section ‹Convex sets, functions and related things›
theory Convex_Euclidean_Space
imports
Connected
"HOL-Library.Set_Algebras"
begin
lemma swap_continuous:
assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
shows "continuous_on (cbox (c,a) (d,b)) (λ(x, y). f y x)"
proof -
have "(λ(x, y). f y x) = (λ(x, y). f x y) ∘ prod.swap"
by auto
then show ?thesis
apply (rule ssubst)
apply (rule continuous_on_compose)
apply (simp add: split_def)
apply (rule continuous_intros | simp add: assms)+
done
qed
lemma substdbasis_expansion_unique:
assumes d: "d ⊆ Basis"
shows "(∑i∈d. f i *⇩R i) = (x::'a::euclidean_space) ⟷
(∀i∈Basis. (i ∈ d ⟶ f i = x ∙ i) ∧ (i ∉ d ⟶ x ∙ i = 0))"
proof -
have *: "⋀x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
by auto
have **: "finite d"
by (auto intro: finite_subset[OF assms])
have ***: "⋀i. i ∈ Basis ⟹ (∑i∈d. f i *⇩R i) ∙ i = (∑x∈d. if x = i then f x else 0)"
using d
by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
show ?thesis
unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
qed
lemma independent_substdbasis: "d ⊆ Basis ⟹ independent d"
by (rule independent_mono[OF independent_Basis])
lemma dim_cball:
assumes "e > 0"
shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
proof -
{
fix x :: "'n::euclidean_space"
define y where "y = (e / norm x) *⇩R x"
then have "y ∈ cball 0 e"
using assms by auto
moreover have *: "x = (norm x / e) *⇩R y"
using y_def assms by simp
moreover from * have "x = (norm x/e) *⇩R y"
by auto
ultimately have "x ∈ span (cball 0 e)"
using span_scale[of y "cball 0 e" "norm x/e"]
span_superset[of "cball 0 e"]
by (simp add: span_base)
}
then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
by auto
then show ?thesis
using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
qed
lemma sum_not_0: "sum f A ≠ 0 ⟹ ∃a ∈ A. f a ≠ 0"
by (rule ccontr) auto
lemma subset_translation_eq [simp]:
fixes a :: "'a::real_vector" shows "(+) a ` s ⊆ (+) a ` t ⟷ s ⊆ t"
by auto
lemma translate_inj_on:
fixes A :: "'a::ab_group_add set"
shows "inj_on (λx. a + x) A"
unfolding inj_on_def by auto
lemma translation_assoc:
fixes a b :: "'a::ab_group_add"
shows "(λx. b + x) ` ((λx. a + x) ` S) = (λx. (a + b) + x) ` S"
by auto
lemma translation_invert:
fixes a :: "'a::ab_group_add"
assumes "(λx. a + x) ` A = (λx. a + x) ` B"
shows "A = B"
proof -
have "(λx. -a + x) ` ((λx. a + x) ` A) = (λx. - a + x) ` ((λx. a + x) ` B)"
using assms by auto
then show ?thesis
using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
qed
lemma translation_galois:
fixes a :: "'a::ab_group_add"
shows "T = ((λx. a + x) ` S) ⟷ S = ((λx. (- a) + x) ` T)"
using translation_assoc[of "-a" a S]
apply auto
using translation_assoc[of a "-a" T]
apply auto
done
lemma translation_inverse_subset:
assumes "((λx. - a + x) ` V) ≤ (S :: 'n::ab_group_add set)"
shows "V ≤ ((λx. a + x) ` S)"
proof -
{
fix x
assume "x ∈ V"
then have "x-a ∈ S" using assms by auto
then have "x ∈ {a + v |v. v ∈ S}"
apply auto
apply (rule exI[of _ "x-a"], simp)
done
then have "x ∈ ((λx. a+x) ` S)" by auto
}
then show ?thesis by auto
qed
subsection ‹Convexity›
definition%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"
apply (rule convexD)
using assms
apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
done
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%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_head_Suc[of "Suc (Suc 0)" 2 "λ j. (1 - μ) *⇩R y"] by auto
from sum_head_Suc[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)"
unfolding convex_explicit
apply safe
subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
subgoal for t u
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
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"]] * show "(∑x∈t. u x *⇩R x) ∈ s"
by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
qed
done
subsection ‹Functions that are convex on a set›
definition%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)"
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%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:
assumes "convex S"
shows "convex ((λx. a + x) ` S)"
proof -
have "(⋃ x∈ {a}. ⋃y ∈ S. {x + y}) = (λx. a + x) ` S"
by auto
then show ?thesis
using convex_sums[OF convex_singleton[of a] assms] by auto
qed
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 pos_is_convex: "convex {0 :: real <..}"
unfolding convex_alt
proof safe
fix y x μ :: real
assume *: "y > 0" "x > 0" "μ ≥ 0" "μ ≤ 1"
{
assume "μ = 0"
then have "μ *⇩R x + (1 - μ) *⇩R y = y"
by simp
then have "μ *⇩R x + (1 - μ) *⇩R y > 0"
using * by simp
}
moreover
{
assume "μ = 1"
then have "μ *⇩R x + (1 - μ) *⇩R y > 0"
using * by simp
}
moreover
{
assume "μ ≠ 1" "μ ≠ 0"
then have "μ > 0" "(1 - μ) > 0"
using * by auto
then have "μ *⇩R x + (1 - μ) *⇩R y > 0"
using * by (auto simp: add_pos_pos)
}
ultimately show "(1 - μ) *⇩R y + μ *⇩R x > 0"
by fastforce
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"
assumes "convex C"
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[OF assms(1)]
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"
using convex_on_alt 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 pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
show ?thesis
by auto
qed
subsection%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: divide_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 divide_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 divide_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_supp_sum:
assumes "convex S" and 1: "supp_sum u I = 1"
and "⋀i. i ∈ I ⟹ 0 ≤ u i ∧ (u i = 0 ∨ f i ∈ S)"
shows "supp_sum (λi. u i *⇩R f i) I ∈ S"
proof -
have fin: "finite {i ∈ I. u i ≠ 0}"
using 1 sum.infinite by (force simp: supp_sum_def support_on_def)
then have eq: "supp_sum (λi. u i *⇩R f i) I = sum (λi. u i *⇩R f i) {i ∈ I. u i ≠ 0}"
by (force intro: sum.mono_neutral_left simp: supp_sum_def support_on_def)
show ?thesis
apply (simp add: eq)
apply (rule convex_sum [OF fin ‹convex S›])
using 1 assms apply (auto simp: supp_sum_def support_on_def)
done
qed
lemma convex_translation_eq [simp]: "convex ((λx. a + x) ` s) ⟷ convex s"
by (metis convex_translation translation_galois)
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 closure_bounded_linear_image_subset:
assumes f: "bounded_linear f"
shows "f ` closure S ⊆ closure (f ` S)"
using linear_continuous_on [OF f] closed_closure closure_subset
by (rule image_closure_subset)
lemma closure_linear_image_subset:
fixes f :: "'m::euclidean_space ⇒ 'n::real_normed_vector"
assumes "linear f"
shows "f ` (closure S) ⊆ closure (f ` S)"
using assms unfolding linear_conv_bounded_linear
by (rule closure_bounded_linear_image_subset)
lemma closed_injective_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes S: "closed S" and f: "linear f" "inj f"
shows "closed (f ` S)"
proof -
obtain g where g: "linear g" "g ∘ f = id"
using linear_injective_left_inverse [OF f] by blast
then have confg: "continuous_on (range f) g"
using linear_continuous_on linear_conv_bounded_linear by blast
have [simp]: "g ` f ` S = S"
using g by (simp add: image_comp)
have cgf: "closed (g ` f ` S)"
by (simp add: ‹g ∘ f = id› S image_comp)
have [simp]: "(range f ∩ g -` S) = f ` S"
using g unfolding o_def id_def image_def by auto metis+
show ?thesis
proof (rule closedin_closed_trans [of "range f"])
show "closedin (subtopology euclidean (range f)) (f ` S)"
using continuous_closedin_preimage [OF confg cgf] by simp
show "closed (range f)"
apply (rule closed_injective_image_subspace)
using f apply (auto simp: linear_linear linear_injective_0)
done
qed
qed
lemma closed_injective_linear_image_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "linear f" "inj f"
shows "(closed(image f s) ⟷ closed s)"
by (metis closed_injective_linear_image closure_eq closure_linear_image_subset closure_subset_eq f(1) f(2) inj_image_subset_iff)
lemma closure_injective_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "⟦linear f; inj f⟧ ⟹ f ` (closure S) = closure (f ` S)"
apply (rule subset_antisym)
apply (simp add: closure_linear_image_subset)
by (simp add: closure_minimal closed_injective_linear_image closure_subset image_mono)
lemma closure_bounded_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "⟦linear f; bounded S⟧ ⟹ f ` (closure S) = closure (f ` S)"
apply (rule subset_antisym, simp add: closure_linear_image_subset)
apply (rule closure_minimal, simp add: closure_subset image_mono)
by (meson bounded_closure closed_closure compact_continuous_image compact_eq_bounded_closed linear_continuous_on linear_conv_bounded_linear)
lemma closure_scaleR:
fixes S :: "'a::real_normed_vector set"
shows "(( *⇩R) c) ` (closure S) = closure ((( *⇩R) c) ` S)"
proof
show "(( *⇩R) c) ` (closure S) ⊆ closure ((( *⇩R) c) ` S)"
using bounded_linear_scaleR_right
by (rule closure_bounded_linear_image_subset)
show "closure ((( *⇩R) c) ` S) ⊆ (( *⇩R) c) ` (closure S)"
by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
qed
lemma fst_linear: "linear fst"
unfolding linear_iff by (simp add: algebra_simps)
lemma snd_linear: "linear snd"
unfolding linear_iff by (simp add: algebra_simps)
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 sphere_eq_empty [simp]:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "sphere a r = {} ⟷ r < 0"
by (auto simp: sphere_def dist_norm) (metis dist_norm le_less_linear vector_choose_dist)
lemma sum_delta_notmem:
assumes "x ∉ s"
shows "sum (λy. if (y = x) then P x else Q y) s = sum Q s"
and "sum (λy. if (x = y) then P x else Q y) s = sum Q s"
and "sum (λy. if (y = x) then P y else Q y) s = sum Q s"
and "sum (λy. if (x = y) then P y else Q y) s = sum Q s"
apply (rule_tac [!] sum.cong)
using assms
apply auto
done
lemma sum_delta'':
fixes s::"'a::real_vector set"
assumes "finite s"
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 if_smult: "(if P then x else (y::real)) *⇩R v = (if P then x *⇩R v else y *⇩R v)"
by (fact if_distrib)
lemma dist_triangle_eq:
fixes x y z :: "'a::real_inner"
shows "dist x z = dist x y + dist y z ⟷
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 ‹Affine set and affine hull›
definition%important affine :: "'a::real_vector set ⇒ bool"
where "affine s ⟷ (∀x∈s. ∀y∈s. ∀u v. u + v = 1 ⟶ u *⇩R x + v *⇩R y ∈ s)"
lemma affine_alt: "affine s ⟷ (∀x∈s. ∀y∈s. ∀u::real. (1 - u) *⇩R x + u *⇩R y ∈ s)"
unfolding affine_def by (metis eq_diff_eq')
lemma affine_empty [iff]: "affine {}"
unfolding affine_def by auto
lemma affine_sing [iff]: "affine {x}"
unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
lemma affine_UNIV [iff]: "affine UNIV"
unfolding affine_def by auto
lemma affine_Inter [intro]: "(⋀s. s∈f ⟹ affine s) ⟹ affine (⋂f)"
unfolding affine_def by auto
lemma affine_Int[intro]: "affine s ⟹ affine t ⟹ affine (s ∩ t)"
unfolding affine_def by auto
lemma affine_scaling: "affine s ⟹ affine (image (λx. c *⇩R x) s)"
apply (clarsimp simp add: affine_def)
apply (rule_tac x="u *⇩R x + v *⇩R y" in image_eqI)
apply (auto simp: algebra_simps)
done
lemma affine_affine_hull [simp]: "affine(affine hull s)"
unfolding hull_def
using affine_Inter[of "{t. affine t ∧ s ⊆ t}"] by auto
lemma affine_hull_eq[simp]: "(affine hull s = s) ⟷ affine s"
by (metis affine_affine_hull hull_same)
lemma affine_hyperplane: "affine {x. a ∙ x = b}"
by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
subsubsection%unimportant ‹Some explicit formulations (from Lars Schewe)›
lemma affine:
fixes V::"'a::real_vector set"
shows "affine V ⟷
(∀S u. finite S ∧ S ≠ {} ∧ S ⊆ V ∧ sum u S = 1 ⟶ (∑x∈S. u x *⇩R x) ∈ V)"
proof -
have "u *⇩R x + v *⇩R y ∈ V" if "x ∈ V" "y ∈ V" "u + v = (1::real)"
and *: "⋀S u. ⟦finite S; S ≠ {}; S ⊆ V; sum u S = 1⟧ ⟹ (∑x∈S. u x *⇩R x) ∈ V" for x y u v
proof (cases "x = y")
case True
then show ?thesis
using that by (metis scaleR_add_left scaleR_one)
next
case False
then show ?thesis
using that *[of "{x,y}" "λw. if w = x then u else v"] by auto
qed
moreover have "(∑x∈S. u x *⇩R x) ∈ V"
if *: "⋀x y u v. ⟦x∈V; y∈V; u + v = 1⟧ ⟹ u *⇩R x + v *⇩R y ∈ V"
and "finite S" "S ≠ {}" "S ⊆ V" "sum u S = 1" for S u
proof -
define n where "n = card S"
consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
then show "(∑x∈S. u x *⇩R x) ∈ V"
proof cases
assume "card S = 1"
then obtain a where "S={a}"
by (auto simp: card_Suc_eq)
then show ?thesis
using that by simp
next
assume "card S = 2"
then obtain a b where "S = {a, b}"
by (metis Suc_1 card_1_singletonE card_Suc_eq)
then show ?thesis
using *[of a b] that
by (auto simp: sum_clauses(2))
next
assume "card S > 2"
then show ?thesis using that n_def
proof (induct n arbitrary: u S)
case 0
then show ?case by auto
next
case (Suc n u S)
have "sum u S = card S" if "¬ (∃x∈S. u x ≠ 1)"
using that unfolding card_eq_sum by auto
with Suc.prems obtain x where "x ∈ S" and x: "u x ≠ 1" by force
have c: "card (S - {x}) = card S - 1"
by (simp add: Suc.prems(3) ‹x ∈ S›)
have "sum u (S - {x}) = 1 - u x"
by (simp add: Suc.prems sum_diff1_ring ‹x ∈ S›)
with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
by auto
have inV: "(∑y∈S - {x}. (inverse (1 - u x) * u y) *⇩R y) ∈ V"
proof (cases "card (S - {x}) > 2")
case True
then have S: "S - {x} ≠ {}" "card (S - {x}) = n"
using Suc.prems c by force+
show ?thesis
proof (rule Suc.hyps)
show "(∑a∈S - {x}. inverse (1 - u x) * u a) = 1"
by (auto simp: eq1 sum_distrib_left[symmetric])
qed (use S Suc.prems True in auto)
next
case False
then have "card (S - {x}) = Suc (Suc 0)"
using Suc.prems c by auto
then obtain a b where ab: "(S - {x}) = {a, b}" "a≠b"
unfolding card_Suc_eq by auto
then show ?thesis
using eq1 ‹S ⊆ V›
by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
qed
have "u x + (1 - u x) = 1 ⟹
u x *⇩R x + (1 - u x) *⇩R ((∑y∈S - {x}. u y *⇩R y) /⇩R (1 - u x)) ∈ V"
by (rule Suc.prems) (use ‹x ∈ S› Suc.prems inV in ‹auto simp: scaleR_right.sum›)
moreover have "(∑a∈S. u a *⇩R a) = u x *⇩R x + (∑a∈S - {x}. u a *⇩R a)"
by (meson Suc.prems(3) sum.remove ‹x ∈ S›)
ultimately show "(∑x∈S. u x *⇩R x) ∈ V"
by (simp add: x)
qed
qed (use ‹S≠{}› ‹finite S› in auto)
qed
ultimately show ?thesis
unfolding affine_def by meson
qed
lemma affine_hull_explicit:
"affine hull p = {y. ∃S u. finite S ∧ S ≠ {} ∧ S ⊆ p ∧ sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
(is "_ = ?rhs")
proof (rule hull_unique)
show "p ⊆ ?rhs"
proof (intro subsetI CollectI exI conjI)
show "⋀x. sum (λz. 1) {x} = 1"
by auto
qed auto
show "?rhs ⊆ T" if "p ⊆ T" "affine T" for T
using that unfolding affine by blast
show "affine ?rhs"
unfolding affine_def
proof clarify
fix u v :: real and sx ux sy uy
assume uv: "u + v = 1"
and x: "finite sx" "sx ≠ {}" "sx ⊆ p" "sum ux sx = (1::real)"
and y: "finite sy" "sy ≠ {}" "sy ⊆ p" "sum uy sy = (1::real)"
have **: "(sx ∪ sy) ∩ sx = sx" "(sx ∪ sy) ∩ sy = sy"
by auto
show "∃S w. finite S ∧ S ≠ {} ∧ S ⊆ p ∧
sum w S = 1 ∧ (∑v∈S. w v *⇩R v) = u *⇩R (∑v∈sx. ux v *⇩R v) + v *⇩R (∑v∈sy. uy v *⇩R v)"
proof (intro exI conjI)
show "finite (sx ∪ sy)"
using x y by auto
show "sum (λi. (if i∈sx then u * ux i else 0) + (if i∈sy then v * uy i else 0)) (sx ∪ sy) = 1"
using x y uv
by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
have "(∑i∈sx ∪ sy. ((if i ∈ sx then u * ux i else 0) + (if i ∈ sy then v * uy i else 0)) *⇩R i)
= (∑i∈sx. (u * ux i) *⇩R i) + (∑i∈sy. (v * uy i) *⇩R i)"
using x y
unfolding scaleR_left_distrib scaleR_zero_left if_smult
by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] **)
also have "… = u *⇩R (∑v∈sx. ux v *⇩R v) + v *⇩R (∑v∈sy. uy v *⇩R v)"
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
finally show "(∑i∈sx ∪ sy. ((if i ∈ sx then u * ux i else 0) + (if i ∈ sy then v * uy i else 0)) *⇩R i)
= u *⇩R (∑v∈sx. ux v *⇩R v) + v *⇩R (∑v∈sy. uy v *⇩R v)" .
qed (use x y in auto)
qed
qed
lemma affine_hull_finite:
assumes "finite S"
shows "affine hull S = {y. ∃u. sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
proof -
have *: "∃h. sum h S = 1 ∧ (∑v∈S. h v *⇩R v) = x"
if "F ⊆ S" "finite F" "F ≠ {}" and sum: "sum u F = 1" and x: "(∑v∈F. u v *⇩R v) = x" for x F u
proof -
have "S ∩ F = F"
using that by auto
show ?thesis
proof (intro exI conjI)
show "(∑x∈S. if x ∈ F then u x else 0) = 1"
by (metis (mono_tags, lifting) ‹S ∩ F = F› assms sum.inter_restrict sum)
show "(∑v∈S. (if v ∈ F then u v else 0) *⇩R v) = x"
by (simp add: if_smult cong: if_cong) (metis (no_types) ‹S ∩ F = F› assms sum.inter_restrict x)
qed
qed
show ?thesis
unfolding affine_hull_explicit using assms
by (fastforce dest: *)
qed
subsubsection%unimportant ‹Stepping theorems and hence small special cases›
lemma affine_hull_empty[simp]: "affine hull {} = {}"
by simp
lemma affine_hull_finite_step:
fixes y :: "'a::real_vector"
shows "finite S ⟹
(∃u. sum u (insert a S) = w ∧ sum (λx. u x *⇩R x) (insert a S) = y) ⟷
(∃v u. sum u S = w - v ∧ sum (λx. u x *⇩R x) S = y - v *⇩R a)" (is "_ ⟹ ?lhs = ?rhs")
proof -
assume fin: "finite S"
show "?lhs = ?rhs"
proof
assume ?lhs
then obtain u where u: "sum u (insert a S) = w ∧ (∑x∈insert a S. u x *⇩R x) = y"
by auto
show ?rhs
proof (cases "a ∈ S")
case True
then show ?thesis
using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
next
case False
show ?thesis
by (rule exI [where x="u a"]) (use u fin False in auto)
qed
next
assume ?rhs
then obtain v u where vu: "sum u S = w - v" "(∑x∈S. u x *⇩R x) = y - v *⇩R a"
by auto
have *: "⋀x M. (if x = a then v else M) *⇩R x = (if x = a then v *⇩R x else M *⇩R x)"
by auto
show ?lhs
proof (cases "a ∈ S")
case True
show ?thesis
by (rule exI [where x="λx. (if x=a then v else 0) + u x"])
(simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
next
case False
then show ?thesis
apply (rule_tac x="λx. if x=a then v else u x" in exI)
apply (simp add: vu sum_clauses(2)[OF fin] *)
by (simp add: sum_delta_notmem(3) vu)
qed
qed
qed
lemma affine_hull_2:
fixes a b :: "'a::real_vector"
shows "affine hull {a,b} = {u *⇩R a + v *⇩R b| u v. (u + v = 1)}"
(is "?lhs = ?rhs")
proof -
have *:
"⋀x y z. z = x - y ⟷ y + z = (x::real)"
"⋀x y z. z = x - y ⟷ y + z = (x::'a)" by auto
have "?lhs = {y. ∃u. sum u {a, b} = 1 ∧ (∑v∈{a, b}. u v *⇩R v) = y}"
using affine_hull_finite[of "{a,b}"] by auto
also have "… = {y. ∃v u. u b = 1 - v ∧ u b *⇩R b = y - v *⇩R a}"
by (simp add: affine_hull_finite_step[of "{b}" a])
also have "… = ?rhs" unfolding * by auto
finally show ?thesis by auto
qed
lemma affine_hull_3:
fixes a b c :: "'a::real_vector"
shows "affine hull {a,b,c} = { u *⇩R a + v *⇩R b + w *⇩R c| u v w. u + v + w = 1}"
proof -
have *:
"⋀x y z. z = x - y ⟷ y + z = (x::real)"
"⋀x y z. z = x - y ⟷ y + z = (x::'a)" by auto
show ?thesis
apply (simp add: affine_hull_finite affine_hull_finite_step)
unfolding *
apply safe
apply (metis add.assoc)
apply (rule_tac x=u in exI, force)
done
qed
lemma mem_affine:
assumes "affine S" "x ∈ S" "y ∈ S" "u + v = 1"
shows "u *⇩R x + v *⇩R y ∈ S"
using assms affine_def[of S] by auto
lemma mem_affine_3:
assumes "affine S" "x ∈ S" "y ∈ S" "z ∈ S" "u + v + w = 1"
shows "u *⇩R x + v *⇩R y + w *⇩R z ∈ S"
proof -
have "u *⇩R x + v *⇩R y + w *⇩R z ∈ affine hull {x, y, z}"
using affine_hull_3[of x y z] assms by auto
moreover
have "affine hull {x, y, z} ⊆ affine hull S"
using hull_mono[of "{x, y, z}" "S"] assms by auto
moreover
have "affine hull S = S"
using assms affine_hull_eq[of S] by auto
ultimately show ?thesis by auto
qed
lemma mem_affine_3_minus:
assumes "affine S" "x ∈ S" "y ∈ S" "z ∈ S"
shows "x + v *⇩R (y-z) ∈ S"
using mem_affine_3[of S x y z 1 v "-v"] assms
by (simp add: algebra_simps)
corollary mem_affine_3_minus2:
"⟦affine S; x ∈ S; y ∈ S; z ∈ S⟧ ⟹ x - v *⇩R (y-z) ∈ S"
by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
subsubsection%unimportant ‹Some relations between affine hull and subspaces›
lemma affine_hull_insert_subset_span:
"affine hull (insert a S) ⊆ {a + v| v . v ∈ span {x - a | x . x ∈ S}}"
proof -
have "∃v T u. x = a + v ∧ (finite T ∧ T ⊆ {x - a |x. x ∈ S} ∧ (∑v∈T. u v *⇩R v) = v)"
if "finite F" "F ≠ {}" "F ⊆ insert a S" "sum u F = 1" "(∑v∈F. u v *⇩R v) = x"
for x F u
proof -
have *: "(λx. x - a) ` (F - {a}) ⊆ {x - a |x. x ∈ S}"
using that by auto
show ?thesis
proof (intro exI conjI)
show "finite ((λx. x - a) ` (F - {a}))"
by (simp add: that(1))
show "(∑v∈(λx. x - a) ` (F - {a}). u(v+a) *⇩R v) = x-a"
by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
qed (use ‹F ⊆ insert a S› in auto)
qed
then show ?thesis
unfolding affine_hull_explicit span_explicit by blast
qed
lemma affine_hull_insert_span:
assumes "a ∉ S"
shows "affine hull (insert a S) = {a + v | v . v ∈ span {x - a | x. x ∈ S}}"
proof -
have *: "∃G u. finite G ∧ G ≠ {} ∧ G ⊆ insert a S ∧ sum u G = 1 ∧ (∑v∈G. u v *⇩R v) = y"
if "v ∈ span {x - a |x. x ∈ S}" "y = a + v" for y v
proof -
from that
obtain T u where u: "finite T" "T ⊆ {x - a |x. x ∈ S}" "a + (∑v∈T. u v *⇩R v) = y"
unfolding span_explicit by auto
define F where "F = (λx. x + a) ` T"
have F: "finite F" "F ⊆ S" "(∑v∈F. u (v - a) *⇩R (v - a)) = y - a"
unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
have *: "F ∩ {a} = {}" "F ∩ - {a} = F"
using F assms by auto
show "∃G u. finite G ∧ G ≠ {} ∧ G ⊆ insert a S ∧ sum u G = 1 ∧ (∑v∈G. u v *⇩R v) = y"
apply (rule_tac x = "insert a F" in exI)
apply (rule_tac x = "λx. if x=a then 1 - sum (λx. u (x - a)) F else u (x - a)" in exI)
using assms F
apply (auto simp: sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
done
qed
show ?thesis
by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
qed
lemma affine_hull_span:
assumes "a ∈ S"
shows "affine hull S = {a + v | v. v ∈ span {x - a | x. x ∈ S - {a}}}"
using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
subsubsection%unimportant ‹Parallel affine sets›
definition affine_parallel :: "'a::real_vector set ⇒ 'a::real_vector set ⇒ bool"
where "affine_parallel S T ⟷ (∃a. T = (λx. a + x) ` S)"
lemma affine_parallel_expl_aux:
fixes S T :: "'a::real_vector set"
assumes "⋀x. x ∈ S ⟷ a + x ∈ T"
shows "T = (λx. a + x) ` S"
proof -
have "x ∈ ((λx. a + x) ` S)" if "x ∈ T" for x
using that
by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
moreover have "T ≥ (λx. a + x) ` S"
using assms by auto
ultimately show ?thesis by auto
qed
lemma affine_parallel_expl: "affine_parallel S T ⟷ (∃a. ∀x. x ∈ S ⟷ a + x ∈ T)"
unfolding affine_parallel_def
using affine_parallel_expl_aux[of S _ T] by auto
lemma affine_parallel_reflex: "affine_parallel S S"
unfolding affine_parallel_def
using image_add_0 by blast
lemma affine_parallel_commut:
assumes "affine_parallel A B"
shows "affine_parallel B A"
proof -
from assms obtain a where B: "B = (λx. a + x) ` A"
unfolding affine_parallel_def by auto
have [simp]: "(λx. x - a) = plus (- a)" by (simp add: fun_eq_iff)
from B show ?thesis
using translation_galois [of B a A]
unfolding affine_parallel_def by auto
qed
lemma affine_parallel_assoc:
assumes "affine_parallel A B"
and "affine_parallel B C"
shows "affine_parallel A C"
proof -
from assms obtain ab where "B = (λx. ab + x) ` A"
unfolding affine_parallel_def by auto
moreover
from assms obtain bc where "C = (λx. bc + x) ` B"
unfolding affine_parallel_def by auto
ultimately show ?thesis
using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
qed
lemma affine_translation_aux:
fixes a :: "'a::real_vector"
assumes "affine ((λx. a + x) ` S)"
shows "affine S"
proof -
{
fix x y u v
assume xy: "x ∈ S" "y ∈ S" "(u :: real) + v = 1"
then have "(a + x) ∈ ((λx. a + x) ` S)" "(a + y) ∈ ((λx. a + x) ` S)"
by auto
then have h1: "u *⇩R (a + x) + v *⇩R (a + y) ∈ (λx. a + x) ` S"
using xy assms unfolding affine_def by auto
have "u *⇩R (a + x) + v *⇩R (a + y) = (u + v) *⇩R a + (u *⇩R x + v *⇩R y)"
by (simp add: algebra_simps)
also have "… = a + (u *⇩R x + v *⇩R y)"
using ‹u + v = 1› by auto
ultimately have "a + (u *⇩R x + v *⇩R y) ∈ (λx. a + x) ` S"
using h1 by auto
then have "u *⇩R x + v *⇩R y ∈ S" by auto
}
then show ?thesis unfolding affine_def by auto
qed
lemma affine_translation:
fixes a :: "'a::real_vector"
shows "affine S ⟷ affine ((λx. a + x) ` S)"
proof -
have "affine S ⟹ affine ((λx. a + x) ` S)"
using affine_translation_aux[of "-a" "((λx. a + x) ` S)"]
using translation_assoc[of "-a" a S] by auto
then show ?thesis using affine_translation_aux by auto
qed
lemma parallel_is_affine:
fixes S T :: "'a::real_vector set"
assumes "affine S" "affine_parallel S T"
shows "affine T"
proof -
from assms obtain a where "T = (λx. a + x) ` S"
unfolding affine_parallel_def by auto
then show ?thesis
using affine_translation assms by auto
qed
lemma subspace_imp_affine: "subspace s ⟹ affine s"
unfolding subspace_def affine_def by auto
subsubsection%unimportant ‹Subspace parallel to an affine set›
lemma subspace_affine: "subspace S ⟷ affine S ∧ 0 ∈ S"
proof -
have h0: "subspace S ⟹ affine S ∧ 0 ∈ S"
using subspace_imp_affine[of S] subspace_0 by auto
{
assume assm: "affine S ∧ 0 ∈ S"
{
fix c :: real
fix x
assume x: "x ∈ S"
have "c *⇩R x = (1-c) *⇩R 0 + c *⇩R x" by auto
moreover
have "(1 - c) *⇩R 0 + c *⇩R x ∈ S"
using affine_alt[of S] assm x by auto
ultimately have "c *⇩R x ∈ S" by auto
}
then have h1: "∀c. ∀x ∈ S. c *⇩R x ∈ S" by auto
{
fix x y
assume xy: "x ∈ S" "y ∈ S"
define u where "u = (1 :: real)/2"
have "(1/2) *⇩R (x+y) = (1/2) *⇩R (x+y)"
by auto
moreover
have "(1/2) *⇩R (x+y)=(1/2) *⇩R x + (1-(1/2)) *⇩R y"
by (simp add: algebra_simps)
moreover
have "(1 - u) *⇩R x + u *⇩R y ∈ S"
using affine_alt[of S] assm xy by auto
ultimately
have "(1/2) *⇩R (x+y) ∈ S"
using u_def by auto
moreover
have "x + y = 2 *⇩R ((1/2) *⇩R (x+y))"
by auto
ultimately
have "x + y ∈ S"
using h1[rule_format, of "(1/2) *⇩R (x+y)" "2"] by auto
}
then have "∀x ∈ S. ∀y ∈ S. x + y ∈ S"
by auto
then have "subspace S"
using h1 assm unfolding subspace_def by auto
}
then show ?thesis using h0 by metis
qed
lemma affine_diffs_subspace:
assumes "affine S" "a ∈ S"
shows "subspace ((λx. (-a)+x) ` S)"
proof -
have [simp]: "(λx. x - a) = plus (- a)" by (simp add: fun_eq_iff)
have "affine ((λx. (-a)+x) ` S)"
using affine_translation assms by auto
moreover have "0 ∈ ((λx. (-a)+x) ` S)"
using assms exI[of "(λx. x∈S ∧ -a+x = 0)" a] by auto
ultimately show ?thesis using subspace_affine by auto
qed
lemma parallel_subspace_explicit:
assumes "affine S"
and "a ∈ S"
assumes "L ≡ {y. ∃x ∈ S. (-a) + x = y}"
shows "subspace L ∧ affine_parallel S L"
proof -
from assms have "L = plus (- a) ` S" by auto
then have par: "affine_parallel S L"
unfolding affine_parallel_def ..
then have "affine L" using assms parallel_is_affine by auto
moreover have "0 ∈ L"
using assms by auto
ultimately show ?thesis
using subspace_affine par by auto
qed
lemma parallel_subspace_aux:
assumes "subspace A"
and "subspace B"
and "affine_parallel A B"
shows "A ⊇ B"
proof -
from assms obtain a where a: "∀x. x ∈ A ⟷ a + x ∈ B"
using affine_parallel_expl[of A B] by auto
then have "-a ∈ A"
using assms subspace_0[of B] by auto
then have "a ∈ A"
using assms subspace_neg[of A "-a"] by auto
then show ?thesis
using assms a unfolding subspace_def by auto
qed
lemma parallel_subspace:
assumes "subspace A"
and "subspace B"
and "affine_parallel A B"
shows "A = B"
proof
show "A ⊇ B"
using assms parallel_subspace_aux by auto
show "A ⊆ B"
using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
qed
lemma affine_parallel_subspace:
assumes "affine S" "S ≠ {}"
shows "∃!L. subspace L ∧ affine_parallel S L"
proof -
have ex: "∃L. subspace L ∧ affine_parallel S L"
using assms parallel_subspace_explicit by auto
{
fix L1 L2
assume ass: "subspace L1 ∧ affine_parallel S L1" "subspace L2 ∧ affine_parallel S L2"
then have "affine_parallel L1 L2"
using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
then have "L1 = L2"
using ass parallel_subspace by auto
}
then show ?thesis using ex by auto
qed
subsection ‹Cones›
definition%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"
apply (rule hull_eq)
using cone_Inter
unfolding subset_eq
apply auto
done
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"
proof -
{
assume "S ≠ {}"
then obtain a where "a ∈ S" by auto
then have "0 ∈ S"
using assms mem_cone[of S a 0] by auto
}
then show ?thesis by auto
qed
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 ‹cone S› ‹c > 0›
unfolding cone_def image_def ‹c > 0› by auto
}
ultimately have "(( *⇩R) c) ` S = S" by auto
}
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"
apply auto
apply (rule_tac x = 1 in exI, auto)
done
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 cone_closure:
fixes S :: "'a::real_normed_vector set"
assumes "cone S"
shows "cone (closure 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
then have "0 ∈ closure S ∧ (∀c. c > 0 ⟶ ( *⇩R) c ` closure S = closure S)"
using closure_subset by (auto simp: closure_scaleR)
then show ?thesis
using False cone_iff[of "closure S"] by auto
qed
subsection ‹Affine dependence and consequential theorems (from Lars Schewe)›
definition%important affine_dependent :: "'a::real_vector set ⇒ bool"
where "affine_dependent s ⟷ (∃x∈s. x ∈ affine hull (s - {x}))"
lemma affine_dependent_subset:
"⟦affine_dependent s; s ⊆ t⟧ ⟹ affine_dependent t"
apply (simp add: affine_dependent_def Bex_def)
apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
done
lemma affine_independent_subset:
shows "⟦~ affine_dependent t; s ⊆ t⟧ ⟹ ~ affine_dependent s"
by (metis affine_dependent_subset)
lemma affine_independent_Diff:
"~ affine_dependent s ⟹ ~ affine_dependent(s - t)"
by (meson Diff_subset affine_dependent_subset)
proposition affine_dependent_explicit:
"affine_dependent p ⟷
(∃S u. finite S ∧ S ⊆ p ∧ sum u S = 0 ∧ (∃v∈S. u v ≠ 0) ∧ sum (λv. u v *⇩R v) S = 0)"
proof -
have "∃S u. finite S ∧ S ⊆ p ∧ sum u S = 0 ∧ (∃v∈S. u v ≠ 0) ∧ (∑w∈S. u w *⇩R w) = 0"
if "(∑w∈S. u w *⇩R w) = x" "x ∈ p" "finite S" "S ≠ {}" "S ⊆ p - {x}" "sum u S = 1" for x S u
proof (intro exI conjI)
have "x ∉ S"
using that by auto
then show "(∑v ∈ insert x S. if v = x then - 1 else u v) = 0"
using that by (simp add: sum_delta_notmem)
show "(∑w ∈ insert x S. (if w = x then - 1 else u w) *⇩R w) = 0"
using that ‹x ∉ S› by (simp add: if_smult sum_delta_notmem cong: if_cong)
qed (use that in auto)
moreover have "∃x∈p. ∃S u. finite S ∧ S ≠ {} ∧ S ⊆ p - {x} ∧ sum u S = 1 ∧ (∑v∈S. u v *⇩R v) = x"
if "(∑v∈S. u v *⇩R v) = 0" "finite S" "S ⊆ p" "sum u S = 0" "v ∈ S" "u v ≠ 0" for S u v
proof (intro bexI exI conjI)
have "S ≠ {v}"
using that by auto
then show "S - {v} ≠ {}"
using that by auto
show "(∑x ∈ S - {v}. - (1 / u v) * u x) = 1"
unfolding sum_distrib_left[symmetric] sum_diff1[OF ‹finite S›] by (simp add: that)
show "(∑x∈S - {v}. (- (1 / u v) * u x) *⇩R x) = v"
unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
scaleR_right.sum [symmetric] sum_diff1[OF ‹finite S›]
using that by auto
show "S - {v} ⊆ p - {v}"
using that by auto
qed (use that in auto)
ultimately show ?thesis
unfolding affine_dependent_def affine_hull_explicit by auto
qed
lemma affine_dependent_explicit_finite:
fixes S :: "'a::real_vector set"
assumes "finite S"
shows "affine_dependent S ⟷
(∃u. sum u S = 0 ∧ (∃v∈S. u v ≠ 0) ∧ sum (λv. u v *⇩R v) S = 0)"
(is "?lhs = ?rhs")
proof
have *: "⋀vt u v. (if vt then u v else 0) *⇩R v = (if vt then (u v) *⇩R v else 0::'a)"
by auto
assume ?lhs
then obtain t u v where
"finite t" "t ⊆ S" "sum u t = 0" "v∈t" "u v ≠ 0" "(∑v∈t. u v *⇩R v) = 0"
unfolding affine_dependent_explicit by auto
then show ?rhs
apply (rule_tac x="λx. if x∈t then u x else 0" in exI)
apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF ‹t⊆S›])
done
next
assume ?rhs
then obtain u v where "sum u S = 0" "v∈S" "u v ≠ 0" "(∑v∈S. u v *⇩R v) = 0"
by auto
then show ?lhs unfolding affine_dependent_explicit
using assms by auto
qed
subsection%unimportant ‹Connectedness of convex sets›
lemma connectedD:
"connected S ⟹ open A ⟹ open B ⟹ S ⊆ A ∪ B ⟹ A ∩ B ∩ S = {} ⟹ A ∩ S = {} ∨ B ∩ S = {}"
by (rule Topological_Spaces.topological_space_class.connectedD)
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 connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
by (simp add: convex_connected)
corollary component_complement_connected:
fixes S :: "'a::real_normed_vector set"
assumes "connected S" "C ∈ components (-S)"
shows "connected(-C)"
using component_diff_connected [of S UNIV] assms
by (auto simp: Compl_eq_Diff_UNIV)
proposition clopen:
fixes S :: "'a :: real_normed_vector set"
shows "closed S ∧ open S ⟷ S = {} ∨ S = UNIV"
by (force intro!: connected_UNIV [unfolded connected_clopen, rule_format])
corollary compact_open:
fixes S :: "'a :: euclidean_space set"
shows "compact S ∧ open S ⟷ S = {}"
by (auto simp: compact_eq_bounded_closed clopen)
corollary finite_imp_not_open:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "⟦finite S; open S⟧ ⟹ S={}"
using clopen [of S] finite_imp_closed not_bounded_UNIV by blast
corollary empty_interior_finite:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "finite S ⟹ interior S = {}"
by (metis interior_subset finite_subset open_interior [of S] finite_imp_not_open)
text ‹Balls, being convex, are 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 add: atLeast_def[symmetric] convex_real_interval)
lemma convex_local_global_minimum:
fixes s :: "'a::real_normed_vector set"
assumes "e > 0"
and "convex_on s f"
and "ball x e ⊆ s"
and "∀y∈ball x e. f x ≤ f y"
shows "∀y∈s. f x ≤ f y"
proof (rule ccontr)
have "x ∈ s" using assms(1,3) by auto
assume "¬ ?thesis"
then obtain y where "y∈s" and y: "f x > f y" by auto
then have xy: "0 < dist x y" by auto
then obtain u where "0 < u" "u ≤ 1" and u: "u < e / dist x y"
using field_lbound_gt_zero[of 1 "e / dist x y"] xy ‹e>0› by auto
then have "f ((1-u) *⇩R x + u *⇩R y) ≤ (1-u) * f x + u * f y"
using ‹x∈s› ‹y∈s›
using assms(2)[unfolded convex_on_def,
THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
by auto
moreover
have *: "x - ((1 - u) *⇩R x + u *⇩R y) = u *⇩R (x - y)"
by (simp add: algebra_simps)
have "(1 - u) *⇩R x + u *⇩R y ∈ ball x e"
unfolding mem_ball dist_norm
unfolding * and norm_scaleR and abs_of_pos[OF ‹0<u›]
unfolding dist_norm[symmetric]
using u
unfolding pos_less_divide_eq[OF xy]
by auto
then have "f x ≤ f ((1 - u) *⇩R x + u *⇩R y)"
using assms(4) by auto
ultimately show False
using mult_strict_left_mono[OF y ‹u>0›]
unfolding left_diff_distrib
by auto
qed
lemma convex_ball [iff]:
fixes x :: "'a::real_normed_vector"
shows "convex (ball x e)"
proof (auto simp: convex_def)
fix y z
assume yz: "dist x y < e" "dist x z < e"
fix u v :: real
assume uv: "0 ≤ u" "0 ≤ v" "u + v = 1"
have "dist x (u *⇩R y + v *⇩R z) ≤ u * dist x y + v * dist x z"
using uv yz
using convex_on_dist [of "ball x e" x, unfolded convex_on_def,
THEN bspec[where x=y], THEN bspec[where x=z]]
by auto
then show "dist x (u *⇩R y + v *⇩R z) < e"
using convex_bound_lt[OF yz uv] by auto
qed
lemma convex_cball [iff]:
fixes x :: "'a::real_normed_vector"
shows "convex (cball x e)"
proof -
{
fix y z
assume yz: "dist x y ≤ e" "dist x z ≤ e"
fix u v :: real
assume uv: "0 ≤ u" "0 ≤ v" "u + v = 1"
have "dist x (u *⇩R y + v *⇩R z) ≤ u * dist x y + v * dist x z"
using uv yz
using convex_on_dist [of "cball x e" x, unfolded convex_on_def,
THEN bspec[where x=y], THEN bspec[where x=z]]
by auto
then have "dist x (u *⇩R y + v *⇩R z) ≤ e"
using convex_bound_le[OF yz uv] by auto
}
then show ?thesis by (auto simp: convex_def Ball_def)
qed
lemma connected_ball [iff]:
fixes x :: "'a::real_normed_vector"
shows "connected (ball x e)"
using convex_connected convex_ball by auto
lemma connected_cball [iff]:
fixes x :: "'a::real_normed_vector"
shows "connected (cball x e)"
using convex_connected convex_cball by auto
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: convex_convex_hull subset_hull)
lemma convex_hull_eq: "convex hull s = s ⟷ convex s"
by (metis convex_convex_hull hull_same)
lemma bounded_convex_hull:
fixes s :: "'a::real_normed_vector set"
assumes "bounded s"
shows "bounded (convex hull s)"
proof -
from assms obtain B where B: "∀x∈s. norm x ≤ B"
unfolding bounded_iff by auto
show ?thesis
apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
unfolding subset_hull[of convex, OF convex_cball]
unfolding subset_eq mem_cball dist_norm using B
apply auto
done
qed
lemma finite_imp_bounded_convex_hull:
fixes s :: "'a::real_normed_vector set"
shows "finite s ⟹ bounded (convex hull s)"
using bounded_convex_hull finite_imp_bounded
by auto
subsubsection%unimportant ‹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%unimportant ‹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
by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
subsubsection%unimportant ‹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%unimportant ‹Another formulation from 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)
apply simp
apply (rule sum_nonneg)
using obt(1)
apply auto
done
}
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}"
apply auto
using f(1)[unfolded inj_on_def]
by (metis One_nat_def atLeastAtMost_iff)
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%unimportant ‹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 (rule, case_tac[!] "a∈S")
assume "a ∈ S"
then have *: "insert a S = S" by auto
assume ?lhs
then show ?rhs
unfolding * by (rule_tac x=0 in exI, auto)
next
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
assume "a ∉ S"
then show ?rhs
apply (rule_tac x="u a" in exI)
using u(1)[THEN bspec[where x=a]]
apply simp
apply (rule_tac x=u in exI)
using u[unfolded sum_clauses(2)[OF assms]] and ‹a∉S›
apply auto
done
next
assume "a ∈ S"
then have *: "insert a S = S" by auto
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
show ?lhs
apply (rule_tac x = "λx. (if a = x then v else 0) + u x" in exI)
unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
unfolding sum_clauses(2)[OF assms]
using uv and uv(2)[THEN bspec[where x=a]] and ‹a∈S›
apply auto
done
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 assume "a ∉ S"
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 ‹a ∉ S›
by (auto simp: intro!: sum.cong)
ultimately show ?lhs
by (rule_tac x="λx. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
qed
subsubsection%unimportant ‹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}"
proof -
have *: "⋀u. (∀x∈{a, b}. 0 ≤ u x) ⟷ 0 ≤ u a ∧ 0 ≤ u b"
by auto
have **: "finite {b}" by auto
show ?thesis
apply (simp add: convex_hull_finite)
unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
apply auto
apply (rule_tac x=v in exI)
apply (rule_tac x="1 - v" in exI, simp)
apply (rule_tac x=u in exI, simp)
apply (rule_tac x="λx. v" in exI, simp)
done
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)"
unfolding *
apply auto
apply (rule_tac[!] x=u in exI)
apply (auto simp: algebra_simps)
done
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%unimportant ‹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 affine_hull_subset_span: "(affine hull s) ⊆ (span s)"
by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
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 affine_dependent_imp_dependent: "affine_dependent s ⟹ dependent s"
unfolding affine_dependent_def dependent_def
using affine_hull_subset_span by auto
lemma dependent_imp_affine_dependent:
assumes "dependent {x - a| x . x ∈ s}"
and "a ∉ s"
shows "affine_dependent (insert a s)"
proof -
from assms(1)[unfolded dependent_explicit] obtain S u v
where obt: "finite S" "S ⊆ {x - a |x. x ∈ s}" "v∈S" "u v ≠ 0" "(∑v∈S. u v *⇩R v) = 0"
by auto
define t where "t = (λx. x + a) ` S"
have inj: "inj_on (λx. x + a) S"
unfolding inj_on_def by auto
have "0 ∉ S"
using obt(2) assms(2) unfolding subset_eq by auto
have fin: "finite t" and "t ⊆ s"
unfolding t_def using obt(1,2) by auto
then have "finite (insert a t)" and "insert a t ⊆ insert a s"
by auto
moreover have *: "⋀P Q. (∑x∈t. (if x = a then P x else Q x)) = (∑x∈t. Q x)"
apply (rule sum.cong)
using ‹a∉s› ‹t⊆s›
apply auto
done
have "(∑x∈insert a t. if x = a then - (∑x∈t. u (x - a)) else u (x - a)) = 0"
unfolding sum_clauses(2)[OF fin] * using ‹a∉s› ‹t⊆s› by auto
moreover have "∃v∈insert a t. (if v = a then - (∑x∈t. u (x - a)) else u (v - a)) ≠ 0"
using obt(3,4) ‹0∉S›
by (rule_tac x="v + a" in bexI) (auto simp: t_def)
moreover have *: "⋀P Q. (∑x∈t. (if x = a then P x else Q x) *⇩R x) = (∑x∈t. Q x *⇩R x)"
using ‹a∉s› ‹t⊆s› by (auto intro!: sum.cong)
have "(∑x∈t. u (x - a)) *⇩R a = (∑v∈t. u (v - a) *⇩R v)"
unfolding scaleR_left.sum
unfolding t_def and sum.reindex[OF inj] and o_def
using obt(5)
by (auto simp: sum.distrib scaleR_right_distrib)
then have "(∑v∈insert a t. (if v = a then - (∑x∈t. u (x - a)) else u (v - a)) *⇩R v) = 0"
unfolding sum_clauses(2)[OF fin]
using ‹a∉s› ‹t⊆s›
by (auto simp: *)
ultimately show ?thesis
unfolding affine_dependent_explicit
apply (rule_tac x="insert a t" in exI, auto)
done
qed
lemma convex_cone:
"convex s ∧ 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
lemma affine_dependent_biggerset:
fixes s :: "'a::euclidean_space set"
assumes "finite s" "card s ≥ DIM('a) + 2"
shows "affine_dependent s"
proof -
have "s ≠ {}" using assms by auto
then obtain a where "a∈s" by auto
have *: "{x - a |x. x ∈ s - {a}} = (λx. x - a) ` (s - {a})"
by auto
have "card {x - a |x. x ∈ s - {a}} = card (s - {a})"
unfolding * by (simp add: card_image inj_on_def)
also have "… > DIM('a)" using assms(2)
unfolding card_Diff_singleton[OF assms(1) ‹a∈s›] by auto
finally show ?thesis
apply (subst insert_Diff[OF ‹a∈s›, symmetric])
apply (rule dependent_imp_affine_dependent)
apply (rule dependent_biggerset, auto)
done
qed
lemma affine_dependent_biggerset_general:
assumes "finite (S :: 'a::euclidean_space set)"
and "card S ≥ dim S + 2"
shows "affine_dependent S"
proof -
from assms(2) have "S ≠ {}" by auto
then obtain a where "a∈S" by auto
have *: "{x - a |x. x ∈ S - {a}} = (λx. x - a) ` (S - {a})"
by auto
have **: "card {x - a |x. x ∈ S - {a}} = card (S - {a})"
by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
have "dim {x - a |x. x ∈ S - {a}} ≤ dim S"
using ‹a∈S› by (auto simp: span_base span_diff intro: subset_le_dim)
also have "… < dim S + 1" by auto
also have "… ≤ card (S - {a})"
using assms
using card_Diff_singleton[OF assms(1) ‹a∈S›]
by auto
finally show ?thesis
apply (subst insert_Diff[OF ‹a∈S›, symmetric])
apply (rule dependent_imp_affine_dependent)
apply (rule dependent_biggerset_general)
unfolding **
apply auto
done
qed
subsection%unimportant ‹Some Properties of Affine Dependent Sets›
lemma affine_independent_0 [simp]: "¬ affine_dependent {}"
by (simp add: affine_dependent_def)
lemma affine_independent_1 [simp]: "¬ affine_dependent {a}"
by (simp add: affine_dependent_def)
lemma affine_independent_2 [simp]: "¬ affine_dependent {a,b}"
by (simp add: affine_dependent_def insert_Diff_if hull_same)
lemma affine_hull_translation: "affine hull ((λx. a + x) ` S) = (λx. a + x) ` (affine hull S)"
proof -
have "affine ((λx. a + x) ` (affine hull S))"
using affine_translation affine_affine_hull by blast
moreover have "(λx. a + x) ` S ⊆ (λx. a + x) ` (affine hull S)"
using hull_subset[of S] by auto
ultimately have h1: "affine hull ((λx. a + x) ` S) ⊆ (λx. a + x) ` (affine hull S)"
by (metis hull_minimal)
have "affine((λx. -a + x) ` (affine hull ((λx. a + x) ` S)))"
using affine_translation affine_affine_hull by blast
moreover have "(λx. -a + x) ` (λx. a + x) ` S ⊆ (λx. -a + x) ` (affine hull ((λx. a + x) ` S))"
using hull_subset[of "(λx. a + x) ` S"] by auto
moreover have "S = (λx. -a + x) ` (λx. a + x) ` S"
using translation_assoc[of "-a" a] by auto
ultimately have "(λx. -a + x) ` (affine hull ((λx. a + x) ` S)) >= (affine hull S)"
by (metis hull_minimal)
then have "affine hull ((λx. a + x) ` S) >= (λx. a + x) ` (affine hull S)"
by auto
then show ?thesis using h1 by auto
qed
lemma affine_dependent_translation:
assumes "affine_dependent S"
shows "affine_dependent ((λx. a + x) ` S)"
proof -
obtain x where x: "x ∈ S ∧ x ∈ affine hull (S - {x})"
using assms affine_dependent_def by auto
have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
by auto
then have "a + x ∈ affine hull ((λx. a + x) ` S - {a + x})"
using affine_hull_translation[of a "S - {x}"] x by auto
moreover have "a + x ∈ (λx. a + x) ` S"
using x by auto
ultimately show ?thesis
unfolding affine_dependent_def by auto
qed
lemma affine_dependent_translation_eq:
"affine_dependent S ⟷ affine_dependent ((λx. a + x) ` S)"
proof -
{
assume "affine_dependent ((λx. a + x) ` S)"
then have "affine_dependent S"
using affine_dependent_translation[of "((λx. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
by auto
}
then show ?thesis
using affine_dependent_translation by auto
qed
lemma affine_hull_0_dependent:
assumes "0 ∈ affine hull S"
shows "dependent S"
proof -
obtain s u where s_u: "finite s ∧ s ≠ {} ∧ s ⊆ S ∧ sum u s = 1 ∧ (∑v∈s. u v *⇩R v) = 0"
using assms affine_hull_explicit[of S] by auto
then have "∃v∈s. u v ≠ 0"
using sum_not_0[of "u" "s"] by auto
then have "finite s ∧ s ⊆ S ∧ (∃v∈s. u v ≠ 0 ∧ (∑v∈s. u v *⇩R v) = 0)"
using s_u by auto
then show ?thesis
unfolding dependent_explicit[of S] by auto
qed
lemma affine_dependent_imp_dependent2:
assumes "affine_dependent (insert 0 S)"
shows "dependent S"
proof -
obtain x where x: "x ∈ insert 0 S ∧ x ∈ affine hull (insert 0 S - {x})"
using affine_dependent_def[of "(insert 0 S)"] assms by blast
then have "x ∈ span (insert 0 S - {x})"
using affine_hull_subset_span by auto
moreover have "span (insert 0 S - {x}) = span (S - {x})"
using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
ultimately have "x ∈ span (S - {x})" by auto
then have "x ≠ 0 ⟹ dependent S"
using x dependent_def by auto
moreover
{
assume "x = 0"
then have "0 ∈ affine hull S"
using x hull_mono[of "S - {0}" S] by auto
then have "dependent S"
using affine_hull_0_dependent by auto
}
ultimately show ?thesis by auto
qed
lemma affine_dependent_iff_dependent:
assumes "a ∉ S"
shows "affine_dependent (insert a S) ⟷ dependent ((λx. -a + x) ` S)"
proof -
have "((+) (- a) ` S) = {x - a| x . x ∈ S}" by auto
then show ?thesis
using affine_dependent_translation_eq[of "(insert a S)" "-a"]
affine_dependent_imp_dependent2 assms
dependent_imp_affine_dependent[of a S]
by (auto simp del: uminus_add_conv_diff)
qed
lemma affine_dependent_iff_dependent2:
assumes "a ∈ S"
shows "affine_dependent S ⟷ dependent ((λx. -a + x) ` (S-{a}))"
proof -
have "insert a (S - {a}) = S"
using assms by auto
then show ?thesis
using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
qed
lemma affine_hull_insert_span_gen:
"affine hull (insert a s) = (λx. a + x) ` span ((λx. - a + x) ` s)"
proof -
have h1: "{x - a |x. x ∈ s} = ((λx. -a+x) ` s)"
by auto
{
assume "a ∉ s"
then have ?thesis
using affine_hull_insert_span[of a s] h1 by auto
}
moreover
{
assume a1: "a ∈ s"
have "∃x. x ∈ s ∧ -a+x=0"
apply (rule exI[of _ a])
using a1
apply auto
done
then have "insert 0 ((λx. -a+x) ` (s - {a})) = (λx. -a+x) ` s"
by auto
then have "span ((λx. -a+x) ` (s - {a}))=span ((λx. -a+x) ` s)"
using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
moreover have "{x - a |x. x ∈ (s - {a})} = ((λx. -a+x) ` (s - {a}))"
by auto
moreover have "insert a (s - {a}) = insert a s"
by auto
ultimately have ?thesis
using affine_hull_insert_span[of "a" "s-{a}"] by auto
}
ultimately show ?thesis by auto
qed
lemma affine_hull_span2:
assumes "a ∈ s"
shows "affine hull s = (λx. a+x) ` span ((λx. -a+x) ` (s-{a}))"
using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
by auto
lemma affine_hull_span_gen:
assumes "a ∈ affine hull s"
shows "affine hull s = (λx. a+x) ` span ((λx. -a+x) ` s)"
proof -
have "affine hull (insert a s) = affine hull s"
using hull_redundant[of a affine s] assms by auto
then show ?thesis
using affine_hull_insert_span_gen[of a "s"] by auto
qed
lemma affine_hull_span_0:
assumes "0 ∈ affine hull S"
shows "affine hull S = span S"
using affine_hull_span_gen[of "0" S] assms by auto
lemma extend_to_affine_basis_nonempty:
fixes S V :: "'n::euclidean_space set"
assumes "¬ affine_dependent S" "S ⊆ V" "S ≠ {}"
shows "∃T. ¬ affine_dependent T ∧ S ⊆ T ∧ T ⊆ V ∧ affine hull T = affine hull V"
proof -
obtain a where a: "a ∈ S"
using assms by auto
then have h0: "independent ((λx. -a + x) ` (S-{a}))"
using affine_dependent_iff_dependent2 assms by auto
obtain B where B:
"(λx. -a+x) ` (S - {a}) ⊆ B ∧ B ⊆ (λx. -a+x) ` V ∧ independent B ∧ (λx. -a+x) ` V ⊆ span B"
using assms
by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(λx. -a + x) ` V"])
define T where "T = (λx. a+x) ` insert 0 B"
then have "T = insert a ((λx. a+x) ` B)"
by auto
then have "affine hull T = (λx. a+x) ` span B"
using affine_hull_insert_span_gen[of a "((λx. a+x) ` B)"] translation_assoc[of "-a" a B]
by auto
then have "V ⊆ affine hull T"
using B assms translation_inverse_subset[of a V "span B"]
by auto
moreover have "T ⊆ V"
using T_def B a assms by auto
ultimately have "affine hull T = affine hull V"
by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
moreover have "S ⊆ T"
using T_def B translation_inverse_subset[of a "S-{a}" B]
by auto
moreover have "¬ affine_dependent T"
using T_def affine_dependent_translation_eq[of "insert 0 B"]
affine_dependent_imp_dependent2 B
by auto
ultimately show ?thesis using ‹T ⊆ V› by auto
qed
lemma affine_basis_exists:
fixes V :: "'n::euclidean_space set"
shows "∃B. B ⊆ V ∧ ¬ affine_dependent B ∧ affine hull V = affine hull B"
proof (cases "V = {}")
case True
then show ?thesis
using affine_independent_0 by auto
next
case False
then obtain x where "x ∈ V" by auto
then show ?thesis
using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
by auto
qed
proposition extend_to_affine_basis:
fixes S V :: "'n::euclidean_space set"
assumes "¬ affine_dependent S" "S ⊆ V"
obtains T where "¬ affine_dependent T" "S ⊆ T" "T ⊆ V" "affine hull T = affine hull V"
proof (cases "S = {}")
case True then show ?thesis
using affine_basis_exists by (metis empty_subsetI that)
next
case False
then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
qed
subsection ‹Affine Dimension of a Set›
definition%important aff_dim :: "('a::euclidean_space) set ⇒ int"
where "aff_dim V =
(SOME d :: int.
∃B. affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ of_nat (card B) = d + 1)"
lemma aff_dim_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "∃B. affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ of_nat (card B) = aff_dim V + 1"
proof -
obtain B where "¬ affine_dependent B ∧ affine hull B = affine hull V"
using affine_basis_exists[of V] by auto
then show ?thesis
unfolding aff_dim_def
some_eq_ex[of "λd. ∃B. affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ of_nat (card B) = d + 1"]
apply auto
apply (rule exI[of _ "int (card B) - (1 :: int)"])
apply (rule exI[of _ "B"], auto)
done
qed
lemma affine_hull_nonempty: "S ≠ {} ⟷ affine hull S ≠ {}"
proof -
have "S = {} ⟹ affine hull S = {}"
using affine_hull_empty by auto
moreover have "affine hull S = {} ⟹ S = {}"
unfolding hull_def by auto
ultimately show ?thesis by blast
qed
lemma aff_dim_parallel_subspace_aux:
fixes B :: "'n::euclidean_space set"
assumes "¬ affine_dependent B" "a ∈ B"
shows "finite B ∧ ((card B) - 1 = dim (span ((λx. -a+x) ` (B-{a}))))"
proof -
have "independent ((λx. -a + x) ` (B-{a}))"
using affine_dependent_iff_dependent2 assms by auto
then have fin: "dim (span ((λx. -a+x) ` (B-{a}))) = card ((λx. -a + x) ` (B-{a}))"
"finite ((λx. -a + x) ` (B - {a}))"
using indep_card_eq_dim_span[of "(λx. -a+x) ` (B-{a})"] by auto
show ?thesis
proof (cases "(λx. -a + x) ` (B - {a}) = {}")
case True
have "B = insert a ((λx. a + x) ` (λx. -a + x) ` (B - {a}))"
using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
then have "B = {a}" using True by auto
then show ?thesis using assms fin by auto
next
case False
then have "card ((λx. -a + x) ` (B - {a})) > 0"
using fin by auto
moreover have h1: "card ((λx. -a + x) ` (B-{a})) = card (B-{a})"
by (rule card_image) (use translate_inj_on in blast)
ultimately have "card (B-{a}) > 0" by auto
then have *: "finite (B - {a})"
using card_gt_0_iff[of "(B - {a})"] by auto
then have "card (B - {a}) = card B - 1"
using card_Diff_singleton assms by auto
with * show ?thesis using fin h1 by auto
qed
qed
lemma aff_dim_parallel_subspace:
fixes V L :: "'n::euclidean_space set"
assumes "V ≠ {}"
and "subspace L"
and "affine_parallel (affine hull V) L"
shows "aff_dim V = int (dim L)"
proof -
obtain B where
B: "affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ int (card B) = aff_dim V + 1"
using aff_dim_basis_exists by auto
then have "B ≠ {}"
using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
by auto
then obtain a where a: "a ∈ B" by auto
define Lb where "Lb = span ((λx. -a+x) ` (B-{a}))"
moreover have "affine_parallel (affine hull B) Lb"
using Lb_def B assms affine_hull_span2[of a B] a
affine_parallel_commut[of "Lb" "(affine hull B)"]
unfolding affine_parallel_def
by auto
moreover have "subspace Lb"
using Lb_def subspace_span by auto
moreover have "affine hull B ≠ {}"
using assms B affine_hull_nonempty[of V] by auto
ultimately have "L = Lb"
using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
by auto
then have "dim L = dim Lb"
by auto
moreover have "card B - 1 = dim Lb" and "finite B"
using Lb_def aff_dim_parallel_subspace_aux a B by auto
ultimately show ?thesis
using B ‹B ≠ {}› card_gt_0_iff[of B] by auto
qed
lemma aff_independent_finite:
fixes B :: "'n::euclidean_space set"
assumes "¬ affine_dependent B"
shows "finite B"
proof -
{
assume "B ≠ {}"
then obtain a where "a ∈ B" by auto
then have ?thesis
using aff_dim_parallel_subspace_aux assms by auto
}
then show ?thesis by auto
qed
lemmas independent_finite = independent_imp_finite
lemma span_substd_basis:
assumes d: "d ⊆ Basis"
shows "span d = {x. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
(is "_ = ?B")
proof -
have "d ⊆ ?B"
using d by (auto simp: inner_Basis)
moreover have s: "subspace ?B"
using subspace_substandard[of "λi. i ∉ d"] .
ultimately have "span d ⊆ ?B"
using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
moreover have *: "card d ≤ dim (span d)"
using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
span_superset[of d]
by auto
moreover from * have "dim ?B ≤ dim (span d)"
using dim_substandard[OF assms] by auto
ultimately show ?thesis
using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
qed
lemma basis_to_substdbasis_subspace_isomorphism:
fixes B :: "'a::euclidean_space set"
assumes "independent B"
shows "∃f d::'a set. card d = card B ∧ linear f ∧ f ` B = d ∧
f ` span B = {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0} ∧ inj_on f (span B) ∧ d ⊆ Basis"
proof -
have B: "card B = dim B"
using dim_unique[of B B "card B"] assms span_superset[of B] by auto
have "dim B ≤ card (Basis :: 'a set)"
using dim_subset_UNIV[of B] by simp
from ex_card[OF this] obtain d :: "'a set" where d: "d ⊆ Basis" and t: "card d = dim B"
by auto
let ?t = "{x::'a::euclidean_space. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
have "∃f. linear f ∧ f ` B = d ∧ f ` span B = ?t ∧ inj_on f (span B)"
proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
show "d ⊆ {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0}"
using d inner_not_same_Basis by blast
qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
with t ‹card B = dim B› d show ?thesis by auto
qed
lemma aff_dim_empty:
fixes S :: "'n::euclidean_space set"
shows "S = {} ⟷ aff_dim S = -1"
proof -
obtain B where *: "affine hull B = affine hull S"
and "¬ affine_dependent B"
and "int (card B) = aff_dim S + 1"
using aff_dim_basis_exists by auto
moreover
from * have "S = {} ⟷ B = {}"
using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
ultimately show ?thesis
using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
qed
lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
by (simp add: aff_dim_empty [symmetric])
lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
unfolding aff_dim_def using hull_hull[of _ S] by auto
lemma aff_dim_affine_hull2:
assumes "affine hull S = affine hull T"
shows "aff_dim S = aff_dim T"
unfolding aff_dim_def using assms by auto
lemma aff_dim_unique:
fixes B V :: "'n::euclidean_space set"
assumes "affine hull B = affine hull V ∧ ¬ affine_dependent B"
shows "of_nat (card B) = aff_dim V + 1"
proof (cases "B = {}")
case True
then have "V = {}"
using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
by auto
then have "aff_dim V = (-1::int)"
using aff_dim_empty by auto
then show ?thesis
using ‹B = {}› by auto
next
case False
then obtain a where a: "a ∈ B" by auto
define Lb where "Lb = span ((λx. -a+x) ` (B-{a}))"
have "affine_parallel (affine hull B) Lb"
using Lb_def affine_hull_span2[of a B] a
affine_parallel_commut[of "Lb" "(affine hull B)"]
unfolding affine_parallel_def by auto
moreover have "subspace Lb"
using Lb_def subspace_span by auto
ultimately have "aff_dim B = int(dim Lb)"
using aff_dim_parallel_subspace[of B Lb] ‹B ≠ {}› by auto
moreover have "(card B) - 1 = dim Lb" "finite B"
using Lb_def aff_dim_parallel_subspace_aux a assms by auto
ultimately have "of_nat (card B) = aff_dim B + 1"
using ‹B ≠ {}› card_gt_0_iff[of B] by auto
then show ?thesis
using aff_dim_affine_hull2 assms by auto
qed
lemma aff_dim_affine_independent:
fixes B :: "'n::euclidean_space set"
assumes "¬ affine_dependent B"
shows "of_nat (card B) = aff_dim B + 1"
using aff_dim_unique[of B B] assms by auto
lemma affine_independent_iff_card:
fixes s :: "'a::euclidean_space set"
shows "~ affine_dependent s ⟷ finite s ∧ aff_dim s = int(card s) - 1"
apply (rule iffI)
apply (simp add: aff_dim_affine_independent aff_independent_finite)
by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
lemma aff_dim_sing [simp]:
fixes a :: "'n::euclidean_space"
shows "aff_dim {a} = 0"
using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
proof (clarsimp)
assume "a ≠ b"
then have "aff_dim{a,b} = card{a,b} - 1"
using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
also have "… = 1"
using ‹a ≠ b› by simp
finally show "aff_dim {a, b} = 1" .
qed
lemma aff_dim_inner_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "∃B. B ⊆ V ∧ affine hull B = affine hull V ∧
¬ affine_dependent B ∧ of_nat (card B) = aff_dim V + 1"
proof -
obtain B where B: "¬ affine_dependent B" "B ⊆ V" "affine hull B = affine hull V"
using affine_basis_exists[of V] by auto
then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
with B show ?thesis by auto
qed
lemma aff_dim_le_card:
fixes V :: "'n::euclidean_space set"
assumes "finite V"
shows "aff_dim V ≤ of_nat (card V) - 1"
proof -
obtain B where B: "B ⊆ V" "of_nat (card B) = aff_dim V + 1"
using aff_dim_inner_basis_exists[of V] by auto
then have "card B ≤ card V"
using assms card_mono by auto
with B show ?thesis by auto
qed
lemma aff_dim_parallel_eq:
fixes S T :: "'n::euclidean_space set"
assumes "affine_parallel (affine hull S) (affine hull T)"
shows "aff_dim S = aff_dim T"
proof -
{
assume "T ≠ {}" "S ≠ {}"
then obtain L where L: "subspace L ∧ affine_parallel (affine hull T) L"
using affine_parallel_subspace[of "affine hull T"]
affine_affine_hull[of T] affine_hull_nonempty
by auto
then have "aff_dim T = int (dim L)"
using aff_dim_parallel_subspace ‹T ≠ {}› by auto
moreover have *: "subspace L ∧ affine_parallel (affine hull S) L"
using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
moreover from * have "aff_dim S = int (dim L)"
using aff_dim_parallel_subspace ‹S ≠ {}› by auto
ultimately have ?thesis by auto
}
moreover
{
assume "S = {}"
then have "S = {}" and "T = {}"
using assms affine_hull_nonempty
unfolding affine_parallel_def
by auto
then have ?thesis using aff_dim_empty by auto
}
moreover
{
assume "T = {}"
then have "S = {}" and "T = {}"
using assms affine_hull_nonempty
unfolding affine_parallel_def
by auto
then have ?thesis
using aff_dim_empty by auto
}
ultimately show ?thesis by blast
qed
lemma aff_dim_translation_eq:
fixes a :: "'n::euclidean_space"
shows "aff_dim ((λx. a + x) ` S) = aff_dim S"
proof -
have "affine_parallel (affine hull S) (affine hull ((λx. a + x) ` S))"
unfolding affine_parallel_def
apply (rule exI[of _ "a"])
using affine_hull_translation[of a S]
apply auto
done
then show ?thesis
using aff_dim_parallel_eq[of S "(λx. a + x) ` S"] by auto
qed
lemma aff_dim_affine:
fixes S L :: "'n::euclidean_space set"
assumes "S ≠ {}"
and "affine S"
and "subspace L"
and "affine_parallel S L"
shows "aff_dim S = int (dim L)"
proof -
have *: "affine hull S = S"
using assms affine_hull_eq[of S] by auto
then have "affine_parallel (affine hull S) L"
using assms by (simp add: *)
then show ?thesis
using assms aff_dim_parallel_subspace[of S L] by blast
qed
lemma dim_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "dim (affine hull S) = dim S"
proof -
have "dim (affine hull S) ≥ dim S"
using dim_subset by auto
moreover have "dim (span S) ≥ dim (affine hull S)"
using dim_subset affine_hull_subset_span by blast
moreover have "dim (span S) = dim S"
using dim_span by auto
ultimately show ?thesis by auto
qed
lemma aff_dim_subspace:
fixes S :: "'n::euclidean_space set"
assumes "subspace S"
shows "aff_dim S = int (dim S)"
proof (cases "S={}")
case True with assms show ?thesis
by (simp add: subspace_affine)
next
case False
with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
show ?thesis by auto
qed
lemma aff_dim_zero:
fixes S :: "'n::euclidean_space set"
assumes "0 ∈ affine hull S"
shows "aff_dim S = int (dim S)"
proof -
have "subspace (affine hull S)"
using subspace_affine[of "affine hull S"] affine_affine_hull assms
by auto
then have "aff_dim (affine hull S) = int (dim (affine hull S))"
using assms aff_dim_subspace[of "affine hull S"] by auto
then show ?thesis
using aff_dim_affine_hull[of S] dim_affine_hull[of S]
by auto
qed
lemma aff_dim_eq_dim:
fixes S :: "'n::euclidean_space set"
assumes "a ∈ affine hull S"
shows "aff_dim S = int (dim ((λx. -a+x) ` S))"
proof -
have "0 ∈ affine hull ((λx. -a+x) ` S)"
unfolding Convex_Euclidean_Space.affine_hull_translation
using assms by (simp add: ab_group_add_class.ab_left_minus image_iff)
with aff_dim_zero show ?thesis
by (metis aff_dim_translation_eq)
qed
lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
dim_UNIV[where 'a="'n::euclidean_space"]
by auto
lemma aff_dim_geq:
fixes V :: "'n::euclidean_space set"
shows "aff_dim V ≥ -1"
proof -
obtain B where "affine hull B = affine hull V"
and "¬ affine_dependent B"
and "int (card B) = aff_dim V + 1"
using aff_dim_basis_exists by auto
then show ?thesis by auto
qed
lemma aff_dim_negative_iff [simp]:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S < 0 ⟷S = {}"
by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
lemma aff_lowdim_subset_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes "aff_dim S < DIM('a)"
obtains a b where "a ≠ 0" "S ⊆ {x. a ∙ x = b}"
proof (cases "S={}")
case True
moreover
have "(SOME b. b ∈ Basis) ≠ 0"
by (metis norm_some_Basis norm_zero zero_neq_one)
ultimately show ?thesis
using that by blast
next
case False
then obtain c S' where "c ∉ S'" "S = insert c S'"
by (meson equals0I mk_disjoint_insert)
have "dim ((+) (-c) ` S) < DIM('a)"
by (metis ‹S = insert c S'› aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
then obtain a where "a ≠ 0" "span ((+) (-c) ` S) ⊆ {x. a ∙ x = 0}"
using lowdim_subset_hyperplane by blast
moreover
have "a ∙ w = a ∙ c" if "span ((+) (- c) ` S) ⊆ {x. a ∙ x = 0}" "w ∈ S" for w
proof -
have "w-c ∈ span ((+) (- c) ` S)"
by (simp add: span_base ‹w ∈ S›)
with that have "w-c ∈ {x. a ∙ x = 0}"
by blast
then show ?thesis
by (auto simp: algebra_simps)
qed
ultimately have "S ⊆ {x. a ∙ x = a ∙ c}"
by blast
then show ?thesis
by (rule that[OF ‹a ≠ 0›])
qed
lemma affine_independent_card_dim_diffs:
fixes S :: "'a :: euclidean_space set"
assumes "~ affine_dependent S" "a ∈ S"
shows "card S = dim {x - a|x. x ∈ S} + 1"
proof -
have 1: "{b - a|b. b ∈ (S - {a})} ⊆ {x - a|x. x ∈ S}" by auto
have 2: "x - a ∈ span {b - a |b. b ∈ S - {a}}" if "x ∈ S" for x
proof (cases "x = a")
case True then show ?thesis by (simp add: span_clauses)
next
case False then show ?thesis
using assms by (blast intro: span_base that)
qed
have "¬ affine_dependent (insert a S)"
by (simp add: assms insert_absorb)
then have 3: "independent {b - a |b. b ∈ S - {a}}"
using dependent_imp_affine_dependent by fastforce
have "{b - a |b. b ∈ S - {a}} = (λb. b-a) ` (S - {a})"
by blast
then have "card {b - a |b. b ∈ S - {a}} = card ((λb. b-a) ` (S - {a}))"
by simp
also have "… = card (S - {a})"
by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
also have "… = card S - 1"
by (simp add: aff_independent_finite assms)
finally have 4: "card {b - a |b. b ∈ S - {a}} = card S - 1" .
have "finite S"
by (meson assms aff_independent_finite)
with ‹a ∈ S› have "card S ≠ 0" by auto
moreover have "dim {x - a |x. x ∈ S} = card S - 1"
using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
ultimately show ?thesis
by auto
qed
lemma independent_card_le_aff_dim:
fixes B :: "'n::euclidean_space set"
assumes "B ⊆ V"
assumes "¬ affine_dependent B"
shows "int (card B) ≤ aff_dim V + 1"
proof -
obtain T where T: "¬ affine_dependent T ∧ B ⊆ T ∧ T ⊆ V ∧ affine hull T = affine hull V"
by (metis assms extend_to_affine_basis[of B V])
then have "of_nat (card T) = aff_dim V + 1"
using aff_dim_unique by auto
then show ?thesis
using T card_mono[of T B] aff_independent_finite[of T] by auto
qed
lemma aff_dim_subset:
fixes S T :: "'n::euclidean_space set"
assumes "S ⊆ T"
shows "aff_dim S ≤ aff_dim T"
proof -
obtain B where B: "¬ affine_dependent B" "B ⊆ S" "affine hull B = affine hull S"
"of_nat (card B) = aff_dim S + 1"
using aff_dim_inner_basis_exists[of S] by auto
then have "int (card B) ≤ aff_dim T + 1"
using assms independent_card_le_aff_dim[of B T] by auto
with B show ?thesis by auto
qed
lemma aff_dim_le_DIM:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S ≤ int (DIM('n))"
proof -
have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
using aff_dim_UNIV by auto
then show "aff_dim (S:: 'n::euclidean_space set) ≤ int(DIM('n))"
using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
qed
lemma affine_dim_equal:
fixes S :: "'n::euclidean_space set"
assumes "affine S" "affine T" "S ≠ {}" "S ⊆ T" "aff_dim S = aff_dim T"
shows "S = T"
proof -
obtain a where "a ∈ S" using assms by auto
then have "a ∈ T" using assms by auto
define LS where "LS = {y. ∃x ∈ S. (-a) + x = y}"
then have ls: "subspace LS" "affine_parallel S LS"
using assms parallel_subspace_explicit[of S a LS] ‹a ∈ S› by auto
then have h1: "int(dim LS) = aff_dim S"
using assms aff_dim_affine[of S LS] by auto
have "T ≠ {}" using assms by auto
define LT where "LT = {y. ∃x ∈ T. (-a) + x = y}"
then have lt: "subspace LT ∧ affine_parallel T LT"
using assms parallel_subspace_explicit[of T a LT] ‹a ∈ T› by auto
then have "int(dim LT) = aff_dim T"
using assms aff_dim_affine[of T LT] ‹T ≠ {}› by auto
then have "dim LS = dim LT"
using h1 assms by auto
moreover have "LS ≤ LT"
using LS_def LT_def assms by auto
ultimately have "LS = LT"
using subspace_dim_equal[of LS LT] ls lt by auto
moreover have "S = {x. ∃y ∈ LS. a+y=x}"
using LS_def by auto
moreover have "T = {x. ∃y ∈ LT. a+y=x}"
using LT_def by auto
ultimately show ?thesis by auto
qed
lemma aff_dim_eq_0:
fixes S :: "'a::euclidean_space set"
shows "aff_dim S = 0 ⟷ (∃a. S = {a})"
proof (cases "S = {}")
case True
then show ?thesis
by auto
next
case False
then obtain a where "a ∈ S" by auto
show ?thesis
proof safe
assume 0: "aff_dim S = 0"
have "~ {a,b} ⊆ S" if "b ≠ a" for b
by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
then show "∃a. S = {a}"
using ‹a ∈ S› by blast
qed auto
qed
lemma affine_hull_UNIV:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
proof -
have "S ≠ {}"
using assms aff_dim_empty[of S] by auto
have h0: "S ⊆ affine hull S"
using hull_subset[of S _] by auto
have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
using aff_dim_UNIV assms by auto
then have h2: "aff_dim (affine hull S) ≤ aff_dim (UNIV :: ('n::euclidean_space) set)"
using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
have h3: "aff_dim S ≤ aff_dim (affine hull S)"
using h0 aff_dim_subset[of S "affine hull S"] assms by auto
then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
using h0 h1 h2 by auto
then show ?thesis
using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
affine_affine_hull[of S] affine_UNIV assms h4 h0 ‹S ≠ {}›
by auto
qed
lemma disjoint_affine_hull:
fixes s :: "'n::euclidean_space set"
assumes "~ affine_dependent s" "t ⊆ s" "u ⊆ s" "t ∩ u = {}"
shows "(affine hull t) ∩ (affine hull u) = {}"
proof -
have "finite s" using assms by (simp add: aff_independent_finite)
then have "finite t" "finite u" using assms finite_subset by blast+
{ fix y
assume yt: "y ∈ affine hull t" and yu: "y ∈ affine hull u"
then obtain a b
where a1 [simp]: "sum a t = 1" and [simp]: "sum (λv. a v *⇩R v) t = y"
and [simp]: "sum b u = 1" "sum (λv. b v *⇩R v) u = y"
by (auto simp: affine_hull_finite ‹finite t› ‹finite u›)
define c where "c x = (if x ∈ t then a x else if x ∈ u then -(b x) else 0)" for x
have [simp]: "s ∩ t = t" "s ∩ - t ∩ u = u" using assms by auto
have "sum c s = 0"
by (simp add: c_def comm_monoid_add_class.sum.If_cases ‹finite s› sum_negf)
moreover have "~ (∀v∈s. c v = 0)"
by (metis (no_types) IntD1 ‹s ∩ t = t› a1 c_def sum_not_0 zero_neq_one)
moreover have "(∑v∈s. c v *⇩R v) = 0"
by (simp add: c_def if_smult sum_negf
comm_monoid_add_class.sum.If_cases ‹finite s›)
ultimately have False
using assms ‹finite s› by (auto simp: affine_dependent_explicit)
}
then show ?thesis by blast
qed
lemma aff_dim_convex_hull:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (convex hull S) = aff_dim S"
using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
aff_dim_subset[of "convex hull S" "affine hull S"]
by auto
lemma aff_dim_cball:
fixes a :: "'n::euclidean_space"
assumes "e > 0"
shows "aff_dim (cball a e) = int (DIM('n))"
proof -
have "(λx. a + x) ` (cball 0 e) ⊆ cball a e"
unfolding cball_def dist_norm by auto
then have "aff_dim (cball (0 :: 'n::euclidean_space) e) ≤ aff_dim (cball a e)"
using aff_dim_translation_eq[of a "cball 0 e"]
aff_dim_subset[of "(+) a ` cball 0 e" "cball a e"]
by auto
moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
ultimately show ?thesis
using aff_dim_le_DIM[of "cball a e"] by auto
qed
lemma aff_dim_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
and "S ≠ {}"
shows "aff_dim S = int (DIM('n))"
proof -
obtain x where "x ∈ S"
using assms by auto
then obtain e where e: "e > 0" "cball x e ⊆ S"
using open_contains_cball[of S] assms by auto
then have "aff_dim (cball x e) ≤ aff_dim S"
using aff_dim_subset by auto
with e show ?thesis
using aff_dim_cball[of e x] aff_dim_le_DIM[of S] by auto
qed
lemma low_dim_interior:
fixes S :: "'n::euclidean_space set"
assumes "¬ aff_dim S = int (DIM('n))"
shows "interior S = {}"
proof -
have "aff_dim(interior S) ≤ aff_dim S"
using interior_subset aff_dim_subset[of "interior S" S] by auto
then show ?thesis
using aff_dim_open[of "interior S"] aff_dim_le_DIM[of S] assms by auto
qed
corollary empty_interior_lowdim:
fixes S :: "'n::euclidean_space set"
shows "dim S < DIM ('n) ⟹ interior S = {}"
by (metis low_dim_interior affine_hull_UNIV dim_affine_hull less_not_refl dim_UNIV)
corollary aff_dim_nonempty_interior:
fixes S :: "'a::euclidean_space set"
shows "interior S ≠ {} ⟹ aff_dim S = DIM('a)"
by (metis low_dim_interior)
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"
apply (rule_tac ex_least_nat_le, auto)
done
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"
apply (rule_tac sum_nonneg, auto)
done
then have "sum w s > 0"
unfolding sum.remove[OF obt(1) ‹v∈s›]
using as[THEN bspec[where x=v]] ‹v∈s› ‹w v ≠ 0› by auto
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› unfolding t_def i_def
apply (rule_tac Min_le)
using obt(1) apply auto
done
then show ?thesis
unfolding real_0_le_add_iff
using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
by auto
qed
qed
obtain a where "a ∈ 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
show "?lhs ⊆ ?rhs"
apply (subst convex_hull_caratheodory_aff_dim, clarify)
apply (rule_tac x=s in exI)
apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
done
next
show "?rhs ⊆ ?lhs"
using hull_mono by blast
qed
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"
apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
apply (erule ex_forward)+
using aff_dim_le_DIM [of p]
apply simp
done
next
fix x
assume "x ∈ ?rhs" then show "x ∈ ?lhs"
by (auto simp: convex_hull_explicit)
qed
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"
apply (rule_tac x=s in exI)
using hull_subset[of s convex]
using convex_convex_hull[simplified convex_explicit, of s,
THEN spec[where x=s], THEN spec[where x=u]]
apply auto
done
next
fix x s
assume "finite s" "s ⊆ p" "card s ≤ DIM('a) + 1" "x ∈ convex hull s"
then show "x ∈ convex hull p"
using hull_mono[OF ‹s⊆p›] by auto
qed
subsection ‹Relative interior of a set›
definition%important "rel_interior S =
{x. ∃T. openin (subtopology euclidean (affine hull S)) T ∧ x ∈ T ∧ T ⊆ S}"
lemma rel_interior_mono:
"⟦S ⊆ T; affine hull S = affine hull T⟧
⟹ (rel_interior S) ⊆ (rel_interior T)"
by (auto simp: rel_interior_def)
lemma rel_interior_maximal:
"⟦T ⊆ S; openin(subtopology euclidean (affine hull S)) T⟧ ⟹ T ⊆ (rel_interior S)"
by (auto simp: rel_interior_def)
lemma rel_interior:
"rel_interior S = {x ∈ S. ∃T. open T ∧ x ∈ T ∧ T ∩ affine hull S ⊆ S}"
unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
apply auto
proof -
fix x T
assume *: "x ∈ S" "open T" "x ∈ T" "T ∩ affine hull S ⊆ S"
then have **: "x ∈ T ∩ affine hull S"
using hull_inc by auto
show "∃Tb. (∃Ta. open Ta ∧ Tb = affine hull S ∩ Ta) ∧ x ∈ Tb ∧ Tb ⊆ S"
apply (rule_tac x = "T ∩ (affine hull S)" in exI)
using * **
apply auto
done
qed
lemma mem_rel_interior: "x ∈ rel_interior S ⟷ (∃T. open T ∧ x ∈ T ∩ S ∧ T ∩ affine hull S ⊆ S)"
by (auto simp: rel_interior)
lemma mem_rel_interior_ball:
"x ∈ rel_interior S ⟷ x ∈ S ∧ (∃e. e > 0 ∧ ball x e ∩ affine hull S ⊆ S)"
apply (simp add: rel_interior, safe)
apply (force simp: open_contains_ball)
apply (rule_tac x = "ball x e" in exI, simp)
done
lemma rel_interior_ball:
"rel_interior S = {x ∈ S. ∃e. e > 0 ∧ ball x e ∩ affine hull S ⊆ S}"
using mem_rel_interior_ball [of _ S] by auto
lemma mem_rel_interior_cball:
"x ∈ rel_interior S ⟷ x ∈ S ∧ (∃e. e > 0 ∧ cball x e ∩ affine hull S ⊆ S)"
apply (simp add: rel_interior, safe)
apply (force simp: open_contains_cball)
apply (rule_tac x = "ball x e" in exI)
apply (simp add: subset_trans [OF ball_subset_cball], auto)
done
lemma rel_interior_cball:
"rel_interior S = {x ∈ S. ∃e. e > 0 ∧ cball x e ∩ affine hull S ⊆ S}"
using mem_rel_interior_cball [of _ S] by auto
lemma rel_interior_empty [simp]: "rel_interior {} = {}"
by (auto simp: rel_interior_def)
lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
by (metis affine_hull_eq affine_sing)
lemma rel_interior_sing [simp]:
fixes a :: "'n::euclidean_space" shows "rel_interior {a} = {a}"
apply (auto simp: rel_interior_ball)
apply (rule_tac x=1 in exI, force)
done
lemma subset_rel_interior:
fixes S T :: "'n::euclidean_space set"
assumes "S ⊆ T"
and "affine hull S = affine hull T"
shows "rel_interior S ⊆ rel_interior T"
using assms by (auto simp: rel_interior_def)
lemma rel_interior_subset: "rel_interior S ⊆ S"
by (auto simp: rel_interior_def)
lemma rel_interior_subset_closure: "rel_interior S ⊆ closure S"
using rel_interior_subset by (auto simp: closure_def)
lemma interior_subset_rel_interior: "interior S ⊆ rel_interior S"
by (auto simp: rel_interior interior_def)
lemma interior_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "rel_interior S = interior S"
proof -
have "affine hull S = UNIV"
using assms affine_hull_UNIV[of S] by auto
then show ?thesis
unfolding rel_interior interior_def by auto
qed
lemma rel_interior_interior:
fixes S :: "'n::euclidean_space set"
assumes "affine hull S = UNIV"
shows "rel_interior S = interior S"
using assms unfolding rel_interior interior_def by auto
lemma rel_interior_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
shows "rel_interior S = S"
by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
lemma interior_ball [simp]: "interior (ball x e) = ball x e"
by (simp add: interior_open)
lemma interior_rel_interior_gen:
fixes S :: "'n::euclidean_space set"
shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
by (metis interior_rel_interior low_dim_interior)
lemma rel_interior_nonempty_interior:
fixes S :: "'n::euclidean_space set"
shows "interior S ≠ {} ⟹ rel_interior S = interior S"
by (metis interior_rel_interior_gen)
lemma affine_hull_nonempty_interior:
fixes S :: "'n::euclidean_space set"
shows "interior S ≠ {} ⟹ affine hull S = UNIV"
by (metis affine_hull_UNIV interior_rel_interior_gen)
lemma rel_interior_affine_hull [simp]:
fixes S :: "'n::euclidean_space set"
shows "rel_interior (affine hull S) = affine hull S"
proof -
have *: "rel_interior (affine hull S) ⊆ affine hull S"
using rel_interior_subset by auto
{
fix x
assume x: "x ∈ affine hull S"
define e :: real where "e = 1"
then have "e > 0" "ball x e ∩ affine hull (affine hull S) ⊆ affine hull S"
using hull_hull[of _ S] by auto
then have "x ∈ rel_interior (affine hull S)"
using x rel_interior_ball[of "affine hull S"] by auto
}
then show ?thesis using * by auto
qed
lemma rel_interior_UNIV [simp]: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
by (metis open_UNIV rel_interior_open)
lemma rel_interior_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c ∈ rel_interior S"
and "x ∈ S"
and "0 < e"
and "e ≤ 1"
shows "x - e *⇩R (x - c) ∈ rel_interior S"
proof -
obtain d where "d > 0" and d: "ball c d ∩ affine hull S ⊆ S"
using assms(2) unfolding mem_rel_interior_ball by auto
{
fix y
assume as: "dist (x - e *⇩R (x - c)) y < e * d" "y ∈ affine hull S"
have *: "y = (1 - (1 - e)) *⇩R ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) + (1 - e) *⇩R x"
using ‹e > 0› by (auto simp: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "x ∈ affine hull S"
using assms hull_subset[of S] by auto
moreover have "1 / e + - ((1 - e) / e) = 1"
using ‹e > 0› left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
ultimately have **: "(1 / e) *⇩R y - ((1 - e) / e) *⇩R x ∈ affine hull S"
using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
by (simp add: algebra_simps)
have "dist c ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) = ¦1/e¦ * norm (e *⇩R c - y + (1 - e) *⇩R x)"
unfolding dist_norm norm_scaleR[symmetric]
apply (rule arg_cong[where f=norm])
using ‹e > 0›
apply (auto simp: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
done
also have "… = ¦1/e¦ * norm (x - e *⇩R (x - c) - y)"
by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
also have "… < d"
using as[unfolded dist_norm] and ‹e > 0›
by (auto simp:pos_divide_less_eq[OF ‹e > 0›] mult.commute)
finally have "y ∈ S"
apply (subst *)
apply (rule assms(1)[unfolded convex_alt,rule_format])
apply (rule d[THEN subsetD])
unfolding mem_ball
using assms(3-5) **
apply auto
done
}
then have "ball (x - e *⇩R (x - c)) (e*d) ∩ affine hull S ⊆ S"
by auto
moreover have "e * d > 0"
using ‹e > 0› ‹d > 0› by simp
moreover have c: "c ∈ S"
using assms rel_interior_subset by auto
moreover from c have "x - e *⇩R (x - c) ∈ S"
using convexD_alt[of S x c e]
apply (simp add: algebra_simps)
using assms
apply auto
done
ultimately show ?thesis
using mem_rel_interior_ball[of "x - e *⇩R (x - c)" S] ‹e > 0› by auto
qed
lemma interior_real_semiline:
fixes a :: real
shows "interior {a..} = {a<..}"
proof -
{
fix y
assume "a < y"
then have "y ∈ interior {a..}"
apply (simp add: mem_interior)
apply (rule_tac x="(y-a)" in exI)
apply (auto simp: dist_norm)
done
}
moreover
{
fix y
assume "y ∈ interior {a..}"
then obtain e where e: "e > 0" "cball y e ⊆ {a..}"
using mem_interior_cball[of y "{a..}"] by auto
moreover from e have "y - e ∈ cball y e"
by (auto simp: cball_def dist_norm)
ultimately have "a ≤ y - e" by blast
then have "a < y" using e by auto
}
ultimately show ?thesis by auto
qed
lemma continuous_ge_on_Ioo:
assumes "continuous_on {c..d} g" "⋀x. x ∈ {c<..<d} ⟹ g x ≥ a" "c < d" "x ∈ {c..d}"
shows "g (x::real) ≥ (a::real)"
proof-
from assms(3) have "{c..d} = closure {c<..<d}" by (rule closure_greaterThanLessThan[symmetric])
also from assms(2) have "{c<..<d} ⊆ (g -` {a..} ∩ {c..d})" by auto
hence "closure {c<..<d} ⊆ closure (g -` {a..} ∩ {c..d})" by (rule closure_mono)
also from assms(1) have "closed (g -` {a..} ∩ {c..d})"
by (auto simp: continuous_on_closed_vimage)
hence "closure (g -` {a..} ∩ {c..d}) = g -` {a..} ∩ {c..d}" by simp
finally show ?thesis using ‹x ∈ {c..d}› by auto
qed
lemma interior_real_semiline':
fixes a :: real
shows "interior {..a} = {..<a}"
proof -
{
fix y
assume "a > y"
then have "y ∈ interior {..a}"
apply (simp add: mem_interior)
apply (rule_tac x="(a-y)" in exI)
apply (auto simp: dist_norm)
done
}
moreover
{
fix y
assume "y ∈ interior {..a}"
then obtain e where e: "e > 0" "cball y e ⊆ {..a}"
using mem_interior_cball[of y "{..a}"] by auto
moreover from e have "y + e ∈ cball y e"
by (auto simp: cball_def dist_norm)
ultimately have "a ≥ y + e" by auto
then have "a > y" using e by auto
}
ultimately show ?thesis by auto
qed
lemma interior_atLeastAtMost_real [simp]: "interior {a..b} = {a<..<b :: real}"
proof-
have "{a..b} = {a..} ∩ {..b}" by auto
also have "interior … = {a<..} ∩ {..<b}"
by (simp add: interior_real_semiline interior_real_semiline')
also have "… = {a<..<b}" by auto
finally show ?thesis .
qed
lemma interior_atLeastLessThan [simp]:
fixes a::real shows "interior {a..<b} = {a<..<b}"
by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost_real interior_Int interior_interior interior_real_semiline)
lemma interior_lessThanAtMost [simp]:
fixes a::real shows "interior {a<..b} = {a<..<b}"
by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost_real interior_Int
interior_interior interior_real_semiline)
lemma interior_greaterThanLessThan_real [simp]: "interior {a<..<b} = {a<..<b :: real}"
by (metis interior_atLeastAtMost_real interior_interior)
lemma frontier_real_Iic [simp]:
fixes a :: real
shows "frontier {..a} = {a}"
unfolding frontier_def by (auto simp: interior_real_semiline')
lemma rel_interior_real_box [simp]:
fixes a b :: real
assumes "a < b"
shows "rel_interior {a .. b} = {a <..< b}"
proof -
have "box a b ≠ {}"
using assms
unfolding set_eq_iff
by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
then show ?thesis
using interior_rel_interior_gen[of "cbox a b", symmetric]
by (simp split: if_split_asm del: box_real add: box_real[symmetric] interior_cbox)
qed
lemma rel_interior_real_semiline [simp]:
fixes a :: real
shows "rel_interior {a..} = {a<..}"
proof -
have *: "{a<..} ≠ {}"
unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
then show ?thesis using interior_real_semiline interior_rel_interior_gen[of "{a..}"]
by (auto split: if_split_asm)
qed
subsubsection ‹Relative open sets›
definition%important "rel_open S ⟷ rel_interior S = S"
lemma rel_open: "rel_open S ⟷ openin (subtopology euclidean (affine hull S)) S"
unfolding rel_open_def rel_interior_def
apply auto
using openin_subopen[of "subtopology euclidean (affine hull S)" S]
apply auto
done
lemma openin_rel_interior: "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
apply (simp add: rel_interior_def)
apply (subst openin_subopen, blast)
done
lemma openin_set_rel_interior:
"openin (subtopology euclidean S) (rel_interior S)"
by (rule openin_subset_trans [OF openin_rel_interior rel_interior_subset hull_subset])
lemma affine_rel_open:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "rel_open S"
unfolding rel_open_def
using assms rel_interior_affine_hull[of S] affine_hull_eq[of S]
by metis
lemma affine_closed:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "closed S"
proof -
{
assume "S ≠ {}"
then obtain L where L: "subspace L" "affine_parallel S L"
using assms affine_parallel_subspace[of S] by auto
then obtain a where a: "S = ((+) a ` L)"
using affine_parallel_def[of L S] affine_parallel_commut by auto
from L have "closed L" using closed_subspace by auto
then have "closed S"
using closed_translation a by auto
}
then show ?thesis by auto
qed
lemma closure_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "closure S ⊆ affine hull S"
by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
lemma closure_same_affine_hull [simp]:
fixes S :: "'n::euclidean_space set"
shows "affine hull (closure S) = affine hull S"
proof -
have "affine hull (closure S) ⊆ affine hull S"
using hull_mono[of "closure S" "affine hull S" "affine"]
closure_affine_hull[of S] hull_hull[of "affine" S]
by auto
moreover have "affine hull (closure S) ⊇ affine hull S"
using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
ultimately show ?thesis by auto
qed
lemma closure_aff_dim [simp]:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (closure S) = aff_dim S"
proof -
have "aff_dim S ≤ aff_dim (closure S)"
using aff_dim_subset closure_subset by auto
moreover have "aff_dim (closure S) ≤ aff_dim (affine hull S)"
using aff_dim_subset closure_affine_hull by blast
moreover have "aff_dim (affine hull S) = aff_dim S"
using aff_dim_affine_hull by auto
ultimately show ?thesis by auto
qed
lemma rel_interior_closure_convex_shrink:
fixes S :: "_::euclidean_space set"
assumes "convex S"
and "c ∈ rel_interior S"
and "x ∈ closure S"
and "e > 0"
and "e ≤ 1"
shows "x - e *⇩R (x - c) ∈ rel_interior S"
proof -
obtain d where "d > 0" and d: "ball c d ∩ affine hull S ⊆ S"
using assms(2) unfolding mem_rel_interior_ball by auto
have "∃y ∈ S. norm (y - x) * (1 - e) < e * d"
proof (cases "x ∈ S")
case True
then show ?thesis using ‹e > 0› ‹d > 0›
apply (rule_tac bexI[where x=x], auto)
done
next
case False
then have x: "x islimpt S"
using assms(3)[unfolded closure_def] by auto
show ?thesis
proof (cases "e = 1")
case True
obtain y where "y ∈ S" "y ≠ x" "dist y x < 1"
using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
then show ?thesis
apply (rule_tac x=y in bexI)
unfolding True
using ‹d > 0›
apply auto
done
next
case False
then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
using ‹e ≤ 1› ‹e > 0› ‹d > 0› by auto
then obtain y where "y ∈ S" "y ≠ x" "dist y x < e * d / (1 - e)"
using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
then show ?thesis
apply (rule_tac x=y in bexI)
unfolding dist_norm
using pos_less_divide_eq[OF *]
apply auto
done
qed
qed
then obtain y where "y ∈ S" and y: "norm (y - x) * (1 - e) < e * d"
by auto
define z where "z = c + ((1 - e) / e) *⇩R (x - y)"
have *: "x - e *⇩R (x - c) = y - e *⇩R (y - z)"
unfolding z_def using ‹e > 0›
by (auto simp: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
have zball: "z ∈ ball c d"
using mem_ball z_def dist_norm[of c]
using y and assms(4,5)
by (auto simp:field_simps norm_minus_commute)
have "x ∈ affine hull S"
using closure_affine_hull assms by auto
moreover have "y ∈ affine hull S"
using ‹y ∈ S› hull_subset[of S] by auto
moreover have "c ∈ affine hull S"
using assms rel_interior_subset hull_subset[of S] by auto
ultimately have "z ∈ affine hull S"
using z_def affine_affine_hull[of S]
mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
assms
by (auto simp: field_simps)
then have "z ∈ S" using d zball by auto
obtain d1 where "d1 > 0" and d1: "ball z d1 ≤ ball c d"
using zball open_ball[of c d] openE[of "ball c d" z] by auto
then have "ball z d1 ∩ affine hull S ⊆ ball c d ∩ affine hull S"
by auto
then have "ball z d1 ∩ affine hull S ⊆ S"
using d by auto
then have "z ∈ rel_interior S"
using mem_rel_interior_ball using ‹d1 > 0› ‹z ∈ S› by auto
then have "y - e *⇩R (y - z) ∈ rel_interior S"
using rel_interior_convex_shrink[of S z y e] assms ‹y ∈ S› by auto
then show ?thesis using * by auto
qed
lemma rel_interior_eq:
"rel_interior s = s ⟷ openin(subtopology euclidean (affine hull s)) s"
using rel_open rel_open_def by blast
lemma rel_interior_openin:
"openin(subtopology euclidean (affine hull s)) s ⟹ rel_interior s = s"
by (simp add: rel_interior_eq)
lemma rel_interior_affine:
fixes S :: "'n::euclidean_space set"
shows "affine S ⟹ rel_interior S = S"
using affine_rel_open rel_open_def by auto
lemma rel_interior_eq_closure:
fixes S :: "'n::euclidean_space set"
shows "rel_interior S = closure S ⟷ affine S"
proof (cases "S = {}")
case True
then show ?thesis
by auto
next
case False show ?thesis
proof
assume eq: "rel_interior S = closure S"
have "S = {} ∨ S = affine hull S"
apply (rule connected_clopen [THEN iffD1, rule_format])
apply (simp add: affine_imp_convex convex_connected)
apply (rule conjI)
apply (metis eq closure_subset openin_rel_interior rel_interior_subset subset_antisym)
apply (metis closed_subset closure_subset_eq eq hull_subset rel_interior_subset)
done
with False have "affine hull S = S"
by auto
then show "affine S"
by (metis affine_hull_eq)
next
assume "affine S"
then show "rel_interior S = closure S"
by (simp add: rel_interior_affine affine_closed)
qed
qed
subsubsection%unimportant‹Relative interior preserves under linear transformations›
lemma rel_interior_translation_aux:
fixes a :: "'n::euclidean_space"
shows "((λx. a + x) ` rel_interior S) ⊆ rel_interior ((λx. a + x) ` S)"
proof -
{
fix x
assume x: "x ∈ rel_interior S"
then obtain T where "open T" "x ∈ T ∩ S" "T ∩ affine hull S ⊆ S"
using mem_rel_interior[of x S] by auto
then have "open ((λx. a + x) ` T)"
and "a + x ∈ ((λx. a + x) ` T) ∩ ((λx. a + x) ` S)"
and "((λx. a + x) ` T) ∩ affine hull ((λx. a + x) ` S) ⊆ (λx. a + x) ` S"
using affine_hull_translation[of a S] open_translation[of T a] x by auto
then have "a + x ∈ rel_interior ((λx. a + x) ` S)"
using mem_rel_interior[of "a+x" "((λx. a + x) ` S)"] by auto
}
then show ?thesis by auto
qed
lemma rel_interior_translation:
fixes a :: "'n::euclidean_space"
shows "rel_interior ((λx. a + x) ` S) = (λx. a + x) ` rel_interior S"
proof -
have "(λx. (-a) + x) ` rel_interior ((λx. a + x) ` S) ⊆ rel_interior S"
using rel_interior_translation_aux[of "-a" "(λx. a + x) ` S"]
translation_assoc[of "-a" "a"]
by auto
then have "((λx. a + x) ` rel_interior S) ⊇ rel_interior ((λx. a + x) ` S)"
using translation_inverse_subset[of a "rel_interior ((+) a ` S)" "rel_interior S"]
by auto
then show ?thesis
using rel_interior_translation_aux[of a S] by auto
qed
lemma affine_hull_linear_image:
assumes "bounded_linear f"
shows "f ` (affine hull s) = affine hull f ` s"
proof -
interpret f: bounded_linear f by fact
have "affine {x. f x ∈ affine hull f ` s}"
unfolding affine_def
by (auto simp: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
moreover have "affine {x. x ∈ f ` (affine hull s)}"
using affine_affine_hull[unfolded affine_def, of s]
unfolding affine_def by (auto simp: f.scaleR [symmetric] f.add [symmetric])
ultimately show ?thesis
by (auto simp: hull_inc elim!: hull_induct)
qed
lemma rel_interior_injective_on_span_linear_image:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
and S :: "'m::euclidean_space set"
assumes "bounded_linear f"
and "inj_on f (span S)"
shows "rel_interior (f ` S) = f ` (rel_interior S)"
proof -
{
fix z
assume z: "z ∈ rel_interior (f ` S)"
then have "z ∈ f ` S"
using rel_interior_subset[of "f ` S"] by auto
then obtain x where x: "x ∈ S" "f x = z" by auto
obtain e2 where e2: "e2 > 0" "cball z e2 ∩ affine hull (f ` S) ⊆ (f ` S)"
using z rel_interior_cball[of "f ` S"] by auto
obtain K where K: "K > 0" "⋀x. norm (f x) ≤ norm x * K"
using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
define e1 where "e1 = 1 / K"
then have e1: "e1 > 0" "⋀x. e1 * norm (f x) ≤ norm x"
using K pos_le_divide_eq[of e1] by auto
define e where "e = e1 * e2"
then have "e > 0" using e1 e2 by auto
{
fix y
assume y: "y ∈ cball x e ∩ affine hull S"
then have h1: "f y ∈ affine hull (f ` S)"
using affine_hull_linear_image[of f S] assms by auto
from y have "norm (x-y) ≤ e1 * e2"
using cball_def[of x e] dist_norm[of x y] e_def by auto
moreover have "f x - f y = f (x - y)"
using assms linear_diff[of f x y] linear_conv_bounded_linear[of f] by auto
moreover have "e1 * norm (f (x-y)) ≤ norm (x - y)"
using e1 by auto
ultimately have "e1 * norm ((f x)-(f y)) ≤ e1 * e2"
by auto
then have "f y ∈ cball z e2"
using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
then have "f y ∈ f ` S"
using y e2 h1 by auto
then have "y ∈ S"
using assms y hull_subset[of S] affine_hull_subset_span
inj_on_image_mem_iff [OF ‹inj_on f (span S)›]
by (metis Int_iff span_superset subsetCE)
}
then have "z ∈ f ` (rel_interior S)"
using mem_rel_interior_cball[of x S] ‹e > 0› x by auto
}
moreover
{
fix x
assume x: "x ∈ rel_interior S"
then obtain e2 where e2: "e2 > 0" "cball x e2 ∩ affine hull S ⊆ S"
using rel_interior_cball[of S] by auto
have "x ∈ S" using x rel_interior_subset by auto
then have *: "f x ∈ f ` S" by auto
have "∀x∈span S. f x = 0 ⟶ x = 0"
using assms subspace_span linear_conv_bounded_linear[of f]
linear_injective_on_subspace_0[of f "span S"]
by auto
then obtain e1 where e1: "e1 > 0" "∀x ∈ span S. e1 * norm x ≤ norm (f x)"
using assms injective_imp_isometric[of "span S" f]
subspace_span[of S] closed_subspace[of "span S"]
by auto
define e where "e = e1 * e2"
hence "e > 0" using e1 e2 by auto
{
fix y
assume y: "y ∈ cball (f x) e ∩ affine hull (f ` S)"
then have "y ∈ f ` (affine hull S)"
using affine_hull_linear_image[of f S] assms by auto
then obtain xy where xy: "xy ∈ affine hull S" "f xy = y" by auto
with y have "norm (f x - f xy) ≤ e1 * e2"
using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
moreover have "f x - f xy = f (x - xy)"
using assms linear_diff[of f x xy] linear_conv_bounded_linear[of f] by auto
moreover have *: "x - xy ∈ span S"
using subspace_diff[of "span S" x xy] subspace_span ‹x ∈ S› xy
affine_hull_subset_span[of S] span_superset
by auto
moreover from * have "e1 * norm (x - xy) ≤ norm (f (x - xy))"
using e1 by auto
ultimately have "e1 * norm (x - xy) ≤ e1 * e2"
by auto
then have "xy ∈ cball x e2"
using cball_def[of x e2] dist_norm[of x xy] e1 by auto
then have "y ∈ f ` S"
using xy e2 by auto
}
then have "f x ∈ rel_interior (f ` S)"
using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * ‹e > 0› by auto
}
ultimately show ?thesis by auto
qed
lemma rel_interior_injective_linear_image:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "bounded_linear f"
and "inj f"
shows "rel_interior (f ` S) = f ` (rel_interior S)"
using assms rel_interior_injective_on_span_linear_image[of f S]
subset_inj_on[of f "UNIV" "span S"]
by auto
subsection%unimportant‹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%unimportant ‹Openness and compactness are preserved by convex hull operation›
lemma open_convex_hull[intro]:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open (convex hull S)"
proof (clarsimp simp: open_contains_cball convex_hull_explicit)
fix T and u :: "'a⇒real"
assume obt: "finite T" "T⊆S" "∀x∈T. 0 ≤ u x" "sum u T = 1"
from assms[unfolded open_contains_cball] obtain b
where b: "⋀x. x∈S ⟹ 0 < b x ∧ cball x (b x) ⊆ S" by metis
have "b ` T ≠ {}"
using obt by auto
define i where "i = b ` T"
let ?Φ = "λy. ∃F. finite F ∧ F ⊆ S ∧ (∃u. (∀x∈F. 0 ≤ u x) ∧ sum u F = 1 ∧ (∑v∈F. u v *⇩R v) = y)"
let ?a = "∑v∈T. u v *⇩R v"
show "∃e > 0. cball ?a e ⊆ {y. ?Φ y}"
proof (intro exI subsetI conjI)
show "0 < Min i"
unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] ‹b ` T≠{}›]
using b ‹T⊆S› by auto
next
fix y
assume "y ∈ cball ?a (Min i)"
then have y: "norm (?a - y) ≤ Min i"
unfolding dist_norm[symmetric] by auto
{ fix x
assume "x ∈ T"
then have "Min i ≤ b x"
by (simp add: i_def obt(1))
then have "x + (y - ?a) ∈ cball x (b x)"
using y unfolding mem_cball dist_norm by auto
moreover have "x ∈ S"
using ‹x∈T› ‹T⊆S› by auto
ultimately have "x + (y - ?a) ∈ S"
using y b by blast
}
moreover
have *: "inj_on (λv. v + (y - ?a)) T"
unfolding inj_on_def by auto
have "(∑v∈(λv. v + (y - ?a)) ` T. u (v - (y - ?a)) *⇩R v) = y"
unfolding sum.reindex[OF *] o_def using obt(4)
by (simp add: sum.distrib sum_subtractf scaleR_left.sum[symmetric] scaleR_right_distrib)
ultimately show "y ∈ {y. ?Φ y}"
proof (intro CollectI exI conjI)
show "finite ((λv. v + (y - ?a)) ` T)"
by (simp add: obt(1))
show "sum (λv. u (v - (y - ?a))) ((λv. v + (y - ?a)) ` T) = 1"
unfolding sum.reindex[OF *] o_def using obt(4) by auto
qed (use obt(1, 3) in auto)
qed
qed
lemma compact_convex_combinations:
fixes S T :: "'a::real_normed_vector set"
assumes "compact S" "compact T"
shows "compact { (1 - u) *⇩R x + u *⇩R y | x y u. 0 ≤ u ∧ u ≤ 1 ∧ x ∈ S ∧ y ∈ T}"
proof -
let ?X = "{0..1} × S × T"
let ?h = "(λz. (1 - fst z) *⇩R fst (snd z) + fst z *⇩R snd (snd z))"
have *: "{ (1 - u) *⇩R x + u *⇩R y | x y u. 0 ≤ u ∧ u ≤ 1 ∧ x ∈ S ∧ y ∈ T} = ?h ` ?X"
by force
have "continuous_on ?X (λz. (1 - fst z) *⇩R fst (snd z) + fst z *⇩R snd (snd z))"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
with assms show ?thesis
by (simp add: * compact_Times compact_continuous_image)
qed
lemma finite_imp_compact_convex_hull:
fixes S :: "'a::real_normed_vector set"
assumes "finite S"
shows "compact (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis by simp
next
case False
with assms show ?thesis
proof (induct rule: finite_ne_induct)
case (singleton x)
show ?case by simp
next
case (insert x A)
let ?f = "λ(u, y::'a). u *⇩R x + (1 - u) *⇩R y"
let ?T = "{0..1::real} × (convex hull A)"
have "continuous_on ?T ?f"
unfolding split_def continuous_on by (intro ballI tendsto_intros)
moreover have "compact ?T"
by (intro compact_Times compact_Icc insert)
ultimately have "compact (?f ` ?T)"
by (rule compact_continuous_image)
also have "?f ` ?T = convex hull (insert x A)"
unfolding convex_hull_insert [OF ‹A ≠ {}›]
apply safe
apply (rule_tac x=a in exI, simp)
apply (rule_tac x="1 - a" in exI, simp, fast)
apply (rule_tac x="(u, b)" in image_eqI, simp_all)
done
finally show "compact (convex hull (insert x A))" .
qed
qed
lemma compact_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "compact S"
shows "compact (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis using compact_empty by simp
next
case False
then obtain w where "w ∈ S" by auto
show ?thesis
unfolding caratheodory[of S]
proof (induct ("DIM('a) + 1"))
case 0
have *: "{x.∃sa. finite sa ∧ sa ⊆ S ∧ card sa ≤ 0 ∧ x ∈ convex hull sa} = {}"
using compact_empty by auto
from 0 show ?case unfolding * by simp
next
case (Suc n)
show ?case
proof (cases "n = 0")
case True
have "{x. ∃T. finite T ∧ T ⊆ S ∧ card T ≤ Suc n ∧ x ∈ convex hull T} = S"
unfolding set_eq_iff and mem_Collect_eq
proof (rule, rule)
fix x
assume "∃T. finite T ∧ T ⊆ S ∧ card T ≤ Suc n ∧ x ∈ convex hull T"
then obtain T where T: "finite T" "T ⊆ S" "card T ≤ Suc n" "x ∈ convex hull T"
by auto
show "x ∈ S"
proof (cases "card T = 0")
case True
then show ?thesis
using T(4) unfolding card_0_eq[OF T(1)] by simp
next
case False
then have "card T = Suc 0" using T(3) ‹n=0› by auto
then obtain a where "T = {a}" unfolding card_Suc_eq by auto
then show ?thesis using T(2,4) by simp
qed
next
fix x assume "x∈S"
then show "∃T. finite T ∧ T ⊆ S ∧ card T ≤ Suc n ∧ x ∈ convex hull T"
apply (rule_tac x="{x}" in exI)
unfolding convex_hull_singleton
apply auto
done
qed
then show ?thesis using assms by simp
next
case False
have "{x. ∃T. finite T ∧ T ⊆ S ∧ card T ≤ Suc n ∧ x ∈ convex hull T} =
{(1 - u) *⇩R x + u *⇩R y | x y u.
0 ≤ u ∧ u ≤ 1 ∧ x ∈ S ∧ y ∈ {x. ∃T. finite T ∧ T ⊆ S ∧ card T ≤ n ∧ x ∈ convex hull T}}"
unfolding set_eq_iff and mem_Collect_eq
proof (rule, rule)
fix x
assume "∃u v c. x = (1 - c) *⇩R u + c *⇩R v ∧
0 ≤ c ∧ c ≤ 1 ∧ u ∈ S ∧ (∃T. finite T ∧ T ⊆ S ∧ card T ≤ n ∧ v ∈ convex hull T)"
then obtain u v c T where obt: "x = (1 - c) *⇩R u + c *⇩R v"
"0 ≤ c ∧ c ≤ 1" "u ∈ S" "finite T" "T ⊆ S" "card T ≤ n" "v ∈ convex hull T"
by auto
moreover have "(1 - c) *⇩R u + c *⇩R v ∈ convex hull insert u T"
apply (rule convexD_alt)
using obt(2) and convex_convex_hull and hull_subset[of "insert u T" convex]
using obt(7) and hull_mono[of T "insert u T"]
apply auto
done
ultimately show "∃T. finite T ∧ T ⊆ S ∧ card T ≤ Suc n ∧ x ∈ convex hull T"
apply (rule_tac x="insert u T" in exI)
apply (auto simp: card_insert_if)
done
next
fix x
assume "∃T. finite T ∧ T ⊆ S ∧ card T ≤ Suc n ∧ x ∈ convex hull T"
then obtain T where T: "finite T" "T ⊆ S" "card T ≤ Suc n" "x ∈ convex hull T"
by auto
show "∃u v c. x = (1 - c) *⇩R u + c *⇩R v ∧
0 ≤ c ∧ c ≤ 1 ∧ u ∈ S ∧ (∃T. finite T ∧ T ⊆ S ∧ card T ≤ n ∧ v ∈ convex hull T)"
proof (cases "card T = Suc n")
case False
then have "card T ≤ n" using T(3) by auto
then show ?thesis
apply (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI)
using ‹w∈S› and T
apply (auto intro!: exI[where x=T])
done
next
case True
then obtain a u where au: "T = insert a u" "a∉u"
apply (drule_tac card_eq_SucD, auto)
done
show ?thesis
proof (cases "u = {}")
case True
then have "x = a" using T(4)[unfolded au] by auto
show ?thesis unfolding ‹x = a›
apply (rule_tac x=a in exI)
apply (rule_tac x=a in exI)
apply (rule_tac x=1 in exI)
using T and ‹n ≠ 0›
unfolding au
apply (auto intro!: exI[where x="{a}"])
done
next
case False
obtain ux vx b where obt: "ux≥0" "vx≥0" "ux + vx = 1"
"b ∈ convex hull u" "x = ux *⇩R a + vx *⇩R b"
using T(4)[unfolded au convex_hull_insert[OF False]]
by auto
have *: "1 - vx = ux" using obt(3) by auto
show ?thesis
apply (rule_tac x=a in exI)
apply (rule_tac x=b in exI)
apply (rule_tac x=vx in exI)
using obt and T(1-3)
unfolding au and * using card_insert_disjoint[OF _ au(2)]
apply (auto intro!: exI[where x=u])
done
qed
qed
qed
then show ?thesis
using compact_convex_combinations[OF assms Suc] by simp
qed
qed
qed
subsection%unimportant ‹Extremal points of a simplex are some vertices›
lemma dist_increases_online:
fixes a b d :: "'a::real_inner"
assumes "d ≠ 0"
shows "dist a (b + d) > dist a b ∨ dist a (b - d) > dist a b"
proof (cases "inner a d - inner b d > 0")
case True
then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
apply (rule_tac add_pos_pos)
using assms
apply auto
done
then show ?thesis
apply (rule_tac disjI2)
unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
apply (simp add: algebra_simps inner_commute)
done
next
case False
then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
apply (rule_tac add_pos_nonneg)
using assms
apply auto
done
then show ?thesis
apply (rule_tac disjI1)
unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
apply (simp add: algebra_simps inner_commute)
done
qed
lemma norm_increases_online:
fixes d :: "'a::real_inner"
shows "d ≠ 0 ⟹ norm (a + d) > norm a ∨ norm(a - d) > norm a"
using dist_increases_online[of d a 0] unfolding dist_norm by auto
lemma simplex_furthest_lt:
fixes S :: "'a::real_inner set"
assumes "finite S"
shows "∀x ∈ convex hull S. x ∉ S ⟶ (∃y ∈ convex hull S. norm (x - a) < norm(y - a))"
using assms
proof induct
fix x S
assume as: "finite S" "x∉S" "∀x∈convex hull S. x ∉ S ⟶ (∃y∈convex hull S. norm (x - a) < norm (y - a))"
show "∀xa∈convex hull insert x S. xa ∉ insert x S ⟶
(∃y∈convex hull insert x S. norm (xa - a) < norm (y - a))"
proof (intro impI ballI, cases "S = {}")
case False
fix y
assume y: "y ∈ convex hull insert x S" "y ∉ insert x S"
obtain u v b where obt: "u≥0" "v≥0" "u + v = 1" "b ∈ convex hull S" "y = u *⇩R x + v *⇩R b"
using y(1)[unfolded convex_hull_insert[OF False]] by auto
show "∃z∈convex hull insert x S. norm (y - a) < norm (z - a)"
proof (cases "y ∈ convex hull S")
case True
then obtain z where "z ∈ convex hull S" "norm (y - a) < norm (z - a)"
using as(3)[THEN bspec[where x=y]] and y(2) by auto
then show ?thesis
apply (rule_tac x=z in bexI)
unfolding convex_hull_insert[OF False]
apply auto
done
next
case False
show ?thesis
using obt(3)
proof (cases "u = 0", case_tac[!] "v = 0")
assume "u = 0" "v ≠ 0"
then have "y = b" using obt by auto
then show ?thesis using False and obt(4) by auto
next
assume "u ≠ 0" "v = 0"
then have "y = x" using obt by auto
then show ?thesis using y(2) by auto
next
assume "u ≠ 0" "v ≠ 0"
then obtain w where w: "w>0" "w<u" "w<v"
using field_lbound_gt_zero[of u v] and obt(1,2) by auto
have "x ≠ b"
proof
assume "x = b"
then have "y = b" unfolding obt(5)
using obt(3) by (auto simp: scaleR_left_distrib[symmetric])
then show False using obt(4) and False by simp
qed
then have *: "w *⇩R (x - b) ≠ 0" using w(1) by auto
show ?thesis
using dist_increases_online[OF *, of a y]
proof (elim disjE)
assume "dist a y < dist a (y + w *⇩R (x - b))"
then have "norm (y - a) < norm ((u + w) *⇩R x + (v - w) *⇩R b - a)"
unfolding dist_commute[of a]
unfolding dist_norm obt(5)
by (simp add: algebra_simps)
moreover have "(u + w) *⇩R x + (v - w) *⇩R b ∈ convex hull insert x S"
unfolding convex_hull_insert[OF ‹S≠{}›]
proof (intro CollectI conjI exI)
show "u + w ≥ 0" "v - w ≥ 0"
using obt(1) w by auto
qed (use obt in auto)
ultimately show ?thesis by auto
next
assume "dist a y < dist a (y - w *⇩R (x - b))"
then have "norm (y - a) < norm ((u - w) *⇩R x + (v + w) *⇩R b - a)"
unfolding dist_commute[of a]
unfolding dist_norm obt(5)
by (simp add: algebra_simps)
moreover have "(u - w) *⇩R x + (v + w) *⇩R b ∈ convex hull insert x S"
unfolding convex_hull_insert[OF ‹S≠{}›]
proof (intro CollectI conjI exI)
show "u - w ≥ 0" "v + w ≥ 0"
using obt(1) w by auto
qed (use obt in auto)
ultimately show ?thesis by auto
qed
qed auto
qed
qed auto
qed (auto simp: assms)
lemma simplex_furthest_le:
fixes S :: "'a::real_inner set"
assumes "finite S"
and "S ≠ {}"
shows "∃y∈S. ∀x∈ convex hull S. norm (x - a) ≤ norm (y - a)"
proof -
have "convex hull S ≠ {}"
using hull_subset[of S convex] and assms(2) by auto
then obtain x where x: "x ∈ convex hull S" "∀y∈convex hull S. norm (y - a) ≤ norm (x - a)"
using distance_attains_sup[OF finite_imp_compact_convex_hull[OF ‹finite S›], of a]
unfolding dist_commute[of a]
unfolding dist_norm
by auto
show ?thesis
proof (cases "x ∈ S")
case False
then obtain y where "y ∈ convex hull S" "norm (x - a) < norm (y - a)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
by auto
then show ?thesis
using x(2)[THEN bspec[where x=y]] by auto
next
case True
with x show ?thesis by auto
qed
qed
lemma simplex_furthest_le_exists:
fixes S :: "('a::real_inner) set"
shows "finite S ⟹ ∀x∈(convex hull S). ∃y∈S. norm (x - a) ≤ norm (y - a)"
using simplex_furthest_le[of S] by (cases "S = {}") auto
lemma simplex_extremal_le:
fixes S :: "'a::real_inner set"
assumes "finite S"
and "S ≠ {}"
shows "∃u∈S. ∃v∈S. ∀x∈convex hull S. ∀y ∈ convex hull S. norm (x - y) ≤ norm (u - v)"
proof -
have "convex hull S ≠ {}"
using hull_subset[of S convex] and assms(2) by auto
then obtain u v where obt: "u ∈ convex hull S" "v ∈ convex hull S"
"∀x∈convex hull S. ∀y∈convex hull S. norm (x - y) ≤ norm (u - v)"
using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
by (auto simp: dist_norm)
then show ?thesis
proof (cases "u∉S ∨ v∉S", elim disjE)
assume "u ∉ S"
then obtain y where "y ∈ convex hull S" "norm (u - v) < norm (y - v)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
by auto
then show ?thesis
using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
by auto
next
assume "v ∉ S"
then obtain y where "y ∈ convex hull S" "norm (v - u) < norm (y - u)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
by auto
then show ?thesis
using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
by (auto simp: norm_minus_commute)
qed auto
qed
lemma simplex_extremal_le_exists:
fixes S :: "'a::real_inner set"
shows "finite S ⟹ x ∈ convex hull S ⟹ y ∈ convex hull S ⟹
∃u∈S. ∃v∈S. norm (x - y) ≤ norm (u - v)"
using convex_hull_empty simplex_extremal_le[of S]
by(cases "S = {}") auto
subsection ‹Closest point of a convex set is unique, with a continuous projection›
definition%important closest_point :: "'a::{real_inner,heine_borel} set ⇒ 'a ⇒ 'a"
where "closest_point S a = (SOME x. x ∈ S ∧ (∀y∈S. dist a x ≤ dist a y))"
lemma closest_point_exists:
assumes "closed S"
and "S ≠ {}"
shows "closest_point S a ∈ S"
and "∀y∈S. dist a (closest_point S a) ≤ dist a y"
unfolding closest_point_def
apply(rule_tac[!] someI2_ex)
apply (auto intro: distance_attains_inf[OF assms(1,2), of a])
done
lemma closest_point_in_set: "closed S ⟹ S ≠ {} ⟹ closest_point S a ∈ S"
by (meson closest_point_exists)
lemma closest_point_le: "closed S ⟹ x ∈ S ⟹ dist a (closest_point S a) ≤ dist a x"
using closest_point_exists[of S] by auto
lemma closest_point_self:
assumes "x ∈ S"
shows "closest_point S x = x"
unfolding closest_point_def
apply (rule some1_equality, rule ex1I[of _ x])
using assms
apply auto
done
lemma closest_point_refl: "closed S ⟹ S ≠ {} ⟹ closest_point S x = x ⟷ x ∈ S"
using closest_point_in_set[of S x] closest_point_self[of x S]
by auto
lemma closer_points_lemma:
assumes "inner y z > 0"
shows "∃u>0. ∀v>0. v ≤ u ⟶ norm(v *⇩R z - y) < norm y"
proof -
have z: "inner z z > 0"
unfolding inner_gt_zero_iff using assms by auto
have "norm (v *⇩R z - y) < norm y"
if "0 < v" and "v ≤ inner y z / inner z z" for v
unfolding norm_lt using z assms that
by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ ‹0<v›])
then show ?thesis
using assms z
by (rule_tac x = "inner y z / inner z z" in exI) auto
qed
lemma closer_point_lemma:
assumes "inner (y - x) (z - x) > 0"
shows "∃u>0. u ≤ 1 ∧ dist (x + u *⇩R (z - x)) y < dist x y"
proof -
obtain u where "u > 0"
and u: "∀v>0. v ≤ u ⟶ norm (v *⇩R (z - x) - (y - x)) < norm (y - x)"
using closer_points_lemma[OF assms] by auto
show ?thesis
apply (rule_tac x="min u 1" in exI)
using u[THEN spec[where x="min u 1"]] and ‹u > 0›
unfolding dist_norm by (auto simp: norm_minus_commute field_simps)
qed
lemma any_closest_point_dot:
assumes "convex S" "closed S" "x ∈ S" "y ∈ S" "∀z∈S. dist a x ≤ dist a z"
shows "inner (a - x) (y - x) ≤ 0"
proof (rule ccontr)
assume "¬ ?thesis"
then obtain u where u: "u>0" "u≤1" "dist (x + u *⇩R (y - x)) a < dist x a"
using closer_point_lemma[of a x y] by auto
let ?z = "(1 - u) *⇩R x + u *⇩R y"
have "?z ∈ S"
using convexD_alt[OF assms(1,3,4), of u] using u by auto
then show False
using assms(5)[THEN bspec[where x="?z"]] and u(3)
by (auto simp: dist_commute algebra_simps)
qed
lemma any_closest_point_unique:
fixes x :: "'a::real_inner"
assumes "convex S" "closed S" "x ∈ S" "y ∈ S"
"∀z∈S. dist a x ≤ dist a z" "∀z∈S. dist a y ≤ dist a z"
shows "x = y"
using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
unfolding norm_pths(1) and norm_le_square
by (auto simp: algebra_simps)
lemma closest_point_unique:
assumes "convex S" "closed S" "x ∈ S" "∀z∈S. dist a x ≤ dist a z"
shows "x = closest_point S a"
using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point S a"]
using closest_point_exists[OF assms(2)] and assms(3) by auto
lemma closest_point_dot:
assumes "convex S" "closed S" "x ∈ S"
shows "inner (a - closest_point S a) (x - closest_point S a) ≤ 0"
apply (rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
using closest_point_exists[OF assms(2)] and assms(3)
apply auto
done
lemma closest_point_lt:
assumes "convex S" "closed S" "x ∈ S" "x ≠ closest_point S a"
shows "dist a (closest_point S a) < dist a x"
apply (rule ccontr)
apply (rule_tac notE[OF assms(4)])
apply (rule closest_point_unique[OF assms(1-3), of a])
using closest_point_le[OF assms(2), of _ a]
apply fastforce
done
lemma closest_point_lipschitz:
assumes "convex S"
and "closed S" "S ≠ {}"
shows "dist (closest_point S x) (closest_point S y) ≤ dist x y"
proof -
have "inner (x - closest_point S x) (closest_point S y - closest_point S x) ≤ 0"
and "inner (y - closest_point S y) (closest_point S x - closest_point S y) ≤ 0"
apply (rule_tac[!] any_closest_point_dot[OF assms(1-2)])
using closest_point_exists[OF assms(2-3)]
apply auto
done
then show ?thesis unfolding dist_norm and norm_le
using inner_ge_zero[of "(x - closest_point S x) - (y - closest_point S y)"]
by (simp add: inner_add inner_diff inner_commute)
qed
lemma continuous_at_closest_point:
assumes "convex S"
and "closed S"
and "S ≠ {}"
shows "continuous (at x) (closest_point S)"
unfolding continuous_at_eps_delta
using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
lemma continuous_on_closest_point:
assumes "convex S"
and "closed S"
and "S ≠ {}"
shows "continuous_on t (closest_point S)"
by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
proposition closest_point_in_rel_interior:
assumes "closed S" "S ≠ {}" and x: "x ∈ affine hull S"
shows "closest_point S x ∈ rel_interior S ⟷ x ∈ rel_interior S"
proof (cases "x ∈ S")
case True
then show ?thesis
by (simp add: closest_point_self)
next
case False
then have "False" if asm: "closest_point S x ∈ rel_interior S"
proof -
obtain e where "e > 0" and clox: "closest_point S x ∈ S"
and e: "cball (closest_point S x) e ∩ affine hull S ⊆ S"
using asm mem_rel_interior_cball by blast
then have clo_notx: "closest_point S x ≠ x"
using ‹x ∉ S› by auto
define y where "y ≡ closest_point S x -
(min 1 (e / norm(closest_point S x - x))) *⇩R (closest_point S x - x)"
have "x - y = (1 - min 1 (e / norm (closest_point S x - x))) *⇩R (x - closest_point S x)"
by (simp add: y_def algebra_simps)
then have "norm (x - y) = abs ((1 - min 1 (e / norm (closest_point S x - x)))) * norm(x - closest_point S x)"
by simp
also have "… < norm(x - closest_point S x)"
using clo_notx ‹e > 0›
by (auto simp: mult_less_cancel_right2 divide_simps)
finally have no_less: "norm (x - y) < norm (x - closest_point S x)" .
have "y ∈ affine hull S"
unfolding y_def
by (meson affine_affine_hull clox hull_subset mem_affine_3_minus2 subsetD x)
moreover have "dist (closest_point S x) y ≤ e"
using ‹e > 0› by (auto simp: y_def min_mult_distrib_right)
ultimately have "y ∈ S"
using subsetD [OF e] by simp
then have "dist x (closest_point S x) ≤ dist x y"
by (simp add: closest_point_le ‹closed S›)
with no_less show False
by (simp add: dist_norm)
qed
moreover have "x ∉ rel_interior S"
using rel_interior_subset False by blast
ultimately show ?thesis by blast
qed
subsubsection%unimportant ‹Various point-to-set separating/supporting hyperplane theorems›
lemma supporting_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex S"
and "closed S"
and "S ≠ {}"
and "z ∉ S"
shows "∃a b. ∃y∈S. inner a z < b ∧ inner a y = b ∧ (∀x∈S. inner a x ≥ b)"
proof -
obtain y where "y ∈ S" and y: "∀x∈S. dist z y ≤ dist z x"
by (metis distance_attains_inf[OF assms(2-3)])
show ?thesis
proof (intro exI bexI conjI ballI)
show "(y - z) ∙ z < (y - z) ∙ y"
by (metis ‹y ∈ S› assms(4) diff_gt_0_iff_gt inner_commute inner_diff_left inner_gt_zero_iff right_minus_eq)
show "(y - z) ∙ y ≤ (y - z) ∙ x" if "x ∈ S" for x
proof (rule ccontr)
have *: "⋀u. 0 ≤ u ∧ u ≤ 1 ⟶ dist z y ≤ dist z ((1 - u) *⇩R y + u *⇩R x)"
using assms(1)[unfolded convex_alt] and y and ‹x∈S› and ‹y∈S› by auto
assume "¬ (y - z) ∙ y ≤ (y - z) ∙ x"
then obtain v where "v > 0" "v ≤ 1" "dist (y + v *⇩R (x - y)) z < dist y z"
using closer_point_lemma[of z y x] by (auto simp: inner_diff)
then show False
using *[of v] by (auto simp: dist_commute algebra_simps)
qed
qed (use ‹y ∈ S› in auto)
qed
lemma separating_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex S"
and "closed S"
and "z ∉ S"
shows "∃a b. inner a z < b ∧ (∀x∈S. inner a x > b)"
proof (cases "S = {}")
case True
then show ?thesis
by (simp add: gt_ex)
next
case False
obtain y where "y ∈ S" and y: "⋀x. x ∈ S ⟹ dist z y ≤ dist z x"
by (metis distance_attains_inf[OF assms(2) False])
show ?thesis
proof (intro exI conjI ballI)
show "(y - z) ∙ z < inner (y - z) z + (norm (y - z))⇧2 / 2"
using ‹y∈S› ‹z∉S› by auto
next
fix x
assume "x ∈ S"
have "False" if *: "0 < inner (z - y) (x - y)"
proof -
obtain u where "u > 0" "u ≤ 1" "dist (y + u *⇩R (x - y)) z < dist y z"
using * closer_point_lemma by blast
then show False using y[of "y + u *⇩R (x - y)"] convexD_alt [OF ‹convex S›]
using ‹x∈S› ‹y∈S› by (auto simp: dist_commute algebra_simps)
qed
moreover have "0 < (norm (y - z))⇧2"
using ‹y∈S› ‹z∉S› by auto
then have "0 < inner (y - z) (y - z)"
unfolding power2_norm_eq_inner by simp
ultimately show "(y - z) ∙ z + (norm (y - z))⇧2 / 2 < (y - z) ∙ x"
by (force simp: field_simps power2_norm_eq_inner inner_commute inner_diff)
qed
qed
lemma separating_hyperplane_closed_0:
assumes "convex (S::('a::euclidean_space) set)"
and "closed S"
and "0 ∉ S"
shows "∃a b. a ≠ 0 ∧ 0 < b ∧ (∀x∈S. inner a x > b)"
proof (cases "S = {}")
case True
have "(SOME i. i∈Basis) ≠ (0::'a)"
by (metis Basis_zero SOME_Basis)
then show ?thesis
using True zero_less_one by blast
next
case False
then show ?thesis
using False using separating_hyperplane_closed_point[OF assms]
by (metis all_not_in_conv inner_zero_left inner_zero_right less_eq_real_def not_le)
qed
subsubsection%unimportant ‹Now set-to-set for closed/compact sets›
lemma separating_hyperplane_closed_compact:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "closed S"
and "convex T"
and "compact T"
and "T ≠ {}"
and "S ∩ T = {}"
shows "∃a b. (∀x∈S. inner a x < b) ∧ (∀x∈T. inner a x > b)"
proof (cases "S = {}")
case True
obtain b where b: "b > 0" "∀x∈T. norm x ≤ b"
using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
obtain z :: 'a where z: "norm z = b + 1"
using vector_choose_size[of "b + 1"] and b(1) by auto
then have "z ∉ T" using b(2)[THEN bspec[where x=z]] by auto
then obtain a b where ab: "inner a z < b" "∀x∈T. b < inner a x"
using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
by auto
then show ?thesis
using True by auto
next
case False
then obtain y where "y ∈ S" by auto
obtain a b where "0 < b" "∀x ∈ (⋃x∈ S. ⋃y ∈ T. {x - y}). b < inner a x"
using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
using closed_compact_differences[OF assms(2,4)]
using assms(6) by auto
then have ab: "∀x∈S. ∀y∈T. b + inner a y < inner a x"
apply -
apply rule
apply rule
apply (erule_tac x="x - y" in ballE)
apply (auto simp: inner_diff)
done
define k where "k = (SUP x:T. a ∙ x)"
show ?thesis
apply (rule_tac x="-a" in exI)
apply (rule_tac x="-(k + b / 2)" in exI)
apply (intro conjI ballI)
unfolding inner_minus_left and neg_less_iff_less
proof -
fix x assume "x ∈ T"
then have "inner a x - b / 2 < k"
unfolding k_def
proof (subst less_cSUP_iff)
show "T ≠ {}" by fact
show "bdd_above ((∙) a ` T)"
using ab[rule_format, of y] ‹y ∈ S›
by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
qed (auto intro!: bexI[of _ x] ‹0<b›)
then show "inner a x < k + b / 2"
by auto
next
fix x
assume "x ∈ S"
then have "k ≤ inner a x - b"
unfolding k_def
apply (rule_tac cSUP_least)
using assms(5)
using ab[THEN bspec[where x=x]]
apply auto
done
then show "k + b / 2 < inner a x"
using ‹0 < b› by auto
qed
qed
lemma separating_hyperplane_compact_closed:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "compact S"
and "S ≠ {}"
and "convex T"
and "closed T"
and "S ∩ T = {}"
shows "∃a b. (∀x∈S. inner a x < b) ∧ (∀x∈T. inner a x > b)"
proof -
obtain a b where "(∀x∈T. inner a x < b) ∧ (∀x∈S. b < inner a x)"
using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6)
by auto
then show ?thesis
apply (rule_tac x="-a" in exI)
apply (rule_tac x="-b" in exI, auto)
done
qed
subsubsection%unimportant ‹General case without assuming closure and getting non-strict separation›
lemma separating_hyperplane_set_0:
assumes "convex S" "(0::'a::euclidean_space) ∉ S"
shows "∃a. a ≠ 0 ∧ (∀x∈S. 0 ≤ inner a x)"
proof -
let ?k = "λc. {x::'a. 0 ≤ inner c x}"
have *: "frontier (cball 0 1) ∩ ⋂f ≠ {}" if as: "f ⊆ ?k ` S" "finite f" for f
proof -
obtain c where c: "f = ?k ` c" "c ⊆ S" "finite c"
using finite_subset_image[OF as(2,1)] by auto
then obtain a b where ab: "a ≠ 0" "0 < b" "∀x∈convex hull c. b < inner a x"
using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
using subset_hull[of convex, OF assms(1), symmetric, of c]
by force
then have "∃x. norm x = 1 ∧ (∀y∈c. 0 ≤ inner y x)"
apply (rule_tac x = "inverse(norm a) *⇩R a" in exI)
using hull_subset[of c convex]
unfolding subset_eq and inner_scaleR
by (auto simp: inner_commute del: ballE elim!: ballE)
then show "frontier (cball 0 1) ∩ ⋂f ≠ {}"
unfolding c(1) frontier_cball sphere_def dist_norm by auto
qed
have "frontier (cball 0 1) ∩ (⋂(?k ` S)) ≠ {}"
apply (rule compact_imp_fip)
apply (rule compact_frontier[OF compact_cball])
using * closed_halfspace_ge
by auto
then obtain x where "norm x = 1" "∀y∈S. x∈?k y"
unfolding frontier_cball dist_norm sphere_def by auto
then show ?thesis
by (metis inner_commute mem_Collect_eq norm_eq_zero zero_neq_one)
qed
lemma separating_hyperplane_sets:
fixes S T :: "'a::euclidean_space set"
assumes "convex S"
and "convex T"
and "S ≠ {}"
and "T ≠ {}"
and "S ∩ T = {}"
shows "∃a b. a ≠ 0 ∧ (∀x∈S. inner a x ≤ b) ∧ (∀x∈T. inner a x ≥ b)"
proof -
from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
obtain a where "a ≠ 0" "∀x∈{x - y |x y. x ∈ T ∧ y ∈ S}. 0 ≤ inner a x"
using assms(3-5) by force
then have *: "⋀x y. x ∈ T ⟹ y ∈ S ⟹ inner a y ≤ inner a x"
by (force simp: inner_diff)
then have bdd: "bdd_above (((∙) a)`S)"
using ‹T ≠ {}› by (auto intro: bdd_aboveI2[OF *])
show ?thesis
using ‹a≠0›
by (intro exI[of _ a] exI[of _ "SUP x:S. a ∙ x"])
(auto intro!: cSUP_upper bdd cSUP_least ‹a ≠ 0› ‹S ≠ {}› *)
qed
subsection%unimportant ‹More convexity generalities›
lemma convex_closure [intro,simp]:
fixes S :: "'a::real_normed_vector set"
assumes "convex S"
shows "convex (closure S)"
apply (rule convexI)
apply (unfold closure_sequential, elim exE)
apply (rule_tac x="λn. u *⇩R xa n + v *⇩R xb n" in exI)
apply (rule,rule)
apply (rule convexD [OF assms])
apply (auto del: tendsto_const intro!: tendsto_intros)
done
lemma convex_interior [intro,simp]:
fixes S :: "'a::real_normed_vector set"
assumes "convex S"
shows "convex (interior S)"
unfolding convex_alt Ball_def mem_interior
proof clarify
fix x y u
assume u: "0 ≤ u" "u ≤ (1::real)"
fix e d
assume ed: "ball x e ⊆ S" "ball y d ⊆ S" "0<d" "0<e"
show "∃e>0. ball ((1 - u) *⇩R x + u *⇩R y) e ⊆ S"
proof (intro exI conjI subsetI)
fix z
assume "z ∈ ball ((1 - u) *⇩R x + u *⇩R y) (min d e)"
then have "(1- u) *⇩R (z - u *⇩R (y - x)) + u *⇩R (z + (1 - u) *⇩R (y - x)) ∈ S"
apply (rule_tac assms[unfolded convex_alt, rule_format])
using ed(1,2) and u
unfolding subset_eq mem_ball Ball_def dist_norm
apply (auto simp: algebra_simps)
done
then show "z ∈ S"
using u by (auto simp: algebra_simps)
qed(insert u ed(3-4), auto)
qed
lemma convex_hull_eq_empty[simp]: "convex hull S = {} ⟷ S = {}"
using hull_subset[of S convex] convex_hull_empty by auto
subsection%unimportant ‹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(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
subsection%unimportant ‹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%unimportant ‹Convex set as intersection of halfspaces›
lemma convex_halfspace_intersection:
fixes s :: "('a::euclidean_space) set"
assumes "closed s" "convex s"
shows "s = ⋂{h. s ⊆ h ∧ (∃a b. h = {x. inner a x ≤ b})}"
apply (rule set_eqI, rule)
unfolding Inter_iff Ball_def mem_Collect_eq
apply (rule,rule,erule conjE)
proof -
fix x
assume "∀xa. s ⊆ xa ∧ (∃a b. xa = {x. inner a x ≤ b}) ⟶ x ∈ xa"
then have "∀a b. s ⊆ {x. inner a x ≤ b} ⟶ x ∈ {x. inner a x ≤ b}"
by blast
then show "x ∈ s"
apply (rule_tac ccontr)
apply (drule separating_hyperplane_closed_point[OF assms(2,1)])
apply (erule exE)+
apply (erule_tac x="-a" in allE)
apply (erule_tac x="-b" in allE, auto)
done
qed auto
subsection ‹Radon's theorem (from 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 and sum.inter_restrict[OF assms(1), symmetric]
apply (auto simp: Int_absorb1)
done
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"
apply (rule_tac sum_mono, auto)
done
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
apply (rule_tac conjI)
apply (rule_tac sum_nonneg, auto)
done
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)
apply (rule_tac[!] sum.mono_neutral_left, auto)
done
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"
apply rule
apply (rule mult_nonneg_nonneg)
using *
apply auto
done
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] and z_def
apply (auto simp: sum_negf sum_distrib_left[symmetric])
done
moreover have "∀x∈{v ∈ c. 0 < u v}. 0 ≤ inverse (sum u {x ∈ c. 0 < u x}) * u x"
apply rule
apply (rule mult_nonneg_nonneg)
using *
apply auto
done
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] and z_def
using *
apply (auto simp: sum_negf sum_distrib_left[symmetric])
done
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
then show ?thesis
apply (rule_tac that[of p m])
using s
apply auto
done
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 mp(1)
unfolding gh
using inj_on_image_Int[OF True gh(3,4)]
by auto
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 ≠ {}"
apply (rule helly_induct)
using assms
apply auto
done
subsection ‹Epigraphs of convex functions›
definition%important "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
apply (clarsimp simp: convex_def convex_on_def epigraph_def)
apply (erule_tac x=x in allE)
apply (erule_tac x="f x" in allE, safe)
apply (erule_tac x=y in allE)
apply (erule_tac x="f y" in allE)
apply (auto simp: )
done
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%unimportant ‹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})"
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
unfolding fst_sum snd_sum fst_scaleR snd_scaleR
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
using assms[unfolded convex] apply simp
apply (rule_tac y="∑i = 1..k. u i * f (fst (x i))" in order_trans, force)
apply (rule sum_mono)
apply (erule_tac x=i in allE)
unfolding real_scaleR_def
apply (rule mult_left_mono)
using assms[unfolded convex] apply auto
done
subsection%unimportant ‹Convexity of general and special intervals›
lemma is_interval_convex:
fixes S :: "'a::euclidean_space set"
assumes "is_interval S"
shows "convex S"
proof (rule convexI)
fix x y and u v :: real
assume as: "x ∈ S" "y ∈ S" "0 ≤ u" "0 ≤ v" "u + v = 1"
then have *: "u = 1 - v" "1 - v ≥ 0" and **: "v = 1 - u" "1 - u ≥ 0"
by auto
{
fix a b
assume "¬ b ≤ u * a + v * b"
then have "u * a < (1 - v) * b"
unfolding not_le using as(4) by (auto simp: field_simps)
then have "a < b"
unfolding * using as(4) *(2)
apply (rule_tac mult_left_less_imp_less[of "1 - v"])
apply (auto simp: field_simps)
done
then have "a ≤ u * a + v * b"
unfolding * using as(4)
by (auto simp: field_simps intro!:mult_right_mono)
}
moreover
{
fix a b
assume "¬ u * a + v * b ≤ a"
then have "v * b > (1 - u) * a"
unfolding not_le using as(4) by (auto simp: field_simps)
then have "a < b"
unfolding * using as(4)
apply (rule_tac mult_left_less_imp_less)
apply (auto simp: field_simps)
done
then have "u * a + v * b ≤ b"
unfolding **
using **(2) as(3)
by (auto simp: field_simps intro!:mult_right_mono)
}
ultimately show "u *⇩R x + v *⇩R y ∈ S"
apply -
apply (rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
using as(3-) DIM_positive[where 'a='a]
apply (auto simp: inner_simps)
done
qed
lemma is_interval_connected:
fixes S :: "'a::euclidean_space set"
shows "is_interval S ⟹ connected S"
using is_interval_convex convex_connected by auto
lemma convex_box [simp]: "convex (cbox a b)" "convex (box a (b::'a::euclidean_space))"
apply (rule_tac[!] is_interval_convex)+
using is_interval_box is_interval_cbox
apply auto
done
text‹A non-singleton connected set is perfect (i.e. has no isolated points). ›
lemma connected_imp_perfect:
fixes a :: "'a::metric_space"
assumes "connected S" "a ∈ S" and S: "⋀x. S ≠ {x}"
shows "a islimpt S"
proof -
have False if "a ∈ T" "open T" "⋀y. ⟦y ∈ S; y ∈ T⟧ ⟹ y = a" for T
proof -
obtain e where "e > 0" and e: "cball a e ⊆ T"
using ‹open T› ‹a ∈ T› by (auto simp: open_contains_cball)
have "openin (subtopology euclidean S) {a}"
unfolding openin_open using that ‹a ∈ S› by blast
moreover have "closedin (subtopology euclidean S) {a}"
by (simp add: assms)
ultimately show "False"
using ‹connected S› connected_clopen S by blast
qed
then show ?thesis
unfolding islimpt_def by blast
qed
lemma connected_imp_perfect_aff_dim:
"⟦connected S; aff_dim S ≠ 0; a ∈ S⟧ ⟹ a islimpt S"
using aff_dim_sing connected_imp_perfect by blast
subsection%unimportant ‹On ‹real›, ‹is_interval›, ‹convex› and ‹connected› are all equivalent›
lemma mem_is_interval_1_I:
fixes a b c::real
assumes "is_interval S"
assumes "a ∈ S" "c ∈ S"
assumes "a ≤ b" "b ≤ c"
shows "b ∈ S"
using assms is_interval_1 by blast
lemma is_interval_connected_1:
fixes s :: "real set"
shows "is_interval s ⟷ connected s"
apply rule
apply (rule is_interval_connected, assumption)
unfolding is_interval_1
apply rule
apply rule
apply rule
apply rule
apply (erule conjE)
apply (rule ccontr)
proof -
fix a b x
assume as: "connected s" "a ∈ s" "b ∈ s" "a ≤ x" "x ≤ b" "x ∉ s"
then have *: "a < x" "x < b"
unfolding not_le [symmetric] by auto
let ?halfl = "{..<x} "
let ?halfr = "{x<..}"
{
fix y
assume "y ∈ s"
with ‹x ∉ s› have "x ≠ y" by auto
then have "y ∈ ?halfr ∪ ?halfl" by auto
}
moreover have "a ∈ ?halfl" "b ∈ ?halfr" using * by auto
then have "?halfl ∩ s ≠ {}" "?halfr ∩ s ≠ {}"
using as(2-3) by auto
ultimately show False
apply (rule_tac notE[OF as(1)[unfolded connected_def]])
apply (rule_tac x = ?halfl in exI)
apply (rule_tac x = ?halfr in exI, rule)
apply (rule open_lessThan, rule)
apply (rule open_greaterThan, auto)
done
qed
lemma is_interval_convex_1:
fixes s :: "real set"
shows "is_interval s ⟷ convex s"
by (metis is_interval_convex convex_connected is_interval_connected_1)
lemma is_interval_ball_real: "is_interval (ball a b)" for a b::real
by (metis connected_ball is_interval_connected_1)
lemma connected_compact_interval_1:
"connected S ∧ compact S ⟷ (∃a b. S = {a..b::real})"
by (auto simp: is_interval_connected_1 [symmetric] is_interval_compact)
lemma connected_convex_1:
fixes s :: "real set"
shows "connected s ⟷ convex s"
by (metis is_interval_convex convex_connected is_interval_connected_1)
lemma connected_convex_1_gen:
fixes s :: "'a :: euclidean_space set"
assumes "DIM('a) = 1"
shows "connected s ⟷ convex s"
proof -
obtain f:: "'a ⇒ real" where linf: "linear f" and "inj f"
using subspace_isomorphism[OF subspace_UNIV subspace_UNIV,
where 'a='a and 'b=real]
unfolding Euclidean_Space.dim_UNIV
by (auto simp: assms)
then have "f -` (f ` s) = s"
by (simp add: inj_vimage_image_eq)
then show ?thesis
by (metis connected_convex_1 convex_linear_vimage linf convex_connected connected_linear_image)
qed
lemma is_interval_cball_1[intro, simp]: "is_interval (cball a b)" for a b::real
by (simp add: is_interval_convex_1)
subsection%unimportant ‹Another intermediate value theorem formulation›
lemma ivt_increasing_component_on_1:
fixes f :: "real ⇒ 'a::euclidean_space"
assumes "a ≤ b"
and "continuous_on {a..b} f"
and "(f a)∙k ≤ y" "y ≤ (f b)∙k"
shows "∃x∈{a..b}. (f x)∙k = y"
proof -
have "f a ∈ f ` cbox a b" "f b ∈ f ` cbox a b"
apply (rule_tac[!] imageI)
using assms(1)
apply auto
done
then show ?thesis
using connected_ivt_component[of "f ` cbox a b" "f a" "f b" k y]
by (simp add: connected_continuous_image assms)
qed
lemma ivt_increasing_component_1:
fixes f :: "real ⇒ 'a::euclidean_space"
shows "a ≤ b ⟹ ∀x∈{a..b}. continuous (at x) f ⟹
f a∙k ≤ y ⟹ y ≤ f b∙k ⟹ ∃x∈{a..b}. (f x)∙k = y"
by (rule ivt_increasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
lemma ivt_decreasing_component_on_1:
fixes f :: "real ⇒ 'a::euclidean_space"
assumes "a ≤ b"
and "continuous_on {a..b} f"
and "(f b)∙k ≤ y"
and "y ≤ (f a)∙k"
shows "∃x∈{a..b}. (f x)∙k = y"
apply (subst neg_equal_iff_equal[symmetric])
using ivt_increasing_component_on_1[of a b "λx. - f x" k "- y"]
using assms using continuous_on_minus
apply auto
done
lemma ivt_decreasing_component_1:
fixes f :: "real ⇒ 'a::euclidean_space"
shows "a ≤ b ⟹ ∀x∈{a..b}. continuous (at x) f ⟹
f b∙k ≤ y ⟹ y ≤ f a∙k ⟹ ∃x∈{a..b}. (f x)∙k = y"
by (rule ivt_decreasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
subsection%unimportant ‹A bound within a convex hull, and so an interval›
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
obt: "∀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] obt(2) mult_1
apply (drule_tac meta_mp)
apply (rule mult_left_mono)
using assms(2) obt(1)
apply auto
done
then show "f x ≤ b"
using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
unfolding obt(2-3)
using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
by auto
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:
shows "{x. ∀i∈Basis. x∙i ∈ B i} = (∑i∈Basis. image (λx. x *⇩R i) (B i))"
apply (subst set_sum_alt [OF finite_Basis], safe)
apply (fast intro: euclidean_representation [symmetric])
apply (subst inner_sum_left)
apply (rename_tac f)
apply (subgoal_tac "(∑x∈Basis. f x ∙ i) = f i ∙ i")
apply (drule (1) bspec)
apply clarsimp
apply (frule sum.remove [OF finite_Basis])
apply (erule trans, simp)
apply (rule sum.neutral, clarsimp)
apply (frule_tac x=i in bspec, assumption)
apply (drule_tac x=x in bspec, assumption, clarsimp)
apply (cut_tac u=x and v=i in inner_Basis, assumption+)
apply (rule ccontr, simp)
done
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
lemma convex_hull_eq_real_cbox:
fixes x y :: real assumes "x ≤ y"
shows "convex hull {x, y} = cbox x y"
proof (rule hull_unique)
show "{x, y} ⊆ cbox x y" using ‹x ≤ y› by auto
show "convex (cbox x y)"
by (rule convex_box)
next
fix S assume "{x, y} ⊆ S" and "convex S"
then show "cbox x y ⊆ S"
unfolding is_interval_convex_1 [symmetric] is_interval_def Basis_real_def
by - (clarify, simp (no_asm_use), fast)
qed
lemma unit_interval_convex_hull:
"cbox (0::'a::euclidean_space) One = convex hull {x. ∀i∈Basis. (x∙i = 0) ∨ (x∙i = 1)}"
(is "?int = convex hull ?points")
proof -
have One[simp]: "⋀i. i ∈ Basis ⟹ One ∙ i = 1"
by (simp add: inner_sum_left sum.If_cases inner_Basis)
have "?int = {x. ∀i∈Basis. x ∙ i ∈ cbox 0 1}"
by (auto simp: cbox_def)
also have "… = (∑i∈Basis. (λx. x *⇩R i) ` cbox 0 1)"
by (simp only: box_eq_set_sum_Basis)
also have "… = (∑i∈Basis. (λx. x *⇩R i) ` (convex hull {0, 1}))"
by (simp only: convex_hull_eq_real_cbox zero_le_one)
also have "… = (∑i∈Basis. convex hull ((λx. x *⇩R i) ` {0, 1}))"
by (simp add: convex_hull_linear_image)
also have "… = convex hull (∑i∈Basis. (λx. x *⇩R i) ` {0, 1})"
by (simp only: convex_hull_set_sum)
also have "… = convex hull {x. ∀i∈Basis. x∙i ∈ {0, 1}}"
by (simp only: box_eq_set_sum_Basis)
also have "convex hull {x. ∀i∈Basis. x∙i ∈ {0, 1}} = convex hull ?points"
by simp
finally show ?thesis .
qed
text ‹And this is a finite set of vertices.›
lemma unit_cube_convex_hull:
obtains S :: "'a::euclidean_space set"
where "finite S" and "cbox 0 (∑Basis) = convex hull S"
proof
show "finite {x::'a. ∀i∈Basis. x ∙ i = 0 ∨ x ∙ i = 1}"
proof (rule finite_subset, clarify)
show "finite ((λS. ∑i∈Basis. (if i ∈ S then 1 else 0) *⇩R i) ` Pow Basis)"
using finite_Basis by blast
fix x :: 'a
assume as: "∀i∈Basis. x ∙ i = 0 ∨ x ∙ i = 1"
show "x ∈ (λS. ∑i∈Basis. (if i∈S then 1 else 0) *⇩R i) ` Pow Basis"
apply (rule image_eqI[where x="{i. i∈Basis ∧ x∙i = 1}"])
using as
apply (subst euclidean_eq_iff, auto)
done
qed
show "cbox 0 One = convex hull {x. ∀i∈Basis. x ∙ i = 0 ∨ x ∙ i = 1}"
using unit_interval_convex_hull by blast
qed
text ‹Hence any cube (could do any nonempty interval).›
lemma cube_convex_hull:
assumes "d > 0"
obtains S :: "'a::euclidean_space set" where
"finite S" and "cbox (x - (∑i∈Basis. d*⇩Ri)) (x + (∑i∈Basis. d*⇩Ri)) = convex hull S"
proof -
let ?d = "(∑i∈Basis. d *⇩R i)::'a"
have *: "cbox (x - ?d) (x + ?d) = (λy. x - ?d + (2 * d) *⇩R y) ` cbox 0 (∑Basis)"
proof (intro set_eqI iffI)
fix y
assume "y ∈ cbox (x - ?d) (x + ?d)"
then have "inverse (2 * d) *⇩R (y - (x - ?d)) ∈ cbox 0 (∑Basis)"
using assms by (simp add: mem_box field_simps inner_simps)
with ‹0 < d› show "y ∈ (λy. x - sum (( *⇩R) d) Basis + (2 * d) *⇩R y) ` cbox 0 One"
by (auto intro: image_eqI[where x= "inverse (2 * d) *⇩R (y - (x - ?d))"])
next
fix y
assume "y ∈ (λy. x - ?d + (2 * d) *⇩R y) ` cbox 0 One"
then obtain z where z: "z ∈ cbox 0 One" "y = x - ?d + (2*d) *⇩R z"
by auto
then show "y ∈ cbox (x - ?d) (x + ?d)"
using z assms by (auto simp: mem_box inner_simps)
qed
obtain S where "finite S" "cbox 0 (∑Basis::'a) = convex hull S"
using unit_cube_convex_hull by auto
then show ?thesis
by (rule_tac that[of "(λy. x - ?d + (2 * d) *⇩R y)` S"]) (auto simp: convex_hull_affinity *)
qed
subsection%unimportant‹Representation of any interval as a finite convex hull›
lemma image_stretch_interval:
"(λx. ∑k∈Basis. (m k * (x∙k)) *⇩R k) ` cbox a (b::'a::euclidean_space) =
(if (cbox a b) = {} then {} else
cbox (∑k∈Basis. (min (m k * (a∙k)) (m k * (b∙k))) *⇩R k::'a)
(∑k∈Basis. (max (m k * (a∙k)) (m k * (b∙k))) *⇩R k))"
proof cases
assume *: "cbox a b ≠ {}"
show ?thesis
unfolding box_ne_empty if_not_P[OF *]
apply (simp add: cbox_def image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
apply (subst choice_Basis_iff[symmetric])
proof (intro allI ball_cong refl)
fix x i :: 'a assume "i ∈ Basis"
with * have a_le_b: "a ∙ i ≤ b ∙ i"
unfolding box_ne_empty by auto
show "(∃xa. x ∙ i = m i * xa ∧ a ∙ i ≤ xa ∧ xa ≤ b ∙ i) ⟷
min (m i * (a ∙ i)) (m i * (b ∙ i)) ≤ x ∙ i ∧ x ∙ i ≤ max (m i * (a ∙ i)) (m i * (b ∙ i))"
proof (cases "m i = 0")
case True
with a_le_b show ?thesis by auto
next
case False
then have *: "⋀a b. a = m i * b ⟷ b = a / m i"
by (auto simp: field_simps)
from False have
"min (m i * (a ∙ i)) (m i * (b ∙ i)) = (if 0 < m i then m i * (a ∙ i) else m i * (b ∙ i))"
"max (m i * (a ∙ i)) (m i * (b ∙ i)) = (if 0 < m i then m i * (b ∙ i) else m i * (a ∙ i))"
using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
with False show ?thesis using a_le_b
unfolding * by (auto simp: le_divide_eq divide_le_eq ac_simps)
qed
qed
qed simp
lemma interval_image_stretch_interval:
"∃u v. (λx. ∑k∈Basis. (m k * (x∙k))*⇩R k) ` cbox a (b::'a::euclidean_space) = cbox u (v::'a::euclidean_space)"
unfolding image_stretch_interval by auto
lemma cbox_translation: "cbox (c + a) (c + b) = image (λx. c + x) (cbox a b)"
using image_affinity_cbox [of 1 c a b]
using box_ne_empty [of "a+c" "b+c"] box_ne_empty [of a b]
by (auto simp: inner_left_distrib add.commute)
lemma cbox_image_unit_interval:
fixes a :: "'a::euclidean_space"
assumes "cbox a b ≠ {}"
shows "cbox a b =
(+) a ` (λx. ∑k∈Basis. ((b ∙ k - a ∙ k) * (x ∙ k)) *⇩R k) ` cbox 0 One"
using assms
apply (simp add: box_ne_empty image_stretch_interval cbox_translation [symmetric])
apply (simp add: min_def max_def algebra_simps sum_subtractf euclidean_representation)
done
lemma closed_interval_as_convex_hull:
fixes a :: "'a::euclidean_space"
obtains S where "finite S" "cbox a b = convex hull S"
proof (cases "cbox a b = {}")
case True with convex_hull_empty that show ?thesis
by blast
next
case False
obtain S::"'a set" where "finite S" and eq: "cbox 0 One = convex hull S"
by (blast intro: unit_cube_convex_hull)
have lin: "linear (λx. ∑k∈Basis. ((b ∙ k - a ∙ k) * (x ∙ k)) *⇩R k)"
by (rule linear_compose_sum) (auto simp: algebra_simps linearI)
have "finite ((+) a ` (λx. ∑k∈Basis. ((b ∙ k - a ∙ k) * (x ∙ k)) *⇩R k) ` S)"
by (rule finite_imageI ‹finite S›)+
then show ?thesis
apply (rule that)
apply (simp add: convex_hull_translation convex_hull_linear_image [OF lin, symmetric])
apply (simp add: eq [symmetric] cbox_image_unit_interval [OF False])
done
qed
subsection%unimportant ‹Bounded convex function on open set is continuous›
lemma convex_on_bounded_continuous:
fixes S :: "('a::real_normed_vector) set"
assumes "open S"
and "convex_on S f"
and "∀x∈S. ¦f x¦ ≤ b"
shows "continuous_on S f"
apply (rule continuous_at_imp_continuous_on)
unfolding continuous_at_real_range
proof (rule,rule,rule)
fix x and e :: real
assume "x ∈ S" "e > 0"
define B where "B = ¦b¦ + 1"
then have B: "0 < B""⋀x. x∈S ⟹ ¦f x¦ ≤ B"
using assms(3) by auto
obtain k where "k > 0" and k: "cball x k ⊆ S"
using ‹x ∈ S› assms(1) open_contains_cball_eq by blast
show "∃d>0. ∀x'. norm (x' - x) < d ⟶ ¦f x' - f x¦ < e"
proof (intro exI conjI allI impI)
fix y
assume as: "norm (y - x) < min (k / 2) (e / (2 * B) * k)"
show "¦f y - f x¦ < e"
proof (cases "y = x")
case False
define t where "t = k / norm (y - x)"
have "2 < t" "0<t"
unfolding t_def using as False and ‹k>0›
by (auto simp:field_simps)
have "y ∈ S"
apply (rule k[THEN subsetD])
unfolding mem_cball dist_norm
apply (rule order_trans[of _ "2 * norm (x - y)"])
using as
by (auto simp: field_simps norm_minus_commute)
{
define w where "w = x + t *⇩R (y - x)"
have "w ∈ S"
using ‹k>0› by (auto simp: dist_norm t_def w_def k[THEN subsetD])
have "(1 / t) *⇩R x + - x + ((t - 1) / t) *⇩R x = (1 / t - 1 + (t - 1) / t) *⇩R x"
by (auto simp: algebra_simps)
also have "… = 0"
using ‹t > 0› by (auto simp:field_simps)
finally have w: "(1 / t) *⇩R w + ((t - 1) / t) *⇩R x = y"
unfolding w_def using False and ‹t > 0›
by (auto simp: algebra_simps)
have 2: "2 * B < e * t"
unfolding t_def using ‹0 < e› ‹0 < k› ‹B > 0› and as and False
by (auto simp:field_simps)
have "f y - f x ≤ (f w - f x) / t"
using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
using ‹0 < t› ‹2 < t› and ‹x ∈ S› ‹w ∈ S›
by (auto simp:field_simps)
also have "... < e"
using B(2)[OF ‹w∈S›] and B(2)[OF ‹x∈S›] 2 ‹t > 0› by (auto simp: field_simps)
finally have th1: "f y - f x < e" .
}
moreover
{
define w where "w = x - t *⇩R (y - x)"
have "w ∈ S"
using ‹k > 0› by (auto simp: dist_norm t_def w_def k[THEN subsetD])
have "(1 / (1 + t)) *⇩R x + (t / (1 + t)) *⇩R x = (1 / (1 + t) + t / (1 + t)) *⇩R x"
by (auto simp: algebra_simps)
also have "… = x"
using ‹t > 0› by (auto simp:field_simps)
finally have w: "(1 / (1+t)) *⇩R w + (t / (1 + t)) *⇩R y = x"
unfolding w_def using False and ‹t > 0›
by (auto simp: algebra_simps)
have "2 * B < e * t"
unfolding t_def
using ‹0 < e› ‹0 < k› ‹B > 0› and as and False
by (auto simp:field_simps)
then have *: "(f w - f y) / t < e"
using B(2)[OF ‹w∈S›] and B(2)[OF ‹y∈S›]
using ‹t > 0›
by (auto simp:field_simps)
have "f x ≤ 1 / (1 + t) * f w + (t / (1 + t)) * f y"
using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
using ‹0 < t› ‹2 < t› and ‹y ∈ S› ‹w ∈ S›
by (auto simp:field_simps)
also have "… = (f w + t * f y) / (1 + t)"
using ‹t > 0› by (auto simp: divide_simps)
also have "… < e + f y"
using ‹t > 0› * ‹e > 0› by (auto simp: field_simps)
finally have "f x - f y < e" by auto
}
ultimately show ?thesis by auto
qed (insert ‹0<e›, auto)
qed (insert ‹0<e› ‹0<k› ‹0<B›, auto simp: field_simps)
qed
subsection%unimportant ‹Upper bound on a ball implies upper and lower bounds›
lemma convex_bounds_lemma:
fixes x :: "'a::real_normed_vector"
assumes "convex_on (cball x e) f"
and "∀y ∈ cball x e. f y ≤ b"
shows "∀y ∈ cball x e. ¦f y¦ ≤ b + 2 * ¦f x¦"
apply rule
proof (cases "0 ≤ e")
case True
fix y
assume y: "y ∈ cball x e"
define z where "z = 2 *⇩R x - y"
have *: "x - (2 *⇩R x - y) = y - x"
by (simp add: scaleR_2)
have z: "z ∈ cball x e"
using y unfolding z_def mem_cball dist_norm * by (auto simp: norm_minus_commute)
have "(1 / 2) *⇩R y + (1 / 2) *⇩R z = x"
unfolding z_def by (auto simp: algebra_simps)
then show "¦f y¦ ≤ b + 2 * ¦f x¦"
using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z]
by (auto simp:field_simps)
next
case False
fix y
assume "y ∈ cball x e"
then have "dist x y < 0"
using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
then show "¦f y¦ ≤ b + 2 * ¦f x¦"
using zero_le_dist[of x y] by auto
qed
subsubsection%unimportant ‹Hence a convex function on an open set is continuous›
lemma real_of_nat_ge_one_iff: "1 ≤ real (n::nat) ⟷ 1 ≤ n"
by auto
lemma convex_on_continuous:
assumes "open (s::('a::euclidean_space) set)" "convex_on s f"
shows "continuous_on s f"
unfolding continuous_on_eq_continuous_at[OF assms(1)]
proof
note dimge1 = DIM_positive[where 'a='a]
fix x
assume "x ∈ s"
then obtain e where e: "cball x e ⊆ s" "e > 0"
using assms(1) unfolding open_contains_cball by auto
define d where "d = e / real DIM('a)"
have "0 < d"
unfolding d_def using ‹e > 0› dimge1 by auto
let ?d = "(∑i∈Basis. d *⇩R i)::'a"
obtain c
where c: "finite c"
and c1: "convex hull c ⊆ cball x e"
and c2: "cball x d ⊆ convex hull c"
proof
define c where "c = (∑i∈Basis. (λa. a *⇩R i) ` {x∙i - d, x∙i + d})"
show "finite c"
unfolding c_def by (simp add: finite_set_sum)
have 1: "convex hull c = {a. ∀i∈Basis. a ∙ i ∈ cbox (x ∙ i - d) (x ∙ i + d)}"
unfolding box_eq_set_sum_Basis
unfolding c_def convex_hull_set_sum
apply (subst convex_hull_linear_image [symmetric])
apply (simp add: linear_iff scaleR_add_left)
apply (rule sum.cong [OF refl])
apply (rule image_cong [OF _ refl])
apply (rule convex_hull_eq_real_cbox)
apply (cut_tac ‹0 < d›, simp)
done
then have 2: "convex hull c = {a. ∀i∈Basis. a ∙ i ∈ cball (x ∙ i) d}"
by (simp add: dist_norm abs_le_iff algebra_simps)
show "cball x d ⊆ convex hull c"
unfolding 2
by (clarsimp simp: dist_norm) (metis inner_commute inner_diff_right norm_bound_Basis_le)
have e': "e = (∑(i::'a)∈Basis. d)"
by (simp add: d_def DIM_positive)
show "convex hull c ⊆ cball x e"
unfolding 2
apply clarsimp
apply (subst euclidean_dist_l2)
apply (rule order_trans [OF L2_set_le_sum])
apply (rule zero_le_dist)
unfolding e'
apply (rule sum_mono, simp)
done
qed
define k where "k = Max (f ` c)"
have "convex_on (convex hull c) f"
apply(rule convex_on_subset[OF assms(2)])
apply(rule subset_trans[OF c1 e(1)])
done
then have k: "∀y∈convex hull c. f y ≤ k"
apply (rule_tac convex_on_convex_hull_bound, assumption)
by (simp add: k_def c)
have "e ≤ e * real DIM('a)"
using e(2) real_of_nat_ge_one_iff by auto
then have "d ≤ e"
by (simp add: d_def divide_simps)
then have dsube: "cball x d ⊆ cball x e"
by (rule subset_cball)
have conv: "convex_on (cball x d) f"
using ‹convex_on (convex hull c) f› c2 convex_on_subset by blast
then have "∀y∈cball x d. ¦f y¦ ≤ k + 2 * ¦f x¦"
by (rule convex_bounds_lemma) (use c2 k in blast)
then have "continuous_on (ball x d) f"
apply (rule_tac convex_on_bounded_continuous)
apply (rule open_ball, rule convex_on_subset[OF conv])
apply (rule ball_subset_cball, force)
done
then show "continuous (at x) f"
unfolding continuous_on_eq_continuous_at[OF open_ball]
using ‹d > 0› by auto
qed
end