# Theory Convex_Euclidean_Space

theory Convex_Euclidean_Space
imports Topology_Euclidean_Space Convex Set_Algebras
```(*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
Author:     Robert Himmelmann, TU Muenchen
Author:     Bogdan Grechuk, University of Edinburgh
*)

section ‹Convex sets, functions and related things.›

theory Convex_Euclidean_Space
imports
Topology_Euclidean_Space
"~~/src/HOL/Library/Convex"
"~~/src/HOL/Library/Set_Algebras"
begin

lemma independent_injective_on_span_image:
assumes iS: "independent S"
and lf: "linear f"
and fi: "inj_on f (span S)"
shows "independent (f ` S)"
proof -
{
fix a
assume a: "a ∈ S" "f a ∈ span (f ` S - {f a})"
have eq: "f ` S - {f a} = f ` (S - {a})"
using fi a span_inc by (auto simp add: inj_on_def)
from a have "f a ∈ f ` span (S -{a})"
unfolding eq span_linear_image [OF lf, of "S - {a}"] by blast
moreover have "span (S - {a}) ⊆ span S"
using span_mono[of "S - {a}" S] by auto
ultimately have "a ∈ span (S - {a})"
using fi a span_inc by (auto simp add: inj_on_def)
with a(1) iS have False
}
then show ?thesis
unfolding dependent_def by blast
qed

lemma dim_image_eq:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes lf: "linear f"
and fi: "inj_on f (span S)"
shows "dim (f ` S) = dim (S::'n::euclidean_space set)"
proof -
obtain B where B: "B ⊆ S" "independent B" "S ⊆ span B" "card B = dim S"
using basis_exists[of S] by auto
then have "span S = span B"
using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
then have "independent (f ` B)"
using independent_injective_on_span_image[of B f] B assms by auto
moreover have "card (f ` B) = card B"
using assms card_image[of f B] subset_inj_on[of f "span S" B] B span_inc by auto
moreover have "(f ` B) ⊆ (f ` S)"
using B by auto
ultimately have "dim (f ` S) ≥ dim S"
using independent_card_le_dim[of "f ` B" "f ` S"] B by auto
then show ?thesis
using dim_image_le[of f S] assms by auto
qed

lemma linear_injective_on_subspace_0:
assumes lf: "linear f"
and "subspace S"
shows "inj_on f S ⟷ (∀x ∈ S. f x = 0 ⟶ x = 0)"
proof -
have "inj_on f S ⟷ (∀x ∈ S. ∀y ∈ S. f x = f y ⟶ x = y)"
also have "… ⟷ (∀x ∈ S. ∀y ∈ S. f x - f y = 0 ⟶ x - y = 0)"
by simp
also have "… ⟷ (∀x ∈ S. ∀y ∈ S. f (x - y) = 0 ⟶ x - y = 0)"
also have "… ⟷ (∀x ∈ S. f x = 0 ⟶ x = 0)"
using ‹subspace S› subspace_def[of S] subspace_sub[of S] by auto
finally show ?thesis .
qed

lemma subspace_Inter: "∀s ∈ f. subspace s ⟹ subspace (⋂f)"
unfolding subspace_def by auto

lemma span_eq[simp]: "span s = s ⟷ subspace s"
unfolding span_def by (rule hull_eq) (rule subspace_Inter)

lemma 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!: setsum.cong simp: inner_Basis inner_setsum_left)
show ?thesis
unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum.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"
def y ≡ "(e / norm x) *⇩R x"
then have "y ∈ cball 0 e"
using cball_def dist_norm[of 0 y] assms by auto
moreover have *: "x = (norm x / e) *⇩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_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
}
then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
by auto
then show ?thesis
using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
qed

lemma indep_card_eq_dim_span:
fixes B :: "'n::euclidean_space set"
assumes "independent B"
shows "finite B ∧ card B = dim (span B)"
using assms basis_card_eq_dim[of B "span B"] span_inc by auto

lemma setsum_not_0: "setsum f A ≠ 0 ⟹ ∃a ∈ A. f a ≠ 0"
by (rule ccontr) auto

lemma subset_translation_eq [simp]:
fixes a :: "'a::real_vector" shows "op + a ` s ⊆ op + a ` t ⟷ s ⊆ t"
by auto

lemma translate_inj_on:
shows "inj_on (λx. a + x) A"
unfolding inj_on_def by auto

lemma translation_assoc:
shows "(λx. b + x) ` ((λx. a + x) ` S) = (λx. (a + b) + x) ` S"
by auto

lemma translation_invert:
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:
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 convex_translation_eq [simp]: "convex ((λx. a + x) ` s) ⟷ convex s"
by (metis convex_translation translation_galois)

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"])
apply simp
done
then have "x ∈ ((λx. a+x) ` S)" by auto
}
then show ?thesis by auto
qed

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 basis_to_basis_subspace_isomorphism:
assumes s: "subspace (S:: ('n::euclidean_space) set)"
and t: "subspace (T :: ('m::euclidean_space) set)"
and d: "dim S = dim T"
and B: "B ⊆ S" "independent B" "S ⊆ span B" "card B = dim S"
and C: "C ⊆ T" "independent C" "T ⊆ span C" "card C = dim T"
shows "∃f. linear f ∧ f ` B = C ∧ f ` S = T ∧ inj_on f S"
proof -
from B independent_bound have fB: "finite B"
by blast
from C independent_bound have fC: "finite C"
by blast
from B(4) C(4) card_le_inj[of B C] d obtain f where
f: "f ` B ⊆ C" "inj_on f B" using ‹finite B› ‹finite C› by auto
from linear_independent_extend[OF B(2)] obtain g where
g: "linear g" "∀x ∈ B. g x = f x" by blast
from inj_on_iff_eq_card[OF fB, of f] f(2)
have "card (f ` B) = card B" by simp
with B(4) C(4) have ceq: "card (f ` B) = card C" using d
by simp
have "g ` B = f ` B" using g(2)
also have "… = C" using card_subset_eq[OF fC f(1) ceq] .
finally have gBC: "g ` B = C" .
have gi: "inj_on g B" using f(2) g(2)
note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
{
fix x y
assume x: "x ∈ S" and y: "y ∈ S" and gxy: "g x = g y"
from B(3) x y have x': "x ∈ span B" and y': "y ∈ span B"
by blast+
from gxy have th0: "g (x - y) = 0"
have th1: "x - y ∈ span B" using x' y'
by (metis span_sub)
have "x = y" using g0[OF th1 th0] by simp
}
then have giS: "inj_on g S" unfolding inj_on_def by blast
from span_subspace[OF B(1,3) s]
have "g ` S = span (g ` B)"
also have "… = span C"
unfolding gBC ..
also have "… = T"
using span_subspace[OF C(1,3) t] .
finally have gS: "g ` S = T" .
from g(1) gS giS gBC show ?thesis
by blast
qed

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]: "{x ∈ range f. g x ∈ S} = f ` S"
using g by (simp add: o_def id_def image_def) metis
show ?thesis
apply (rule closedin_closed_trans [of "range f"])
apply (rule continuous_closedin_preimage [OF confg cgf, simplified])
apply (rule closed_injective_image_subspace)
using f
apply (auto simp: linear_linear linear_injective_0)
done
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)
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 "(op *⇩R c) ` (closure S) = closure ((op *⇩R c) ` S)"
proof
show "(op *⇩R c) ` (closure S) ⊆ closure ((op *⇩R c) ` S)"
using bounded_linear_scaleR_right
by (rule closure_bounded_linear_image_subset)
show "closure ((op *⇩R c) ` S) ⊆ (op *⇩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 scaleR_2:
fixes x :: "'a::real_vector"
shows "scaleR 2 x = x + x"
unfolding one_add_one [symmetric] scaleR_left_distrib by simp

lemma vector_choose_size:
"0 ≤ c ⟹ ∃x::'a::euclidean_space. norm x = c"
apply (rule exI [where x="c *⇩R (SOME i. i ∈ Basis)"])
apply (auto simp: SOME_Basis)
done

lemma setsum_delta_notmem:
assumes "x ∉ s"
shows "setsum (λy. if (y = x) then P x else Q y) s = setsum Q s"
and "setsum (λy. if (x = y) then P x else Q y) s = setsum Q s"
and "setsum (λy. if (y = x) then P y else Q y) s = setsum Q s"
and "setsum (λy. if (x = y) then P y else Q y) s = setsum Q s"
apply (rule_tac [!] setsum.cong)
using assms
apply auto
done

lemma setsum_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 setsum.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 *]
qed

lemma norm_minus_eqI: "x = - y ⟹ norm x = norm y" by auto

lemma Min_grI:
assumes "finite A" "A ≠ {}" "∀a∈A. x < a"
shows "x < Min A"
unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto

lemma norm_lt: "norm x < norm y ⟷ inner x x < inner y y"
unfolding norm_eq_sqrt_inner by simp

lemma norm_le: "norm x ≤ norm y ⟷ inner x x ≤ inner y y"
unfolding norm_eq_sqrt_inner by simp

subsection ‹Affine set and affine hull›

definition 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[intro]: "affine {}"
unfolding affine_def by auto

lemma affine_sing[intro]: "affine {x}"
unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])

lemma affine_UNIV[intro]: "affine UNIV"
unfolding affine_def by auto

lemma affine_Inter[intro]: "(∀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_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)

subsubsection ‹Some explicit formulations (from Lars Schewe)›

lemma affine:
fixes V::"'a::real_vector set"
shows "affine V ⟷
(∀s u. finite s ∧ s ≠ {} ∧ s ⊆ V ∧ setsum u s = 1 ⟶ (setsum (λx. (u x) *⇩R x)) s ∈ V)"
unfolding affine_def
apply rule
apply(rule, rule, rule)
apply(erule conjE)+
defer
apply (rule, rule, rule, rule, rule)
proof -
fix x y u v
assume as: "x ∈ V" "y ∈ V" "u + v = (1::real)"
"∀s u. finite s ∧ s ≠ {} ∧ s ⊆ V ∧ setsum u s = 1 ⟶ (∑x∈s. u x *⇩R x) ∈ V"
then show "u *⇩R x + v *⇩R y ∈ V"
apply (cases "x = y")
using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="λw. if w = x then u else v"]]
and as(1-3)
done
next
fix s u
assume as: "∀x∈V. ∀y∈V. ∀u v. u + v = 1 ⟶ u *⇩R x + v *⇩R y ∈ V"
"finite s" "s ≠ {}" "s ⊆ V" "setsum u s = (1::real)"
def n ≡ "card s"
have "card s = 0 ∨ card s = 1 ∨ card s = 2 ∨ card s > 2" by auto
then show "(∑x∈s. u x *⇩R x) ∈ V"
proof (auto simp only: disjE)
assume "card s = 2"
then have "card s = Suc (Suc 0)"
by auto
then obtain a b where "s = {a, b}"
unfolding card_Suc_eq by auto
then show ?thesis
using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
next
assume "card s > 2"
then show ?thesis using as and n_def
proof (induct n arbitrary: u s)
case 0
then show ?case by auto
next
case (Suc n)
fix s :: "'a set" and u :: "'a ⇒ real"
assume IA:
"⋀u s.  ⟦2 < card s; ∀x∈V. ∀y∈V. ∀u v. u + v = 1 ⟶ u *⇩R x + v *⇩R y ∈ V; finite s;
s ≠ {}; s ⊆ V; setsum u s = 1; n = card s ⟧ ⟹ (∑x∈s. u x *⇩R x) ∈ V"
and as:
"Suc n = card s" "2 < card s" "∀x∈V. ∀y∈V. ∀u v. u + v = 1 ⟶ u *⇩R x + v *⇩R y ∈ V"
"finite s" "s ≠ {}" "s ⊆ V" "setsum u s = 1"
have "∃x∈s. u x ≠ 1"
proof (rule ccontr)
assume "¬ ?thesis"
then have "setsum u s = real_of_nat (card s)"
unfolding card_eq_setsum by auto
then show False
using as(7) and ‹card s > 2›
by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
qed
then obtain x where x:"x ∈ s" "u x ≠ 1" by auto

have c: "card (s - {x}) = card s - 1"
apply (rule card_Diff_singleton)
using ‹x∈s› as(4)
apply auto
done
have *: "s = insert x (s - {x})" "finite (s - {x})"
using ‹x∈s› and as(4) by auto
have **: "setsum u (s - {x}) = 1 - u x"
using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"
unfolding ** using ‹u x ≠ 1› by auto
have "(∑xa∈s - {x}. (inverse (1 - u x) * u xa) *⇩R xa) ∈ V"
proof (cases "card (s - {x}) > 2")
case True
then have "s - {x} ≠ {}" "card (s - {x}) = n"
unfolding c and as(1)[symmetric]
proof (rule_tac ccontr)
assume "¬ s - {x} ≠ {}"
then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
then show False using True by auto
qed auto
then show ?thesis
apply (rule_tac IA[of "s - {x}" "λy. (inverse (1 - u x) * u y)"])
unfolding setsum_right_distrib[symmetric]
using as and *** and True
apply auto
done
next
case False
then have "card (s - {x}) = Suc (Suc 0)"
using as(2) and c by auto
then obtain a b where "(s - {x}) = {a, b}" "a≠b"
unfolding card_Suc_eq by auto
then show ?thesis
using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
using *** *(2) and ‹s ⊆ V›
unfolding setsum_right_distrib
qed
then have "u x + (1 - u x) = 1 ⟹
u x *⇩R x + (1 - u x) *⇩R ((∑xa∈s - {x}. u xa *⇩R xa) /⇩R (1 - u x)) ∈ V"
apply -
apply (rule as(3)[rule_format])
unfolding  Real_Vector_Spaces.scaleR_right.setsum
using x(1) as(6)
apply auto
done
then show "(∑x∈s. u x *⇩R x) ∈ V"
unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
apply (subst *)
unfolding setsum_clauses(2)[OF *(2)]
using ‹u x ≠ 1›
apply auto
done
qed
next
assume "card s = 1"
then obtain a where "s={a}"
then show ?thesis
using as(4,5) by simp
qed (insert ‹s≠{}› ‹finite s›, auto)
qed

lemma affine_hull_explicit:
"affine hull p =
{y. ∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ setsum (λv. (u v) *⇩R v) s = y}"
apply (rule hull_unique)
apply (subst subset_eq)
prefer 3
apply rule
unfolding mem_Collect_eq
apply (erule exE)+
apply (erule conjE)+
prefer 2
apply rule
proof -
fix x
assume "x∈p"
then show "∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"
apply (rule_tac x="{x}" in exI)
apply (rule_tac x="λx. 1" in exI)
apply auto
done
next
fix t x s u
assume as: "p ⊆ t" "affine t" "finite s" "s ≠ {}"
"s ⊆ p" "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"
then show "x ∈ t"
using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]]
by auto
next
show "affine {y. ∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = y}"
unfolding affine_def
apply (rule, rule, rule, rule, rule)
unfolding mem_Collect_eq
proof -
fix u v :: real
assume uv: "u + v = 1"
fix x
assume "∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"
then obtain sx ux where
x: "finite sx" "sx ≠ {}" "sx ⊆ p" "setsum ux sx = 1" "(∑v∈sx. ux v *⇩R v) = x"
by auto
fix y
assume "∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = y"
then obtain sy uy where
y: "finite sy" "sy ≠ {}" "sy ⊆ p" "setsum uy sy = 1" "(∑v∈sy. uy v *⇩R v) = y" by auto
have xy: "finite (sx ∪ sy)"
using x(1) y(1) by auto
have **: "(sx ∪ sy) ∩ sx = sx" "(sx ∪ sy) ∩ sy = sy"
by auto
show "∃s ua. finite s ∧ s ≠ {} ∧ s ⊆ p ∧
setsum ua s = 1 ∧ (∑v∈s. ua v *⇩R v) = u *⇩R x + v *⇩R y"
apply (rule_tac x="sx ∪ sy" in exI)
apply (rule_tac x="λa. (if a∈sx then u * ux a else 0) + (if a∈sy then v * uy a else 0)" in exI)
unfolding scaleR_left_distrib setsum.distrib if_smult scaleR_zero_left
** setsum.inter_restrict[OF xy, symmetric]
unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
and setsum_right_distrib[symmetric]
unfolding x y
using x(1-3) y(1-3) uv
apply simp
done
qed
qed

lemma affine_hull_finite:
assumes "finite s"
shows "affine hull s = {y. ∃u. setsum u s = 1 ∧ setsum (λv. u v *⇩R v) s = y}"
unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq
apply (rule, rule)
apply (erule exE)+
apply (erule conjE)+
defer
apply (erule exE)
apply (erule conjE)
proof -
fix x u
assume "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"
then show "∃sa u. finite sa ∧
¬ (∀x. (x ∈ sa) = (x ∈ {})) ∧ sa ⊆ s ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = x"
apply (rule_tac x=s in exI, rule_tac x=u in exI)
using assms
apply auto
done
next
fix x t u
assume "t ⊆ s"
then have *: "s ∩ t = t"
by auto
assume "finite t" "¬ (∀x. (x ∈ t) = (x ∈ {}))" "setsum u t = 1" "(∑v∈t. u v *⇩R v) = x"
then show "∃u. setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"
apply (rule_tac x="λx. if x∈t then u x else 0" in exI)
unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms, symmetric] and *
apply auto
done
qed

subsubsection ‹Stepping theorems and hence small special cases›

lemma affine_hull_empty[simp]: "affine hull {} = {}"
by (rule hull_unique) auto

lemma affine_hull_finite_step:
fixes y :: "'a::real_vector"
shows
"(∃u. setsum u {} = w ∧ setsum (λx. u x *⇩R x) {} = y) ⟷ w = 0 ∧ y = 0" (is ?th1)
and
"finite s ⟹
(∃u. setsum u (insert a s) = w ∧ setsum (λx. u x *⇩R x) (insert a s) = y) ⟷
(∃v u. setsum u s = w - v ∧ setsum (λx. u x *⇩R x) s = y - v *⇩R a)" (is "_ ⟹ ?lhs = ?rhs")
proof -
show ?th1 by simp
assume fin: "finite s"
show "?lhs = ?rhs"
proof
assume ?lhs
then obtain u where u: "setsum 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 have *: "insert a s = s" by auto
show ?thesis
using u[unfolded *]
apply(rule_tac x=0 in exI)
apply auto
done
next
case False
then show ?thesis
apply (rule_tac x="u a" in exI)
using u and fin
apply auto
done
qed
next
assume ?rhs
then obtain v u where vu: "setsum 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
then show ?thesis
apply (rule_tac x="λx. (if x=a then v else 0) + u x" in exI)
unfolding setsum_clauses(2)[OF fin]
apply simp
unfolding scaleR_left_distrib and setsum.distrib
unfolding vu and * and scaleR_zero_left
apply (auto simp add: setsum.delta[OF fin])
done
next
case False
then have **:
"⋀x. x ∈ s ⟹ u x = (if x = a then v else u x)"
"⋀x. x ∈ s ⟹ u x *⇩R x = (if x = a then v *⇩R x else u x *⇩R x)" by auto
from False show ?thesis
apply (rule_tac x="λx. if x=a then v else u x" in exI)
unfolding setsum_clauses(2)[OF fin] and * using vu
using setsum.cong [of s _ "λx. u x *⇩R x" "λx. if x = a then v *⇩R x else u x *⇩R x", OF _ **(2)]
using setsum.cong [of s _ u "λx. if x = a then v else u x", OF _ **(1)]
apply auto
done
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. setsum 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(2)[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
unfolding *
apply auto
apply (rule_tac x=v in exI)
apply (rule_tac x=va in exI)
apply auto
apply (rule_tac x=u in exI)
apply 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

corollary mem_affine_3_minus2:
"⟦affine S; x ∈ S; y ∈ S; z ∈ S⟧ ⟹ x - v *⇩R (y-z) ∈ S"

subsubsection ‹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}}"
unfolding subset_eq Ball_def
unfolding affine_hull_explicit span_explicit mem_Collect_eq
apply (rule, rule)
apply (erule exE)+
apply (erule conjE)+
proof -
fix x t u
assume as: "finite t" "t ≠ {}" "t ⊆ insert a s" "setsum u t = 1" "(∑v∈t. u v *⇩R v) = x"
have "(λx. x - a) ` (t - {a}) ⊆ {x - a |x. x ∈ s}"
using as(3) by auto
then show "∃v. x = a + v ∧ (∃S u. finite S ∧ S ⊆ {x - a |x. x ∈ s} ∧ (∑v∈S. u v *⇩R v) = v)"
apply (rule_tac x="x - a" in exI)
apply (rule conjI, simp)
apply (rule_tac x="(λx. x - a) ` (t - {a})" in exI)
apply (rule_tac x="λx. u (x + a)" in exI)
apply (rule conjI) using as(1) apply simp
apply (erule conjI)
using as(1)
apply (simp add: setsum.reindex[unfolded inj_on_def] scaleR_right_diff_distrib
setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)
unfolding as
apply simp
done
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}}"
apply (rule, rule affine_hull_insert_subset_span)
unfolding subset_eq Ball_def
unfolding affine_hull_explicit and mem_Collect_eq
proof (rule, rule, erule exE, erule conjE)
fix y v
assume "y = a + v" "v ∈ span {x - a |x. x ∈ s}"
then obtain t u where obt: "finite t" "t ⊆ {x - a |x. x ∈ s}" "a + (∑v∈t. u v *⇩R v) = y"
unfolding span_explicit by auto
def 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 obt by (auto simp add: setsum.reindex[unfolded inj_on_def])
have *: "f ∩ {a} = {}" "f ∩ - {a} = f"
using f(2) assms by auto
show "∃sa u. finite sa ∧ sa ≠ {} ∧ sa ⊆ insert a s ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = y"
apply (rule_tac x = "insert a f" in exI)
apply (rule_tac x = "λx. if x=a then 1 - setsum (λx. u (x - a)) f else u (x - a)" in exI)
using assms and f
unfolding setsum_clauses(2)[OF f(1)] and if_smult
unfolding setsum.If_cases[OF f(1), of "λx. x = a"]
apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)
done
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 ‹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 -
{
fix x
assume "x ∈ T"
then have "( - a) + x ∈ S"
using assms by auto
then have "x ∈ ((λx. a + x) ` S)"
using imageI[of "-a+x" S "(λx. a+x)"] by auto
}
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
apply (rule exI[of _ "0"])
apply auto
done

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)"
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 ‹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"
def 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"
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 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

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 ⟶ (op *⇩R c) ` S = S)"
proof -
{
assume "cone S"
{
fix c :: real
assume "c > 0"
{
fix x
assume "x ∈ S"
then have "x ∈ (op *⇩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 ∈ (op *⇩R c) ` S"
then have "x ∈ S"
using ‹cone S› ‹c > 0›
unfolding cone_def image_def ‹c > 0› by auto
}
ultimately have "(op *⇩R c) ` S = S" by auto
}
then have "0 ∈ S ∧ (∀c. c > 0 ⟶ (op *⇩R c) ` S = S)"
using ‹cone S› cone_contains_0[of S] assms by auto
}
moreover
{
assume a: "0 ∈ S ∧ (∀c. c > 0 ⟶ (op *⇩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)

lemma 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)
apply 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 ⟶ op *⇩R c ` S = S)"
using cone_iff[of S] assms by auto
then have "0 ∈ closure S ∧ (∀c. c > 0 ⟶ op *⇩R c ` closure S = closure S)"
using closure_subset by (auto simp add: 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 affine_dependent :: "'a::real_vector set ⇒ bool"
where "affine_dependent s ⟷ (∃x∈s. x ∈ affine hull (s - {x}))"

lemma affine_dependent_explicit:
"affine_dependent p ⟷
(∃s u. finite s ∧ s ⊆ p ∧ setsum u s = 0 ∧
(∃v∈s. u v ≠ 0) ∧ setsum (λv. u v *⇩R v) s = 0)"
unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq
apply rule
apply (erule bexE, erule exE, erule exE)
apply (erule conjE)+
defer
apply (erule exE, erule exE)
apply (erule conjE)+
apply (erule bexE)
proof -
fix x s u
assume as: "x ∈ p" "finite s" "s ≠ {}" "s ⊆ p - {x}" "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"
have "x ∉ s" using as(1,4) by auto
show "∃s u. finite s ∧ s ⊆ p ∧ setsum u s = 0 ∧ (∃v∈s. u v ≠ 0) ∧ (∑v∈s. u v *⇩R v) = 0"
apply (rule_tac x="insert x s" in exI, rule_tac x="λv. if v = x then - 1 else u v" in exI)
unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF ‹x∉s›] and as
using as
apply auto
done
next
fix s u v
assume as: "finite s" "s ⊆ p" "setsum u s = 0" "(∑v∈s. u v *⇩R v) = 0" "v ∈ s" "u v ≠ 0"
have "s ≠ {v}"
using as(3,6) by auto
then show "∃x∈p. ∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p - {x} ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"
apply (rule_tac x=v in bexI)
apply (rule_tac x="s - {v}" in exI)
apply (rule_tac x="λx. - (1 / u v) * u x" in exI)
unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]
using as
apply auto
done
qed

lemma affine_dependent_explicit_finite:
fixes s :: "'a::real_vector set"
assumes "finite s"
shows "affine_dependent s ⟷
(∃u. setsum u s = 0 ∧ (∃v∈s. u v ≠ 0) ∧ setsum (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" "setsum 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 unfolding * and setsum.inter_restrict[OF assms, symmetric]
unfolding Int_absorb1[OF ‹t⊆s›]
apply auto
done
next
assume ?rhs
then obtain u v where "setsum 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 ‹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
def f ≡ "λu. u *⇩R a + (1 - u) *⇩R b"
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)"

proposition clopen:
fixes s :: "'a :: real_normed_vector set"
shows "closed s ∧ open s ⟷ s = {} ∨ s = UNIV"
apply (rule iffI)
apply (rule connected_UNIV [unfolded connected_clopen, rule_format])
apply (force simp add: open_openin closed_closedin, force)
done

corollary compact_open:
fixes s :: "'a :: euclidean_space set"
shows "compact s ∧ open s ⟷ s = {}"
by (auto simp: compact_eq_bounded_closed clopen)

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

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 real_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)"
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)"
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 add: 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_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 ‹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∈convex hull s. ∀y∈convex hull t. (x, y) ∈ convex hull (s × t)"
proof (intro hull_induct)
fix x y assume "x ∈ s" and "y ∈ t"
then show "(x, y) ∈ convex hull (s × t)"
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,
also have "?S = {y. (x, y) ∈ convex hull (s × t)}"
by (auto simp add: image_def Bex_def)
finally show "convex {y. (x, y) ∈ convex hull (s × t)}" .
next
show "convex {x. ∀y∈convex hull t. (x, y) ∈ convex hull (s × t)}"
proof (unfold Collect_ball_eq, rule convex_INT [rule_format])
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,
also have "?S = {x. (x, y) ∈ convex hull (s × t)}"
by (auto simp add: 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 .
qed

subsubsection ‹Stepping theorems for convex hulls of finite sets›

lemma convex_hull_empty[simp]: "convex hull {} = {}"
by (rule hull_unique) auto

lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
by (rule hull_unique) auto

lemma convex_hull_insert:
fixes s :: "'a::real_vector set"
assumes "s ≠ {}"
shows "convex hull (insert a s) =
{x. ∃u≥0. ∃v≥0. ∃b. (u + v = 1) ∧ b ∈ (convex hull s) ∧ (x = u *⇩R a + v *⇩R b)}"
(is "_ = ?hull")
apply (rule, rule hull_minimal, rule)
unfolding insert_iff
prefer 3
apply rule
proof -
fix x
assume x: "x = a ∨ x ∈ s"
then show "x ∈ ?hull"
apply rule
unfolding mem_Collect_eq
apply (rule_tac x=1 in exI)
defer
apply (rule_tac x=0 in exI)
using assms hull_subset[of s convex]
apply auto
done
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" "x∈?hull" "y∈?hull"
from as(4) obtain u1 v1 b1 where
obt1: "u1≥0" "v1≥0" "u1 + v1 = 1" "b1 ∈ convex hull s" "x = u1 *⇩R a + v1 *⇩R b1"
by auto
from as(5) obtain u2 v2 b2 where
obt2: "u2≥0" "v2≥0" "u2 + v2 = 1" "b2 ∈ convex hull s" "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"
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"
from True have ***: "u * v1 = 0" "v * v2 = 0"
using 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
unfolding obt1(5) obt2(5) *
using assms hull_subset[of s convex]
by (auto simp add: *** scaleR_right_distrib)
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 add: field_simps)
also have "… = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
also have "… = u * v1 + v * v2"
by simp
finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
have "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
then show ?thesis
unfolding obt1(5) obt2(5)
unfolding * and **
using False
apply (rule_tac
x = "((u * v1) / (u * v1 + v * v2)) *⇩R b1 + ((v * v2) / (u * v1 + v * v2)) *⇩R b2" in bexI)
defer
apply (rule convexD [OF convex_convex_hull])
using obt1(4) obt2(4)
apply (auto simp add: scaleR_left_distrib scaleR_right_distrib)
done
qed
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"
apply (rule_tac [!] mult_right_mono)
using as(1,2) obt1(1,2) obt2(1,2)
apply auto
done
also have "… ≤ 1"
unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
finally show "u *⇩R x + v *⇩R y ∈ ?hull"
unfolding mem_Collect_eq
apply (rule_tac x="u * u1 + v * u2" in exI)
apply (rule conjI)
defer
apply (rule_tac x="1 - u * u1 - v * u2" in exI)
unfolding Bex_def
using as(1,2) obt1(1,2) obt2(1,2) **
done
qed
qed

subsubsection ‹Explicit expression for convex hull›

lemma 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) ∧
(setsum u {1..k} = 1) ∧ (setsum (λi. u i *⇩R x i) {1..k} = y)}"
(is "?xyz = ?hull")
apply (rule hull_unique)
apply rule
defer
apply (rule convexI)
proof -
fix x
assume "x∈s"
then show "x ∈ ?hull"
unfolding mem_Collect_eq
apply (rule_tac x=1 in exI, rule_tac x="λx. 1" in exI)
apply auto
done
next
fix t
assume as: "s ⊆ t" "convex t"
show "?hull ⊆ t"
apply rule
unfolding mem_Collect_eq
apply (elim exE conjE)
proof -
fix x k u y
assume assm:
"∀i∈{1::nat..k}. 0 ≤ u i ∧ y i ∈ s"
"setsum u {1..k} = 1" "(∑i = 1..k. u i *⇩R y i) = x"
show "x∈t"
unfolding assm(3) [symmetric]
apply (rule as(2)[unfolded convex, rule_format])
using assm(1,2) as(1) apply auto
done
qed
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: "∀i∈{1::nat..k1}. 0≤u1 i ∧ x1 i ∈ s" "setsum 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: "∀i∈{1::nat..k2}. 0≤u2 i ∧ x2 i ∈ s" "setsum u2 {Suc 0..k2} = 1" "(∑i = Suc 0..k2. u2 i *⇩R x2 i) = y"
by auto
have *: "⋀P (x1::'a) x2 s1 s2 i.
(if P i then s1 else s2) *⇩R (if P i then x1 else x2) = (if P i then s1 *⇩R x1 else s2 *⇩R x2)"
"{1..k1 + k2} ∩ {1..k1} = {1..k1}" "{1..k1 + k2} ∩ - {1..k1} = (λi. i + k1) ` {1..k2}"
prefer 3
apply (rule, rule)
unfolding image_iff
apply (rule_tac x = "x - k1" in bexI)
done
have inj: "inj_on (λi. i + k1) {1..k2}"
unfolding inj_on_def by auto
show "u *⇩R x + v *⇩R y ∈ ?hull"
apply rule
apply (rule_tac x="k1 + k2" in exI)
apply (rule_tac x="λi. if i ∈ {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
apply (rule_tac x="λi. if i ∈ {1..k1} then x1 i else x2 (i - k1)" in exI)
apply (rule, rule)
defer
apply rule
unfolding * and setsum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
setsum.reindex[OF inj] and o_def Collect_mem_eq
unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]
proof -
fix i
assume i: "i ∈ {1..k1+k2}"
show "0 ≤ (if i ∈ {1..k1} then u * u1 i else v * u2 (i - k1)) ∧
(if i ∈ {1..k1} then x1 i else x2 (i - k1)) ∈ s"
proof (cases "i∈{1..k1}")
case True
then show ?thesis
using uv(1) x(1)[THEN bspec[where x=i]] by auto
next
case False
def j ≡ "i - k1"
from i False have "j ∈ {1..k2}"
unfolding j_def by auto
then show ?thesis
using False uv(2) y(1)[THEN bspec[where x=j]]
by (auto simp: j_def[symmetric])
qed
qed (auto simp add: not_le x(2,3) y(2,3) uv(3))
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) ∧
setsum u s = 1 ∧ setsum (λx. u x *⇩R x) s = y}"
(is "?HULL = ?set")
proof (rule hull_unique, auto simp add: convex_def[of ?set])
fix x
assume "x ∈ s"
then show "∃u. (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ (∑x∈s. u x *⇩R x) = x"
apply (rule_tac x="λy. if x=y then 1 else 0" in exI)
apply auto
unfolding setsum.delta'[OF assms] and setsum_delta''[OF assms]
apply auto
done
next
fix u v :: real
assume uv: "0 ≤ u" "0 ≤ v" "u + v = 1"
fix ux assume ux: "∀x∈s. 0 ≤ ux x" "setsum ux s = (1::real)"
fix uy assume uy: "∀x∈s. 0 ≤ uy x" "setsum uy s = (1::real)"
{
fix x
assume "x∈s"
then have "0 ≤ u * ux x + v * uy x"
using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
by auto
}
moreover
have "(∑x∈s. u * ux x + v * uy x) = 1"
unfolding setsum.distrib and setsum_right_distrib[symmetric] and 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 and setsum.distrib and scaleR_scaleR[symmetric]
and scaleR_right.setsum [symmetric]
by auto
ultimately
show "∃uc. (∀x∈s. 0 ≤ uc x) ∧ setsum 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)"
apply (rule_tac x="λx. u * ux x + v * uy x" in exI)
apply auto
done
next
fix t
assume t: "s ⊆ t" "convex t"
fix u
assume u: "∀x∈s. 0 ≤ u x" "setsum u s = (1::real)"
then show "(∑x∈s. u x *⇩R x) ∈ t"
using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
using assms and t(1) by auto
qed

subsubsection ‹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) ∧ setsum u s = 1 ∧ setsum (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" "setsum 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 ≤ setsum 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 setsum_nonneg)
using obt(1)
apply auto
done
}
moreover
have "(∑v∈y ` {1..k}. setsum u {i ∈ {1..k}. y i = v}) = 1"
unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
moreover have "(∑v∈y ` {1..k}. setsum u {i ∈ {1..k}. y i = v} *⇩R v) = x"
using setsum_image_gen[OF fin, of "λi. u i *⇩R y i" y, symmetric]
unfolding scaleR_left.setsum using obt(3) by auto
ultimately
have "∃s u. finite s ∧ s ⊆ p ∧ (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"
apply (rule_tac x="y ` {1..k}" in exI)
apply (rule_tac x="λv. setsum u {i∈{1..k}. y i = v}" in exI)
apply 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" "setsum 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"
apply (subst f(2)[symmetric])
apply auto
done
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]
apply(erule_tac x=x in ballE)
apply auto
done
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"
}
then have "(∑x = 1..card s. u (f x)) = 1" "(∑i = 1..card s. u (f i) *⇩R f i) = y"
unfolding setsum_image_gen[OF *(1), of "λx. u (f x) *⇩R f x" f]
and setsum_image_gen[OF *(1), of "λx. u (f x)" f]
unfolding f
using setsum.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 setsum.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) ∧ setsum 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)
apply fastforce
done
then have "y ∈ ?lhs"
unfolding convex_hull_indexed by auto
}
ultimately show ?thesis
unfolding set_eq_iff by blast
qed

subsubsection ‹A stepping theorem for that expansion›

lemma convex_hull_finite_step:
fixes s :: "'a::real_vector set"
assumes "finite s"
shows
"(∃u. (∀x∈insert a s. 0 ≤ u x) ∧ setsum u (insert a s) = w ∧ setsum (λx. u x *⇩R x) (insert a s) = y)
⟷ (∃v≥0. ∃u. (∀x∈s. 0 ≤ u x) ∧ setsum u s = w - v ∧ setsum (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 *
apply (rule_tac x=0 in exI)
apply auto
done
next
assume ?lhs
then obtain u where
u: "∀x∈insert a s. 0 ≤ u x" "setsum 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 setsum_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" "setsum 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 setsum.distrib and setsum_delta''[OF fin] and setsum.delta'[OF fin]
unfolding setsum_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" "setsum 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) = setsum u s"
and "(∑x∈s. (if a = x then v else u x) *⇩R x) = (∑x∈s. u x *⇩R x)"
apply (rule_tac setsum.cong) apply rule
defer
apply (rule_tac setsum.cong) apply rule
using ‹a ∉ s›
apply auto
done
ultimately show ?lhs
apply (rule_tac x="λx. if a = x then v else u x" in exI)
unfolding setsum_clauses(2)[OF assms]
apply auto
done
qed

subsubsection ‹Hence some special cases›

lemma convex_hull_2:
"convex hull {a,b} = {u *⇩R a + v *⇩R b | u v. 0 ≤ u ∧ 0 ≤ v ∧ u + v = 1}"
proof -
have *: "⋀u. (∀x∈{a, b}. 0 ≤ u x) ⟷ 0 ≤ u a ∧ 0 ≤ u b"
by auto
have **: "finite {b}" by auto
show ?thesis
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)
apply simp
apply (rule_tac x=u in exI)
apply simp
apply (rule_tac x="λx. v" in exI)
apply 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)
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"
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)
apply simp
apply auto
apply (rule_tac x=va in exI)
apply (rule_tac x="u c" in exI)
apply simp
apply (rule_tac x="1 - v - w" in exI)
apply simp
apply (rule_tac x=v in exI)
apply simp
apply (rule_tac x="λx. w" in exI)
apply 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 (rule_tac x=v in exI)
apply (rule_tac x=w in exI)
apply (rule_tac x=u in exI)
apply (rule_tac x=v in exI)
done
qed

subsection ‹Relations among closure notions and corresponding hulls›

lemma affine_imp_convex: "affine s ⟹ convex s"
unfolding affine_def convex_def by auto

lemma 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_inc subspace_imp_affine subspace_span)

lemma convex_hull_subset_span: "(convex hull s) ⊆ (span s)"
by (metis hull_minimal span_inc 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
def 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 setsum.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 setsum_clauses(2)[OF fin]
using ‹a∉s› ‹t⊆s›
apply auto
unfolding *
apply auto
done
moreover have "∃v∈insert a t. (if v = a then - (∑x∈t. u (x - a)) else u (v - a)) ≠ 0"
apply (rule_tac x="v + a" in bexI)
using obt(3,4) and ‹0∉S›
unfolding t_def
apply auto
done
moreover have *: "⋀P Q. (∑x∈t. (if x = a then P x else Q x) *⇩R x) = (∑x∈t. Q x *⇩R x)"
apply (rule setsum.cong)
using ‹a∉s› ‹t⊆s›
apply auto
done
have "(∑x∈t. u (x - a)) *⇩R a = (∑v∈t. u (v - a) *⇩R v)"
unfolding scaleR_left.setsum
unfolding t_def and setsum.reindex[OF inj] and o_def
using obt(5)
by (auto simp add: setsum.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 setsum_clauses(2)[OF fin]
using ‹a∉s› ‹t⊆s›
ultimately show ?thesis
unfolding affine_dependent_explicit
apply (rule_tac x="insert a t" in exI)
apply 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)
apply simp
apply (erule_tac x="1/2" in allE)
apply 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 *
apply (rule card_image)
unfolding inj_on_def
apply auto
done
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)
apply 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})"
unfolding *
apply (rule card_image)
unfolding inj_on_def
apply auto
done
have "dim {x - a |x. x ∈ s - {a}} ≤ dim s"
apply (rule subset_le_dim)
unfolding subset_eq
using ‹a∈s›
done
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 ‹Some Properties of Affine Dependent Sets›

lemma affine_independent_empty: "¬ affine_dependent {}"

lemma affine_independent_sing: "¬ affine_dependent {a}"

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 "op + a ` (S - {x}) = op + 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 ∧ setsum 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 setsum_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 "(op + (- 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]
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 "op + (- 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"
using assms by auto
ultimately have ?thesis
using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
}
ultimately show ?thesis by auto
qed

lemma affine_hull_span2:
assumes "a ∈ s"
shows "affine hull s = (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:
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
then obtain B where B:
"(λx. -a+x) ` (S - {a}) ⊆ B ∧ B ⊆ (λx. -a+x) ` V ∧ independent B ∧ (λx. -a+x) ` V ⊆ span B"
using maximal_independent_subset_extend[of "(λx. -a+x) ` (S-{a})" "(λx. -a + x) ` V"] assms
by blast
def 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_empty 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[of "{x}" V]
by auto
qed

subsection ‹Affine Dimension of a Set›

definition 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"])
apply 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})"
apply (rule card_image)
using translate_inj_on
done
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
def 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

lemma independent_finite:
fixes B :: "'n::euclidean_space set"
assumes "independent B"
shows "finite B"
using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms
by auto

lemma subspace_dim_equal:
assumes "subspace (S :: ('n::euclidean_space) set)"
and "subspace T"
and "S ⊆ T"
and "dim S ≥ dim T"
shows "S = T"
proof -
obtain B where B: "B ≤ S" "independent B ∧ S ⊆ span B" "card B = dim S"
using basis_exists[of S] by auto
then have "span B ⊆ S"
using span_mono[of B S] span_eq[of S] assms by metis
then have "span B = S"
using B by auto
have "dim S = dim T"
using assms dim_subset[of S T] by auto
then have "T ⊆ span B"
using card_eq_dim[of B T] B independent_finite assms by auto
then show ?thesis
using assms ‹span B = S› by auto
qed

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[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_inc[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_inc[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)"
apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "d"])
apply (rule subspace_span)
apply (rule subspace_substandard)
defer
apply (rule span_inc)
apply (rule assms)
defer
unfolding dim_span[of B]
apply(rule B)
unfolding span_substd_basis[OF d, symmetric]
apply (rule span_inc)
apply (rule independent_substdbasis[OF d])
apply rule
apply assumption
unfolding t[symmetric] span_substd_basis[OF d] dim_substandard[OF d]
apply auto
done
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"

lemma aff_dim_affine_hull: "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
def 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)
by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)

lemma aff_dim_sing:
fixes a :: "'n::euclidean_space"
shows "aff_dim {a} = 0"
using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto

lemma aff_dim_inner_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "∃B. B ⊆ V ∧ affine hull B = affine hull V ∧
¬ affine_dependent B ∧ of_nat (card B) = aff_dim V + 1"
proof -
obtain B where B: "¬ 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 "S ≠ {}"
and "subspace S"
shows "aff_dim S = int (dim S)"
using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S]
by auto

lemma aff_dim_zero:
fixes S :: "'n::euclidean_space set"
assumes "0 ∈ affine hull S"
shows "aff_dim S = int (dim S)"
proof -
have "subspace (affine hull S)"
using subspace_affine[of "affine hull S"] affine_affine_hull assms
by auto
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_univ: "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 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 (cases "B = {}")
case True
then have "-1 ≤ aff_dim V"
using aff_dim_geq by auto
with True show ?thesis by auto
next
case False
then obtain T where T: "¬ affine_dependent T ∧ B ⊆ T ∧ T ⊆ V ∧ affine hull T = affine hull V"
using assms extend_to_affine_basis[of B V] by auto
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_subset_univ:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S ≤ int (DIM('n))"
proof -
have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
using aff_dim_univ by auto
then show "aff_dim (S:: 'n::euclidean_space set) ≤ int(DIM('n))"
using assms aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
qed

lemma affine_dim_equal:
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
def 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
def 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 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_subset_univ[of "affine hull S"] assms h0 by auto
have h3: "aff_dim S ≤ aff_dim (affine hull S)"
using h0 aff_dim_subset[of S "affine hull S"] assms by auto
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 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 "op + a ` cball 0 e" "cball a e"]
by auto
moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
ultimately show ?thesis
using aff_dim_subset_univ[of "cball a e"] by auto
qed

lemma aff_dim_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
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_subset_univ[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_subset_univ[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)

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) ∧ setsum u s = 1 ∧ setsum (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) ∧
setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = y"
assume "∃s u. finite s ∧ s ⊆ p ∧ (∀x∈s. 0 ≤ u x) ∧ setsum 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)
apply 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"
"setsum 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: "setsum 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
def i ≡ "(λv. (u v) / (- w v)) ` {v∈s. w v < 0}"
def 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 "setsum w (s - {v}) ≥ 0"
apply (rule_tac setsum_nonneg)
apply auto
done
then have "setsum w s > 0"
unfolding setsum.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
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. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
unfolding setsum.remove[OF obt(1) ‹a∈s›] by auto
have "(∑v∈s. u v + t * w v) = 1"
unfolding setsum.distrib wv(1) setsum_right_distrib[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 setsum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
ultimately have "?P (n - 1)"
apply (rule_tac x="(s - {a})" in exI)
apply (rule_tac x="λv. u v + t * w v" in exI)
using obt(1-3) and t and a
apply (auto simp add: * 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) ∧ setsum 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)
apply 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) ∧ setsum u s = 1 ∧ setsum (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_subset_univ [of p]
apply simp
done
next
fix x
assume "x ∈ ?rhs" then show "x ∈ ?lhs"
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" "setsum 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[unfolded 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 "rel_interior S =
{x. ∃T. openin (subtopology euclidean (affine hull S)) T ∧ x ∈ T ∧ T ⊆ S}"

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)"

lemma mem_rel_interior_ball:
"x ∈ rel_interior S ⟷ x ∈ S ∧ (∃e. e > 0 ∧ ball x e ∩ affine hull S ⊆ S)"
apply (rule_tac x = "ball x e" in exI)
apply simp
done

lemma rel_interior_ball:
"rel_interior S = {x ∈ S. ∃e. e > 0 ∧ ball x e ∩ 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 (rule_tac x = "ball x e" in exI)
apply (simp add: subset_trans [OF ball_subset_cball])
apply auto
done

lemma rel_interior_cball:
"rel_interior S = {x ∈ S. ∃e. e > 0 ∧ cball x e ∩ affine hull S ⊆ S}"
using mem_rel_interior_cball [of _ S] by auto

lemma rel_interior_empty [simp]: "rel_interior {} = {}"

lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
by (metis affine_hull_eq affine_sing)

lemma rel_interior_sing [simp]: "rel_interior {a :: 'n::euclidean_space} = {a}"
unfolding rel_interior_ball affine_hull_sing
apply auto
apply (rule_tac x = "1 :: real" in exI)
apply simp
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 add: rel_interior_def)

lemma rel_interior_subset: "rel_interior S ⊆ S"

lemma rel_interior_subset_closure: "rel_interior S ⊆ closure S"
using rel_interior_subset by (auto simp add: closure_def)

lemma interior_subset_rel_interior: "interior S ⊆ rel_interior S"
by (auto simp add: rel_interior interior_def)

lemma interior_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "rel_interior S = interior S"
proof -
have "affine hull S = UNIV"
using assms affine_hull_univ[of S] by auto
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"

lemma interior_rel_interior_gen:
fixes S :: "'n::euclidean_space set"
shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
by (metis interior_rel_interior low_dim_interior)

lemma rel_interior_univ:
fixes S :: "'n::euclidean_space set"
shows "rel_interior (affine hull S) = affine hull S"
proof -
have *: "rel_interior (affine hull S) ⊆ affine hull S"
using rel_interior_subset by auto
{
fix x
assume x: "x ∈ affine hull S"
def e ≡ "1::real"
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_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
by (metis open_UNIV rel_interior_open)

lemma rel_interior_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
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 add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "x ∈ affine hull S"
using assms hull_subset[of S] by auto
moreover have "1 / e + - ((1 - e) / e) = 1"
using ‹e > 0› left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
ultimately have **: "(1 / e) *⇩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)"]
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 add: 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 add:pos_divide_less_eq[OF ‹e > 0›] mult.commute)
finally have "y ∈ S"
apply (subst *)
apply (rule assms(1)[unfolded convex_alt,rule_format])
apply (rule d[unfolded subset_eq,rule_format])
unfolding mem_ball
using assms(3-5) **
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]
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 (rule_tac x="(y-a)" in exI)
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 add: 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 (rule_tac x="(a-y)" in exI)
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 add: 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: "interior {a..b} = {a<..<b :: real}"
proof-
have "{a..b} = {a..} ∩ {..b}" by auto
also have "interior ... = {a<..} ∩ {..<b}"
also have "... = {a<..<b}" by auto
finally show ?thesis .
qed

lemma frontier_real_Iic:
fixes a :: real
shows "frontier {..a} = {a}"
unfolding frontier_def by (auto simp add: interior_real_semiline')

lemma rel_interior_real_box:
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: split_if_asm del: box_real add: box_real[symmetric] interior_cbox)
qed

lemma rel_interior_real_semiline:
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: split_if_asm)
qed

subsubsection ‹Relative open sets›

definition "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 opein_rel_interior: "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
apply (subst openin_subopen)
apply blast
done

lemma affine_rel_open:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "rel_open S"
unfolding rel_open_def
using assms rel_interior_univ[of S] affine_hull_eq[of S]
by metis

lemma affine_closed:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "closed S"
proof -
{
assume "S ≠ {}"
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 = (op + 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:
fixes S :: "'n::euclidean_space set"
shows "affine hull (closure S) = affine hull S"
proof -
have "affine hull (closure S) ⊆ affine hull S"
using hull_mono[of "closure S" "affine hull S" "affine"]
closure_affine_hull[of S] hull_hull[of "affine" S]
by auto
moreover have "affine hull (closure S) ⊇ affine hull S"
using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
ultimately show ?thesis by auto
qed

lemma closure_aff_dim:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (closure S) = aff_dim S"
proof -
have "aff_dim S ≤ aff_dim (closure S)"
using aff_dim_subset closure_subset by auto
moreover have "aff_dim (closure S) ≤ aff_dim (affine hull S)"
using aff_dim_subset closure_affine_hull by auto
moreover have "aff_dim (affine hull S) = aff_dim S"
using aff_dim_affine_hull by auto
ultimately show ?thesis by auto
qed

lemma rel_interior_closure_convex_shrink:
fixes S :: "_::euclidean_space set"
assumes "convex S"
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])
apply (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
def 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 add: 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)
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
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

subsubsection‹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 (op + 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"
apply rule
unfolding subset_eq ball_simps
apply (rule_tac[!] hull_induct, rule hull_inc)
prefer 3
apply (erule imageE)
apply (rule_tac x=xa in image_eqI)
apply assumption
apply (rule hull_subset[unfolded subset_eq, rule_format])
apply assumption
proof -
interpret f: bounded_linear f by fact
show "affine {x. f x ∈ affine hull f ` s}"
unfolding affine_def
show "affine {x. x ∈ f ` (affine hull s)}"
using affine_affine_hull[unfolded affine_def, of s]
qed auto

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
def 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
def 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_sub[of f x y] linear_conv_bounded_linear[of f] by auto
moreover have "e1 * norm (f (x-y)) ≤ norm (x - y)"
using e1 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_inc 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
def 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_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
moreover have *: "x - xy ∈ span S"
using subspace_sub[of "span S" x xy] subspace_span ‹x ∈ S› xy
affine_hull_subset_span[of S] span_inc
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‹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. op + (0::'a) ` A = A" "⋀A. op + (- (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 ‹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)"
unfolding open_contains_cball convex_hull_explicit
unfolding mem_Collect_eq ball_simps(8)
proof (rule, rule)
fix a
assume "∃sa u. finite sa ∧ sa ⊆ s ∧ (∀x∈sa. 0 ≤ u x) ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = a"
then obtain t u where obt: "finite t" "t⊆s" "∀x∈t. 0 ≤ u x" "setsum u t = 1" "(∑v∈t. u v *⇩R v) = a"
by auto

from assms[unfolded open_contains_cball] obtain b
where b: "∀x∈s. 0 < b x ∧ cball x (b x) ⊆ s"
using bchoice[of s "λx e. e > 0 ∧ cball x e ⊆ s"] by auto
have "b ` t ≠ {}"
using obt by auto
def i ≡ "b ` t"

show "∃e > 0.
cball a e ⊆ {y. ∃sa u. finite sa ∧ sa ⊆ s ∧ (∀x∈sa. 0 ≤ u x) ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = y}"
apply (rule_tac x = "Min i" in exI)
unfolding subset_eq
apply rule
defer
apply rule
unfolding mem_Collect_eq
proof -
show "0 < Min i"
unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] ‹b ` t≠{}›]
using b
apply simp
apply rule
apply (erule_tac x=x in ballE)
using ‹t⊆s›
apply auto
done
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"
unfolding i_def
apply (rule_tac Min_le)
using obt(1)
apply auto
done
then have "x + (y - a) ∈ cball x (b x)"
using y unfolding mem_cball dist_norm by auto
moreover from ‹x∈t› have "x ∈ s"
using obt(2) by auto
ultimately have "x + (y - a) ∈ s"
using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast
}
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))) = 1"
unfolding setsum.reindex[OF *] o_def using obt(4) by auto
moreover have "(∑v∈(λv. v + (y - a)) ` t. u (v - (y - a)) *⇩R v) = y"
unfolding setsum.reindex[OF *] o_def using obt(4,5)
by (simp add: setsum.distrib setsum_subtractf scaleR_left.setsum[symmetric] scaleR_right_distrib)
ultimately
show "∃sa u. finite sa ∧ (∀x∈sa. x ∈ s) ∧ (∀x∈sa. 0 ≤ u x) ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = y"
apply (rule_tac x="(λv. v + (y - a)) ` t" in exI)
apply (rule_tac x="λv. u (v - (y - a))" in exI)
using obt(1, 3)
apply auto
done
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"
apply (rule set_eqI)
unfolding image_iff mem_Collect_eq
apply rule
apply auto
apply (rule_tac x=u in rev_bexI)
apply simp
apply (erule rev_bexI)
apply (erule rev_bexI)
apply simp
apply auto
done
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)
then show ?thesis
unfolding *
apply (rule compact_continuous_image)
apply (intro compact_Times compact_Icc assms)
done
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)
apply 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)
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)
apply 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 ‹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)"
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
done
next
case False
then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
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
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 (rule, rule, 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 real_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 add: 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)
moreover have "(u + w) *⇩R x + (v - w) *⇩R b ∈ convex hull insert x s"
unfolding convex_hull_insert[OF ‹s≠{}›] and mem_Collect_eq
apply (rule_tac x="u + w" in exI)
apply rule
defer
apply (rule_tac x="v - w" in exI)
using ‹u ≥ 0› and w and obt(3,4)
apply auto
done
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)