(* Title: HOL/Analysis/Convex.thy Author: L C Paulson, University of Cambridge Author: Robert Himmelmann, TU Muenchen Author: Bogdan Grechuk, University of Edinburgh Author: Armin Heller, TU Muenchen Author: Johannes Hoelzl, TU Muenchen *) section ‹Convex Sets and Functions› theory Convex imports Affine "HOL-Library.Set_Algebras" begin subsection ‹Convex Sets› definition✐‹tag important› convex :: "'a::real_vector set ⇒ bool" where "convex s ⟷ (∀x∈s. ∀y∈s. ∀u≥0. ∀v≥0. u + v = 1 ⟶ u *⇩_{R}x + v *⇩_{R}y ∈ s)" lemma convexI: assumes "⋀x y u v. x ∈ s ⟹ y ∈ s ⟹ 0 ≤ u ⟹ 0 ≤ v ⟹ u + v = 1 ⟹ u *⇩_{R}x + v *⇩_{R}y ∈ s" shows "convex s" using assms unfolding convex_def by fast lemma convexD: assumes "convex s" and "x ∈ s" and "y ∈ s" and "0 ≤ u" and "0 ≤ v" and "u + v = 1" shows "u *⇩_{R}x + v *⇩_{R}y ∈ s" using assms unfolding convex_def by fast lemma convex_alt: "convex s ⟷ (∀x∈s. ∀y∈s. ∀u. 0 ≤ u ∧ u ≤ 1 ⟶ ((1 - u) *⇩_{R}x + u *⇩_{R}y) ∈ s)" (is "_ ⟷ ?alt") proof show "convex s" if alt: ?alt proof - { fix x y and u v :: real assume mem: "x ∈ s" "y ∈ s" assume "0 ≤ u" "0 ≤ v" moreover assume "u + v = 1" then have "u = 1 - v" by auto ultimately have "u *⇩_{R}x + v *⇩_{R}y ∈ s" using alt [rule_format, OF mem] by auto } then show ?thesis unfolding convex_def by auto qed show ?alt if "convex s" using that by (auto simp: convex_def) qed lemma convexD_alt: assumes "convex s" "a ∈ s" "b ∈ s" "0 ≤ u" "u ≤ 1" shows "((1 - u) *⇩_{R}a + u *⇩_{R}b) ∈ s" using assms unfolding convex_alt by auto lemma mem_convex_alt: assumes "convex S" "x ∈ S" "y ∈ S" "u ≥ 0" "v ≥ 0" "u + v > 0" shows "((u/(u+v)) *⇩_{R}x + (v/(u+v)) *⇩_{R}y) ∈ S" using assms by (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric]) lemma convex_empty[intro,simp]: "convex {}" unfolding convex_def by simp lemma convex_singleton[intro,simp]: "convex {a}" unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric]) lemma convex_UNIV[intro,simp]: "convex UNIV" unfolding convex_def by auto lemma convex_Inter: "(⋀s. s∈f ⟹ convex s) ⟹ convex(⋂f)" unfolding convex_def by auto lemma convex_Int: "convex s ⟹ convex t ⟹ convex (s ∩ t)" unfolding convex_def by auto lemma convex_INT: "(⋀i. i ∈ A ⟹ convex (B i)) ⟹ convex (⋂i∈A. B i)" unfolding convex_def by auto lemma convex_Times: "convex s ⟹ convex t ⟹ convex (s × t)" unfolding convex_def by auto lemma convex_halfspace_le: "convex {x. inner a x ≤ b}" unfolding convex_def by (auto simp: inner_add intro!: convex_bound_le) lemma convex_halfspace_ge: "convex {x. inner a x ≥ b}" proof - have *: "{x. inner a x ≥ b} = {x. inner (-a) x ≤ -b}" by auto show ?thesis unfolding * using convex_halfspace_le[of "-a" "-b"] by auto qed lemma convex_halfspace_abs_le: "convex {x. ¦inner a x¦ ≤ b}" proof - have *: "{x. ¦inner a x¦ ≤ b} = {x. inner a x ≤ b} ∩ {x. -b ≤ inner a x}" by auto show ?thesis unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le) qed lemma convex_hyperplane: "convex {x. inner a x = b}" proof - have *: "{x. inner a x = b} = {x. inner a x ≤ b} ∩ {x. inner a x ≥ b}" by auto show ?thesis using convex_halfspace_le convex_halfspace_ge by (auto intro!: convex_Int simp: *) qed lemma convex_halfspace_lt: "convex {x. inner a x < b}" unfolding convex_def by (auto simp: convex_bound_lt inner_add) lemma convex_halfspace_gt: "convex {x. inner a x > b}" using convex_halfspace_lt[of "-a" "-b"] by auto lemma convex_halfspace_Re_ge: "convex {x. Re x ≥ b}" using convex_halfspace_ge[of b "1::complex"] by simp lemma convex_halfspace_Re_le: "convex {x. Re x ≤ b}" using convex_halfspace_le[of "1::complex" b] by simp lemma convex_halfspace_Im_ge: "convex {x. Im x ≥ b}" using convex_halfspace_ge[of b 𝗂] by simp lemma convex_halfspace_Im_le: "convex {x. Im x ≤ b}" using convex_halfspace_le[of 𝗂 b] by simp lemma convex_halfspace_Re_gt: "convex {x. Re x > b}" using convex_halfspace_gt[of b "1::complex"] by simp lemma convex_halfspace_Re_lt: "convex {x. Re x < b}" using convex_halfspace_lt[of "1::complex" b] by simp lemma convex_halfspace_Im_gt: "convex {x. Im x > b}" using convex_halfspace_gt[of b 𝗂] by simp lemma convex_halfspace_Im_lt: "convex {x. Im x < b}" using convex_halfspace_lt[of 𝗂 b] by simp lemma convex_real_interval [iff]: fixes a b :: "real" shows "convex {a..}" and "convex {..b}" and "convex {a<..}" and "convex {..<b}" and "convex {a..b}" and "convex {a<..b}" and "convex {a..<b}" and "convex {a<..<b}" proof - have "{a..} = {x. a ≤ inner 1 x}" by auto then show 1: "convex {a..}" by (simp only: convex_halfspace_ge) have "{..b} = {x. inner 1 x ≤ b}" by auto then show 2: "convex {..b}" by (simp only: convex_halfspace_le) have "{a<..} = {x. a < inner 1 x}" by auto then show 3: "convex {a<..}" by (simp only: convex_halfspace_gt) have "{..<b} = {x. inner 1 x < b}" by auto then show 4: "convex {..<b}" by (simp only: convex_halfspace_lt) have "{a..b} = {a..} ∩ {..b}" by auto then show "convex {a..b}" by (simp only: convex_Int 1 2) have "{a<..b} = {a<..} ∩ {..b}" by auto then show "convex {a<..b}" by (simp only: convex_Int 3 2) have "{a..<b} = {a..} ∩ {..<b}" by auto then show "convex {a..<b}" by (simp only: convex_Int 1 4) have "{a<..<b} = {a<..} ∩ {..<b}" by auto then show "convex {a<..<b}" by (simp only: convex_Int 3 4) qed lemma convex_Reals: "convex ℝ" by (simp add: convex_def scaleR_conv_of_real) subsection✐‹tag unimportant› ‹Explicit expressions for convexity in terms of arbitrary sums› lemma convex_sum: fixes C :: "'a::real_vector set" assumes "finite S" and "convex C" and "(∑ i ∈ S. a i) = 1" assumes "⋀i. i ∈ S ⟹ a i ≥ 0" and "⋀i. i ∈ S ⟹ y i ∈ C" shows "(∑ j ∈ S. a j *⇩_{R}y j) ∈ C" using assms(1,3,4,5) proof (induct arbitrary: a set: finite) case empty then show ?case by simp next case (insert i S) note IH = this(3) have "a i + sum a S = 1" and "0 ≤ a i" and "∀j∈S. 0 ≤ a j" and "y i ∈ C" and "∀j∈S. y j ∈ C" using insert.hyps(1,2) insert.prems by simp_all then have "0 ≤ sum a S" by (simp add: sum_nonneg) have "a i *⇩_{R}y i + (∑j∈S. a j *⇩_{R}y j) ∈ C" proof (cases "sum a S = 0") case True with ‹a i + sum a S = 1› have "a i = 1" by simp from sum_nonneg_0 [OF ‹finite S› _ True] ‹∀j∈S. 0 ≤ a j› have "∀j∈S. a j = 0" by simp show ?thesis using ‹a i = 1› and ‹∀j∈S. a j = 0› and ‹y i ∈ C› by simp next case False with ‹0 ≤ sum a S› have "0 < sum a S" by simp then have "(∑j∈S. (a j / sum a S) *⇩_{R}y j) ∈ C" using ‹∀j∈S. 0 ≤ a j› and ‹∀j∈S. y j ∈ C› by (simp add: IH sum_divide_distrib [symmetric]) from ‹convex C› and ‹y i ∈ C› and this and ‹0 ≤ a i› and ‹0 ≤ sum a S› and ‹a i + sum a S = 1› have "a i *⇩_{R}y i + sum a S *⇩_{R}(∑j∈S. (a j / sum a S) *⇩_{R}y j) ∈ C" by (rule convexD) then show ?thesis by (simp add: scaleR_sum_right False) qed then show ?case using ‹finite S› and ‹i ∉ S› by simp qed lemma convex: "convex S ⟷ (∀(k::nat) u x. (∀i. 1≤i ∧ i≤k ⟶ 0 ≤ u i ∧ x i ∈S) ∧ (sum u {1..k} = 1) ⟶ sum (λi. u i *⇩_{R}x i) {1..k} ∈ S)" proof safe fix k :: nat fix u :: "nat ⇒ real" fix x assume "convex S" "∀i. 1 ≤ i ∧ i ≤ k ⟶ 0 ≤ u i ∧ x i ∈ S" "sum u {1..k} = 1" with convex_sum[of "{1 .. k}" S] show "(∑j∈{1 .. k}. u j *⇩_{R}x j) ∈ S" by auto next assume *: "∀k u x. (∀ i :: nat. 1 ≤ i ∧ i ≤ k ⟶ 0 ≤ u i ∧ x i ∈ S) ∧ sum u {1..k} = 1 ⟶ (∑i = 1..k. u i *⇩_{R}(x i :: 'a)) ∈ S" { fix μ :: real fix x y :: 'a assume xy: "x ∈ S" "y ∈ S" assume mu: "μ ≥ 0" "μ ≤ 1" let ?u = "λi. if (i :: nat) = 1 then μ else 1 - μ" let ?x = "λi. if (i :: nat) = 1 then x else y" have "{1 :: nat .. 2} ∩ - {x. x = 1} = {2}" by auto then have card: "card ({1 :: nat .. 2} ∩ - {x. x = 1}) = 1" by simp then have "sum ?u {1 .. 2} = 1" using sum.If_cases[of "{(1 :: nat) .. 2}" "λ x. x = 1" "λ x. μ" "λ x. 1 - μ"] by auto with *[rule_format, of "2" ?u ?x] have S: "(∑j ∈ {1..2}. ?u j *⇩_{R}?x j) ∈ S" using mu xy by auto have grarr: "(∑j ∈ {Suc (Suc 0)..2}. ?u j *⇩_{R}?x j) = (1 - μ) *⇩_{R}y" using sum.atLeast_Suc_atMost[of "Suc (Suc 0)" 2 "λ j. (1 - μ) *⇩_{R}y"] by auto from sum.atLeast_Suc_atMost[of "Suc 0" 2 "λ j. ?u j *⇩_{R}?x j", simplified this] have "(∑j ∈ {1..2}. ?u j *⇩_{R}?x j) = μ *⇩_{R}x + (1 - μ) *⇩_{R}y" by auto then have "(1 - μ) *⇩_{R}y + μ *⇩_{R}x ∈ S" using S by (auto simp: add.commute) } then show "convex S" unfolding convex_alt by auto qed lemma convex_explicit: fixes S :: "'a::real_vector set" shows "convex S ⟷ (∀t u. finite t ∧ t ⊆ S ∧ (∀x∈t. 0 ≤ u x) ∧ sum u t = 1 ⟶ sum (λx. u x *⇩_{R}x) t ∈ S)" proof safe fix t fix u :: "'a ⇒ real" assume "convex S" and "finite t" and "t ⊆ S" "∀x∈t. 0 ≤ u x" "sum u t = 1" then show "(∑x∈t. u x *⇩_{R}x) ∈ S" using convex_sum[of t S u "λ x. x"] by auto next assume *: "∀t. ∀ u. finite t ∧ t ⊆ S ∧ (∀x∈t. 0 ≤ u x) ∧ sum u t = 1 ⟶ (∑x∈t. u x *⇩_{R}x) ∈ S" show "convex S" unfolding convex_alt proof safe fix x y fix μ :: real assume **: "x ∈ S" "y ∈ S" "0 ≤ μ" "μ ≤ 1" show "(1 - μ) *⇩_{R}x + μ *⇩_{R}y ∈ S" proof (cases "x = y") case False then show ?thesis using *[rule_format, of "{x, y}" "λ z. if z = x then 1 - μ else μ"] ** by auto next case True then show ?thesis using *[rule_format, of "{x, y}" "λ z. 1"] ** by (auto simp: field_simps real_vector.scale_left_diff_distrib) qed qed qed lemma convex_finite: assumes "finite S" shows "convex S ⟷ (∀u. (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ⟶ sum (λx. u x *⇩_{R}x) S ∈ S)" (is "?lhs = ?rhs") proof { have if_distrib_arg: "⋀P f g x. (if P then f else g) x = (if P then f x else g x)" by simp fix T :: "'a set" and u :: "'a ⇒ real" assume sum: "∀u. (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ⟶ (∑x∈S. u x *⇩_{R}x) ∈ S" assume *: "∀x∈T. 0 ≤ u x" "sum u T = 1" assume "T ⊆ S" then have "S ∩ T = T" by auto with sum[THEN spec[where x="λx. if x∈T then u x else 0"]] * have "(∑x∈T. u x *⇩_{R}x) ∈ S" by (auto simp: assms sum.If_cases if_distrib if_distrib_arg) } moreover assume ?rhs ultimately show ?lhs unfolding convex_explicit by auto qed (auto simp: convex_explicit assms) subsection ‹Convex Functions on a Set› definition✐‹tag important› convex_on :: "'a::real_vector set ⇒ ('a ⇒ real) ⇒ bool" where "convex_on S f ⟷ (∀x∈S. ∀y∈S. ∀u≥0. ∀v≥0. u + v = 1 ⟶ f (u *⇩_{R}x + v *⇩_{R}y) ≤ u * f x + v * f y)" definition✐‹tag important› concave_on :: "'a::real_vector set ⇒ ('a ⇒ real) ⇒ bool" where "concave_on S f ≡ convex_on S (λx. - f x)" lemma concave_on_iff: "concave_on S f ⟷ (∀x∈S. ∀y∈S. ∀u≥0. ∀v≥0. u + v = 1 ⟶ f (u *⇩_{R}x + v *⇩_{R}y) ≥ u * f x + v * f y)" by (auto simp: concave_on_def convex_on_def algebra_simps) lemma convex_onI [intro?]: assumes "⋀t x y. t > 0 ⟹ t < 1 ⟹ x ∈ A ⟹ y ∈ A ⟹ f ((1 - t) *⇩_{R}x + t *⇩_{R}y) ≤ (1 - t) * f x + t * f y" shows "convex_on A f" unfolding convex_on_def proof clarify fix x y fix u v :: real assume A: "x ∈ A" "y ∈ A" "u ≥ 0" "v ≥ 0" "u + v = 1" from A(5) have [simp]: "v = 1 - u" by (simp add: algebra_simps) from A(1-4) show "f (u *⇩_{R}x + v *⇩_{R}y) ≤ u * f x + v * f y" using assms[of u y x] by (cases "u = 0 ∨ u = 1") (auto simp: algebra_simps) qed lemma convex_on_linorderI [intro?]: fixes A :: "('a::{linorder,real_vector}) set" assumes "⋀t x y. t > 0 ⟹ t < 1 ⟹ x ∈ A ⟹ y ∈ A ⟹ x < y ⟹ f ((1 - t) *⇩_{R}x + t *⇩_{R}y) ≤ (1 - t) * f x + t * f y" shows "convex_on A f" proof fix x y fix t :: real assume A: "x ∈ A" "y ∈ A" "t > 0" "t < 1" with assms [of t x y] assms [of "1 - t" y x] show "f ((1 - t) *⇩_{R}x + t *⇩_{R}y) ≤ (1 - t) * f x + t * f y" by (cases x y rule: linorder_cases) (auto simp: algebra_simps) qed lemma convex_onD: assumes "convex_on A f" shows "⋀t x y. t ≥ 0 ⟹ t ≤ 1 ⟹ x ∈ A ⟹ y ∈ A ⟹ f ((1 - t) *⇩_{R}x + t *⇩_{R}y) ≤ (1 - t) * f x + t * f y" using assms by (auto simp: convex_on_def) lemma convex_onD_Icc: assumes "convex_on {x..y} f" "x ≤ (y :: _ :: {real_vector,preorder})" shows "⋀t. t ≥ 0 ⟹ t ≤ 1 ⟹ f ((1 - t) *⇩_{R}x + t *⇩_{R}y) ≤ (1 - t) * f x + t * f y" using assms(2) by (intro convex_onD [OF assms(1)]) simp_all lemma convex_on_subset: "convex_on t f ⟹ S ⊆ t ⟹ convex_on S f" unfolding convex_on_def by auto lemma convex_on_add [intro]: assumes "convex_on S f" and "convex_on S g" shows "convex_on S (λx. f x + g x)" proof - { fix x y assume "x ∈ S" "y ∈ S" moreover fix u v :: real assume "0 ≤ u" "0 ≤ v" "u + v = 1" ultimately have "f (u *⇩_{R}x + v *⇩_{R}y) + g (u *⇩_{R}x + v *⇩_{R}y) ≤ (u * f x + v * f y) + (u * g x + v * g y)" using assms unfolding convex_on_def by (auto simp: add_mono) then have "f (u *⇩_{R}x + v *⇩_{R}y) + g (u *⇩_{R}x + v *⇩_{R}y) ≤ u * (f x + g x) + v * (f y + g y)" by (simp add: field_simps) } then show ?thesis unfolding convex_on_def by auto qed lemma convex_on_cmul [intro]: fixes c :: real assumes "0 ≤ c" and "convex_on S f" shows "convex_on S (λx. c * f x)" proof - have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" for u c fx v fy :: real by (simp add: field_simps) show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)] unfolding convex_on_def and * by auto qed lemma convex_lower: assumes "convex_on S f" and "x ∈ S" and "y ∈ S" and "0 ≤ u" and "0 ≤ v" and "u + v = 1" shows "f (u *⇩_{R}x + v *⇩_{R}y) ≤ max (f x) (f y)" proof - let ?m = "max (f x) (f y)" have "u * f x + v * f y ≤ u * max (f x) (f y) + v * max (f x) (f y)" using assms(4,5) by (auto simp: mult_left_mono add_mono) also have "… = max (f x) (f y)" using assms(6) by (simp add: distrib_right [symmetric]) finally show ?thesis using assms unfolding convex_on_def by fastforce qed lemma convex_on_dist [intro]: fixes S :: "'a::real_normed_vector set" shows "convex_on S (λx. dist a x)" proof (auto simp: convex_on_def dist_norm) fix x y assume "x ∈ S" "y ∈ S" fix u v :: real assume "0 ≤ u" assume "0 ≤ v" assume "u + v = 1" have "a = u *⇩_{R}a + v *⇩_{R}a" unfolding scaleR_left_distrib[symmetric] and ‹u + v = 1› by simp then have *: "a - (u *⇩_{R}x + v *⇩_{R}y) = (u *⇩_{R}(a - x)) + (v *⇩_{R}(a - y))" by (auto simp: algebra_simps) show "norm (a - (u *⇩_{R}x + v *⇩_{R}y)) ≤ u * norm (a - x) + v * norm (a - y)" unfolding * using norm_triangle_ineq[of "u *⇩_{R}(a - x)" "v *⇩_{R}(a - y)"] using ‹0 ≤ u› ‹0 ≤ v› by auto qed subsection✐‹tag unimportant› ‹Arithmetic operations on sets preserve convexity› lemma convex_linear_image: assumes "linear f" and "convex S" shows "convex (f ` S)" proof - interpret f: linear f by fact from ‹convex S› show "convex (f ` S)" by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric]) qed lemma convex_linear_vimage: assumes "linear f" and "convex S" shows "convex (f -` S)" proof - interpret f: linear f by fact from ‹convex S› show "convex (f -` S)" by (simp add: convex_def f.add f.scaleR) qed lemma convex_scaling: assumes "convex S" shows "convex ((λx. c *⇩_{R}x) ` S)" proof - have "linear (λx. c *⇩_{R}x)" by (simp add: linearI scaleR_add_right) then show ?thesis using ‹convex S› by (rule convex_linear_image) qed lemma convex_scaled: assumes "convex S" shows "convex ((λx. x *⇩_{R}c) ` S)" proof - have "linear (λx. x *⇩_{R}c)" by (simp add: linearI scaleR_add_left) then show ?thesis using ‹convex S› by (rule convex_linear_image) qed lemma convex_negations: assumes "convex S" shows "convex ((λx. - x) ` S)" proof - have "linear (λx. - x)" by (simp add: linearI) then show ?thesis using ‹convex S› by (rule convex_linear_image) qed lemma convex_sums: assumes "convex S" and "convex T" shows "convex (⋃x∈ S. ⋃y ∈ T. {x + y})" proof - have "linear (λ(x, y). x + y)" by (auto intro: linearI simp: scaleR_add_right) with assms have "convex ((λ(x, y). x + y) ` (S × T))" by (intro convex_linear_image convex_Times) also have "((λ(x, y). x + y) ` (S × T)) = (⋃x∈ S. ⋃y ∈ T. {x + y})" by auto finally show ?thesis . qed lemma convex_differences: assumes "convex S" "convex T" shows "convex (⋃x∈ S. ⋃y ∈ T. {x - y})" proof - have "{x - y| x y. x ∈ S ∧ y ∈ T} = {x + y |x y. x ∈ S ∧ y ∈ uminus ` T}" by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff) then show ?thesis using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto qed lemma convex_translation: "convex ((+) a ` S)" if "convex S" proof - have "(⋃ x∈ {a}. ⋃y ∈ S. {x + y}) = (+) a ` S" by auto then show ?thesis using convex_sums [OF convex_singleton [of a] that] by auto qed lemma convex_translation_subtract: "convex ((λb. b - a) ` S)" if "convex S" using convex_translation [of S "- a"] that by (simp cong: image_cong_simp) lemma convex_affinity: assumes "convex S" shows "convex ((λx. a + c *⇩_{R}x) ` S)" proof - have "(λx. a + c *⇩_{R}x) ` S = (+) a ` (*⇩_{R}) c ` S" by auto then show ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed lemma convex_on_sum: fixes a :: "'a ⇒ real" and y :: "'a ⇒ 'b::real_vector" and f :: "'b ⇒ real" assumes "finite s" "s ≠ {}" and "convex_on C f" and "convex C" and "(∑ i ∈ s. a i) = 1" and "⋀i. i ∈ s ⟹ a i ≥ 0" and "⋀i. i ∈ s ⟹ y i ∈ C" shows "f (∑ i ∈ s. a i *⇩_{R}y i) ≤ (∑ i ∈ s. a i * f (y i))" using assms proof (induct s arbitrary: a rule: finite_ne_induct) case (singleton i) then have ai: "a i = 1" by auto then show ?case by auto next case (insert i s) then have "convex_on C f" by simp from this[unfolded convex_on_def, rule_format] have conv: "⋀x y μ. x ∈ C ⟹ y ∈ C ⟹ 0 ≤ μ ⟹ μ ≤ 1 ⟹ f (μ *⇩_{R}x + (1 - μ) *⇩_{R}y) ≤ μ * f x + (1 - μ) * f y" by simp show ?case proof (cases "a i = 1") case True then have "(∑ j ∈ s. a j) = 0" using insert by auto then have "⋀j. j ∈ s ⟹ a j = 0" using insert by (fastforce simp: sum_nonneg_eq_0_iff) then show ?thesis using insert by auto next case False from insert have yai: "y i ∈ C" "a i ≥ 0" by auto have fis: "finite (insert i s)" using insert by auto then have ai1: "a i ≤ 1" using sum_nonneg_leq_bound[of "insert i s" a] insert by simp then have "a i < 1" using False by auto then have i0: "1 - a i > 0" by auto let ?a = "λj. a j / (1 - a i)" have a_nonneg: "?a j ≥ 0" if "j ∈ s" for j using i0 insert that by fastforce have "(∑ j ∈ insert i s. a j) = 1" using insert by auto then have "(∑ j ∈ s. a j) = 1 - a i" using sum.insert insert by fastforce then have "(∑ j ∈ s. a j) / (1 - a i) = 1" using i0 by auto then have a1: "(∑ j ∈ s. ?a j) = 1" unfolding sum_divide_distrib by simp have "convex C" using insert by auto then have asum: "(∑ j ∈ s. ?a j *⇩_{R}y j) ∈ C" using insert convex_sum [OF ‹finite s› ‹convex C› a1 a_nonneg] by auto have asum_le: "f (∑ j ∈ s. ?a j *⇩_{R}y j) ≤ (∑ j ∈ s. ?a j * f (y j))" using a_nonneg a1 insert by blast have "f (∑ j ∈ insert i s. a j *⇩_{R}y j) = f ((∑ j ∈ s. a j *⇩_{R}y j) + a i *⇩_{R}y i)" using sum.insert[of s i "λ j. a j *⇩_{R}y j", OF ‹finite s› ‹i ∉ s›] insert by (auto simp only: add.commute) also have "… = f (((1 - a i) * inverse (1 - a i)) *⇩_{R}(∑ j ∈ s. a j *⇩_{R}y j) + a i *⇩_{R}y i)" using i0 by auto also have "… = f ((1 - a i) *⇩_{R}(∑ j ∈ s. (a j * inverse (1 - a i)) *⇩_{R}y j) + a i *⇩_{R}y i)" using scaleR_right.sum[of "inverse (1 - a i)" "λ j. a j *⇩_{R}y j" s, symmetric] by (auto simp: algebra_simps) also have "… = f ((1 - a i) *⇩_{R}(∑ j ∈ s. ?a j *⇩_{R}y j) + a i *⇩_{R}y i)" by (auto simp: divide_inverse) also have "… ≤ (1 - a i) *⇩_{R}f ((∑ j ∈ s. ?a j *⇩_{R}y j)) + a i * f (y i)" using conv[of "y i" "(∑ j ∈ s. ?a j *⇩_{R}y j)" "a i", OF yai(1) asum yai(2) ai1] by (auto simp: add.commute) also have "… ≤ (1 - a i) * (∑ j ∈ s. ?a j * f (y j)) + a i * f (y i)" using add_right_mono [OF mult_left_mono [of _ _ "1 - a i", OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp also have "… = (∑ j ∈ s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)" unfolding sum_distrib_left[of "1 - a i" "λ j. ?a j * f (y j)"] using i0 by auto also have "… = (∑ j ∈ s. a j * f (y j)) + a i * f (y i)" using i0 by auto also have "… = (∑ j ∈ insert i s. a j * f (y j))" using insert by auto finally show ?thesis by simp qed qed lemma convex_on_alt: fixes C :: "'a::real_vector set" shows "convex_on C f ⟷ (∀x ∈ C. ∀ y ∈ C. ∀ μ :: real. μ ≥ 0 ∧ μ ≤ 1 ⟶ f (μ *⇩_{R}x + (1 - μ) *⇩_{R}y) ≤ μ * f x + (1 - μ) * f y)" proof safe fix x y fix μ :: real assume *: "convex_on C f" "x ∈ C" "y ∈ C" "0 ≤ μ" "μ ≤ 1" from this[unfolded convex_on_def, rule_format] have "0 ≤ u ⟹ 0 ≤ v ⟹ u + v = 1 ⟹ f (u *⇩_{R}x + v *⇩_{R}y) ≤ u * f x + v * f y" for u v by auto from this [of "μ" "1 - μ", simplified] * show "f (μ *⇩_{R}x + (1 - μ) *⇩_{R}y) ≤ μ * f x + (1 - μ) * f y" by auto next assume *: "∀x∈C. ∀y∈C. ∀μ. 0 ≤ μ ∧ μ ≤ 1 ⟶ f (μ *⇩_{R}x + (1 - μ) *⇩_{R}y) ≤ μ * f x + (1 - μ) * f y" { fix x y fix u v :: real assume **: "x ∈ C" "y ∈ C" "u ≥ 0" "v ≥ 0" "u + v = 1" then have[simp]: "1 - u = v" by auto from *[rule_format, of x y u] have "f (u *⇩_{R}x + v *⇩_{R}y) ≤ u * f x + v * f y" using ** by auto } then show "convex_on C f" unfolding convex_on_def by auto qed lemma convex_on_diff: fixes f :: "real ⇒ real" assumes f: "convex_on I f" and I: "x ∈ I" "y ∈ I" and t: "x < t" "t < y" shows "(f x - f t) / (x - t) ≤ (f x - f y) / (x - y)" and "(f x - f y) / (x - y) ≤ (f t - f y) / (t - y)" proof - define a where "a ≡ (t - y) / (x - y)" with t have "0 ≤ a" "0 ≤ 1 - a" by (auto simp: field_simps) with f ‹x ∈ I› ‹y ∈ I› have cvx: "f (a * x + (1 - a) * y) ≤ a * f x + (1 - a) * f y" by (auto simp: convex_on_def) have "a * x + (1 - a) * y = a * (x - y) + y" by (simp add: field_simps) also have "… = t" unfolding a_def using ‹x < t› ‹t < y› by simp finally have "f t ≤ a * f x + (1 - a) * f y" using cvx by simp also have "… = a * (f x - f y) + f y" by (simp add: field_simps) finally have "f t - f y ≤ a * (f x - f y)" by simp with t show "(f x - f t) / (x - t) ≤ (f x - f y) / (x - y)" by (simp add: le_divide_eq divide_le_eq field_simps a_def) with t show "(f x - f y) / (x - y) ≤ (f t - f y) / (t - y)" by (simp add: le_divide_eq divide_le_eq field_simps) qed lemma pos_convex_function: fixes f :: "real ⇒ real" assumes "convex C" and leq: "⋀x y. x ∈ C ⟹ y ∈ C ⟹ f' x * (y - x) ≤ f y - f x" shows "convex_on C f" unfolding convex_on_alt using assms proof safe fix x y μ :: real let ?x = "μ *⇩_{R}x + (1 - μ) *⇩_{R}y" assume *: "convex C" "x ∈ C" "y ∈ C" "μ ≥ 0" "μ ≤ 1" then have "1 - μ ≥ 0" by auto then have xpos: "?x ∈ C" using * unfolding convex_alt by fastforce have geq: "μ * (f x - f ?x) + (1 - μ) * (f y - f ?x) ≥ μ * f' ?x * (x - ?x) + (1 - μ) * f' ?x * (y - ?x)" using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] ‹μ ≥ 0›] mult_left_mono [OF leq [OF xpos *(3)] ‹1 - μ ≥ 0›]] by auto then have "μ * f x + (1 - μ) * f y - f ?x ≥ 0" by (auto simp: field_simps) then show "f (μ *⇩_{R}x + (1 - μ) *⇩_{R}y) ≤ μ * f x + (1 - μ) * f y" by auto qed lemma atMostAtLeast_subset_convex: fixes C :: "real set" assumes "convex C" and "x ∈ C" "y ∈ C" "x < y" shows "{x .. y} ⊆ C" proof safe fix z assume z: "z ∈ {x .. y}" have less: "z ∈ C" if *: "x < z" "z < y" proof - let ?μ = "(y - z) / (y - x)" have "0 ≤ ?μ" "?μ ≤ 1" using assms * by (auto simp: field_simps) then have comb: "?μ * x + (1 - ?μ) * y ∈ C" using assms iffD1[OF convex_alt, rule_format, of C y x ?μ] by (simp add: algebra_simps) have "?μ * x + (1 - ?μ) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y" by (auto simp: field_simps) also have "… = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)" using assms by (simp only: add_divide_distrib) (auto simp: field_simps) also have "… = z" using assms by (auto simp: field_simps) finally show ?thesis using comb by auto qed show "z ∈ C" using z less assms by (auto simp: le_less) qed lemma f''_imp_f': fixes f :: "real ⇒ real" assumes "convex C" and f': "⋀x. x ∈ C ⟹ DERIV f x :> (f' x)" and f'': "⋀x. x ∈ C ⟹ DERIV f' x :> (f'' x)" and pos: "⋀x. x ∈ C ⟹ f'' x ≥ 0" and x: "x ∈ C" and y: "y ∈ C" shows "f' x * (y - x) ≤ f y - f x" using assms proof - have less_imp: "f y - f x ≥ f' x * (y - x)" "f' y * (x - y) ≤ f x - f y" if *: "x ∈ C" "y ∈ C" "y > x" for x y :: real proof - from * have ge: "y - x > 0" "y - x ≥ 0" by auto from * have le: "x - y < 0" "x - y ≤ 0" by auto then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1" using subsetD[OF atMostAtLeast_subset_convex[OF ‹convex C› ‹x ∈ C› ‹y ∈ C› ‹x < y›], THEN f', THEN MVT2[OF ‹x < y›, rule_format, unfolded atLeastAtMost_iff[symmetric]]] by auto then have "z1 ∈ C" using atMostAtLeast_subset_convex ‹convex C› ‹x ∈ C› ‹y ∈ C› ‹x < y› by fastforce from z1 have z1': "f x - f y = (x - y) * f' z1" by (simp add: field_simps) obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2" using subsetD[OF atMostAtLeast_subset_convex[OF ‹convex C› ‹x ∈ C› ‹z1 ∈ C› ‹x < z1›], THEN f'', THEN MVT2[OF ‹x < z1›, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1 by auto obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3" using subsetD[OF atMostAtLeast_subset_convex[OF ‹convex C› ‹z1 ∈ C› ‹y ∈ C› ‹z1 < y›], THEN f'', THEN MVT2[OF ‹z1 < y›, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1 by auto have "f' y - (f x - f y) / (x - y) = f' y - f' z1" using * z1' by auto also have "… = (y - z1) * f'' z3" using z3 by auto finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3" by simp have A': "y - z1 ≥ 0" using z1 by auto have "z3 ∈ C" using z3 * atMostAtLeast_subset_convex ‹convex C› ‹x ∈ C› ‹z1 ∈ C› ‹x < z1› by fastforce then have B': "f'' z3 ≥ 0" using assms by auto from A' B' have "(y - z1) * f'' z3 ≥ 0" by auto from cool' this have "f' y - (f x - f y) / (x - y) ≥ 0" by auto from mult_right_mono_neg[OF this le(2)] have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) ≤ 0 * (x - y)" by (simp add: algebra_simps) then have "f' y * (x - y) - (f x - f y) ≤ 0" using le by auto then have res: "f' y * (x - y) ≤ f x - f y" by auto have "(f y - f x) / (y - x) - f' x = f' z1 - f' x" using * z1 by auto also have "… = (z1 - x) * f'' z2" using z2 by auto finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2" by simp have A: "z1 - x ≥ 0" using z1 by auto have "z2 ∈ C" using z2 z1 * atMostAtLeast_subset_convex ‹convex C› ‹z1 ∈ C› ‹y ∈ C› ‹z1 < y› by fastforce then have B: "f'' z2 ≥ 0" using assms by auto from A B have "(z1 - x) * f'' z2 ≥ 0" by auto with cool have "(f y - f x) / (y - x) - f' x ≥ 0" by auto from mult_right_mono[OF this ge(2)] have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) ≥ 0 * (y - x)" by (simp add: algebra_simps) then have "f y - f x - f' x * (y - x) ≥ 0" using ge by auto then show "f y - f x ≥ f' x * (y - x)" "f' y * (x - y) ≤ f x - f y" using res by auto qed show ?thesis proof (cases "x = y") case True with x y show ?thesis by auto next case False with less_imp x y show ?thesis by (auto simp: neq_iff) qed qed lemma f''_ge0_imp_convex: fixes f :: "real ⇒ real" assumes conv: "convex C" and f': "⋀x. x ∈ C ⟹ DERIV f x :> (f' x)" and f'': "⋀x. x ∈ C ⟹ DERIV f' x :> (f'' x)" and 0: "⋀x. x ∈ C ⟹ f'' x ≥ 0" shows "convex_on C f" using f''_imp_f'[OF conv f' f'' 0] assms pos_convex_function by fastforce lemma f''_le0_imp_concave: fixes f :: "real ⇒ real" assumes "convex C" and "⋀x. x ∈ C ⟹ DERIV f x :> (f' x)" and "⋀x. x ∈ C ⟹ DERIV f' x :> (f'' x)" and "⋀x. x ∈ C ⟹ f'' x ≤ 0" shows "concave_on C f" unfolding concave_on_def by (rule assms f''_ge0_imp_convex derivative_eq_intros | simp)+ lemma log_concave: fixes b :: real assumes "b > 1" shows "concave_on {0<..} (λ x. log b x)" using assms by (intro f''_le0_imp_concave derivative_eq_intros | simp)+ lemma ln_concave: "concave_on {0<..} ln" unfolding log_ln by (simp add: log_concave) lemma minus_log_convex: fixes b :: real assumes "b > 1" shows "convex_on {0 <..} (λ x. - log b x)" using assms concave_on_def log_concave by blast lemma powr_convex: assumes "p ≥ 1" shows "convex_on {0<..} (λx. x powr p)" using assms by (intro f''_ge0_imp_convex derivative_eq_intros | simp)+ lemma exp_convex: "convex_on UNIV exp" by (intro f''_ge0_imp_convex derivative_eq_intros | simp)+ subsection✐‹tag unimportant› ‹Convexity of real functions› lemma convex_on_realI: assumes "connected A" and "⋀x. x ∈ A ⟹ (f has_real_derivative f' x) (at x)" and "⋀x y. x ∈ A ⟹ y ∈ A ⟹ x ≤ y ⟹ f' x ≤ f' y" shows "convex_on A f" proof (rule convex_on_linorderI) fix t x y :: real assume t: "t > 0" "t < 1" assume xy: "x ∈ A" "y ∈ A" "x < y" define z where "z = (1 - t) * x + t * y" with ‹connected A› and xy have ivl: "{x..y} ⊆ A" using connected_contains_Icc by blast from xy t have xz: "z > x" by (simp add: z_def algebra_simps) have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps) also from xy t have "… > 0" by (intro mult_pos_pos) simp_all finally have yz: "z < y" by simp from assms xz yz ivl t have "∃ξ. ξ > x ∧ ξ < z ∧ f z - f x = (z - x) * f' ξ" by (intro MVT2) (auto intro!: assms(2)) then obtain ξ where ξ: "ξ > x" "ξ < z" "f' ξ = (f z - f x) / (z - x)" by auto from assms xz yz ivl t have "∃η. η > z ∧ η < y ∧ f y - f z = (y - z) * f' η" by (intro MVT2) (auto intro!: assms(2)) then obtain η where η: "η > z" "η < y" "f' η = (f y - f z) / (y - z)" by auto from η(3) have "(f y - f z) / (y - z) = f' η" .. also from ξ η ivl have "ξ ∈ A" "η ∈ A" by auto with ξ η have "f' η ≥ f' ξ" by (intro assms(3)) auto also from ξ(3) have "f' ξ = (f z - f x) / (z - x)" . finally have "(f y - f z) * (z - x) ≥ (f z - f x) * (y - z)" using xz yz by (simp add: field_simps) also have "z - x = t * (y - x)" by (simp add: z_def algebra_simps) also have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps) finally have "(f y - f z) * t ≥ (f z - f x) * (1 - t)" using xy by simp then show "(1 - t) * f x + t * f y ≥ f ((1 - t) *⇩_{R}x + t *⇩_{R}y)" by (simp add: z_def algebra_simps) qed lemma convex_on_inverse: assumes "A ⊆ {0<..}" shows "convex_on A (inverse :: real ⇒ real)" proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "λx. -inverse (x^2)"]) fix u v :: real assume "u ∈ {0<..}" "v ∈ {0<..}" "u ≤ v" with assms show "-inverse (u^2) ≤ -inverse (v^2)" by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all) qed (insert assms, auto intro!: derivative_eq_intros simp: field_split_simps power2_eq_square) lemma convex_onD_Icc': assumes "convex_on {x..y} f" "c ∈ {x..y}" defines "d ≡ y - x" shows "f c ≤ (f y - f x) / d * (c - x) + f x" proof (cases x y rule: linorder_cases) case less then have d: "d > 0" by (simp add: d_def) from assms(2) less have A: "0 ≤ (c - x) / d" "(c - x) / d ≤ 1" by (simp_all add: d_def field_split_simps) have "f c = f (x + (c - x) * 1)" by simp also from less have "1 = ((y - x) / d)" by (simp add: d_def) also from d have "x + (c - x) * … = (1 - (c - x) / d) *⇩_{R}x + ((c - x) / d) *⇩_{R}y" by (simp add: field_simps) also have "f … ≤ (1 - (c - x) / d) * f x + (c - x) / d * f y" using assms less by (intro convex_onD_Icc) simp_all also from d have "… = (f y - f x) / d * (c - x) + f x" by (simp add: field_simps) finally show ?thesis . qed (insert assms(2), simp_all) lemma convex_onD_Icc'': assumes "convex_on {x..y} f" "c ∈ {x..y}" defines "d ≡ y - x" shows "f c ≤ (f x - f y) / d * (y - c) + f y" proof (cases x y rule: linorder_cases) case less then have d: "d > 0" by (simp add: d_def) from assms(2) less have A: "0 ≤ (y - c) / d" "(y - c) / d ≤ 1" by (simp_all add: d_def field_split_simps) have "f c = f (y - (y - c) * 1)" by simp also from less have "1 = ((y - x) / d)" by (simp add: d_def) also from d have "y - (y - c) * … = (1 - (1 - (y - c) / d)) *⇩_{R}x + (1 - (y - c) / d) *⇩_{R}y" by (simp add: field_simps) also have "f … ≤ (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y" using assms less by (intro convex_onD_Icc) (simp_all add: field_simps) also from d have "… = (f x - f y) / d * (y - c) + f y" by (simp add: field_simps) finally show ?thesis . qed (insert assms(2), simp_all) subsection ‹Some inequalities› lemma Youngs_inequality_0: fixes a::real assumes "0 ≤ α" "0 ≤ β" "α+β = 1" "a>0" "b>0" shows "a powr α * b powr β ≤ α*a + β*b" proof - have "α * ln a + β * ln b ≤ ln (α * a + β * b)" using assms ln_concave by (simp add: concave_on_iff) moreover have "0 < α * a + β * b" using assms by (smt (verit) mult_pos_pos split_mult_pos_le) ultimately show ?thesis using assms by (simp add: powr_def mult_exp_exp flip: ln_ge_iff) qed lemma Youngs_inequality: fixes p::real assumes "p>1" "q>1" "1/p + 1/q = 1" "a≥0" "b≥0" shows "a * b ≤ a powr p / p + b powr q / q" proof (cases "a=0 ∨ b=0") case False then show ?thesis using Youngs_inequality_0 [of "1/p" "1/q" "a powr p" "b powr q"] assms by (simp add: powr_powr) qed (use assms in auto) lemma Cauchy_Schwarz_ineq_sum: fixes a :: "'a ⇒ 'b::linordered_field" shows "(∑i∈I. a i * b i)⇧^{2}≤ (∑i∈I. (a i)⇧^{2}) * (∑i∈I. (b i)⇧^{2})" proof (cases "(∑i∈I. (b i)⇧^{2}) > 0") case False then consider "⋀i. i∈I ⟹ b i = 0" | "infinite I" by (metis (mono_tags, lifting) sum_pos2 zero_le_power2 zero_less_power2) thus ?thesis by fastforce next case True define r where "r ≡ (∑i∈I. a i * b i) / (∑i∈I. (b i)⇧^{2})" with True have *: "(∑i∈I. a i * b i) = r * (∑i∈I. (b i)⇧^{2})" by simp have "0 ≤ (∑i∈I. (a i - r * b i)⇧^{2})" by (meson sum_nonneg zero_le_power2) also have "... = (∑i∈I. (a i)⇧^{2}) - 2 * r * (∑i∈I. a i * b i) + r⇧^{2}* (∑i∈I. (b i)⇧^{2})" by (simp add: algebra_simps power2_eq_square sum_distrib_left flip: sum.distrib) also have "… = (∑i∈I. (a i)⇧^{2}) - (∑i∈I. a i * b i) * r" by (simp add: * power2_eq_square) also have "… = (∑i∈I. (a i)⇧^{2}) - ((∑i∈I. a i * b i))⇧^{2}/ (∑i∈I. (b i)⇧^{2})" by (simp add: r_def power2_eq_square) finally have "0 ≤ (∑i∈I. (a i)⇧^{2}) - ((∑i∈I. a i * b i))⇧^{2}/ (∑i∈I. (b i)⇧^{2})" . hence "((∑i∈I. a i * b i))⇧^{2}/ (∑i∈I. (b i)⇧^{2}) ≤ (∑i∈I. (a i)⇧^{2})" by (simp add: le_diff_eq) thus "((∑i∈I. a i * b i))⇧^{2}≤ (∑i∈I. (a i)⇧^{2}) * (∑i∈I. (b i)⇧^{2})" by (simp add: pos_divide_le_eq True) qed subsection ‹Misc related lemmas› lemma convex_translation_eq [simp]: "convex ((+) a ` s) ⟷ convex s" by (metis convex_translation translation_galois) lemma convex_translation_subtract_eq [simp]: "convex ((λb. b - a) ` s) ⟷ convex s" using convex_translation_eq [of "- a"] by (simp cong: image_cong_simp) lemma convex_linear_image_eq [simp]: fixes f :: "'a::real_vector ⇒ 'b::real_vector" shows "⟦linear f; inj f⟧ ⟹ convex (f ` s) ⟷ convex s" by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq) lemma vector_choose_size: assumes "0 ≤ c" obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c" proof - obtain a::'a where "a ≠ 0" using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce then show ?thesis by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms) qed lemma vector_choose_dist: assumes "0 ≤ c" obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c" by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size) lemma sum_delta'': fixes s::"'a::real_vector set" assumes "finite s" shows "(∑x∈s. (if y = x then f x else 0) *⇩_{R}x) = (if y∈s then (f y) *⇩_{R}y else 0)" proof - have *: "⋀x y. (if y = x then f x else (0::real)) *⇩_{R}x = (if x=y then (f x) *⇩_{R}x else 0)" by auto show ?thesis unfolding * using sum.delta[OF assms, of y "λx. f x *⇩_{R}x"] by auto qed subsection ‹Cones› definition✐‹tag important› cone :: "'a::real_vector set ⇒ bool" where "cone s ⟷ (∀x∈s. ∀c≥0. c *⇩_{R}x ∈ s)" lemma cone_empty[intro, simp]: "cone {}" unfolding cone_def by auto lemma cone_univ[intro, simp]: "cone UNIV" unfolding cone_def by auto lemma cone_Inter[intro]: "∀s∈f. cone s ⟹ cone (⋂f)" unfolding cone_def by auto lemma subspace_imp_cone: "subspace S ⟹ cone S" by (simp add: cone_def subspace_scale) subsubsection ‹Conic hull› lemma cone_cone_hull: "cone (cone hull S)" unfolding hull_def by auto lemma cone_hull_eq: "cone hull S = S ⟷ cone S" by (metis cone_cone_hull hull_same) lemma mem_cone: assumes "cone S" "x ∈ S" "c ≥ 0" shows "c *⇩_{R}x ∈ S" using assms cone_def[of S] by auto lemma cone_contains_0: assumes "cone S" shows "S ≠ {} ⟷ 0 ∈ S" using assms mem_cone by fastforce lemma cone_0: "cone {0}" unfolding cone_def by auto lemma cone_Union[intro]: "(∀s∈f. cone s) ⟶ cone (⋃f)" unfolding cone_def by blast lemma cone_iff: assumes "S ≠ {}" shows "cone S ⟷ 0 ∈ S ∧ (∀c. c > 0 ⟶ ((*⇩_{R}) c) ` S = S)" proof - { assume "cone S" { fix c :: real assume "c > 0" { fix x assume "x ∈ S" then have "x ∈ ((*⇩_{R}) c) ` S" unfolding image_def using ‹cone S› ‹c>0› mem_cone[of S x "1/c"] exI[of "(λt. t ∈ S ∧ x = c *⇩_{R}t)" "(1 / c) *⇩_{R}x"] by auto } moreover { fix x assume "x ∈ ((*⇩_{R}) c) ` S" then have "x ∈ S" using ‹0 < c› ‹cone S› mem_cone by fastforce } ultimately have "((*⇩_{R}) c) ` S = S" by blast } then have "0 ∈ S ∧ (∀c. c > 0 ⟶ ((*⇩_{R}) c) ` S = S)" using ‹cone S› cone_contains_0[of S] assms by auto } moreover { assume a: "0 ∈ S ∧ (∀c. c > 0 ⟶ ((*⇩_{R}) c) ` S = S)" { fix x assume "x ∈ S" fix c1 :: real assume "c1 ≥ 0" then have "c1 = 0 ∨ c1 > 0" by auto then have "c1 *⇩_{R}x ∈ S" using a ‹x ∈ S› by auto } then have "cone S" unfolding cone_def by auto } ultimately show ?thesis by blast qed lemma cone_hull_empty: "cone hull {} = {}" by (metis cone_empty cone_hull_eq) lemma cone_hull_empty_iff: "S = {} ⟷ cone hull S = {}" by (metis bot_least cone_hull_empty hull_subset xtrans(5)) lemma cone_hull_contains_0: "S ≠ {} ⟷ 0 ∈ cone hull S" using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] by auto lemma mem_cone_hull: assumes "x ∈ S" "c ≥ 0" shows "c *⇩_{R}x ∈ cone hull S" by (metis assms cone_cone_hull hull_inc mem_cone) proposition cone_hull_expl: "cone hull S = {c *⇩_{R}x | c x. c ≥ 0 ∧ x ∈ S}" (is "?lhs = ?rhs") proof - { fix x assume "x ∈ ?rhs" then obtain cx :: real and xx where x: "x = cx *⇩_{R}xx" "cx ≥ 0" "xx ∈ S" by auto fix c :: real assume c: "c ≥ 0" then have "c *⇩_{R}x = (c * cx) *⇩_{R}xx" using x by (simp add: algebra_simps) moreover have "c * cx ≥ 0" using c x by auto ultimately have "c *⇩_{R}x ∈ ?rhs" using x by auto } 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 *⇩_{R}x ∈ ?rhs" using zero_le_one by blast then have "x ∈ ?rhs" by auto } then have "S ⊆ ?rhs" by auto then have "?lhs ⊆ ?rhs" using ‹?rhs ∈ Collect cone› hull_minimal[of S "?rhs" "cone"] by auto moreover { fix x assume "x ∈ ?rhs" then obtain cx :: real and xx where x: "x = cx *⇩_{R}xx" "cx ≥ 0" "xx ∈ S" by auto then have "xx ∈ cone hull S" using hull_subset[of S] by auto then have "x ∈ ?lhs" using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto } ultimately show ?thesis by auto qed lemma convex_cone: "convex s ∧ cone s ⟷ (∀x∈s. ∀y∈s. (x + y) ∈ s) ∧ (∀x∈s. ∀c≥0. (c *⇩_{R}x) ∈ s)" (is "?lhs = ?rhs") proof - { fix x y assume "x∈s" "y∈s" and ?lhs then have "2 *⇩_{R}x ∈s" "2 *⇩_{R}y ∈ s" unfolding cone_def by auto then have "x + y ∈ s" using ‹?lhs›[unfolded convex_def, THEN conjunct1] apply (erule_tac x="2*⇩_{R}x" in ballE) apply (erule_tac x="2*⇩_{R}y" in ballE) apply (erule_tac x="1/2" in allE, simp) apply (erule_tac x="1/2" in allE, auto) done } then show ?thesis unfolding convex_def cone_def by blast qed subsection✐‹tag unimportant› ‹Connectedness of convex sets› lemma convex_connected: fixes S :: "'a::real_normed_vector set" assumes "convex S" shows "connected S" proof (rule connectedI) fix A B assume "open A" "open B" "A ∩ B ∩ S = {}" "S ⊆ A ∪ B" moreover assume "A ∩ S ≠ {}" "B ∩ S ≠ {}" then obtain a b where a: "a ∈ A" "a ∈ S" and b: "b ∈ B" "b ∈ S" by auto define f where [abs_def]: "f u = u *⇩_{R}a + (1 - u) *⇩_{R}b" for u then have "continuous_on {0 .. 1} f" by (auto intro!: continuous_intros) then have "connected (f ` {0 .. 1})" by (auto intro!: connected_continuous_image) note connectedD[OF this, of A B] moreover have "a ∈ A ∩ f ` {0 .. 1}" using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def) moreover have "b ∈ B ∩ f ` {0 .. 1}" using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def) moreover have "f ` {0 .. 1} ⊆ S" using ‹convex S› a b unfolding convex_def f_def by auto ultimately show False by auto qed corollary%unimportant connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)" by (simp add: convex_connected) lemma convex_prod: assumes "⋀i. i ∈ Basis ⟹ convex {x. P i x}" shows "convex {x. ∀i∈Basis. P i (x∙i)}" using assms unfolding convex_def by (auto simp: inner_add_left) lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (∀i∈Basis. 0 ≤ x∙i)}" by (rule convex_prod) (simp flip: atLeast_def) subsection ‹Convex hull› lemma convex_convex_hull [iff]: "convex (convex hull s)" unfolding hull_def using convex_Inter[of "{t. convex t ∧ s ⊆ t}"] by auto lemma convex_hull_subset: "s ⊆ convex hull t ⟹ convex hull s ⊆ convex hull t" by (simp add: subset_hull) lemma convex_hull_eq: "