Theory Convex
section ‹Convex Sets and Functions›
theory Convex
imports
Affine
"HOL-Library.Set_Algebras"
begin
subsection ‹Convex Sets›
definition 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 ‹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 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)"
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 ‹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 pos: "⋀x. x ∈ C ⟹ f'' x ≥ 0"
shows "convex_on C f"
using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
by fastforce
lemma minus_log_convex:
fixes b :: real
assumes "b > 1"
shows "convex_on {0 <..} (λ x. - log b x)"
proof -
have "⋀z. z > 0 ⟹ DERIV (log b) z :> 1 / (ln b * z)"
using DERIV_log by auto
then have f': "⋀z. z > 0 ⟹ DERIV (λ z. - log b z) z :> - 1 / (ln b * z)"
by (auto simp: DERIV_minus)
have "⋀z::real. z > 0 ⟹ DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
have "⋀z::real. z > 0 ⟹
DERIV (λ z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
by auto
then have f''0: "⋀z::real. z > 0 ⟹
DERIV (λ z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
unfolding inverse_eq_divide by (auto simp: mult.assoc)
have f''_ge0: "⋀z::real. z > 0 ⟹ 1 / (ln b * z * z) ≥ 0"
using ‹b > 1› by (auto intro!: less_imp_le)
from f''_ge0_imp_convex[OF convex_real_interval(3), unfolded greaterThan_iff, OF f' f''0 f''_ge0]
show ?thesis
by auto
qed
subsection ‹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)
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 fst_snd_linear: "linear (λ(x,y). x + y)"
unfolding linear_iff by (simp add: algebra_simps)
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
lemma dist_triangle_eq:
fixes x y z :: "'a::real_inner"
shows "dist x z = dist x y + dist y z ⟷
norm (x - y) *⇩R (y - z) = norm (y - z) *⇩R (x - y)"
proof -
have *: "x - y + (y - z) = x - z" by auto
show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
by (auto simp:norm_minus_commute)
qed
subsection ‹Cones›
definition 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 ‹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: "convex hull s = s ⟷ convex s"
by (metis convex_convex_hull hull_same)
subsubsection ‹Convex hull is "preserved" by a linear function›
lemma convex_hull_linear_image:
assumes f: "linear f"
shows "f ` (convex hull s) = convex hull (f ` s)"
proof
show "convex hull (f ` s) ⊆ f ` (convex hull s)"
by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
show "f ` (convex hull s) ⊆ convex hull (f ` s)"
proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
show "s ⊆ f -` (convex hull (f ` s))"
by (fast intro: hull_inc)
show "convex (f -` (convex hull (f ` s)))"
by (intro convex_linear_vimage [OF f] convex_convex_hull)
qed
qed
lemma in_convex_hull_linear_image:
assumes "linear f"
and "x ∈ convex hull s"
shows "f x ∈ convex hull (f ` s)"
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
lemma convex_hull_Times:
"convex hull (s × t) = (convex hull s) × (convex hull t)"
proof
show "convex hull (s × t) ⊆ (convex hull s) × (convex hull t)"
by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
have "(x, y) ∈ convex hull (s × t)" if x: "x ∈ convex hull s" and y: "y ∈ convex hull t" for x y
proof (rule hull_induct [OF x], rule hull_induct [OF y])
fix x y assume "x ∈ s" and "y ∈ t"
then show "(x, y) ∈ convex hull (s × t)"
by (simp add: hull_inc)
next
fix x let ?S = "((λy. (0, y)) -` (λp. (- x, 0) + p) ` (convex hull s × t))"
have "convex ?S"
by (intro convex_linear_vimage convex_translation convex_convex_hull,
simp add: linear_iff)
also have "?S = {y. (x, y) ∈ convex hull (s × t)}"
by (auto simp: image_def Bex_def)
finally show "convex {y. (x, y) ∈ convex hull (s × t)}" .
next
show "convex {x. (x, y) ∈ convex hull s × t}"
proof -
fix y let ?S = "((λx. (x, 0)) -` (λp. (0, - y) + p) ` (convex hull s × t))"
have "convex ?S"
by (intro convex_linear_vimage convex_translation convex_convex_hull,
simp add: linear_iff)
also have "?S = {x. (x, y) ∈ convex hull (s × t)}"
by (auto simp: image_def Bex_def)
finally show "convex {x. (x, y) ∈ convex hull (s × t)}" .
qed
qed
then show "(convex hull s) × (convex hull t) ⊆ convex hull (s × t)"
unfolding subset_eq split_paired_Ball_Sigma by blast
qed
subsubsection ‹Stepping theorems for convex hulls of finite sets›
lemma convex_hull_empty[simp]: "convex hull {} = {}"
by (rule hull_unique) auto
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
by (rule hull_unique) auto
lemma convex_hull_insert:
fixes S :: "'a::real_vector set"
assumes "S ≠ {}"
shows "convex hull (insert a S) =
{x. ∃u≥0. ∃v≥0. ∃b. (u + v = 1) ∧ b ∈ (convex hull S) ∧ (x = u *⇩R a + v *⇩R b)}"
(is "_ = ?hull")
proof (intro equalityI hull_minimal subsetI)
fix x
assume "x ∈ insert a S"
then have "∃u≥0. ∃v≥0. u + v = 1 ∧ (∃b. b ∈ convex hull S ∧ x = u *⇩R a + v *⇩R b)"
unfolding insert_iff
proof
assume "x = a"
then show ?thesis
by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
next
assume "x ∈ S"
with hull_subset[of S convex] show ?thesis
by force
qed
then show "x ∈ ?hull"
by simp
next
fix x
assume "x ∈ ?hull"
then obtain u v b where obt: "u≥0" "v≥0" "u + v = 1" "b ∈ convex hull S" "x = u *⇩R a + v *⇩R b"
by auto
have "a ∈ convex hull insert a S" "b ∈ convex hull insert a S"
using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
by auto
then show "x ∈ convex hull insert a S"
unfolding obt(5) using obt(1-3)
by (rule convexD [OF convex_convex_hull])
next
show "convex ?hull"
proof (rule convexI)
fix x y u v
assume as: "(0::real) ≤ u" "0 ≤ v" "u + v = 1" and x: "x ∈ ?hull" and y: "y ∈ ?hull"
from x obtain u1 v1 b1 where
obt1: "u1≥0" "v1≥0" "u1 + v1 = 1" "b1 ∈ convex hull S" and xeq: "x = u1 *⇩R a + v1 *⇩R b1"
by auto
from y obtain u2 v2 b2 where
obt2: "u2≥0" "v2≥0" "u2 + v2 = 1" "b2 ∈ convex hull S" and yeq: "y = u2 *⇩R a + v2 *⇩R b2"
by auto
have *: "⋀(x::'a) s1 s2. x - s1 *⇩R x - s2 *⇩R x = ((1::real) - (s1 + s2)) *⇩R x"
by (auto simp: algebra_simps)
have "∃b ∈ convex hull S. u *⇩R x + v *⇩R y =
(u * u1) *⇩R a + (v * u2) *⇩R a + (b - (u * u1) *⇩R b - (v * u2) *⇩R b)"
proof (cases "u * v1 + v * v2 = 0")
case True
have *: "⋀(x::'a) s1 s2. x - s1 *⇩R x - s2 *⇩R x = ((1::real) - (s1 + s2)) *⇩R x"
by (auto simp: algebra_simps)
have eq0: "u * v1 = 0" "v * v2 = 0"
using True mult_nonneg_nonneg[OF ‹u≥0› ‹v1≥0›] mult_nonneg_nonneg[OF ‹v≥0› ‹v2≥0›]
by arith+
then have "u * u1 + v * u2 = 1"
using as(3) obt1(3) obt2(3) by auto
then show ?thesis
using "*" eq0 as obt1(4) xeq yeq by auto
next
case False
have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
also have "… = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
also have "… = u * v1 + v * v2"
by simp
finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
let ?b = "((u * v1) / (u * v1 + v * v2)) *⇩R b1 + ((v * v2) / (u * v1 + v * v2)) *⇩R b2"
have zeroes: "0 ≤ u * v1 + v * v2" "0 ≤ u * v1" "0 ≤ u * v1 + v * v2" "0 ≤ v * v2"
using as(1,2) obt1(1,2) obt2(1,2) by auto
show ?thesis
proof
show "u *⇩R x + v *⇩R y = (u * u1) *⇩R a + (v * u2) *⇩R a + (?b - (u * u1) *⇩R ?b - (v * u2) *⇩R ?b)"
unfolding xeq yeq * **
using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
show "?b ∈ convex hull S"
using False zeroes obt1(4) obt2(4)
by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib add_divide_distrib[symmetric] zero_le_divide_iff)
qed
qed
then obtain b where b: "b ∈ convex hull S"
"u *⇩R x + v *⇩R y = (u * u1) *⇩R a + (v * u2) *⇩R a + (b - (u * u1) *⇩R b - (v * u2) *⇩R b)" ..
have u1: "u1 ≤ 1"
unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
have u2: "u2 ≤ 1"
unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
have "u1 * u + u2 * v ≤ max u1 u2 * u + max u1 u2 * v"
proof (rule add_mono)
show "u1 * u ≤ max u1 u2 * u" "u2 * v ≤ max u1 u2 * v"
by (simp_all add: as mult_right_mono)
qed
also have "… ≤ 1"
unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
finally have le1: "u1 * u + u2 * v ≤ 1" .
show "u *⇩R x + v *⇩R y ∈ ?hull"
proof (intro CollectI exI conjI)
show "0 ≤ u * u1 + v * u2"
by (simp add: as(1) as(2) obt1(1) obt2(1))
show "0 ≤ 1 - u * u1 - v * u2"
by (simp add: le1 diff_diff_add mult.commute)
qed (use b in ‹auto simp: algebra_simps›)
qed
qed
lemma convex_hull_insert_alt:
"convex hull (insert a S) =
(if S = {} then {a}
else {(1 - u) *⇩R a + u *⇩R x |x u. 0 ≤ u ∧ u ≤ 1 ∧ x ∈ convex hull S})"
apply (auto simp: convex_hull_insert)
using diff_eq_eq apply fastforce
using diff_add_cancel diff_ge_0_iff_ge by blast
subsubsection ‹Explicit expression for convex hull›
proposition convex_hull_indexed:
fixes S :: "'a::real_vector set"
shows "convex hull S =
{y. ∃k u x. (∀i∈{1::nat .. k}. 0 ≤ u i ∧ x i ∈ S) ∧
(sum u {1..k} = 1) ∧ (∑i = 1..k. u i *⇩R x i) = y}"
(is "?xyz = ?hull")
proof (rule hull_unique [OF _ convexI])
show "S ⊆ ?hull"
by (clarsimp, rule_tac x=1 in exI, rule_tac x="λx. 1" in exI, auto)
next
fix T
assume "S ⊆ T" "convex T"
then show "?hull ⊆ T"
by (blast intro: convex_sum)
next
fix x y u v
assume uv: "0 ≤ u" "0 ≤ v" "u + v = (1::real)"
assume xy: "x ∈ ?hull" "y ∈ ?hull"
from xy obtain k1 u1 x1 where
x [rule_format]: "∀i∈{1::nat..k1}. 0≤u1 i ∧ x1 i ∈ S"
"sum u1 {Suc 0..k1} = 1" "(∑i = Suc 0..k1. u1 i *⇩R x1 i) = x"
by auto
from xy obtain k2 u2 x2 where
y [rule_format]: "∀i∈{1::nat..k2}. 0≤u2 i ∧ x2 i ∈ S"
"sum u2 {Suc 0..k2} = 1" "(∑i = Suc 0..k2. u2 i *⇩R x2 i) = y"
by auto
have *: "⋀P (x::'a) y s t i. (if P i then s else t) *⇩R (if P i then x else y) = (if P i then s *⇩R x else t *⇩R y)"
"{1..k1 + k2} ∩ {1..k1} = {1..k1}" "{1..k1 + k2} ∩ - {1..k1} = (λi. i + k1) ` {1..k2}"
by auto
have inj: "inj_on (λi. i + k1) {1..k2}"
unfolding inj_on_def by auto
let ?uu = "λi. if i ∈ {1..k1} then u * u1 i else v * u2 (i - k1)"
let ?xx = "λi. if i ∈ {1..k1} then x1 i else x2 (i - k1)"
show "u *⇩R x + v *⇩R y ∈ ?hull"
proof (intro CollectI exI conjI ballI)
show "0 ≤ ?uu i" "?xx i ∈ S" if "i ∈ {1..k1+k2}" for i
using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
show "(∑i = 1..k1 + k2. ?uu i) = 1" "(∑i = 1..k1 + k2. ?uu i *⇩R ?xx i) = u *⇩R x + v *⇩R y"
unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
sum.reindex[OF inj] Collect_mem_eq o_def
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
by (simp_all add: sum_distrib_left[symmetric] x(2,3) y(2,3) uv(3))
qed
qed
lemma convex_hull_finite:
fixes S :: "'a::real_vector set"
assumes "finite S"
shows "convex hull S = {y. ∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ sum (λx. u x *⇩R x) S = y}"
(is "?HULL = _")
proof (rule hull_unique [OF _ convexI]; clarify)
fix x
assume "x ∈ S"
then show "∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ (∑x∈S. u x *⇩R x) = x"
by (rule_tac x="λy. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
next
fix u v :: real
assume uv: "0 ≤ u" "0 ≤ v" "u + v = 1"
fix ux assume ux [rule_format]: "∀x∈S. 0 ≤ ux x" "sum ux S = (1::real)"
fix uy assume uy [rule_format]: "∀x∈S. 0 ≤ uy x" "sum uy S = (1::real)"
have "0 ≤ u * ux x + v * uy x" if "x∈S" for x
by (simp add: that uv ux(1) uy(1))
moreover
have "(∑x∈S. u * ux x + v * uy x) = 1"
unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
using uv(3) by auto
moreover
have "(∑x∈S. (u * ux x + v * uy x) *⇩R x) = u *⇩R (∑x∈S. ux x *⇩R x) + v *⇩R (∑x∈S. uy x *⇩R x)"
unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
by auto
ultimately
show "∃uc. (∀x∈S. 0 ≤ uc x) ∧ sum uc S = 1 ∧
(∑x∈S. uc x *⇩R x) = u *⇩R (∑x∈S. ux x *⇩R x) + v *⇩R (∑x∈S. uy x *⇩R x)"
by (rule_tac x="λx. u * ux x + v * uy x" in exI, auto)
qed (use assms in ‹auto simp: convex_explicit›)
subsubsection ‹Another formulation›
text "Formalized by Lars Schewe."
lemma convex_hull_explicit:
fixes p :: "'a::real_vector set"
shows "convex hull p =
{y. ∃S u. finite S ∧ S ⊆ p ∧ (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x∈?lhs"
then obtain k u y where
obt: "∀i∈{1::nat..k}. 0 ≤ u i ∧ y i ∈ p" "sum u {1..k} = 1" "(∑i = 1..k. u i *⇩R y i) = x"
unfolding convex_hull_indexed by auto
have fin: "finite {1..k}" by auto
have fin': "⋀v. finite {i ∈ {1..k}. y i = v}" by auto
{
fix j
assume "j∈{1..k}"
then have "y j ∈ p ∧ 0 ≤ sum u {i. Suc 0 ≤ i ∧ i ≤ k ∧ y i = y j}"
using obt(1)[THEN bspec[where x=j]] and obt(2)
by (metis (no_types, lifting) One_nat_def atLeastAtMost_iff mem_Collect_eq obt(1) sum_nonneg)
}
moreover
have "(∑v∈y ` {1..k}. sum u {i ∈ {1..k}. y i = v}) = 1"
unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
moreover have "(∑v∈y ` {1..k}. sum u {i ∈ {1..k}. y i = v} *⇩R v) = x"
using sum.image_gen[OF fin, of "λi. u i *⇩R y i" y, symmetric]
unfolding scaleR_left.sum using obt(3) by auto
ultimately
have "∃S u. finite S ∧ S ⊆ p ∧ (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ (∑v∈S. u v *⇩R v) = x"
apply (rule_tac x="y ` {1..k}" in exI)
apply (rule_tac x="λv. sum u {i∈{1..k}. y i = v}" in exI, auto)
done
then have "x∈?rhs" by auto
}
moreover
{
fix y
assume "y∈?rhs"
then obtain S u where
obt: "finite S" "S ⊆ p" "∀x∈S. 0 ≤ u x" "sum u S = 1" "(∑v∈S. u v *⇩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∈{1..card S}"
then have "f i ∈ S"
using f(2) by blast
then have "0 ≤ u (f i)" "f i ∈ p" using obt(2,3) by auto
}
moreover have *: "finite {1..card S}" by auto
{
fix y
assume "y∈S"
then obtain i where "i∈{1..card S}" "f i = y"
using f using image_iff[of y f "{1..card S}"]
by auto
then have "{x. Suc 0 ≤ x ∧ x ≤ card S ∧ f x = y} = {i}"
using f(1) inj_onD by fastforce
then have "card {x. Suc 0 ≤ x ∧ x ≤ card S ∧ f x = y} = 1" by auto
then have "(∑x∈{x ∈ {1..card S}. f x = y}. u (f x)) = u y"
"(∑x∈{x ∈ {1..card S}. f x = y}. u (f x) *⇩R f x) = u y *⇩R y"
by (auto simp: sum_constant_scaleR)
}
then have "(∑x = 1..card S. u (f x)) = 1" "(∑i = 1..card S. u (f i) *⇩R f i) = y"
unfolding sum.image_gen[OF *(1), of "λx. u (f x) *⇩R f x" f]
and sum.image_gen[OF *(1), of "λx. u (f x)" f]
unfolding f
using sum.cong [of S S "λy. (∑x∈{x ∈ {1..card S}. f x = y}. u (f x) *⇩R f x)" "λv. u v *⇩R v"]
using sum.cong [of S S "λy. (∑x∈{x ∈ {1..card S}. f x = y}. u (f x))" u]
unfolding obt(4,5)
by auto
ultimately
have "∃k u x. (∀i∈{1..k}. 0 ≤ u i ∧ x i ∈ p) ∧ sum u {1..k} = 1 ∧
(∑i::nat = 1..k. u i *⇩R x i) = y"
apply (rule_tac x="card S" in exI)
apply (rule_tac x="u ∘ f" in exI)
apply (rule_tac x=f in exI, fastforce)
done
then have "y ∈ ?lhs"
unfolding convex_hull_indexed by auto
}
ultimately show ?thesis
unfolding set_eq_iff by blast
qed
subsubsection ‹A stepping theorem for that expansion›
lemma convex_hull_finite_step:
fixes S :: "'a::real_vector set"
assumes "finite S"
shows
"(∃u. (∀x∈insert a S. 0 ≤ u x) ∧ sum u (insert a S) = w ∧ sum (λx. u x *⇩R x) (insert a S) = y)
⟷ (∃v≥0. ∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S = w - v ∧ sum (λx. u x *⇩R x) S = y - v *⇩R a)"
(is "?lhs = ?rhs")
proof (cases "a ∈ S")
case True
then have *: "insert a S = S" by auto
show ?thesis
proof
assume ?lhs
then show ?rhs
unfolding * by force
next
have fin: "finite (insert a S)" using assms by auto
assume ?rhs
then obtain v u where uv: "v≥0" "∀x∈S. 0 ≤ u x" "sum u S = w - v" "(∑x∈S. u x *⇩R x) = y - v *⇩R a"
by auto
then show ?lhs
using uv True assms
apply (rule_tac x = "λx. (if a = x then v else 0) + u x" in exI)
apply (auto simp: sum_clauses scaleR_left_distrib sum.distrib sum_delta''[OF fin])
done
qed
next
case False
show ?thesis
proof
assume ?lhs
then obtain u where u: "∀x∈insert a S. 0 ≤ u x" "sum u (insert a S) = w" "(∑x∈insert a S. u x *⇩R x) = y"
by auto
then show ?rhs
using u ‹a∉S› by (rule_tac x="u a" in exI) (auto simp: sum_clauses assms)
next
assume ?rhs
then obtain v u where uv: "v≥0" "∀x∈S. 0 ≤ u x" "sum u S = w - v" "(∑x∈S. u x *⇩R x) = y - v *⇩R a"
by auto
moreover
have "(∑x∈S. if a = x then v else u x) = sum u S" "(∑x∈S. (if a = x then v else u x) *⇩R x) = (∑x∈S. u x *⇩R x)"
using False by (auto intro!: sum.cong)
ultimately show ?lhs
using False by (rule_tac x="λx. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
qed
qed
subsubsection ‹Hence some special cases›
lemma convex_hull_2: "convex hull {a,b} = {u *⇩R a + v *⇩R b | u v. 0 ≤ u ∧ 0 ≤ v ∧ u + v = 1}"
(is "?lhs = ?rhs")
proof -
have **: "finite {b}" by auto
have "⋀x v u. ⟦0 ≤ v; v ≤ 1; (1 - v) *⇩R b = x - v *⇩R a⟧
⟹ ∃u v. x = u *⇩R a + v *⇩R b ∧ 0 ≤ u ∧ 0 ≤ v ∧ u + v = 1"
by (metis add.commute diff_add_cancel diff_ge_0_iff_ge)
moreover
have "⋀u v. ⟦0 ≤ u; 0 ≤ v; u + v = 1⟧
⟹ ∃p≥0. ∃q. 0 ≤ q b ∧ q b = 1 - p ∧ q b *⇩R b = u *⇩R a + v *⇩R b - p *⇩R a"
apply (rule_tac x=u in exI, simp)
apply (rule_tac x="λx. v" in exI, simp)
done
ultimately show ?thesis
using convex_hull_finite_step[OF **, of a 1]
by (auto simp add: convex_hull_finite)
qed
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *⇩R (b - a) | u. 0 ≤ u ∧ u ≤ 1}"
unfolding convex_hull_2
proof (rule Collect_cong)
have *: "⋀x y ::real. x + y = 1 ⟷ x = 1 - y"
by auto
fix x
show "(∃v u. x = v *⇩R a + u *⇩R b ∧ 0 ≤ v ∧ 0 ≤ u ∧ v + u = 1) ⟷
(∃u. x = a + u *⇩R (b - a) ∧ 0 ≤ u ∧ u ≤ 1)"
apply (simp add: *)
by (rule ex_cong1) (auto simp: algebra_simps)
qed
lemma convex_hull_3:
"convex hull {a,b,c} = { u *⇩R a + v *⇩R b + w *⇩R c | u v w. 0 ≤ u ∧ 0 ≤ v ∧ 0 ≤ w ∧ u + v + w = 1}"
proof -
have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
by auto
have *: "⋀x y z ::real. x + y + z = 1 ⟷ x = 1 - y - z"
by (auto simp: field_simps)
show ?thesis
unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
unfolding convex_hull_finite_step[OF fin(3)]
apply (rule Collect_cong, simp)
apply auto
apply (rule_tac x=va in exI)
apply (rule_tac x="u c" in exI, simp)
apply (rule_tac x="1 - v - w" in exI, simp)
apply (rule_tac x=v in exI, simp)
apply (rule_tac x="λx. w" in exI, simp)
done
qed
lemma convex_hull_3_alt:
"convex hull {a,b,c} = {a + u *⇩R (b - a) + v *⇩R (c - a) | u v. 0 ≤ u ∧ 0 ≤ v ∧ u + v ≤ 1}"
proof -
have *: "⋀x y z ::real. x + y + z = 1 ⟷ x = 1 - y - z"
by auto
show ?thesis
unfolding convex_hull_3
apply (auto simp: *)
apply (rule_tac x=v in exI)
apply (rule_tac x=w in exI)
apply (simp add: algebra_simps)
apply (rule_tac x=u in exI)
apply (rule_tac x=v in exI)
apply (simp add: algebra_simps)
done
qed
subsection ‹Relations among closure notions and corresponding hulls›
lemma affine_imp_convex: "affine s ⟹ convex s"
unfolding affine_def convex_def by auto
lemma convex_affine_hull [simp]: "convex (affine hull S)"
by (simp add: affine_imp_convex)
lemma subspace_imp_convex: "subspace s ⟹ convex s"
using subspace_imp_affine affine_imp_convex by auto
lemma convex_hull_subset_span: "(convex hull s) ⊆ (span s)"
by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
lemma convex_hull_subset_affine_hull: "(convex hull s) ⊆ (affine hull s)"
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
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
subsection ‹Caratheodory's theorem›
lemma convex_hull_caratheodory_aff_dim:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p =
{y. ∃S u. finite S ∧ S ⊆ p ∧ card S ≤ aff_dim p + 1 ∧
(∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
proof (intro allI iffI)
fix y
let ?P = "λn. ∃S u. finite S ∧ card S = n ∧ S ⊆ p ∧ (∀x∈S. 0 ≤ u x) ∧
sum u S = 1 ∧ (∑v∈S. u v *⇩R v) = y"
assume "∃S u. finite S ∧ S ⊆ p ∧ (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ (∑v∈S. u v *⇩R v) = y"
then obtain N where "?P N" by auto
then have "∃n≤N. (∀k<n. ¬ ?P k) ∧ ?P n"
by (rule_tac ex_least_nat_le, auto)
then obtain n where "?P n" and smallest: "∀k<n. ¬ ?P k"
by blast
then obtain S u where obt: "finite S" "card S = n" "S⊆p" "∀x∈S. 0 ≤ u x"
"sum u S = 1" "(∑v∈S. u v *⇩R v) = y" by auto
have "card S ≤ aff_dim p + 1"
proof (rule ccontr, simp only: not_le)
assume "aff_dim p + 1 < card S"
then have "affine_dependent S"
using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
by blast
then obtain w v where wv: "sum w S = 0" "v∈S" "w v ≠ 0" "(∑v∈S. w v *⇩R v) = 0"
using affine_dependent_explicit_finite[OF obt(1)] by auto
define i where "i = (λv. (u v) / (- w v)) ` {v∈S. w v < 0}"
define t where "t = Min i"
have "∃x∈S. w x < 0"
proof (rule ccontr, simp add: not_less)
assume as:"∀x∈S. 0 ≤ w x"
then have "sum w (S - {v}) ≥ 0"
by (meson Diff_iff sum_nonneg)
then have "sum w S > 0"
using as obt(1) sum_nonneg_eq_0_iff wv by blast
then show False using wv(1) by auto
qed
then have "i ≠ {}" unfolding i_def by auto
then have "t ≥ 0"
using Min_ge_iff[of i 0] and obt(1)
unfolding t_def i_def
using obt(4)[unfolded le_less]
by (auto simp: divide_le_0_iff)
have t: "∀v∈S. u v + t * w v ≥ 0"
proof
fix v
assume "v ∈ S"
then have v: "0 ≤ u v"
using obt(4)[THEN bspec[where x=v]] by auto
show "0 ≤ u v + t * w v"
proof (cases "w v < 0")
case False
thus ?thesis using v ‹t≥0› by auto
next
case True
then have "t ≤ u v / (- w v)"
using ‹v∈S› obt unfolding t_def i_def by (auto intro: Min_le)
then show ?thesis
unfolding real_0_le_add_iff
using True neg_le_minus_divide_eq by auto
qed
qed
obtain a where "a ∈ S" and "t = (λv. (u v) / (- w v)) a" and "w a < 0"
using Min_in[OF _ ‹i≠{}›] and obt(1) unfolding i_def t_def by auto
then have a: "a ∈ S" "u a + t * w a = 0" by auto
have *: "⋀f. sum f (S - {a}) = sum f S - ((f a)::'b::ab_group_add)"
unfolding sum.remove[OF obt(1) ‹a∈S›] by auto
have "(∑v∈S. u v + t * w v) = 1"
unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
moreover have "(∑v∈S. u v *⇩R v + (t * w v) *⇩R v) - (u a *⇩R a + (t * w a) *⇩R a) = y"
unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [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="λv. u v + t * w v" in exI)
using obt(1-3) and t and a
apply (auto simp: * scaleR_left_distrib)
done
then show False
using smallest[THEN spec[where x="n - 1"]] by auto
qed
then show "∃S u. finite S ∧ S ⊆ p ∧ card S ≤ aff_dim p + 1 ∧
(∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ (∑v∈S. u v *⇩R v) = y"
using obt by auto
qed auto
lemma caratheodory_aff_dim:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p = {x. ∃S. finite S ∧ S ⊆ p ∧ card S ≤ aff_dim p + 1 ∧ x ∈ convex hull S}"
(is "?lhs = ?rhs")
proof
have "⋀x S u. ⟦finite S; S ⊆ p; int (card S) ≤ aff_dim p + 1; ∀x∈S. 0 ≤ u x; sum u S = 1⟧
⟹ (∑v∈S. u v *⇩R v) ∈ convex hull S"
by (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
then show "?lhs ⊆ ?rhs"
by (subst convex_hull_caratheodory_aff_dim, auto)
qed (use hull_mono in auto)
lemma convex_hull_caratheodory:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p =
{y. ∃S u. finite S ∧ S ⊆ p ∧ card S ≤ DIM('a) + 1 ∧
(∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
(is "?lhs = ?rhs")
proof (intro set_eqI iffI)
fix x
assume "x ∈ ?lhs" then show "x ∈ ?rhs"
unfolding convex_hull_caratheodory_aff_dim
using aff_dim_le_DIM [of p] by fastforce
qed (auto simp: convex_hull_explicit)
theorem caratheodory:
"convex hull p =
{x::'a::euclidean_space. ∃S. finite S ∧ S ⊆ p ∧ card S ≤ DIM('a) + 1 ∧ x ∈ convex hull S}"
proof safe
fix x
assume "x ∈ convex hull p"
then obtain S u where "finite S" "S ⊆ p" "card S ≤ DIM('a) + 1"
"∀x∈S. 0 ≤ u x" "sum u S = 1" "(∑v∈S. u v *⇩R v) = x"
unfolding convex_hull_caratheodory by auto
then show "∃S. finite S ∧ S ⊆ p ∧ card S ≤ DIM('a) + 1 ∧ x ∈ convex hull S"
using convex_hull_finite by fastforce
qed (use hull_mono in force)
subsection‹Some Properties of subset of standard basis›
lemma affine_hull_substd_basis:
assumes "d ⊆ Basis"
shows "affine hull (insert 0 d) = {x::'a::euclidean_space. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
(is "affine hull (insert 0 ?A) = ?B")
proof -
have *: "⋀A. (+) (0::'a) ` A = A" "⋀A. (+) (- (0::'a)) ` A = A"
by auto
show ?thesis
unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
qed
lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
subsection ‹Moving and scaling convex hulls›
lemma convex_hull_set_plus:
"convex hull (S + T) = convex hull S + convex hull T"
unfolding set_plus_image
apply (subst convex_hull_linear_image [symmetric])
apply (simp add: linear_iff scaleR_right_distrib)
apply (simp add: convex_hull_Times)
done
lemma translation_eq_singleton_plus: "(λx. a + x) ` T = {a} + T"
unfolding set_plus_def by auto
lemma convex_hull_translation:
"convex hull ((λx. a + x) ` S) = (λx. a + x) ` (convex hull S)"
unfolding translation_eq_singleton_plus
by (simp only: convex_hull_set_plus convex_hull_singleton)
lemma convex_hull_scaling:
"convex hull ((λx. c *⇩R x) ` S) = (λx. c *⇩R x) ` (convex hull S)"
using linear_scaleR by (rule convex_hull_linear_image [symmetric])
lemma convex_hull_affinity:
"convex hull ((λx. a + c *⇩R x) ` S) = (λx. a + c *⇩R x) ` (convex hull S)"
by (metis convex_hull_scaling convex_hull_translation image_image)
subsection ‹Convexity of cone hulls›
lemma convex_cone_hull:
assumes "convex S"
shows "convex (cone hull S)"
proof (rule convexI)
fix x y
assume xy: "x ∈ cone hull S" "y ∈ cone hull S"
then have "S ≠ {}"
using cone_hull_empty_iff[of S] by auto
fix u v :: real
assume uv: "u ≥ 0" "v ≥ 0" "u + v = 1"
then have *: "u *⇩R x ∈ cone hull S" "v *⇩R y ∈ cone hull S"
using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
from * obtain cx :: real and xx where x: "u *⇩R x = cx *⇩R xx" "cx ≥ 0" "xx ∈ S"
using cone_hull_expl[of S] by auto
from * obtain cy :: real and yy where y: "v *⇩R y = cy *⇩R yy" "cy ≥ 0" "yy ∈ S"
using cone_hull_expl[of S] by auto
{
assume "cx + cy ≤ 0"
then have "u *⇩R x = 0" and "v *⇩R y = 0"
using x y by auto
then have "u *⇩R x + v *⇩R y = 0"
by auto
then have "u *⇩R x + v *⇩R y ∈ cone hull S"
using cone_hull_contains_0[of S] ‹S ≠ {}› by auto
}
moreover
{
assume "cx + cy > 0"
then have "(cx / (cx + cy)) *⇩R xx + (cy / (cx + cy)) *⇩R yy ∈ S"
using assms mem_convex_alt[of S xx yy cx cy] x y by auto
then have "cx *⇩R xx + cy *⇩R yy ∈ cone hull S"
using mem_cone_hull[of "(cx/(cx+cy)) *⇩R xx + (cy/(cx+cy)) *⇩R yy" S "cx+cy"] ‹cx+cy>0›
by (auto simp: scaleR_right_distrib)
then have "u *⇩R x + v *⇩R y ∈ cone hull S"
using x y by auto
}
moreover have "cx + cy ≤ 0 ∨ cx + cy > 0" by auto
ultimately show "u *⇩R x + v *⇩R y ∈ cone hull S" by blast
qed
lemma cone_convex_hull:
assumes "cone S"
shows "cone (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis by auto
next
case False
then have *: "0 ∈ S ∧ (∀c. c > 0 ⟶ (*⇩R) c ` S = S)"
using cone_iff[of S] assms by auto
{
fix c :: real
assume "c > 0"
then have "(*⇩R) c ` (convex hull S) = convex hull ((*⇩R) c ` S)"
using convex_hull_scaling[of _ S] by auto
also have "… = convex hull S"
using * ‹c > 0› by auto
finally have "(*⇩R) c ` (convex hull S) = convex hull S"
by auto
}
then have "0 ∈ convex hull S" "⋀c. c > 0 ⟹ ((*⇩R) c ` (convex hull S)) = (convex hull S)"
using * hull_subset[of S convex] by auto
then show ?thesis
using ‹S ≠ {}› cone_iff[of "convex hull S"] by auto
qed
subsection ‹Radon's theorem›
text "Formalized by Lars Schewe."
lemma Radon_ex_lemma:
assumes "finite c" "affine_dependent c"
shows "∃u. sum u c = 0 ∧ (∃v∈c. u v ≠ 0) ∧ sum (λv. u v *⇩R v) c = 0"
proof -
from assms(2)[unfolded affine_dependent_explicit]
obtain S u where
"finite S" "S ⊆ c" "sum u S = 0" "∃v∈S. u v ≠ 0" "(∑v∈S. u v *⇩R v) = 0"
by blast
then show ?thesis
apply (rule_tac x="λv. if v∈S then u v else 0" in exI)
unfolding if_smult scaleR_zero_left
by (auto simp: Int_absorb1 sum.inter_restrict[OF ‹finite c›, symmetric])
qed
lemma Radon_s_lemma:
assumes "finite S"
and "sum f S = (0::real)"
shows "sum f {x∈S. 0 < f x} = - sum f {x∈S. f x < 0}"
proof -
have *: "⋀x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
by auto
show ?thesis
unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
and sum.distrib[symmetric] and *
using assms(2)
by assumption
qed
lemma Radon_v_lemma:
assumes "finite S"
and "sum f S = 0"
and "∀x. g x = (0::real) ⟶ f x = (0::'a::euclidean_space)"
shows "(sum f {x∈S. 0 < g x}) = - sum f {x∈S. g x < 0}"
proof -
have *: "⋀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 sum.inter_filter[OF assms(1)]
and sum.distrib[symmetric] and *
using assms(2)
apply assumption
done
qed
lemma Radon_partition:
assumes "finite C" "affine_dependent C"
shows "∃m p. m ∩ p = {} ∧ m ∪ p = C ∧ (convex hull m) ∩ (convex hull p) ≠ {}"
proof -
obtain u v where uv: "sum u C = 0" "v∈C" "u v ≠ 0" "(∑v∈C. u v *⇩R v) = 0"
using Radon_ex_lemma[OF assms] by auto
have fin: "finite {x ∈ C. 0 < u x}" "finite {x ∈ C. 0 > u x}"
using assms(1) by auto
define z where "z = inverse (sum u {x∈C. u x > 0}) *⇩R sum (λx. u x *⇩R x) {x∈C. u x > 0}"
have "sum u {x ∈ C. 0 < u x} ≠ 0"
proof (cases "u v ≥ 0")
case False
then have "u v < 0" by auto
then show ?thesis
proof (cases "∃w∈{x ∈ C. 0 < u x}. u w > 0")
case True
then show ?thesis
using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
next
case False
then have "sum u C ≤ sum (λx. if x=v then u v else 0) C"
by (rule_tac sum_mono, auto)
then show ?thesis
unfolding sum.delta[OF assms(1)] using uv(2) and ‹u v < 0› and uv(1) by auto
qed
qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
then have *: "sum u {x∈C. u x > 0} > 0"
unfolding less_le by (metis (no_types, lifting) mem_Collect_eq sum_nonneg)
moreover have "sum u ({x ∈ C. 0 < u x} ∪ {x ∈ C. u x < 0}) = sum u C"
"(∑x∈{x ∈ C. 0 < u x} ∪ {x ∈ C. u x < 0}. u x *⇩R x) = (∑x∈C. u x *⇩R x)"
using assms(1)
by (rule_tac[!] sum.mono_neutral_left, auto)
then have "sum u {x ∈ C. 0 < u x} = - sum u {x ∈ C. 0 > u x}"
"(∑x∈{x ∈ C. 0 < u x}. u x *⇩R x) = - (∑x∈{x ∈ C. 0 > u x}. u x *⇩R x)"
unfolding eq_neg_iff_add_eq_0
using uv(1,4)
by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
moreover have "∀x∈{v ∈ C. u v < 0}. 0 ≤ inverse (sum u {x ∈ C. 0 < u x}) * - u x"
using * by (fastforce intro: mult_nonneg_nonneg)
ultimately have "z ∈ convex hull {v ∈ C. u v ≤ 0}"
unfolding convex_hull_explicit mem_Collect_eq
apply (rule_tac x="{v ∈ C. u v < 0}" in exI)
apply (rule_tac x="λy. inverse (sum u {x∈C. u x > 0}) * - u y" in exI)
using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
moreover have "∀x∈{v ∈ C. 0 < u v}. 0 ≤ inverse (sum u {x ∈ C. 0 < u x}) * u x"
using * by (fastforce intro: mult_nonneg_nonneg)
then have "z ∈ convex hull {v ∈ C. u v > 0}"
unfolding convex_hull_explicit mem_Collect_eq
apply (rule_tac x="{v ∈ C. 0 < u v}" in exI)
apply (rule_tac x="λy. inverse (sum u {x∈C. u x > 0}) * u y" in exI)
using assms(1)
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
using * by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
ultimately show ?thesis
apply (rule_tac x="{v∈C. u v ≤ 0}" in exI)
apply (rule_tac x="{v∈C. u v > 0}" in exI, auto)
done
qed
theorem Radon:
assumes "affine_dependent c"
obtains m p where "m ⊆ c" "p ⊆ c" "m ∩ p = {}" "(convex hull m) ∩ (convex hull p) ≠ {}"
proof -
from assms[unfolded affine_dependent_explicit]
obtain S u where
"finite S" "S ⊆ c" "sum u S = 0" "∃v∈S. u v ≠ 0" "(∑v∈S. u v *⇩R v) = 0"
by blast
then have *: "finite S" "affine_dependent S" and S: "S ⊆ c"
unfolding affine_dependent_explicit by auto
from Radon_partition[OF *]
obtain m p where "m ∩ p = {}" "m ∪ p = S" "convex hull m ∩ convex hull p ≠ {}"
by blast
with S show ?thesis
by (force intro: that[of p m])
qed
subsection ‹Helly's theorem›
lemma Helly_induct:
fixes f :: "'a::euclidean_space set set"
assumes "card f = n"
and "n ≥ DIM('a) + 1"
and "∀s∈f. convex s" "∀t⊆f. card t = DIM('a) + 1 ⟶ ⋂t ≠ {}"
shows "⋂f ≠ {}"
using assms
proof (induction n arbitrary: f)
case 0
then show ?case by auto
next
case (Suc n)
have "finite f"
using ‹card f = Suc n› by (auto intro: card_ge_0_finite)
show "⋂f ≠ {}"
proof (cases "n = DIM('a)")
case True
then show ?thesis
by (simp add: Suc.prems(1) Suc.prems(4))
next
case False
have "⋂(f - {s}) ≠ {}" if "s ∈ f" for s
proof (rule Suc.IH[rule_format])
show "card (f - {s}) = n"
by (simp add: Suc.prems(1) ‹finite f› that)
show "DIM('a) + 1 ≤ n"
using False Suc.prems(2) by linarith
show "⋀t. ⟦t ⊆ f - {s}; card t = DIM('a) + 1⟧ ⟹ ⋂t ≠ {}"
by (simp add: Suc.prems(4) subset_Diff_insert)
qed (use Suc in auto)
then have "∀s∈f. ∃x. x ∈ ⋂(f - {s})"
by blast
then obtain X where X: "⋀s. s∈f ⟹ X s ∈ ⋂(f - {s})"
by metis
show ?thesis
proof (cases "inj_on X f")
case False
then obtain s t where "s≠t" and st: "s∈f" "t∈f" "X s = X t"
unfolding inj_on_def by auto
then have *: "⋂f = ⋂(f - {s}) ∩ ⋂(f - {t})" by auto
show ?thesis
by (metis "*" X disjoint_iff_not_equal st)
next
case True
then obtain m p where mp: "m ∩ p = {}" "m ∪ p = X ` f" "convex hull m ∩ convex hull p ≠ {}"
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 False
by auto
have "m ⊆ X ` f" "p ⊆ X ` f"
using mp(2) by auto
then obtain g h where gh:"m = X ` g" "p = X ` h" "g ⊆ f" "h ⊆ f"
unfolding subset_image_iff by auto
then have "f ∪ (g ∪ h) = f" by auto
then have f: "f = g ∪ h"
using inj_on_Un_image_eq_iff[of X f "g ∪ h"] and True
unfolding mp(2)[unfolded image_Un[symmetric] gh]
by auto
have *: "g ∩ h = {}"
using gh(1) gh(2) local.mp(1) by blast
have "convex hull (X ` h) ⊆ ⋂g" "convex hull (X ` g) ⊆ ⋂h"
by (rule hull_minimal; use X * f in ‹auto simp: Suc.prems(3) convex_Inter›)+
then show ?thesis
unfolding f using mp(3)[unfolded gh] by blast
qed
qed
qed
theorem Helly:
fixes f :: "'a::euclidean_space set set"
assumes "card f ≥ DIM('a) + 1" "∀s∈f. convex s"
and "⋀t. ⟦t⊆f; card t = DIM('a) + 1⟧ ⟹ ⋂t ≠ {}"
shows "⋂f ≠ {}"
using Helly_induct assms by blast
subsection ‹Epigraphs of convex functions›
definition "epigraph S (f :: _ ⇒ real) = {xy. fst xy ∈ S ∧ f (fst xy) ≤ snd xy}"
lemma mem_epigraph: "(x, y) ∈ epigraph S f ⟷ x ∈ S ∧ f x ≤ y"
unfolding epigraph_def by auto
lemma convex_epigraph: "convex (epigraph S f) ⟷ convex_on S f ∧ convex S"
proof safe
assume L: "convex (epigraph S f)"
then show "convex_on S f"
by (auto simp: convex_def convex_on_def epigraph_def)
show "convex S"
using L by (fastforce simp: convex_def convex_on_def epigraph_def)
next
assume "convex_on S f" "convex S"
then show "convex (epigraph S f)"
unfolding convex_def convex_on_def epigraph_def
apply safe
apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
apply (auto intro!:mult_left_mono add_mono)
done
qed
lemma convex_epigraphI: "convex_on S f ⟹ convex S ⟹ convex (epigraph S f)"
unfolding convex_epigraph by auto
lemma convex_epigraph_convex: "convex S ⟹ convex_on S f ⟷ convex(epigraph S f)"
by (simp add: convex_epigraph)
subsubsection ‹Use this to derive general bound property of convex function›
lemma convex_on:
assumes "convex S"
shows "convex_on S f ⟷
(∀k u x. (∀i∈{1..k::nat}. 0 ≤ u i ∧ x i ∈ S) ∧ sum u {1..k} = 1 ⟶
f (sum (λi. u i *⇩R x i) {1..k}) ≤ sum (λi. u i * f(x i)) {1..k})"
(is "?lhs = (∀k u x. ?rhs k u x)")
proof
assume ?lhs
then have §: "convex {xy. fst xy ∈ S ∧ f (fst xy) ≤ snd xy}"
by (metis assms convex_epigraph epigraph_def)
show "∀k u x. ?rhs k u x"
proof (intro allI)
fix k u x
show "?rhs k u x"
using §
unfolding convex mem_Collect_eq fst_sum snd_sum
apply safe
apply (drule_tac x=k in spec)
apply (drule_tac x=u in spec)
apply (drule_tac x="λi. (x i, f (x i))" in spec)
apply simp
done
qed
next
assume "∀k u x. ?rhs k u x"
then show ?lhs
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq fst_sum snd_sum
using assms[unfolded convex] apply clarsimp
apply (rule_tac y="∑i = 1..k. u i * f (fst (x i))" in order_trans)
by (auto simp add: mult_left_mono intro: sum_mono)
qed
subsection ‹A bound within a convex hull›
lemma convex_on_convex_hull_bound:
assumes "convex_on (convex hull S) f"
and "∀x∈S. f x ≤ b"
shows "∀x∈ convex hull S. f x ≤ b"
proof
fix x
assume "x ∈ convex hull S"
then obtain k u v where
u: "∀i∈{1..k::nat}. 0 ≤ u i ∧ v i ∈ S" "sum u {1..k} = 1" "(∑i = 1..k. u i *⇩R v i) = x"
unfolding convex_hull_indexed mem_Collect_eq by auto
have "(∑i = 1..k. u i * f (v i)) ≤ b"
using sum_mono[of "{1..k}" "λi. u i * f (v i)" "λi. u i * b"]
unfolding sum_distrib_right[symmetric] u(2) mult_1
using assms(2) mult_left_mono u(1) by blast
then show "f x ≤ b"
using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
using hull_inc u by fastforce
qed
lemma inner_sum_Basis[simp]: "i ∈ Basis ⟹ (∑Basis) ∙ i = 1"
by (simp add: inner_sum_left sum.If_cases inner_Basis)
lemma convex_set_plus:
assumes "convex S" and "convex T" shows "convex (S + T)"
proof -
have "convex (⋃x∈ S. ⋃y ∈ T. {x + y})"
using assms by (rule convex_sums)
moreover have "(⋃x∈ S. ⋃y ∈ T. {x + y}) = S + T"
unfolding set_plus_def by auto
finally show "convex (S + T)" .
qed
lemma convex_set_sum:
assumes "⋀i. i ∈ A ⟹ convex (B i)"
shows "convex (∑i∈A. B i)"
proof (cases "finite A")
case True then show ?thesis using assms
by induct (auto simp: convex_set_plus)
qed auto
lemma finite_set_sum:
assumes "finite A" and "∀i∈A. finite (B i)" shows "finite (∑i∈A. B i)"
using assms by (induct set: finite, simp, simp add: finite_set_plus)
lemma box_eq_set_sum_Basis:
"{x. ∀i∈Basis. x∙i ∈ B i} = (∑i∈Basis. (λx. x *⇩R i) ` (B i))" (is "?lhs = ?rhs")
proof -
have "⋀x. ∀i∈Basis. x ∙ i ∈ B i ⟹
∃s. x = sum s Basis ∧ (∀i∈Basis. s i ∈ (λx. x *⇩R i) ` B i)"
by (metis (mono_tags, lifting) euclidean_representation image_iff)
moreover
have "sum f Basis ∙ i ∈ B i" if "i ∈ Basis" and f: "∀i∈Basis. f i ∈ (λx. x *⇩R i) ` B i" for i f
proof -
have "(∑x∈Basis - {i}. f x ∙ i) = 0"
proof (rule sum.neutral, intro strip)
show "f x ∙ i = 0" if "x ∈ Basis - {i}" for x
using that f ‹i ∈ Basis› inner_Basis that by fastforce
qed
then have "(∑x∈Basis. f x ∙ i) = f i ∙ i"
by (metis (no_types) ‹i ∈ Basis› add.right_neutral sum.remove [OF finite_Basis])
then have "(∑x∈Basis. f x ∙ i) ∈ B i"
using f that(1) by auto
then show ?thesis
by (simp add: inner_sum_left)
qed
ultimately show ?thesis
by (subst set_sum_alt [OF finite_Basis]) auto
qed
lemma convex_hull_set_sum:
"convex hull (∑i∈A. B i) = (∑i∈A. convex hull (B i))"
proof (cases "finite A")
assume "finite A" then show ?thesis
by (induct set: finite, simp, simp add: convex_hull_set_plus)
qed simp
end