
 hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
 aff_dim_subset[of "convex hull S" "affine hull S"] by auto
 lemma aff_dim_cball:
 fixes a :: "'n::euclidean_space"
 assumes "0<e"
 shows "aff_dim (cball a e) = int (DIM('n))"
 proof-
 have "(%x. a + x) ` (cball 0 e)<=cball a e" unfolding cball_def dist_norm by auto
 hence "aff_dim (cball (0 :: 'n::euclidean_space) e) <= aff_dim (cball a e)"
 using aff_dim_translation_eq[of a "cball 0 e"]
 aff_dim_subset[of "op + a ` cball 0 e" "cball a e"] by auto
 moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
 using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"] centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
 by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
 ultimately show ?thesis using aff_dim_subset_univ[of "cball a e"] by auto
 qed
 lemma aff_dim_open:
 fixes S :: "('n::euclidean_space) set"
 assumes "open S" "S ~= {}"
 shows "aff_dim S = int (DIM('n))"
 proof-
 obtain x where "x:S" using assms by auto
 from this obtain e where e_def: "e>0 & cball x e <= S" using open_contains_cball[of S] assms by auto
 from this have "aff_dim (cball x e) <= aff_dim S" using aff_dim_subset by auto
 from this show ?thesis using aff_dim_cball[of e x] aff_dim_subset_univ[of S] e_def by auto
 qed
 lemma low_dim_interior:
 fixes S :: "('n::euclidean_space) set"
 assumes "~(aff_dim S = int (DIM('n)))"
 shows "interior S = {}"
 proof-
 have "aff_dim(interior S) <= aff_dim S"
 using interior_subset aff_dim_subset[of "interior S" S] by auto
 from this show ?thesis using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto
 qed
 subsection {* Relative interior of a set *}
 definition "rel_interior S = {x. ? T. openin (subtopology euclidean (affine hull S)) T & x : T & T <= S}"
 show "EX Tb. (EX Ta. open Ta & Tb = affine hull S Int Ta) & x : Tb & Tb <= S"
 apply (rule_tac x="T Int (affine hull S)" in exI)
 using a h1 by auto
 qed
 lemma mem_rel_interior:
 "x : rel_interior S <-> (? T. open T & x : (T Int S) & (T Int (affine hull S)) <= S)"
 by (auto simp add: rel_interior)
 lemma mem_rel_interior_ball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((ball x e) Int (affine hull S)) <= S)"
 apply (simp add: rel_interior, safe)
 apply (force simp add: open_contains_ball)
 apply (rule_tac x="ball x e" in exI)
 apply simp
 done
 lemma rel_interior_ball:
 "rel_interior S = {x : S. ? e. e>0 & ((ball x e) Int (affine hull S)) <= S}"
 using mem_rel_interior_ball [of _ S] by auto
 lemma mem_rel_interior_cball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((cball x e) Int (affine hull S)) <= S)"
 apply (simp add: rel_interior, safe)
 apply (force simp add: open_contains_cball)
 apply (rule_tac x="ball x e" in exI)
 apply (simp add: subset_trans [OF ball_subset_cball])
 apply auto
 done
 lemma rel_interior_cball: "rel_interior S = {x : S. ? e. e>0 & ((cball x e) Int (affine hull S)) <= S}"       using mem_rel_interior_cball [of _ S] by auto
 lemma rel_interior_empty: "rel_interior {} = {}"
 by (auto simp add: rel_interior_def)
 lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"
 by (metis affine_hull_eq affine_sing)
 lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"
 lemma subset_rel_interior:
 fixes S T :: "('n::euclidean_space) set"
 assumes "S<=T" "affine hull S=affine hull T"
 shows "rel_interior S <= rel_interior T"
 using assms by (auto simp add: rel_interior_def)
 lemma rel_interior_subset: "rel_interior S <= S"
 by (auto simp add: rel_interior_def)
 lemma rel_interior_subset_closure: "rel_interior S <= closure S"
 using rel_interior_subset by (auto simp add: closure_def)
 lemma interior_subset_rel_interior: "interior S <= rel_interior S"
 by (auto simp add: rel_interior interior_def)
 lemma interior_rel_interior:
 fixes S :: "('n::euclidean_space) set"
 assumes "aff_dim S = int(DIM('n))"
 shows "rel_interior S = interior S"
 proof -
 have "affine hull S = UNIV" using assms affine_hull_univ[of S] by auto
 from this show ?thesis unfolding rel_interior interior_def by auto
 qed
 lemma rel_interior_open:
 fixes S :: "('n::euclidean_space) set"
 lemma interior_rel_interior_gen:
 fixes S :: "('n::euclidean_space) set"
 shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
 by (metis interior_rel_interior low_dim_interior)
 lemma rel_interior_univ:
 fixes S :: "('n::euclidean_space) set"
 shows "rel_interior (affine hull S) = affine hull S"
 proof-
 have h1: "rel_interior (affine hull S) <= affine hull S" using rel_interior_subset by auto
 { fix x assume x_def: "x : affine hull S"
 obtain e :: real where "e=1" by auto
 hence "e>0 & ball x e Int affine hull (affine hull S) <= affine hull S" using hull_hull[of _ S] by auto
 hence "x : rel_interior (affine hull S)" using x_def rel_interior_ball[of "affine hull S"] by auto
 } from this show ?thesis using h1 by auto
 qed
 lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
 by (metis open_UNIV rel_interior_open)
 lemma rel_interior_convex_shrink:
 fixes S :: "('a::euclidean_space) set"
 assumes "convex S" "c : rel_interior S" "x : S" "0 < e" "e <= 1"
 shows "x - e *\<^sub>R (x - c) : rel_interior S"
 proof-
 (* Proof is a modified copy of the proof of similar lemma mem_interior_convex_shrink
 *)
 obtain d where "d>0" and d:"ball c d Int affine hull S <= S"
 using assms(2) unfolding  mem_rel_interior_ball by auto
 {   fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d & y : affine hull S"
 have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
 have "x : affine hull S" using assms hull_subset[of S] by auto
 moreover have "1 / e + - ((1 - e) / e) = 1"
 using `e>0` left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
 ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x : affine hull S"
 using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"] by (simp add: algebra_simps)
 have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
 unfolding dist_norm unfolding norm_scaleR[symmetric] apply(rule arg_cong[where f=norm]) using `e>0`
 by(auto simp add:euclidean_eq[where 'a='a] field_simps)
 also have "... = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
 also have "... < d" using as[unfolded dist_norm] and `e>0`
 by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
 finally have "y : S" apply(subst *)
 apply(rule assms(1)[unfolded convex_alt,rule_format])
 apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) ** by auto
 } hence "ball (x - e *\<^sub>R (x - c)) (e*d) Int affine hull S <= S" by auto
 moreover have "0 < e*d" using `0<e` `0<d` by (rule mult_pos_pos)
 moreover have "c : S" using assms rel_interior_subset by auto
 moreover hence "x - e *\<^sub>R (x - c) : S"
 using mem_convex[of S x c e] apply (simp add: algebra_simps) using assms by auto
 ultimately show ?thesis
 using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e>0` by auto
 qed
 lemma interior_real_semiline:
 fixes a :: real
 shows "interior {a..} = {a<..}"
 proof-
 { fix y assume "a<y" hence "y : interior {a..}"
 apply (simp add: mem_interior) apply (rule_tac x="(y-a)" in exI) apply (auto simp add: dist_norm)
 done }
 moreover
 { fix y assume "y : interior {a..}" (*hence "a<=y" using interior_subset by auto*)
 from this obtain e where e_def: "e>0 & cball y e \<subseteq> {a..}"
 using mem_interior_cball[of y "{a..}"] by auto
 moreover hence "y-e : cball y e" by (auto simp add: cball_def dist_norm)
 ultimately have "a<=y-e" by auto
 hence "a<y" using e_def by auto
 } ultimately show ?thesis by auto
 qed
 lemma rel_open: "rel_open S <-> openin (subtopology euclidean (affine hull S)) S"
 unfolding rel_open_def rel_interior_def apply auto
 using openin_subopen[of "subtopology euclidean (affine hull S)" S] by auto
 lemma opein_rel_interior:
 "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
 apply (simp add: rel_interior_def)
 apply (subst openin_subopen) by blast
 lemma affine_rel_open:
 fixes S :: "('n::euclidean_space) set"
 assumes "affine S" shows "rel_open S"
 unfolding rel_open_def using assms rel_interior_univ[of S] affine_hull_eq[of S] by metis
 lemma affine_closed:
 fixes S :: "('n::euclidean_space) set"
 assumes "affine S" shows "closed S"
 proof-
 { assume "S ~= {}"
 from this obtain L where L_def: "subspace L & affine_parallel S L"
 using assms affine_parallel_subspace[of S] by auto
 from this obtain "a" where a_def: "S=(op + a ` L)"
 using affine_parallel_def[of L S] affine_parallel_commut by auto
 have "closed L" using L_def closed_subspace by auto
 hence "closed S" using closed_translation a_def by auto
 } from this show ?thesis by auto
 qed
 fixes S :: "('n::euclidean_space) set"
 shows "affine hull (closure S) = affine hull S"
 proof-
 have "affine hull (closure S) <= affine hull S"
 using hull_mono[of "closure S" "affine hull S" "affine"] closure_affine_hull[of S] hull_hull[of "affine" S] by auto
 moreover have "affine hull (closure S) >= affine hull S"
 using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
 ultimately show ?thesis by auto
 qed
 lemma closure_aff_dim:
 fixes S :: "('n::euclidean_space) set"
 shows "aff_dim (closure S) = aff_dim S"
 proof-
 have "aff_dim S <= aff_dim (closure S)" using aff_dim_subset closure_subset by auto
 moreover have "aff_dim (closure S) <= aff_dim (affine hull S)"
 using aff_dim_subset closure_affine_hull by auto
 moreover have "aff_dim (affine hull S) = aff_dim S" using aff_dim_affine_hull by auto
 ultimately show ?thesis by auto
 qed
 lemma rel_interior_closure_convex_shrink:
 fixes S :: "(_::euclidean_space) set"
 assumes "convex S" "c : rel_interior S" "x : closure S" "0 < e" "e <= 1"
 shows "x - e *\<^sub>R (x - c) : rel_interior S"
 proof-
 (* Proof is a modified copy of the proof of similar lemma mem_interior_closure_convex_shrink
 *)
 obtain d where "d>0" and d:"ball c d Int affine hull S <= S"
 using assms(2) unfolding mem_rel_interior_ball by auto
 have "EX y : S. norm (y - x) * (1 - e) < e * d" proof(cases "x : S")
 case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
 case False hence x:"x islimpt S" using assms(3)[unfolded closure_def] by auto
 show ?thesis proof(cases "e=1")
 have zball: "z\<in>ball c d"
 using mem_ball z_def dist_norm[of c] using y and assms(4,5) by (auto simp add:field_simps norm_minus_commute)
 have "x : affine hull S" using closure_affine_hull assms by auto
 moreover have "y : affine hull S" using `y : S` hull_subset[of S] by auto
 moreover have "c : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
 ultimately have "z : affine hull S"
 using z_def affine_affine_hull[of S]
 mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
 assms by (auto simp add: field_simps)
 hence "z : S" using d zball by auto
 obtain d1 where "d1>0" and d1:"ball z d1 <= ball c d"
 using zball open_ball[of c d] openE[of "ball c d" z] by auto
 hence "(ball z d1) Int (affine hull S) <= (ball c d) Int (affine hull S)" by auto
 hence "(ball z d1) Int (affine hull S) <= S" using d by auto
 hence "z : rel_interior S" using mem_rel_interior_ball using `d1>0` `z : S` by auto
 hence "y - e *\<^sub>R (y - z) : rel_interior S" using rel_interior_convex_shrink[of S z y e] assms`y : S` by auto
 thus ?thesis using * by auto
 qed
 subsubsection{* Relative interior preserves under linear transformations *}
 lemma rel_interior_translation_aux:
 fixes a :: "'n::euclidean_space"
 shows "((%x. a + x) ` rel_interior S) <= rel_interior ((%x. a + x) ` S)"
 proof-
 { fix x assume x_def: "x : rel_interior S"
 from this obtain T where T_def: "open T & x : (T Int S) & (T Int (affine hull S)) <= S" using mem_rel_interior[of x S] by auto
 from this have "open ((%x. a + x) ` T)" and
 "(a + x) : (((%x. a + x) ` T) Int ((%x. a + x) ` S))" and
 "(((%x. a + x) ` T) Int (affine hull ((%x. a + x) ` S))) <= ((%x. a + x) ` S)"
 using affine_hull_translation[of a S] open_translation[of T a] x_def by auto
 from this have "(a+x) : rel_interior ((%x. a + x) ` S)"
 using mem_rel_interior[of "a+x" "((%x. a + x) ` S)"] by auto
 } from this show ?thesis by auto
 qed
 lemma rel_interior_translation:
 fixes a :: "'n::euclidean_space"
 shows "rel_interior ((%x. a + x) ` S) = ((%x. a + x) ` rel_interior S)"
 proof-
 have "(%x. (-a) + x) ` rel_interior ((%x. a + x) ` S) <= rel_interior S"
 using rel_interior_translation_aux[of "-a" "(%x. a + x) ` S"]
 translation_assoc[of "-a" "a"] by auto
 hence "((%x. a + x) ` rel_interior S) >= rel_interior ((%x. a + x) ` S)"
 using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"]
 by auto
 from this show ?thesis using  rel_interior_translation_aux[of a S] by auto
 qed
 lemma affine_hull_linear_image:
 assumes "bounded_linear f"
 shows "f ` (affine hull s) = affine hull f ` s"
 (* Proof is a modified copy of the proof of similar lemma convex_hull_linear_image
 *)
 apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3
 apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
 apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
 proof-
 interpret f: bounded_linear f by fact
 show "affine {x. f x : affine hull f ` s}"
 unfolding affine_def by(auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format]) next
 interpret f: bounded_linear f by fact
 show "affine {x. x : f ` (affine hull s)}" using affine_affine_hull[unfolded affine_def, of s]
 unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
 qed auto
 lemma rel_interior_injective_on_span_linear_image:
 shows "rel_interior (f ` S) = f ` (rel_interior S)"
 proof-
 { fix z assume z_def: "z : rel_interior (f ` S)"
 have "z : f ` S" using z_def rel_interior_subset[of "f ` S"] by auto
 from this obtain x where x_def: "x : S & (f x = z)" by auto
 obtain e2 where e2_def: "e2>0 & cball z e2 Int affine hull (f ` S) <= (f ` S)"
 using z_def rel_interior_cball[of "f ` S"] by auto
 obtain K where K_def: "K>0 & (! x. norm (f x) <= norm x * K)"
 using assms RealVector.bounded_linear.pos_bounded[of f] by auto
 def e1 == "1/K" hence e1_def: "e1>0 & (! x. e1 * norm (f x) <= norm x)"
 using K_def pos_le_divide_eq[of e1] by auto
 def e == "e1 * e2" hence "e>0" using e1_def e2_def mult_pos_pos by auto
 { fix y assume y_def: "y : cball x e Int affine hull S"
 from this have h1: "f y : affine hull (f ` S)"
 using affine_hull_linear_image[of f S] assms by auto
 from y_def have "norm (x-y)<=e1 * e2"
 using cball_def[of x e] dist_norm[of x y] e_def by auto
 moreover have "(f x)-(f y)=f (x-y)"
 using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
 moreover have "e1 * norm (f (x-y)) <= norm (x-y)" using e1_def by auto
 ultimately have "e1 * norm ((f x)-(f y)) <= e1 * e2" by auto
 hence "(f y) : (cball z e2)"
 using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1_def x_def by auto
 hence "f y : (f ` S)" using y_def e2_def h1 by auto
 hence "y : S" using assms y_def hull_subset[of S] affine_hull_subset_span
 inj_on_image_mem_iff[of f "span S" S y] by auto
 }
 hence "z : f ` (rel_interior S)" using mem_rel_interior_cball[of x S] `e>0` x_def by auto
 }
 moreover
 { fix x assume x_def: "x : rel_interior S"
 from this obtain e2 where e2_def: "e2>0 & cball x e2 Int affine hull S <= S"
 using rel_interior_cball[of S] by auto
 have "x : S" using x_def rel_interior_subset by auto
 hence *: "f x : f ` S" by auto
 have "! x:span S. f x = 0 --> x = 0"
 using assms subspace_span linear_conv_bounded_linear[of f]
 linear_injective_on_subspace_0[of f "span S"] by auto
 from this obtain e1 where e1_def: "e1>0 & (! x : span S. e1 * norm x <= norm (f x))"
 using assms injective_imp_isometric[of "span S" f]
 subspace_span[of S] closed_subspace[of "span S"] by auto
 def e == "e1 * e2" hence "e>0" using e1_def e2_def mult_pos_pos by auto
 { fix y assume y_def: "y : cball (f x) e Int affine hull (f ` S)"
 from this have "y : f ` (affine hull S)" using affine_hull_linear_image[of f S] assms by auto
 from this obtain xy where xy_def: "xy : affine hull S & (f xy = y)" by auto
 from this y_def have "norm ((f x)-(f xy))<=e1 * e2"
 using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
 moreover have "(f x)-(f xy)=f (x-xy)"
 using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
 moreover have "x-xy : span S"
 using subspace_sub[of "span S" x xy] subspace_span `x : S` xy_def
 affine_hull_subset_span[of S] span_inc by auto
 moreover hence "e1 * norm (x-xy) <= norm (f (x-xy))" using e1_def by auto
 ultimately have "e1 * norm (x-xy) <= e1 * e2" by auto
 hence "xy : (cball x e2)"  using cball_def[of x e2] dist_norm[of x xy] e1_def by auto
 hence "y : (f ` S)" using xy_def e2_def by auto
 }
 hence "(f x) : rel_interior (f ` S)"
 using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e>0` by auto
 }
 ultimately show ?thesis by auto
 qed
 lemma rel_interior_injective_linear_image:
 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
 assumes "bounded_linear f" and "inj f"
 shows "rel_interior (f ` S) = f ` (rel_interior S)"
 using assms rel_interior_injective_on_span_linear_image[of f S]
 subset_inj_on[of f "UNIV" "span S"] by auto
 subsection{* Some Properties of subset of standard basis *}
 lemma affine_hull_substd_basis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
 case True thus ?thesis using compact_empty by simp
 next
 case False then obtain w where "w\<in>s" by auto
 show ?thesis unfolding caratheodory[of s]
 proof(induct ("DIM('a) + 1"))
 have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
 using compact_empty by auto
 case 0 thus ?case unfolding * by simp
 next
 case (Suc n)
 show ?case proof(cases "n=0")
 thus ?thesis using t(2,4) by simp
 qed
 next
 fix x assume "x\<in>s"
 thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
 apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto
 qed thus ?thesis using assms by simp
 next
 case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
 { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
 unfolding set_eq_iff and mem_Collect_eq proof(rule,rule)
 fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
 0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
 then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
 using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
 by(auto intro!: exI[where x=u])
 qed
 qed
 qed
 thus ?thesis using compact_convex_combinations[OF assms Suc] by simp
 qed
 qed
 qed
 subsection {* Extremal points of a simplex are some vertices. *}
 lemma dist_increases_online:
 fixes a b d :: "'a::real_inner"
 assumes "d \<noteq> 0"
 shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
 proof(cases "inner a d - inner b d > 0")
 case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)"
 apply(rule_tac add_pos_pos) using assms by auto
 thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
 by (simp add: algebra_simps inner_commute)
 next
 case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)"
 apply(rule_tac add_pos_nonneg) using assms by auto
 thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
 by (simp add: algebra_simps inner_commute)
 qed
 assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
 thus ?thesis using y(2) by auto
 next
 assume "u\<noteq>0" "v\<noteq>0"
 then obtain w where w:"w>0" "w<u" "w<v" using real_lbound_gt_zero[of u v] and obt(1,2) by auto
 have "x\<noteq>b" proof(rule ccontr)
 assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
 using obt(3) by(auto simp add: scaleR_left_distrib[symmetric])
 thus False using obt(4) and False by simp qed
 hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
 show ?thesis using dist_increases_online[OF *, of a y]
 assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
 hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
 unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
 moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
 unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
 apply(rule_tac x="u + w" in exI) apply rule defer
 apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
 ultimately show ?thesis by auto
 next
 assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
 hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
 unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
 moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
 unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
 apply(rule_tac x="u - w" in exI) apply rule defer
 apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
 ultimately show ?thesis by auto
 qed
 qed auto
 qed
 assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
 using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
 thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
 by (auto simp add: norm_minus_commute)
 qed auto
 qed
 lemma simplex_extremal_le_exists:
 fixes s :: "('a::real_inner) set"
 shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
 \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
 "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
 lemma closest_point_exists:
 assumes "closed s" "s \<noteq> {}"
 shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
 unfolding closest_point_def apply(rule_tac[!] someI2_ex)
 using distance_attains_inf[OF assms(1,2), of a] by auto
 lemma closest_point_in_set:
 "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
 by(meson closest_point_exists)
 "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
 using closest_point_exists[of s] by auto
 lemma closest_point_self:
 assumes "x \<in> s"  shows "closest_point s x = x"
 unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x])
 using assms by auto
 lemma closest_point_refl:
 "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
 using closest_point_in_set[of s x] closest_point_self[of x s] by auto
 by (auto simp add: algebra_simps)
 lemma closest_point_unique:
 assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
 shows "x = closest_point s a"
 using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"]
 using closest_point_exists[OF assms(2)] and assms(3) by auto
 lemma closest_point_dot:
 assumes "convex s" "closed s" "x \<in> s"
 shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
 by (simp add: inner_add inner_diff inner_commute) qed
 lemma continuous_at_closest_point:
 assumes "convex s" "closed s" "s \<noteq> {}"
 shows "continuous (at x) (closest_point s)"
 unfolding continuous_at_eps_delta
 using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
 lemma continuous_on_closest_point:
 assumes "convex s" "closed s" "s \<noteq> {}"
 shows "continuous_on t (closest_point s)"
 from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
 apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
 hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
 fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
 next
 fix x assume "x\<in>s"
 hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
 using ab[THEN bspec[where x=x]] by auto
 thus "k + b / 2 < inner a x" using `0 < b` by auto
 qed
 qed
 lemma separating_hyperplane_sets:
 assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
 shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
 proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
 obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x"
 using assms(3-5) by auto
 hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
 by (force simp add: inner_diff)
 thus ?thesis
 apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
 apply auto
 apply (rule Sup[THEN isLubD2])
 prefer 4
 apply (rule Sup_least)
 using assms(3-5) apply (auto simp add: setle_def)
 apply metis
 done
 qed
 lemma convex_interior:
 fixes s :: "'a::real_normed_vector set"
 assumes "convex s" shows "convex(interior s)"
 unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
 fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
 fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e"
 show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
 apply rule unfolding subset_eq defer apply rule proof-
 fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
 hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
 apply(rule_tac assms[unfolded convex_alt, rule_format])
 lemma convex_hull_bilemma: fixes neg
 assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
 shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
 \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
 using assms by(metis subset_antisym)
 lemma convex_hull_translation:
 "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
 apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
 lemma convex_halfspace_intersection:
 fixes s :: "('a::euclidean_space) set"
 assumes "closed s" "convex s"
 shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
 apply(rule set_eqI, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof-
 fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
 hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
 thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
 apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
 qed auto
 lemma radon_v_lemma:
 assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
 shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
 proof-
 have *:"\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" using assms(3) by auto
 show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[symmetric] and *
 using assms(2) by assumption qed
 lemma radon_partition:
 assumes "finite c" "affine_dependent c"
 obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
 have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
 def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
 have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
 case False hence "u v < 0" by auto
 thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
 case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
 next
 case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
 thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
 qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
 hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
 moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
 "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
 using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
 hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
 "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
 unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, symmetric])
 moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x"
 apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
 ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
 apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
 using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
 by(auto simp add: setsum_negf setsum_right_distrib[symmetric])
 moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x"
 apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
 hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
 apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
 using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def using *
 by(auto simp add: setsum_negf setsum_right_distrib[symmetric])
 ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
 apply(rule, rule X[rule_format]) using X st by auto
 next case True then obtain m p where mp:"m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
 using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
 unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
 have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
 then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto
 hence "f \<union> (g \<union> h) = f" by auto
 hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
 unfolding mp(2)[unfolded image_Un[symmetric] gh] by auto
 have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
 have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
 subsection {* Homeomorphism of all convex compact sets with nonempty interior *}
 lemma compact_frontier_line_lemma:
 fixes s :: "('a::euclidean_space) set"
 assumes "compact s" "0 \<in> s" "x \<noteq> 0"
 obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
 proof-
 obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
 let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
 have A:"?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
 ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
 "y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
 have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[symmetric]] by auto
 { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
 hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)]
 using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
 hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer
 apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI)
 using as(1) `u\<ge>0` by(auto simp add:field_simps)
 hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
 } note u_max = this
 have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[symmetric]
 prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
 assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
 using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
 have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
 apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
 apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_eqI,rule)
 unfolding inj_on_def prefer 3 apply(rule,rule,rule)
 proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
 thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
 next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
 then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
 using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
 thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
 next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
 hence xys:"x\<in>s" "y\<in>s" using fs by auto
 from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto
 from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, symmetric] by auto
 from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto
 have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
 unfolding divide_inverse[symmetric] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
 hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
 using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
 using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
 using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[symmetric])
 thus "x = y" apply(subst injpi[symmetric]) using as(3) by auto
 qed(insert `0 \<notin> frontier s`, auto)
 then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
 "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
 have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
 apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
 { fix x assume as:"x \<in> cball (0::'a) 1"
 have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1")
 case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
 thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
 apply(rule_tac fs[unfolded subset_eq, rule_format])
 unfolding surf(5)[symmetric] by auto
 next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
 hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
 apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[symmetric]] by auto
 have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
 hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
 unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
 moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))"
 unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
 moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
 hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
 using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
 using False `x\<in>s` by(auto simp add:field_simps)
 fix x::"'a" assume as:"x \<in> cball 0 1"
 thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
 case False thus ?thesis apply (intro continuous_intros)
 using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
 next obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
 hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer
 apply(erule_tac x="basis 0" in ballE)
 unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
 by auto
 case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
 apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
 hence "surf (pi x) \<in> frontier s" using surf(5) by auto
 thus False using `0\<notin>frontier s` unfolding as by simp qed
 } note surf_0 = this
 show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
 fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
 thus "x=y" proof(cases "x=0 \<or> y=0")
 case True thus ?thesis using as by(auto elim: surf_0) next
 case False
 hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
 using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
 moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
 ultimately have *:"pi x = pi y" using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]] by auto
 moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
 ultimately show ?thesis using injpi by auto qed qed
 qed auto qed
 lemma homeomorphic_convex_compact_lemma:
 definition "epigraph s (f::_ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
 lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
 (** This might break sooner or later. In fact it did already once. **)
 lemma convex_epigraph:
 "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
 unfolding convex_def convex_on_def
 unfolding Ball_def split_paired_All epigraph_def
 unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
 apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
 { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
 using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
 moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
 hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
 ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
 apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI)
 apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
 by(auto simp add: field_simps) qed
 lemma is_interval_convex_1:
 "is_interval s \<longleftrightarrow> convex (s::(real^1) set)"
 by(metis is_interval_convex convex_connected is_interval_connected_1)
 lemma convex_connected_1:
 "connected s \<longleftrightarrow> convex (s::(real^1) set)"
 by(metis is_interval_convex convex_connected is_interval_connected_1)
 *)
 subsection {* Another intermediate value theorem formulation *}
 lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
 assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
 shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI)
 using assms(1) by auto
 thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
 using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
 using assms by(auto intro!: imageI) qed
 lemma unit_interval_convex_hull:
 "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
 (is "?int = convex hull ?points")
 proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
 { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n"
 hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
 case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
 thus "x\<in>convex hull ?points" using 01 by auto
 next
 case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
 using i'(2-) `x$$i \<noteq> 0` by auto
 show ?thesis proof(cases "x$$i=1")
 case True have "\<forall>j\<in>{i. i<DIM('a) \<and> x$$i \<noteq> 0}. x$$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
 proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
 hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
 hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto
 hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
 thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
 thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
 by auto
 next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
 by(auto simp add:field_simps)
 hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
 moreover have "i\<in>{j. j<DIM('a) \<and> x$$j \<noteq> 0} - {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}"
 using i01 using i'(3) by auto
 hence "{j. j<DIM('a) \<and> x$$j \<noteq> 0} \<noteq> {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}" using i'(3) by blast
 hence **:"{j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<subset> {j. j<DIM('a) \<and> x$$j \<noteq> 0}" apply - apply rule
 by auto
 have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
 using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
 ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
 apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
 unfolding mem_interval using i01 Suc(3) by auto
 qed qed qed } note * = this
 have **:"DIM('a) = card {..<DIM('a)}" by auto
 show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
 apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **)
 apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
 unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
 by auto qed
 text {* And this is a finite set of vertices. *}
 using assms by(auto simp add: field_simps)
 hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
 "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
 hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
 by(auto simp add: field_simps)
 thus "\<exists>z\<in>{0..\<chi>\<chi> i.1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI)
 using assms by auto
 next
 fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z"
 have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
 using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
 apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
 using assms by auto
 thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
 have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
 unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
 obtain k where "k>0"and k:"cball x k \<subseteq> s" using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]] using `x\<in>s` by auto
 show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
 apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
 fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)"
 show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
 case False def t \<equiv> "k / norm (y - x)"
 have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
 have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
 apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute)
 { def w \<equiv> "x + t *\<^sub>R (y - x)"
 have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
 unfolding t_def using `k>0` by auto
 have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x" by (auto simp add: algebra_simps)
 also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps)
 finally have w:"(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
 have  "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps)
 hence "(f w - f x) / t < e"
 using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps)
 hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
 using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
 using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
 moreover
 { def w \<equiv> "x - t *\<^sub>R (y - x)"
 have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
 unfolding t_def using `k>0` by auto
 have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x" by (auto simp add: algebra_simps)
 also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
 finally have w:"(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
 have  "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps)
 hence *:"(f w - f y) / t < e" using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`] using `t>0` by(auto simp add:field_simps)
 have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
 using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
 using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
 also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding divide_inverse by (auto simp add:field_simps)
 also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
 finally have "f x - f y < e" by auto }
 ultimately show ?thesis by auto
 qed(insert `0<e`, auto)
 qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
 subsection {* Upper bound on a ball implies upper and lower bounds *}
 lemma convex_bounds_lemma:
 have *:"x - (2 *\<^sub>R x - y) = y - x" by (simp add: scaleR_2)
 have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
 have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
 thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
 using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
 next case False fix y assume "y\<in>cball x e"
 hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
 thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
 subsubsection {* Hence a convex function on an open set is continuous *}
 lemma convex_on_continuous:
 assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f"
 (* FIXME: generalize to euclidean_space *)
 shows "continuous_on s f"
 unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
 note dimge1 = DIM_positive[where 'a='a]
 fix x assume "x\<in>s"
 then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
 def d \<equiv> "e / real DIM('a)"
 have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto)
 let ?d = "(\<chi>\<chi> i. d)::'a"
 obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
 have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by auto
 hence "c\<noteq>{}" using c by auto
 def k \<equiv> "Max (f ` c)"
 have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
 apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof
 fix z assume z:"z\<in>{x - ?d..x + ?d}"
 have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
 unfolding real_eq_of_nat by auto
 show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
 using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
 have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 by auto
 hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
 have conv:"convex_on (cball x d) f" apply(rule convex_on_subset, rule convex_on_subset[OF assms(2)]) apply(rule e(1)) using dsube by auto
 hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
 fix y assume y:"y\<in>cball x d"
 { fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i"  "y $$ i \<le> x $$ i + d"
 using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by auto  }
 thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
 by auto qed
 hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
 apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
 apply force
 done
 thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
 using `d>0` by auto
 qed
 subsection {* Line segments, Starlike Sets, etc. *}
 (* Use the same overloading tricks as for intervals, so that
 segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
 definition
 midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a" where
 "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
 lemma segment_convex_hull:
 "closed_segment a b = convex hull {a,b}" proof-
 have *:"\<And>x. {x} \<noteq> {}" by auto
 have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
 show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_eqI)
 unfolding mem_Collect_eq apply(rule,erule exE)
 apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
 apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
 lemma convex_segment: "convex (closed_segment a b)"
 unfolding segment_convex_hull by(rule convex_convex_hull)
 fixes x a b :: "'a::euclidean_space"
 assumes "x \<in> closed_segment a b"
 shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
 using segment_furthest_le[OF assms, of a]
 using segment_furthest_le[OF assms, of b]
 by (auto simp add:norm_minus_commute)
 lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
 lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
 unfolding between_def by auto
 lemma between:"between (a,b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
 proof(cases "a = b")
 case True thus ?thesis unfolding between_def split_conv
 by(auto simp add:segment_refl dist_commute) next
 case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto
 have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
 show ?thesis unfolding between_def split_conv closed_segment_def mem_Collect_eq
 apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
 fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
 hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
 unfolding as(1) by(auto simp add:algebra_simps)
 show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
 unfolding norm_minus_commute[of x a] * using as(2,3)
 by(auto simp add: field_simps)
 next assume as:"dist a b = dist a x + dist x b"
 have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2]
 unfolding as[unfolded dist_norm] norm_ge_zero by auto
 thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
 unfolding dist_norm apply(subst euclidean_eq) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
 proof(rule,rule) fix i assume i:"i<DIM('a)"
 have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
 ((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
 using Fal by(auto simp add: field_simps)
 also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
 unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
 apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps)
 finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i"
 by auto
 qed(insert Fal2, auto) qed qed
 lemma between_midpoint: fixes a::"'a::euclidean_space" shows
 "between (a,b) (midpoint a b)" (is ?t1)
 "between (b,a) (midpoint a b)" (is ?t2)
 proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
 show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
 unfolding euclidean_eq[where 'a='a]
 by(auto simp add:field_simps) qed
 apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
 fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
 have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
 have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
 unfolding dist_norm unfolding norm_scaleR[symmetric] apply(rule arg_cong[where f=norm]) using `e>0`
 by(auto simp add: euclidean_eq[where 'a='a] field_simps)
 also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
 also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
 by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
 finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
 apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
 def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
 have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
 have "z\<in>interior s" apply(rule interior_mono[OF d,unfolded subset_eq,rule_format])
 unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
 by(auto simp add:field_simps norm_minus_commute)
 thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink)
 using assms(1,4-5) `y\<in>s` by auto qed
 subsection {* Some obvious but surprisingly hard simplex lemmas *}
 lemma simplex:
 unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
 lemma substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
 shows "convex hull (insert 0 { basis i | i. i : d}) =
 {x::'a::euclidean_space . (!i<DIM('a). 0 <= x$$i) & setsum (%i. x$$i) d <= 1 &
 (!i<DIM('a). i ~: d --> x$$i = 0)}"
 (is "convex hull (insert 0 ?p) = ?s")
 (* Proof is a modified copy of the proof of similar lemma std_simplex in Convex_Euclidean_Space.thy *)
 proof- let ?D = d (*"{..<DIM('a::euclidean_space)}"*)
 have "0 ~: ?p" using assms by (auto simp: image_def)
 have "{(basis i)::'n::euclidean_space |i. i \<in> ?D} = basis ` ?D" by auto
 note sumbas = this setsum_reindex[OF basis_inj_on[of d], unfolded o_def, OF assms]
 show ?thesis unfolding simplex[OF finite_substdbasis `0 ~: ?p`]
 apply(rule set_eqI) unfolding mem_Collect_eq apply rule
 apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
 fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x"
 "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
 have *:"\<forall>i<DIM('a). i:d --> u (basis i) = x$$i" and "(!i<DIM('a). i ~: d --> x $$ i = 0)" using as(3)
 unfolding sumbas unfolding substdbasis_expansion_unique[OF assms] by auto
 hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $$ x) ?D" unfolding sumbas
 apply-apply(rule setsum_cong2) using assms by auto
 have " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1"
 apply - proof(rule,rule,rule)
 fix i assume i:"i<DIM('a)" have "i : d ==> 0 \<le> x$$i" unfolding *[rule_format,OF i,symmetric]
 apply(rule_tac as(1)[rule_format]) by auto
 moreover have "i ~: d ==> 0 \<le> x$$i"
 using `(!i<DIM('a). i ~: d --> x $$ i = 0)`[rule_format, OF i] by auto
 ultimately show "0 \<le> x$$i" by auto
 qed(insert as(2)[unfolded **], auto)
 from this show " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1 & (!i<DIM('a). i ~: d --> x $$ i = 0)"
 using `(!i<DIM('a). i ~: d --> x $$ i = 0)` by auto
 next fix x::"'a::euclidean_space" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
 "(!i<DIM('a). i ~: d --> x $$ i = 0)"
 show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and>
 setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
 fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
 using component_le_norm[of "y - x" i]
 using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
 thus "y $$ i \<le> x $$ i + ?d" by auto qed
 also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant real_eq_of_nat by(auto simp add: Suc_le_eq)
 finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
 proof safe fix i assume i:"i<DIM('a)"
 have "norm (x - y) < x$$i" apply(rule less_le_trans)
 apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
 thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
 qed qed auto qed
 lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
 proof-
 have "finite d" apply(rule finite_subset) using assms by auto
 { assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq[of _ 0] by auto }
 moreover
 { assume "d~={}"
 have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
 using affine_hull_convex_hull affine_hull_substd_basis assms by auto
 have aux: "!x::'n::euclidean_space. !i. ((! i:d. 0 <= x$$i) & (!i. i ~: d --> x$$i = 0))--> 0 <= x$$i" by auto
 { fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"
 from this obtain e where e0: "e>0" and
 "ball x e Int {xa. (!i<DIM('a). i ~: d --> xa$$i = 0)} <= convex hull (insert 0 ?p)"
 using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
 hence as: "ALL xa. (dist x xa < e & (!i<DIM('a). i ~: d --> xa$$i = 0)) -->
 (!i : d. 0 <= xa $$ i) & setsum (op $$ xa) d <= 1"
 unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
 have x0: "(!i<DIM('a). i ~: d --> x$$i = 0)"
 using x_def rel_interior_subset  substd_simplex[OF assms] by auto
 have "(!i : d. 0 < x $$ i) & setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" apply(rule,rule)
 proof-
 fix i::nat assume "i:d"
 hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R basis i) $$ ia" apply-apply(rule as[rule_format,THEN conjunct1])
 unfolding dist_norm using assms `e>0` x0 by auto
 thus "0 < x $$ i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` assms by auto
 next obtain a where a:"a:d" using `d ~= {}` by auto
 have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e"
 using as[THEN spec[where x="x + (e / 2) *\<^sub>R basis a"]] `a:d` using x0
 by(auto elim:allE[where x=a])
 have "setsum (op $$ x) d < setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d" unfolding * setsum_addf
 using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')
 also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R basis a"] by auto
 finally show "setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" using x0 by auto
 qed
 }
 moreover
 {
 fix x::"'a::euclidean_space" assume as: "x : ?s"
 have "!i. ((0<x$$i) | (0=x$$i) --> 0<=x$$i)" by auto
 moreover have "!i. (i:d) | (i ~: d)" by auto
 ultimately
 have "!i. ( (ALL i:d. 0 < x$$i) & (ALL i. i ~: d --> x$$i = 0) ) --> 0 <= x$$i" by metis
 hence h2: "x : convex hull (insert 0 ?p)" using as assms
 unfolding substd_simplex[OF assms] by fastforce
 obtain a where a:"a:d" using `d ~= {}` by auto
 let ?d = "(1 - setsum (op $$ x) d) / real (card d)"
 have "0 < card d" using `d ~={}` `finite d` by (simp add: card_gt_0_iff)
 have "Min ((op $$ x) ` d) > 0" using as `d \<noteq> {}` `finite d` by (simp add: Min_grI)
 moreover have "?d > 0" apply(rule divide_pos_pos) using as using `0 < card d` by auto
 by(auto simp add: norm_minus_commute)
 thus "y $$ i \<le> x $$ i + ?d" by auto qed
 also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant real_eq_of_nat
 using `0 < card d` by auto
 finally show "setsum (op $$ y) d \<le> 1" .
 fix i assume "i<DIM('a)" thus "0 \<le> y$$i"
 proof(cases "i\<in>d") case True
 have "norm (x - y) < x$$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
 using Min_gr_iff[of "op $$ x ` d" "norm (x - y)"] `0 < card d` `i:d`
 by (simp add: card_gt_0_iff)
 thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format] by auto
 have *:"{basis i :: 'a | i. i \<in> ?D} = basis ` ?D" by auto
 have "finite d" apply(rule finite_subset) using assms(2) by auto
 hence d1: "0 < real(card d)" using `d ~={}` by auto
 { fix i assume "i:d" have "?a $$ i = inverse (2 * real (card d))"
 unfolding * setsum_reindex[OF basis_inj_on, OF assms(2)] o_def
 apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
 unfolding euclidean_component_setsum
 apply(rule setsum_cong2)
 using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)
 by (auto simp add: Euclidean_Space.basis_component[of i])}
 note ** = this
 show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
 proof safe fix i assume "i:d"
 have "0 < inverse (2 * real (card d))" using d1 by auto
 also have "...=?a $$ i" using **[of i] `i:d` by auto
 finally show "0 < ?a $$ i" by auto
 next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D"
 by(rule setsum_cong2, rule **)
 also have "\<dots> < 1" unfolding setsum_constant real_eq_of_nat divide_real_def[symmetric]
 by (auto simp add:field_simps)
 finally show "setsum (op $$ ?a) ?D < 1" by auto
 next fix i assume "i<DIM('a)" and "i~:d"
 have "?a : (span {basis i | i. i : d})"
 apply (rule span_setsum[of "{basis i |i. i : d}" "(%b. b /\<^sub>R (2 * real (card d)))" "{basis i |i. i : d}"])
 using finite_substdbasis[of d] apply blast
 proof-
 { fix x assume "(x :: 'a::euclidean_space): {basis i |i. i : d}"
 hence "x : span {basis i |i. i : d}"
 using span_superset[of _ "{basis i |i. i : d}"] by auto
 hence "(x /\<^sub>R (2 * real (card d))) : (span {basis i |i. i : d})"
 using span_mul[of x "{basis i |i. i : d}" "(inverse (real (card d)) / 2)"] by auto
 } thus "\<forall>x\<in>{basis i |i. i \<in> d}. x /\<^sub>R (2 * real (card d)) \<in> span {basis i ::'a |i. i \<in> d}" by auto
 qed
 qed
 qed
 subsection {* Relative interior of convex set *}
 lemma rel_interior_convex_nonempty_aux:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S" and "0 : S"
 shows "rel_interior S ~= {}"
 proof-
 { assume "S = {0}" hence ?thesis using rel_interior_sing by auto }
 moreover {
 assume "S ~= {0}"
 obtain B where B_def: "independent B & B<=S & (S <= span B) & card B = dim S" using basis_exists[of S] by auto
 hence "B~={}" using B_def assms `S ~= {0}` span_empty by auto
 have "insert 0 B <= span B" using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
 hence "span (insert 0 B) <= span B"
 using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
 hence "convex hull insert 0 B <= span B"
 using convex_hull_subset_span[of "insert 0 B"] by auto
 hence "span (convex hull insert 0 B) <= span B"
 using span_span[of B] span_mono[of "convex hull insert 0 B" "span B"] by blast
 hence *: "span (convex hull insert 0 B) = span B"
 using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
 hence "span (convex hull insert 0 B) = span S"
 using B_def span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
 moreover have "0 : affine hull (convex hull insert 0 B)"
 using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
 ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
 using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
 assms  hull_subset[of S] by auto
 obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = {basis i |i. i : (d :: nat set)} &
 f ` span B = {x. ALL i<DIM('n). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B)" and d:"d\<subseteq>{..<DIM('n)}"
 using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by auto
 hence "bounded_linear f" using linear_conv_bounded_linear by auto
 have "d ~={}" using fd B_def `B ~={}` by auto
 have "(insert 0 {basis i |i. i : d}) = f ` (insert 0 B)" using fd linear_0 by auto
 hence "(convex hull (insert 0 {basis i |i. i : d})) = f ` (convex hull (insert 0 B))"
 using convex_hull_linear_image[of f "(insert 0 {basis i |i. i : d})"]
 convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by auto
 moreover have "rel_interior (f ` (convex hull insert 0 B)) =
 f ` rel_interior (convex hull insert 0 B)"
 apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
 using `bounded_linear f` fd * by auto
 ultimately have "rel_interior (convex hull insert 0 B) ~= {}"
 using rel_interior_substd_simplex_nonempty[OF `d~={}` d] apply auto by blast
 moreover have "convex hull (insert 0 B) <= S"
 using B_def assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
 ultimately have ?thesis using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
 } ultimately show ?thesis by auto
 qed
 assumes "convex S"
 shows "rel_interior S = {} <-> S = {}"
 proof-
 { assume "S ~= {}" from this obtain a where "a : S" by auto
 hence "0 : op + (-a) ` S" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
 hence "rel_interior (op + (-a) ` S) ~= {}"
 using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"]
 convex_translation[of S "-a"] assms by auto
 hence "rel_interior S ~= {}" using rel_interior_translation by auto
 } from this show ?thesis using rel_interior_empty by auto
 qed
 lemma convex_rel_interior:
 { fix "x" "y" "u"
 assume assm: "x:rel_interior S" "y:rel_interior S" "0<=u" "(u :: real) <= 1"
 hence "x:S" using rel_interior_subset by auto
 have "x - u *\<^sub>R (x-y) : rel_interior S"
 proof(cases "0=u")
 case False hence "0<u" using assm by auto
 thus ?thesis
 using assm rel_interior_convex_shrink[of S y x u] assms `x:S` by auto
 next
 case True thus ?thesis using assm by auto
 qed
 hence "(1-u) *\<^sub>R x + u *\<^sub>R y : rel_interior S" by (simp add: algebra_simps)
 } from this show ?thesis unfolding convex_alt by auto
 qed
 lemma convex_closure_rel_interior:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S"
 shows "closure(rel_interior S) = closure S"
 proof-
 have h1: "closure(rel_interior S) <= closure S"
 using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
 { assume "S ~= {}" from this obtain a where a_def: "a : rel_interior S"
 using rel_interior_convex_nonempty assms by auto
 { fix x assume x_def: "x : closure S"
 { assume "x=a" hence "x : closure(rel_interior S)" using a_def unfolding closure_def by auto }
 moreover
 { assume "x ~= a"
 { fix e :: real assume e_def: "e>0"
 def e1 == "min 1 (e/norm (x - a))" hence e1_def: "e1>0 & e1<=1 & e1*norm(x-a)<=e"
 using `x ~= a` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(x-a)"] by simp
 hence *: "x - e1 *\<^sub>R (x - a) : rel_interior S"
 using rel_interior_closure_convex_shrink[of S a x e1] assms x_def a_def e1_def by auto
 have "EX y. y:rel_interior S & y ~= x & (dist y x) <= e"
 apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
 using * e1_def dist_norm[of "x - e1 *\<^sub>R (x - a)" x] `x ~= a` by simp
 } hence "x islimpt rel_interior S" unfolding islimpt_approachable_le by auto
 hence "x : closure(rel_interior S)" unfolding closure_def by auto
 } ultimately have "x : closure(rel_interior S)" by auto
 } hence ?thesis using h1 by auto
 }
 moreover
 { assume "S = {}" hence "rel_interior S = {}" using rel_interior_empty by auto
 hence "closure(rel_interior S) = {}" using closure_empty by auto
 hence ?thesis using `S={}` by auto
 } ultimately show ?thesis by blast
 qed
 lemma rel_interior_same_affine_hull:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S"
 shows "affine hull (rel_interior S) = affine hull S"
 by (metis assms closure_same_affine_hull convex_closure_rel_interior)
 lemma rel_interior_aff_dim:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S"
 shows "aff_dim (rel_interior S) = aff_dim S"
 by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
 moreover have "0<e" using e_def assms divide_pos_pos[of a "a+b"] by auto
 moreover have "e<=1" using e_def assms by auto
 ultimately show ?thesis using that[of e] by auto
 qed
 lemma convex_rel_interior_closure:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S"
 shows "rel_interior (closure S) = rel_interior S"
 proof-
 { assume "S={}" hence ?thesis using assms rel_interior_convex_nonempty by auto }
 moreover
 { assume "S ~= {}"
 have "rel_interior (closure S) >= rel_interior S"
 using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset by auto
 moreover
 { fix z assume z_def: "z : rel_interior (closure S)"
 obtain x where x_def: "x : rel_interior S"
 using `S ~= {}` assms rel_interior_convex_nonempty by auto
 { assume "x=z" hence "z : rel_interior S" using x_def by auto }
 moreover
 { assume "x ~= z"
 obtain e where e_def: "e > 0 & cball z e Int affine hull closure S <= closure S"
 using z_def rel_interior_cball[of "closure S"] by auto
 hence *: "0 < e/norm(z-x)" using e_def `x ~= z` divide_pos_pos[of e "norm(z-x)"] by auto
 def y == "z + (e/norm(z-x)) *\<^sub>R (z-x)"
 have yball: "y : cball z e"
 using mem_cball y_def dist_norm[of z y] e_def by auto
 have "x : affine hull closure S"
 using x_def rel_interior_subset_closure hull_inc[of x "closure S"] by auto
 moreover have "z : affine hull closure S"
 using z_def rel_interior_subset hull_subset[of "closure S"] by auto
 ultimately have "y : affine hull closure S"
 using y_def affine_affine_hull[of "closure S"]
 mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
 hence "y : closure S" using e_def yball by auto
 have "(1+(e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
 using y_def by (simp add: algebra_simps)
 from this obtain e1 where "0 < e1 & e1 <= 1 & z = y - e1 *\<^sub>R (y - x)"
 using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
 by (auto simp add: algebra_simps)
 hence "z : rel_interior S"
 using rel_interior_closure_convex_shrink assms x_def `y : closure S` by auto
 } ultimately have "z : rel_interior S" by auto
 } ultimately have ?thesis by auto
 } ultimately show ?thesis by blast
 qed
 lemma convex_interior_closure:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S"
 shows "interior (closure S) = interior S"
 using closure_aff_dim[of S] interior_rel_interior_gen[of S] interior_rel_interior_gen[of "closure S"]
 convex_rel_interior_closure[of S] assms by auto
 lemma closure_eq_rel_interior_eq:
 fixes S1 S2 ::  "('n::euclidean_space) set"
 assumes "convex S1" "convex S2"
 shows "(closure S1 = closure S2) <-> (rel_interior S1 = rel_interior S2)"
 by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
 lemma closure_eq_between:
 fixes S1 S2 ::  "('n::euclidean_space) set"
 assumes "convex S1" "convex S2"
 shows "(closure S1 = closure S2) <->
 ((rel_interior S1 <= S2) & (S2 <= closure S1))" (is "?A <-> ?B")
 proof-
 have "?A --> ?B" by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
 moreover have "?B --> (closure S1 <= closure S2)"
 by (metis assms(1) convex_closure_rel_interior closure_mono)
 moreover have "?B --> (closure S1 >= closure S2)" by (metis closed_closure closure_minimal)
 ultimately show ?thesis by blast
 qed
 lemma open_inter_closure_rel_interior:
 fixes S A ::  "('n::euclidean_space) set"
 assumes "convex S" "open A"
 shows "((A Int closure S) = {}) <-> ((A Int rel_interior S) = {})"
 by (metis assms convex_closure_rel_interior open_inter_closure_eq_empty)
 definition "rel_frontier S = closure S - rel_interior S"
 lemma closed_affine_hull: "closed (affine hull ((S :: ('n::euclidean_space) set)))"
 by (metis affine_affine_hull affine_closed)
 lemma closed_rel_frontier: "closed(rel_frontier (S :: ('n::euclidean_space) set))"
 proof-
 have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"
 apply (rule closedin_diff[of "subtopology euclidean (affine hull S)""closure S" "rel_interior S"])  using closed_closedin_trans[of "affine hull S" "closure S"] closed_affine_hull[of S]
 closure_affine_hull[of S] opein_rel_interior[of S] by auto
 show ?thesis apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
 unfolding rel_frontier_def using * closed_affine_hull by auto
 qed
 lemma convex_rel_frontier_aff_dim:
 fixes S1 S2 ::  "('n::euclidean_space) set"
 assumes "convex S1" "convex S2" "S2 ~= {}"
 assumes "S1 <= rel_frontier S2"
 shows "aff_dim S1 < aff_dim S2"
 proof-
 have "S1 <= closure S2" using assms unfolding rel_frontier_def by auto
 hence *: "affine hull S1 <= affine hull S2"
 using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by auto
 hence "aff_dim S1 <= aff_dim S2" using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
 aff_dim_subset[of "affine hull S1" "affine hull S2"] by auto
 moreover
 { assume eq: "aff_dim S1 = aff_dim S2"
 hence "S1 ~= {}" using aff_dim_empty[of S1] aff_dim_empty[of S2] `S2 ~= {}` by auto
 have **: "affine hull S1 = affine hull S2"
 apply (rule affine_dim_equal) using * affine_affine_hull apply auto
 using `S1 ~= {}` hull_subset[of S1] apply auto
 using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] by auto
 obtain a where a_def: "a : rel_interior S1"
 using  `S1 ~= {}` rel_interior_convex_nonempty assms by auto
 obtain T where T_def: "open T & a : T Int S1 & T Int affine hull S1 <= S1"
 using mem_rel_interior[of a S1] a_def by auto
 hence "a : T Int closure S2" using a_def assms unfolding rel_frontier_def by auto
 from this obtain b where b_def: "b : T Int rel_interior S2"
 using open_inter_closure_rel_interior[of S2 T] assms T_def by auto
 hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto
 hence "b : S1" using T_def b_def by auto
 hence False using b_def assms unfolding rel_frontier_def by auto
 } ultimately show ?thesis using less_le by auto
 qed
 lemma convex_rel_interior_if:
 fixes S ::  "('n::euclidean_space) set"
 assumes "convex S"
 assumes "z : rel_interior S"
 shows "(!x:affine hull S. EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S ))"
 proof-
 obtain e1 where e1_def: "e1>0 & cball z e1 Int affine hull S <= S"
 using mem_rel_interior_cball[of z S] assms by auto
 { fix x assume x_def: "x:affine hull S"
 { assume "x ~= z"
 def m == "1+e1/norm(x-z)"
 hence "m>1" using e1_def `x ~= z` divide_pos_pos[of e1 "norm (x - z)"] by auto
 { fix e assume e_def: "e>1 & e<=m"
 have "z : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
 hence *: "(1-e)*\<^sub>R x+ e *\<^sub>R z : affine hull S"
 using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x_def by auto
 have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x-z))" by (simp add: algebra_simps)
 also have "...= (e - 1) * norm(x-z)" using norm_scaleR e_def by auto
 also have "...<=(m - 1) * norm(x-z)" using e_def mult_right_mono[of _ _ "norm(x-z)"] by auto
 also have "...= (e1 / norm (x - z)) * norm (x - z)" using m_def by auto
 also have "...=e1" using `x ~= z` e1_def by simp
 finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) <= e1" by auto
 have "(1-e)*\<^sub>R x+ e *\<^sub>R z : cball z e1"
 hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S"
 using e1_def x_def `x=z` by (auto simp add: algebra_simps)
 hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def by auto
 } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" using `m>1` by auto
 } ultimately have "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" by auto
 } from this show ?thesis by auto
 qed
 lemma convex_rel_interior_if2:
 fixes S ::  "('n::euclidean_space) set"
 assumes "convex S"
 assumes "z : rel_interior S"
 shows "(!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
 using convex_rel_interior_if[of S z] assms by auto
 lemma convex_rel_interior_only_if:
 fixes S ::  "('n::euclidean_space) set"
 assumes "convex S" "S ~= {}"
 assumes "(!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
 shows "z : rel_interior S"
 proof-
 obtain x where x_def: "x : rel_interior S" using rel_interior_convex_nonempty assms by auto
 hence "x:S" using rel_interior_subset by auto
 from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S" using assms by auto
 def y == "(1 - e) *\<^sub>R x + e *\<^sub>R z" hence "y:S" using e_def by auto
 def e1 == "1/e" hence "0<e1 & e1<1" using e_def by auto
 hence "z=y-(1-e1)*\<^sub>R (y-x)" using e1_def y_def by (auto simp add: algebra_simps)
 from this show ?thesis
 using rel_interior_convex_shrink[of S x y "1-e1"] `0<e1 & e1<1` `y:S` x_def assms by auto
 qed
 lemma convex_rel_interior_iff:
 fixes S ::  "('n::euclidean_space) set"
 assumes "convex S" "S ~= {}"
 shows "z : rel_interior S <-> (!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
 using assms hull_subset[of S "affine"]
 convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z] by auto
 lemma convex_rel_interior_iff2:
 fixes S ::  "('n::euclidean_space) set"
 assumes "convex S" "S ~= {}"
 shows "z : rel_interior S <-> (!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
 using assms hull_subset[of S]
 convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z] by auto
 lemma convex_interior_iff:
 fixes S ::  "('n::euclidean_space) set"
 assumes "convex S"
 shows "z : interior S <-> (!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
 proof-
 { assume a: "~(aff_dim S = int DIM('n))"
 { assume "z : interior S"
 hence x2: "x2 : affine hull S" using e2_def hull_subset[of S] by auto
 have *: "e1/(e1+e2) + e2/(e1+e2) = 1" using add_divide_distrib[of e1 e2 "e1+e2"] e1_def e2_def by simp
 hence "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
 using x1_def x2_def apply (auto simp add: algebra_simps)
 using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z] by auto
 hence z: "z : affine hull S"
 using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
 x1 x2 affine_affine_hull[of S] * by auto
 have "x1-x2 = (e1+e2) *\<^sub>R (x-z)"
 using x1_def x2_def by (auto simp add: algebra_simps)
 hence "x=z+(1/(e1+e2)) *\<^sub>R (x1-x2)" using e1_def e2_def by simp
 hence "x : affine hull S" using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
 x1 x2 z affine_affine_hull[of S] by auto
 } hence "affine hull S = UNIV" by auto
 hence "aff_dim S = int DIM('n)" using aff_dim_affine_hull[of S] by (simp add: aff_dim_univ)
 hence False using a by auto
 } ultimately have ?thesis by auto
 qed
 subsubsection {* Relative interior and closure under common operations *}
 lemma rel_interior_inter_aux: "Inter {rel_interior S |S. S : I} <= Inter I"
 proof-
 { fix y assume "y : Inter {rel_interior S |S. S : I}"
 hence y_def: "!S : I. y : rel_interior S" by auto
 { fix S assume "S : I" hence "y : S" using rel_interior_subset y_def by auto }
 hence "y : Inter I" by auto
 } thus ?thesis by auto
 qed
 lemma closure_inter: "closure (Inter I) <= Inter {closure S |S. S : I}"
 proof-
 { fix y assume "y : Inter I" hence y_def: "!S : I. y : S" by auto
 { fix S assume "S : I" hence "y : closure S" using closure_subset y_def by auto }
 hence "y : Inter {closure S |S. S : I}" by auto
 } hence "Inter I <= Inter {closure S |S. S : I}" by auto
 moreover have "closed (Inter {closure S |S. S : I})"
 unfolding closed_Inter closed_closure by auto
 ultimately show ?thesis using closure_hull[of "Inter I"]
 hull_minimal[of "Inter I" "Inter {closure S |S. S : I}" "closed"] by auto
 qed
 lemma convex_closure_rel_interior_inter:
 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
 assumes "Inter {rel_interior S |S. S : I} ~= {}"
 shows "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
 proof-
 obtain x where x_def: "!S : I. x : rel_interior S" using assms by auto
 { fix y assume "y : Inter {closure S |S. S : I}" hence y_def: "!S : I. y : closure S" by auto
 { assume "y = x"
 hence "y : closure (Inter {rel_interior S |S. S : I})"
 using x_def closure_subset[of "Inter {rel_interior S |S. S : I}"] by auto
 }
 moreover
 { assume "y ~= x"
 { fix e :: real assume e_def: "0 < e"
 def e1 == "min 1 (e/norm (y - x))" hence e1_def: "e1>0 & e1<=1 & e1*norm(y-x)<=e"
 using `y ~= x` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(y-x)"] by simp
 def z == "y - e1 *\<^sub>R (y - x)"
 { fix S assume "S : I"
 hence "z : rel_interior S" using rel_interior_closure_convex_shrink[of S x y e1]
 assms x_def y_def e1_def z_def by auto
 } hence *: "z : Inter {rel_interior S |S. S : I}" by auto
 have "EX z. z:Inter {rel_interior S |S. S : I} & z ~= y & (dist z y) <= e"
 apply (rule_tac x="z" in exI) using `y ~= x` z_def * e1_def e_def dist_norm[of z y] by simp
 } hence "y islimpt Inter {rel_interior S |S. S : I}" unfolding islimpt_approachable_le by blast
 hence "y : closure (Inter {rel_interior S |S. S : I})" unfolding closure_def by auto
 } ultimately have "y : closure (Inter {rel_interior S |S. S : I})" by auto
 } from this show ?thesis by auto
 qed
 lemma convex_closure_inter:
 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
 assumes "Inter {rel_interior S |S. S : I} ~= {}"
 shows "closure (Inter I) = Inter {closure S |S. S : I}"
 proof-
 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
 using convex_closure_rel_interior_inter assms by auto
 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)"
 using rel_interior_inter_aux
 closure_mono[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
 ultimately show ?thesis using closure_inter[of I] by auto
 qed
 lemma convex_inter_rel_interior_same_closure:
 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
 assumes "Inter {rel_interior S |S. S : I} ~= {}"
 shows "closure (Inter {rel_interior S |S. S : I}) = closure (Inter I)"
 proof-
 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
 using convex_closure_rel_interior_inter assms by auto
 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)"
 using rel_interior_inter_aux
 closure_mono[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
 ultimately show ?thesis using closure_inter[of I] by auto
 qed
 lemma convex_rel_interior_inter:
 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
 assumes "Inter {rel_interior S |S. S : I} ~= {}"
 shows "rel_interior (Inter I) <= Inter {rel_interior S |S. S : I}"
 proof-
 have "convex(Inter I)" using assms convex_Inter by auto
 moreover have "convex(Inter {rel_interior S |S. S : I})" apply (rule convex_Inter)
 using assms convex_rel_interior by auto
 ultimately have "rel_interior (Inter {rel_interior S |S. S : I}) = rel_interior (Inter I)"
 using convex_inter_rel_interior_same_closure assms
 closure_eq_rel_interior_eq[of "Inter {rel_interior S |S. S : I}" "Inter I"] by blast
 from this show ?thesis using rel_interior_subset[of "Inter {rel_interior S |S. S : I}"] by auto
 qed
 lemma convex_rel_interior_finite_inter:
 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
 assumes "Inter {rel_interior S |S. S : I} ~= {}"
 assumes "finite I"
 shows "rel_interior (Inter I) = Inter {rel_interior S |S. S : I}"
 proof-
 { assume "I={}" hence ?thesis using Inter_empty rel_interior_univ2 by auto }
 moreover
 { assume "I ~= {}"
 { fix z assume z_def: "z : Inter {rel_interior S |S. S : I}"
 { fix x assume x_def: "x : Inter I"
 { fix S assume S_def: "S : I" hence "z : rel_interior S" "x : S" using z_def x_def by auto
 (*from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S"*)
 hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )"
 using convex_rel_interior_if[of S z] S_def assms hull_subset[of S] by auto
 } from this obtain mS where mS_def: "!S : I. (mS(S) > (1 :: real) &
 (!e. (e>1 & e<=mS(S)) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S))" by metis
 obtain e where e_def: "e=Min (mS ` I)" by auto
 have "e : (mS ` I)" using e_def assms `I ~= {}` by simp
 hence "e>(1 :: real)" using mS_def by auto
 moreover have "!S : I. e<=mS(S)" using e_def assms by auto
 ultimately have "EX e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z : Inter I" using mS_def by auto
 } hence "z : rel_interior (Inter I)" using convex_rel_interior_iff[of "Inter I" z]
 `Inter I ~= {}` `convex (Inter I)` by auto
 } from this have ?thesis using convex_rel_interior_inter[of I] assms by auto
 } ultimately show ?thesis by blast
 qed
 lemma convex_closure_inter_two:
 fixes S T :: "('n::euclidean_space) set"
 assumes "convex S" "convex T"
 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
 shows "closure (S Int T) = (closure S) Int (closure T)"
 using convex_closure_inter[of "{S,T}"] assms by auto
 lemma convex_rel_interior_inter_two:
 fixes S T :: "('n::euclidean_space) set"
 assumes "convex S" "convex T"
 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
 shows "rel_interior (S Int T) = (rel_interior S) Int (rel_interior T)"
 using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
 lemma convex_affine_closure_inter:
 fixes S T :: "('n::euclidean_space) set"
 assumes "convex S" "affine T"
 assumes "(rel_interior S) Int T ~= {}"
 shows "closure (S Int T) = (closure S) Int T"
 proof-
 have "affine hull T = T" using assms by auto
 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
 moreover have "closure T = T" using assms affine_closed[of T] by auto
 ultimately show ?thesis using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
 qed
 lemma convex_affine_rel_interior_inter:
 fixes S T :: "('n::euclidean_space) set"
 assumes "convex S" "affine T"
 assumes "(rel_interior S) Int T ~= {}"
 shows "rel_interior (S Int T) = (rel_interior S) Int T"
 proof-
 have "affine hull T = T" using assms by auto
 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
 moreover have "closure T = T" using assms affine_closed[of T] by auto
 ultimately show ?thesis using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
 qed
 lemma subset_rel_interior_convex:
 fixes S T :: "('n::euclidean_space) set"
 assumes "convex S" "convex T"
 assumes "~(S <= rel_frontier T)"
 shows "rel_interior S <= rel_interior T"
 proof-
 have *: "S Int closure T = S" using assms by auto
 have "~(rel_interior S <= rel_frontier T)"
 using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
 closure_closed convex_closure_rel_interior[of S] closure_subset[of S] assms by auto
 hence "(rel_interior S) Int (rel_interior (closure T)) ~= {}"
 using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T] by auto
 hence "rel_interior S Int rel_interior T = rel_interior (S Int closure T)" using assms convex_closure
 convex_rel_interior_inter_two[of S "closure T"] convex_rel_interior_closure[of T] by auto
 also have "...=rel_interior (S)" using * by auto
 finally show ?thesis by auto
 qed
 { assume "S = {}" hence ?thesis using assms rel_interior_empty rel_interior_convex_nonempty by auto }
 moreover
 { assume "S ~= {}"
 have *: "f ` (rel_interior S) <= f ` S" unfolding image_mono using rel_interior_subset by auto
 have "f ` S <= f ` (closure S)" unfolding image_mono using closure_subset by auto
 also have "... = f ` (closure (rel_interior S))" using convex_closure_rel_interior assms by auto
 also have "... <= closure (f ` (rel_interior S))" using closure_linear_image assms by auto
 finally have "closure (f ` S) = closure (f ` rel_interior S)"
 using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
 closure_mono[of "f ` rel_interior S" "f ` S"] * by auto
 hence "rel_interior (f ` S) = rel_interior (f ` rel_interior S)" using assms convex_rel_interior
 linear_conv_bounded_linear[of f] convex_linear_image[of S] convex_linear_image[of "rel_interior S"]
 closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"] by auto
 hence "rel_interior (f ` S) <= f ` rel_interior S" using rel_interior_subset by auto
 moreover
 { fix z assume z_def: "z : f ` rel_interior S"
 from this obtain z1 where z1_def: "z1 : rel_interior S & (f z1 = z)" by auto
 moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
 using x1_def z1_def `linear f` by (simp add: linear_add_cmul)
 ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
 using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
 hence "EX e. (e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S)" using e_def by auto
 } from this have "z : rel_interior (f ` S)" using convex_rel_interior_iff[of "f ` S" z] `convex S`
 `linear f` `S ~= {}` convex_linear_image[of S f]  linear_conv_bounded_linear[of f] by auto
 } ultimately have ?thesis by auto
 } ultimately show ?thesis by blast
 qed
 assumes "convex S"
 assumes "f -` (rel_interior S) ~= {}"
 shows "rel_interior (f -` S) = f -` (rel_interior S)"
 proof-
 have "S ~= {}" using assms rel_interior_empty by auto
 have nonemp: "f -` S ~= {}" by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
 hence "S Int (range f) ~= {}" by auto
 have conv: "convex (f -` S)" using convex_linear_preimage assms linear_conv_bounded_linear by auto
 hence "convex (S Int (range f))"
 by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
 { fix z assume "z : f -` (rel_interior S)"
 from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) : S"
 using convex_rel_interior_iff[of S "f z"] z_def assms `S ~= {}` by auto
 moreover have "(1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R x + e *\<^sub>R z)"
 using `linear f` by (simp add: linear_def)
 ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R z : f -` S" using e_def by auto
 } hence "z : rel_interior (f -` S)"
 using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
 }
 moreover
 { fix z assume z_def: "z : rel_interior (f -` S)"
 { fix x assume x_def: "x: S Int (range f)"
 from this obtain y where y_def: "(f y = x) & (y : f -` S)" by auto
 from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R y+ e *\<^sub>R z : f -` S"
 using convex_rel_interior_iff[of "f -` S" z] z_def conv by auto
 moreover have "(1-e)*\<^sub>R x+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R y + e *\<^sub>R z)"
 using e_def by auto
 } hence "f z : rel_interior (S Int (range f))" using `convex (S Int (range f))`
 `S Int (range f) ~= {}` convex_rel_interior_iff[of "S Int (range f)" "f z"] by auto
 moreover have "affine (range f)"
 by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
 ultimately have "f z : rel_interior S"
 using convex_affine_rel_interior_inter[of S "range f"] assms by auto
 hence "z : f -` (rel_interior S)" by auto
 }
 ultimately show ?thesis by auto
 qed
 lemma convex_direct_sum:
 fixes S :: "('n::euclidean_space) set"
 fixes T :: "('m::euclidean_space) set"
 assumes "convex S" "convex T"
 fixes T :: "('m::euclidean_space) set"
 shows "convex hull (S <*> T) = (convex hull S) <*> (convex hull T)"
 proof-
 { fix x assume "x : (convex hull S) <*> (convex hull T)"
 from this obtain xs xt where xst_def: "xs : convex hull S & xt : convex hull T & (xs,xt) = x" by auto
 from xst_def obtain sI su where s: "finite sI & sI <= S & (ALL x:sI. 0 <= su x) & setsum su sI = 1
 & (SUM v:sI. su v *\<^sub>R v) = xs" using convex_hull_explicit[of S] by auto
 from xst_def obtain tI tu where t: "finite tI & tI <= T & (ALL x:tI. 0 <= tu x) & setsum tu tI = 1
 & (SUM v:tI. tu v *\<^sub>R v) = xt" using convex_hull_explicit[of T] by auto
 def I == "(sI <*> tI)"
 def u == "(%i. (su (fst i))*(tu(snd i)))"
 have "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
 (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vs)"
 also have "...=(SUM vs:sI. su vs) *\<^sub>R xt" by (simp add: scaleR_left.setsum)
 also have "...=xt" using s by auto
 finally have h2: "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xt" by auto
 from h1 h2 have "(SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v) = x" using xst_def by auto
 moreover have "finite I & (I <= S <*> T)" using s t I_def by auto
 moreover have "!i:I. 0 <= u i" using s t I_def u_def by (simp add: mult_nonneg_nonneg)
 moreover have "setsum u I = 1" using u_def I_def setsum_cartesian_product[of "(% x y. (su x)*(tu y))"]
 s t setsum_product[of su sI tu tI] by (auto simp add: split_def)
 ultimately have "x : convex hull (S <*> T)"
 apply (subst convex_hull_explicit[of "S <*> T"]) apply rule
 apply (rule_tac x="I" in exI) apply (rule_tac x="u" in exI)
 using I_def u_def by auto
 }
 hence "convex hull (S <*> T) >= (convex hull S) <*> (convex hull T)" by auto
 moreover have "convex ((convex hull S) <*> (convex hull T))"
 by (simp add: convex_direct_sum convex_convex_hull)
 ultimately show ?thesis
 using hull_minimal[of "S <*> T" "(convex hull S) <*> (convex hull T)" "convex"]
 hull_subset[of S convex] hull_subset[of T convex] by auto
 qed
 lemma rel_interior_direct_sum:
 fixes S :: "('n::euclidean_space) set"
 moreover {
 assume "S ~={}" "T ~={}"
 hence ri: "rel_interior S ~= {}" "rel_interior T ~= {}" using rel_interior_convex_nonempty assms by auto
 hence "fst -` rel_interior S ~= {}" using fst_vimage_eq_Times[of "rel_interior S"] by auto
 hence "rel_interior ((fst :: 'n * 'm => 'n) -` S) = fst -` rel_interior S"
 using fst_linear `convex S` rel_interior_convex_linear_preimage[of fst S] by auto
 hence s: "rel_interior (S <*> (UNIV :: 'm set)) = rel_interior S <*> UNIV" by (simp add: fst_vimage_eq_Times)
 from ri have "snd -` rel_interior T ~= {}" using snd_vimage_eq_Times[of "rel_interior T"] by auto
 hence "rel_interior ((snd :: 'n * 'm => 'm) -` T) = snd -` rel_interior T"
 using snd_linear `convex T` rel_interior_convex_linear_preimage[of snd T] by auto
 hence t: "rel_interior ((UNIV :: 'n set) <*> T) = UNIV <*> rel_interior T" by (simp add: snd_vimage_eq_Times)
 from s t have *: "rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)
 = rel_interior S <*> rel_interior T" by auto
 have "(S <*> T) = (S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T)" by auto
 hence "rel_interior (S <*> T) = rel_interior ((S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T))" by auto
 also have "...=rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)"
 apply (subst convex_rel_interior_inter_two[of "S <*> (UNIV :: 'm set)" "(UNIV :: 'n set) <*> T"])
 using * ri assms convex_direct_sum by auto
 finally have ?thesis using * by auto
 }
 ultimately show ?thesis by blast
 qed
 lemma rel_interior_scaleR:
 fixes S :: "('n::euclidean_space) set"
 assumes "c ~= 0"
 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
 using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
 linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms by auto
 lemma rel_interior_convex_scaleR:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S"
 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
 by (metis assms linear_scaleR rel_interior_convex_linear_image)
 lemma convex_rel_open_scaleR:
 fixes S :: "('n::euclidean_space) set"
 assumes "convex S" "rel_open S"
 shows "convex ((op *\<^sub>R c) ` S) & rel_open ((op *\<^sub>R c) ` S)"
 by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
 lemma convex_rel_open_finite_inter:
 assumes "!S : I. (convex (S :: ('n::euclidean_space) set) & rel_open S)"
 assumes "finite I"
 shows "convex (Inter I) & rel_open (Inter I)"
 proof-
 { assume "Inter {rel_interior S |S. S : I} = {}"
 lemma convex_rel_open_linear_image:
 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
 assumes "linear f"
 assumes "convex S" "rel_open S"
 shows "convex (f ` S) & rel_open (f ` S)"
 by (metis assms convex_linear_image rel_interior_convex_linear_image
 linear_conv_bounded_linear rel_open_def)
 lemma convex_rel_open_linear_preimage:
 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
 assumes "linear f"
 assumes "convex S" "rel_open S"
 shows "convex (f -` S) & rel_open (f -` S)"
 proof-
 { assume "f -` (rel_interior S) = {}"
 hence "f -` S = {}" using assms unfolding rel_open_def by auto
 hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
 }
 moreover have "x = snd (y,x)" by auto
 ultimately have "x : snd ` (S Int fst -` {y})" by blast
 }
 hence "snd ` (S Int fst -` {y}) = f y" using assms by auto
 hence ***: "rel_interior (f y) = snd ` rel_interior (S Int fst -` {y})"
 using rel_interior_convex_linear_image[of snd "S Int fst -` {y}"] snd_linear conv by auto
 { fix z assume "z : rel_interior (f y)"
 hence "z : snd ` rel_interior (S Int fst -` {y})" using *** by auto
 moreover have "{y} = fst ` rel_interior (S Int fst -` {y})" using * ** rel_interior_subset by auto
 ultimately have "(y,z) : rel_interior (S Int fst -` {y})" by force
 hence "(y,z) : rel_interior S" using ** by auto
 }
 moreover
 { fix z assume "(y,z) : rel_interior S"
 hence "(y,z) : rel_interior (S Int fst -` {y})" using ** by auto
 hence "z : snd ` rel_interior (S Int fst -` {y})" by (metis Range_iff snd_eq_Range)
 hence "z : rel_interior (f y)" using *** by auto
 }
 ultimately have "!!z. (y,z) : rel_interior S <-> z : rel_interior (f y)" by auto
 }
 hence h2: "!!y z. y : rel_interior {t. f t ~= {}} ==> ((y, z) : rel_interior S) = (z : rel_interior (f y))"
 by auto
 { fix y z assume asm: "(y, z) : rel_interior S"
 hence "y : fst ` rel_interior S" by (metis Domain_iff fst_eq_Domain)
 hence "y : rel_interior {t. f t ~= {}}" using h1 by auto
 qed
 lemma rel_interior_convex_cone_aux:
 fixes S :: "('m::euclidean_space) set"
 assumes "convex S"
 shows "(c,x) : rel_interior (cone hull ({(1 :: real)} <*> S)) <->
 c>0 & x : ((op *\<^sub>R c) ` (rel_interior S))"
 proof-
 { assume "S={}" hence ?thesis by (simp add: rel_interior_empty cone_hull_empty) }
 moreover
 { assume "S ~= {}" from this obtain s where "s : S" by auto
 have conv: "convex ({(1 :: real)} <*> S)" using convex_direct_sum[of "{(1 :: real)}" S]
 assms convex_singleton[of "1 :: real"] by auto
 def f == "(%y. {z. (y,z) : cone hull ({(1 :: real)} <*> S)})"
 hence *: "(c, x) : rel_interior (cone hull ({(1 :: real)} <*> S)) =
 (c : rel_interior {y. f y ~= {}} & x : rel_interior (f c))"
 apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} <*> S)" f c x])
 lemma rel_interior_convex_cone:
 fixes S :: "('m::euclidean_space) set"
 assumes "convex S"
 shows "rel_interior (cone hull ({(1 :: real)} <*> S)) =
 {(c,c *\<^sub>R x) |c x. c>0 & x : (rel_interior S)}"
 (is "?lhs=?rhs")
 proof-
 { fix z assume "z:?lhs"
 have *: "z=(fst z,snd z)" by auto
 have "z:?rhs" using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms `z:?lhs` apply auto
 apply (rule_tac x="fst z" in exI) apply (rule_tac x="x" in exI) using * by auto
 }
 moreover
 { fix z assume "z:?rhs" hence "z:?lhs"
 using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms by auto
 }
 ultimately show ?thesis by blast
 qed
 lemma convex_hull_finite_union:
 assumes "finite I"
 assumes "!i:I. (convex (S i) & (S i) ~= {})"
 shows "convex hull (Union (S ` I)) =
 {setsum (%i. c i *\<^sub>R s i) I |c s. (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)}"
 (is "?lhs = ?rhs")
 proof-
 { fix x assume "x : ?rhs"
 from this obtain c s
 where *: "setsum (%i. c i *\<^sub>R s i) I=x" "(setsum c I = 1)"
 "(!i:I. c i >= 0) & (!i:I. s i : S i)" by auto
 hence "!i:I. s i : convex hull (Union (S ` I))" using hull_subset[of "Union (S ` I)" convex] by auto
 hence "x : ?lhs" unfolding *(1)[symmetric]
 apply (subst convex_setsum[of I "convex hull Union (S ` I)" c s])
 using * assms convex_convex_hull by auto
 } hence "?lhs >= ?rhs" by auto
 { fix i assume "i:I"
 from this assms have "EX p. p : S i" by auto
 }
 from this obtain p where p_def: "!i:I. p i : S i" by metis
 { fix i assume "i:I"
 { fix x assume "x : S i"
 def c == "(%j. if (j=i) then (1::real) else 0)"
 hence *: "setsum c I = 1" using `finite I` `i:I` setsum_delta[of I i "(%(j::'a). (1::real))"] by auto
 def s == "(%j. if (j=i) then x else p j)"
 hence "!j. c j *\<^sub>R s j = (if (j=i) then x else 0)" using c_def by (auto simp add: algebra_simps)
 hence "x = setsum (%i. c i *\<^sub>R s i) I"
 using s_def c_def `finite I` `i:I` setsum_delta[of I i "(%(j::'a). x)"] by auto
 hence "x : ?rhs" apply auto
 apply (rule_tac x="c" in exI)
 apply (rule_tac x="s" in exI) using * c_def s_def p_def `x : S i` by auto
 } hence "?rhs >= S i" by auto
 } hence *: "?rhs >= Union (S ` I)" by auto
 { fix u v assume uv: "(u :: real)>=0 & v>=0 & u+v=1"
 fix x y assume xy: "(x : ?rhs) & (y : ?rhs)"
 def e == "(%i. u * (c i)+v * (d i))"
 have ge0: "!i:I. e i >= 0"  using e_def xc yc uv by (simp add: mult_nonneg_nonneg)
 have "setsum (%i. u * c i) I = u * setsum c I" by (simp add: setsum_right_distrib)
 moreover have "setsum (%i. v * d i) I = v * setsum d I" by (simp add: setsum_right_distrib)
 ultimately have sum1: "setsum e I = 1" using e_def xc yc uv by (simp add: setsum_addf)
 def q == "(%i. if (e i = 0) then (p i)
 else (u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))"
 { fix i assume "i:I"
 { assume "e i = 0" hence "q i : S i" using `i:I` p_def q_def by auto }
 moreover
 { assume "e i ~= 0"
 hence "q i : S i" using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
 mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
 assms q_def e_def `i:I` `e i ~= 0` xc yc uv by auto
 } ultimately have "q i : S i" by auto
 } hence qs: "!i:I. q i : S i" by auto
 { fix i assume "i:I"
 { assume "e i = 0"
 have ge: "u * (c i) >= 0 & v * (d i) >= 0" using xc yc uv `i:I` by (simp add: mult_nonneg_nonneg)
 moreover hence "u * (c i) <= 0 & v * (d i) <= 0" using `e i = 0` e_def `i:I` by simp
 ultimately have "u * (c i) = 0 & v * (d i) = 0" by auto
 hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
 using `e i = 0` by auto
 }
 moreover
 using q_def by auto
 hence "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
 = (e i) *\<^sub>R (q i)" by auto
 hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
 using `e i ~= 0` by (simp add: algebra_simps)
 } ultimately have
 "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by blast
 } hence *: "!i:I.
 (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by auto
 have "u *\<^sub>R x + v *\<^sub>R y =
 setsum (%i. (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i)) I"
 using xc yc by (simp add: algebra_simps scaleR_right.setsum setsum_addf)
 also have "...=setsum (%i. (e i) *\<^sub>R (q i)) I" using * by auto
 finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (%i. (e i) *\<^sub>R (q i)) I" by auto
 hence "u *\<^sub>R x + v *\<^sub>R y : ?rhs" using ge0 sum1 qs by auto
 } hence "convex ?rhs" unfolding convex_def by auto
 from this show ?thesis using `?lhs >= ?rhs` *
 hull_minimal[of "Union (S ` I)" "?rhs" "convex"] by blast
 qed
 lemma convex_hull_union_two:
 fixes S T :: "('m::euclidean_space) set"
 have "Union (s ` I) = S Un T" using s_def I_def by auto
 hence "convex hull (Union (s ` I)) = convex hull (S Un T)" by auto
 moreover have "convex hull Union (s ` I) =
 {SUM i:I. c i *\<^sub>R sa i |c sa. (ALL i:I. 0 <= c i) & setsum c I = 1 & (ALL i:I. sa i : s i)}"
 apply (subst convex_hull_finite_union[of I s]) using assms s_def I_def by auto
 moreover have
 "{SUM i:I. c i *\<^sub>R sa i |c sa. (ALL i:I. 0 <= c i) & setsum c I = 1 & (ALL i:I. sa i : s i)} <=
 ?rhs"
 using s_def I_def by auto
 ultimately have "?lhs<=?rhs" by auto
 { fix x assume "x : ?rhs"
 from this obtain u v s t
 where *: "x=u *\<^sub>R s + v *\<^sub>R t & u>=0 & v>=0 & u+v=1 & s:S & t:T" by auto
 hence "x : convex hull {s,t}" using convex_hull_2[of s t] by auto
 hence "x : convex hull (S Un T)" using * hull_mono[of "{s, t}" "S Un T"] by auto
 } hence "?lhs >= ?rhs" by blast
 from this show ?thesis using `?lhs<=?rhs` by auto

changeset 49531	8d68162b7826
parent 49530	7faf67b411b4
child 49962	a8cc904a6820