Theory Convex_Euclidean_Space

Up to index of Isabelle/HOL/HOL-Multivariate_Analysis

theory Convex_Euclidean_Space
imports Topology_Euclidean_Space

(*  Title:      HOL/Library/Convex_Euclidean_Space.thy
Author: Robert Himmelmann, TU Muenchen
*)


header {* Convex sets, functions and related things. *}

theory Convex_Euclidean_Space
imports Topology_Euclidean_Space Convex
begin



(* ------------------------------------------------------------------------- *)
(* To be moved elsewhere *)
(* ------------------------------------------------------------------------- *)

declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]

(*lemma dim1in[intro]:"Suc 0 ∈ {1::nat .. CARD(1)}" by auto*)

lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component

lemma norm_not_0:"(x::real^'n)≠0 ==> norm x ≠ 0" by auto

lemma setsum_delta_notmem: assumes "x∉s"
shows "setsum (λy. if (y = x) then P x else Q y) s = setsum Q s"
"setsum (λy. if (x = y) then P x else Q y) s = setsum Q s"
"setsum (λy. if (y = x) then P y else Q y) s = setsum Q s"
"setsum (λy. if (x = y) then P y else Q y) s = setsum Q s"

apply(rule_tac [!] setsum_cong2) using assms by auto

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 not_disjointI:"x∈A ==> x∈B ==> A ∩ B ≠ {}" by blast

lemma if_smult:"(if P then x else (y::real)) *R v = (if P then x *R v else y *R v)" by auto

lemma image_smult_interval:"(λx. m *R (x::real^'n)) ` {a..b} =
(if {a..b} = {} then {} else if 0 ≤ m then {m *R a..m *R b} else {m *R b..m *R a})"

using image_affinity_interval[of m 0 a b] by auto

lemma dist_triangle_eq:
fixes x y z :: "real ^ _"
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 smult_conv_scaleR *]
by(auto simp add:norm_minus_commute) qed

lemma norm_eqI:"x = y ==> norm x = norm y" by auto
lemma norm_minus_eqI:"(x::real^'n) = - 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 dimindex_ge_1:"CARD(_::finite) ≥ 1"
using one_le_card_finite by auto

lemma real_dimindex_ge_1:"real (CARD('n::finite)) ≥ 1"
by(metis dimindex_ge_1 real_eq_of_nat real_of_nat_1 real_of_nat_le_iff)

lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto

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: "(∀s∈f. affine s) ==> affine (\<Inter> f)"
unfolding affine_def by auto

lemma affine_Int: "affine s ==> affine t ==> affine (s ∩ t)"
unfolding affine_def by auto

lemma affine_affine_hull: "affine(affine hull s)"
unfolding hull_def using affine_Inter[of "{t ∈ affine. s ⊆ t}"]
unfolding mem_def by auto

lemma affine_hull_eq[simp]: "(affine hull s = s) <-> affine s"
by (metis affine_affine_hull hull_same mem_def)

lemma setsum_restrict_set'': assumes "finite A"
shows "setsum f {x ∈ A. P x} = (∑x∈A. if P x then f x else 0)"

unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..

subsection {* 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"

thus "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)
by(auto simp add: scaleR_left_distrib[THEN sym])
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
thus "(∑x∈s. u x *R x) ∈ V" proof(auto simp only: disjE)
assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
by(auto simp add: setsum_clauses(2))
next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
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_tac ccontr)
assume " ¬ (∃x∈s. u x ≠ 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4))
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) by auto
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)[THEN sym] 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 hence "s - {x} ≠ {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr)
assume "¬ s - {x} ≠ {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
thus False using True by auto qed auto
thus ?thesis apply(rule_tac IA[of "s - {x}" "λy. (inverse (1 - u x) * u y)"])
unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
next case False hence "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
thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
using *** *(2) and `s ⊆ V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
thus "(∑x∈s. u x *R x) ∈ V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *R (∑xa∈s - {x}. u xa *R xa)"],
THEN spec[where x="u x"], THEN spec[where x="1 - u x"]] and rev_subsetD[OF `x∈s` `s⊆V`] and `u x ≠ 1`
by auto
qed auto
next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
thus ?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 and mem_def[of _ affine]
apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
fix x assume "x∈p" thus "∃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, rule_tac x="λx. 1" in exI) by auto
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"
thus "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_addf if_smult scaleR_zero_left ** setsum_restrict_set[OF xy, THEN sym]
unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
unfolding x y using x(1-3) y(1-3) uv by simp 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 expand_set_eq 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"
thus "∃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 by auto
next
fix x t u assume "t ⊆ s" hence *:"s ∩ t = t" by auto
assume "finite t" "¬ (∀x. (x ∈ t) = (x ∈ {}))" "setsum u t = 1" "(∑v∈t. u v *R v) = x"
thus "∃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_restrict_set[OF assms, THEN sym] and * by auto qed

subsection {* Stepping theorems and hence small special cases. *}

lemma affine_hull_empty[simp]: "affine hull {} = {}"
apply(rule hull_unique) unfolding mem_def by 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)
"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 "?as ==> (?lhs = ?rhs)")

proof-
show ?th1 by simp
assume ?as
{ assume ?lhs
then obtain u where u:"setsum u (insert a s) = w ∧ (∑x∈insert a s. u x *R x) = y" by auto
have ?rhs proof(cases "a∈s")
case True hence *:"insert a s = s" by auto
show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
next
case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto
qed } moreover
{ 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
have ?lhs proof(cases "a∈s")
case True thus ?thesis
apply(rule_tac x="λx. (if x=a then v else 0) + u x" in exI)
unfolding setsum_clauses(2)[OF `?as`] apply simp
unfolding scaleR_left_distrib and setsum_addf
unfolding vu and * and scaleR_zero_left
by (auto simp add: setsum_delta[OF `?as`])
next
case False
hence **:"!!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 `?as`] and * using vu
using setsum_cong2[of s "λx. u x *R x" "λx. if x = a then v *R x else u x *R x", OF **(2)]
using setsum_cong2[of s u "λx. if x = a then v else u x", OF **(1)] by auto
qed }
ultimately show "?lhs = ?rhs" by blast
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}" (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
show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
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) by(auto intro!: exI)
qed

subsection {* Some relations between affine hull and subspaces. *}

lemma affine_hull_insert_subset_span:
fixes a :: "real ^ _"
shows "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 smult_conv_scaleR
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
thus "∃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[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
unfolding as by simp qed

lemma affine_hull_insert_span:
fixes a :: "real ^ _"
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 smult_conv_scaleR 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_cases[OF f(1), of "λx. x = a"]
by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed

lemma affine_hull_span:
fixes a :: "real ^ _"
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

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(\<Inter> f)"
unfolding cone_def by auto

subsection {* Conic hull. *}

lemma cone_cone_hull: "cone (cone hull s)"
unfolding hull_def using cone_Inter[of "{t ∈ conic. s ⊆ t}"]
by (auto simp add: mem_def)

lemma cone_hull_eq: "(cone hull s = s) <-> cone s"
apply(rule hull_eq[unfolded mem_def])
using cone_Inter unfolding subset_eq by (auto simp add: mem_def)

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 by auto
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
thus "∃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, rule_tac x="s - {v}" in exI, rule_tac x="λx. - (1 / u v) * u x" in exI)
unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
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
thus ?rhs apply(rule_tac x="λx. if x∈t then u x else 0" in exI)
apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
unfolding Int_absorb1[OF `t⊆s`] by auto
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
thus ?lhs unfolding affine_dependent_explicit using assms by auto
qed

subsection {* A general lemma. *}

lemma convex_connected:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "connected s"

proof-
{ fix e1 e2 assume as:"open e1" "open e2" "e1 ∩ e2 ∩ s = {}" "s ⊆ e1 ∪ e2"
assume "e1 ∩ s ≠ {}" "e2 ∩ s ≠ {}"
then obtain x1 x2 where x1:"x1∈e1" "x1∈s" and x2:"x2∈e2" "x2∈s" by auto
hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto

{ fix x e::real assume as:"0 ≤ x" "x ≤ 1" "0 < e"
{ fix y have *:"(1 - x) *R x1 + x *R x2 - ((1 - y) *R x1 + y *R x2) = (y - x) *R x1 - (y - x) *R x2"
by (simp add: algebra_simps)
assume "¦y - x¦ < e / norm (x1 - x2)"
hence "norm ((1 - x) *R x1 + x *R x2 - ((1 - y) *R x1 + y *R x2)) < e"
unfolding * and scaleR_right_diff_distrib[THEN sym]
unfolding less_divide_eq using n by auto }
hence "∃d>0. ∀y. ¦y - x¦ < d --> norm ((1 - x) *R x1 + x *R x2 - ((1 - y) *R x1 + y *R x2)) < e"
apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
apply auto unfolding zero_less_divide_iff using n by simp } note * = this

have "∃x≥0. x ≤ 1 ∧ (1 - x) *R x1 + x *R x2 ∉ e1 ∧ (1 - x) *R x1 + x *R x2 ∉ e2"
apply(rule connected_real_lemma) apply (simp add: `x1∈e1` `x2∈e2` dist_commute)+
using * apply(simp add: dist_norm)
using as(1,2)[unfolded open_dist] apply simp
using as(1,2)[unfolded open_dist] apply simp
using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
using as(3) by auto
then obtain x where "x≥0" "x≤1" "(1 - x) *R x1 + x *R x2 ∉ e1" "(1 - x) *R x1 + x *R x2 ∉ e2" by auto
hence False using as(4)
using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
using x1(2) x2(2) by auto }
thus ?thesis unfolding connected_def by auto
qed

subsection {* One rather trivial consequence. *}

lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
by(simp add: convex_connected convex_UNIV)

subsection {* Balls, being convex, are connected. *}

lemma convex_box:
assumes "!!i. convex {x. P i x}"
shows "convex {x. ∀i. P i (x$i)}"

using assms unfolding convex_def by auto

lemma convex_positive_orthant: "convex {x::real^'n. (∀i. 0 ≤ x$i)}"
by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)

lemma convex_local_global_minimum:
fixes s :: "'a::real_normed_vector set"
assumes "0<e" "convex_on s f" "ball x e ⊆ s" "∀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 "¬ (∀y∈s. f x ≤ f y)"
then obtain y where "y∈s" and y:"f x > f y" by auto
hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])

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"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
hence "f ((1-u) *R x + u *R y) ≤ (1-u) * f x + u * f y" using `x∈s` `y∈s`
using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]] by auto
moreover
have *:"x - ((1 - u) *R x + u *R y) = u *R (x - y)" by (simp add: algebra_simps)
have "(1 - u) *R x + u *R y ∈ ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
using u unfolding pos_less_divide_eq[OF xy] by auto
hence "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:
fixes x :: "'a::real_normed_vector"
shows "convex (ball x e)"

proof(auto simp add: convex_def)
fix y z assume yz:"dist x y < e" "dist x z < e"
fix u v ::real assume uv:"0 ≤ 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_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
thus "dist x (u *R y + v *R z) < e" using convex_bound_lt[OF yz uv] by auto
qed

lemma convex_cball:
fixes x :: "'a::real_normed_vector"
shows "convex(cball x e)"

proof(auto simp add: convex_def Ball_def)
fix y z assume yz:"dist x y ≤ e" "dist x z ≤ e"
fix u v ::real assume uv:" 0 ≤ u" "0 ≤ v" "u + v = 1"
have "dist x (u *R y + v *R z) ≤ u * dist x y + v * dist x z" using uv yz
using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
thus "dist x (u *R y + v *R z) ≤ e" using convex_bound_le[OF yz uv] by auto
qed

lemma connected_ball:
fixes x :: "'a::real_normed_vector"
shows "connected (ball x e)"

using convex_connected convex_ball by auto

lemma connected_cball:
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: "convex(convex hull s)"
unfolding hull_def using convex_Inter[of "{t∈convex. s⊆t}"]
unfolding mem_def by auto

lemma convex_hull_eq: "convex hull s = s <-> convex s"
by (metis convex_convex_hull hull_same mem_def)

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[unfolded mem_def, of convex, OF convex_cball]
unfolding subset_eq mem_cball dist_norm using B by auto 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

subsection {* Stepping theorems for convex hulls of finite sets. *}

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

lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
apply(rule hull_unique) unfolding mem_def by 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 "?xyz = ?hull")

apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
fix x assume x:"x = a ∨ x ∈ s"
thus "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] by auto
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
thus "x∈ convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
next
show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
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" by (auto simp add: algebra_simps)
have "∃b ∈ convex hull s. u *R x + v *R y = (u * u1) *R a + (v * u2) *R a + (b - (u * u1) *R b - (v * u2) *R b)"
proof(cases "u * v1 + v * v2 = 0")
have *:"!!(x::'a) s1 s2. x - s1 *R x - s2 *R x = ((1::real) - (s1 + s2)) *R x" by (auto simp add: algebra_simps)
case True hence **:"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+
hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
next
have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
also have "… = 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
case False have "0 ≤ u * v1 + v * v2" "0 ≤ u * v1" "0 ≤ u * v1 + v * v2" "0 ≤ v * v2" apply -
apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
using as(1,2) obt1(1,2) obt2(1,2) by auto
thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *R b1 + ((v * v2) / (u * v1 + v * v2)) *R b2" in bexI) defer
apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
qed note * = this
have u1:"u1 ≤ 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
have u2:"u2 ≤ 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
have "u1 * u + u2 * v ≤ (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
also have "… ≤ 1" unfolding mult.add_right[THEN sym] 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) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
qed
qed


subsection {* 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) unfolding mem_def[of _ convex] apply(rule) defer
apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
proof-
fix x assume "x∈s"
thus "x ∈ ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="λx. 1" in exI) by auto
next
fix t assume as:"s ⊆ t" "convex t"
show "?hull ⊆ t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule 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)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
using assm(1,2) as(1) by auto qed
next
fix x y u v assume uv:"0≤u" "0≤v" "u+v=(1::real)" and 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) by(auto simp add: not_le)
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, 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_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] 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 thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
next def j "i - k1"
case False with i have "j ∈ {1..k2}" unfolding j_def by auto
thus ?thesis unfolding j_def[symmetric] using False
using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto 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: mem_def[of _ convex] convex_def[of ?set])
fix x assume "x∈s" thus " ∃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] by auto
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"
hence "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, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2)) }
moreover have "(∑x∈s. u * ux x + v * uy x) = 1"
unfolding setsum_addf and setsum_right_distrib[THEN sym] 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_addf and scaleR_scaleR[THEN sym] 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) by auto
next
fix t assume t:"s ⊆ t" "convex t"
fix u assume u:"∀x∈s. 0 ≤ u x" "setsum u s = (1::real)"
thus "(∑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

subsection {* Another formulation from Lars Schewe. *}

lemma setsum_constant_scaleR:
fixes y :: "'a::real_vector"
shows "(∑x∈A. y) = of_nat (card A) *R y"

apply (cases "finite A")
apply (induct set: finite)
apply (simp_all add: algebra_simps)
done

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}"
hence "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) by auto }
moreover
have "(∑v∈y ` {1..k}. setsum u {i ∈ {1..k}. y i = v}) = 1"
unfolding setsum_image_gen[OF fin, THEN sym] 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, THEN sym]
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) by auto
hence "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}"
hence "f i ∈ s" apply(subst f(2)[THEN sym]) by auto
hence "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
hence "{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) by auto
hence "card {x. Suc 0 ≤ x ∧ x ≤ card s ∧ f x = y} = 1" by auto
hence "(∑x∈{x ∈ {1..card s}. f x = y}. u (f x)) = u y"
"(∑x∈{x ∈ {1..card s}. f x = y}. u (f x) *R f x) = u y *R y"

by (auto simp add: setsum_constant_scaleR) }

hence "(∑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_cong2[of s "λy. (∑x∈{x ∈ {1..card s}. f x = y}. u (f x) *R f x)" "λv. u v *R v"]
using setsum_cong2 [of 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 o f" in exI) apply(rule_tac x=f in exI) by fastsimp
hence "y ∈ ?lhs" unfolding convex_hull_indexed by auto }
ultimately show ?thesis unfolding expand_set_eq by blast
qed

subsection {* 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" hence *:"insert a s = s" by auto
assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
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" thus ?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` by auto
next
assume "a∈s" hence *:"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_addf 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` by auto
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" "(∑x∈s. (if a = x then v else u x) *R x) = (∑x∈s. u x *R x)"
apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a∉s` by auto
ultimately show ?lhs apply(rule_tac x="λx. if a = x then v else u x" in exI) unfolding setsum_clauses(2)[OF assms] by auto
qed

subsection {* Hence some special cases. *}

lemma convex_hull_2:
"convex hull {a,b} = {u *R a + v *R b | u v. 0 ≤ u ∧ 0 ≤ v ∧ u + v = 1}"

proof- have *:"!!u. (∀x∈{a, b}. 0 ≤ u x) <-> 0 ≤ u a ∧ 0 ≤ u b" by auto have **:"finite {b}" by auto
show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
apply(rule_tac x=u in exI) apply simp apply(rule_tac x="λx. v" in exI) by simp qed

lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *R (b - a) | u. 0 ≤ u ∧ u ≤ 1}"
unfolding convex_hull_2 unfolding Collect_def
proof(rule ext) 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) by (auto simp add: algebra_simps) qed

lemma convex_hull_3:
"convex hull {a,b,c} = { u *R a + v *R b + w *R c | u v w. 0 ≤ u ∧ 0 ≤ v ∧ 0 ≤ w ∧ u + v + w = 1}"

proof-
have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
have *:"!!x y z ::real. x + y + z = 1 <-> x = 1 - y - z"
"!!x y z ::real^_. x + y + z = 1 <-> x = 1 - y - z"
by (auto simp add: field_simps)
show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) 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) by simp qed

lemma convex_hull_3_alt:
"convex hull {a,b,c} = {a + u *R (b - a) + v *R (c - a) | u v. 0 ≤ u ∧ 0 ≤ v ∧ u + v ≤ 1}"

proof- have *:"!!x y z ::real. x + y + z = 1 <-> x = 1 - y - z" by auto
show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply (simp add: algebra_simps)
apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed

subsection {* Relations among closure notions and corresponding hulls. *}

text {* TODO: Generalize linear algebra concepts defined in @{text
Euclidean_Space.thy} so that we can generalize these lemmas. *}


lemma subspace_imp_affine:
fixes s :: "(real ^ _) set" shows "subspace s ==> affine s"

unfolding subspace_def affine_def smult_conv_scaleR by auto

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

lemma subspace_imp_convex:
fixes s :: "(real ^ _) set" shows "subspace s ==> convex s"

using subspace_imp_affine affine_imp_convex by auto

lemma affine_hull_subset_span:
fixes s :: "(real ^ _) set" shows "(affine hull s) ⊆ (span s)"

by (metis hull_minimal mem_def span_inc subspace_imp_affine subspace_span)

lemma convex_hull_subset_span:
fixes s :: "(real ^ _) set" shows "(convex hull s) ⊆ (span s)"

by (metis hull_minimal mem_def 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 mem_def)


lemma affine_dependent_imp_dependent:
fixes s :: "(real ^ _) set" shows "affine_dependent s ==> dependent s"

unfolding affine_dependent_def dependent_def
using affine_hull_subset_span by auto

lemma dependent_imp_affine_dependent:
fixes s :: "(real ^ _) set"
assumes "dependent {x - a| x . x ∈ s}" "a ∉ s"
shows "affine_dependent (insert a s)"

proof-
from assms(1)[unfolded dependent_explicit smult_conv_scaleR] 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

hence "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_cong2) using `a∉s` `t⊆s` by auto
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 * by auto
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 by auto
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_cong2) using `a∉s` `t⊆s` by auto
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_addf scaleR_right_distrib)
hence "(∑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` by (auto simp add: * vector_smult_lneg)
ultimately show ?thesis unfolding affine_dependent_explicit
apply(rule_tac x="insert a t" in exI) by auto
qed

lemma convex_cone:
"convex s ∧ cone s <-> (∀x∈s. ∀y∈s. (x + y) ∈ s) ∧ (∀x∈s. ∀c≥0. (c *R x) ∈ s)" (is "?lhs = ?rhs")

proof-
{ fix x y assume "x∈s" "y∈s" and ?lhs
hence "2 *R x ∈s" "2 *R y ∈ s" unfolding cone_def by auto
hence "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) by auto }
thus ?thesis unfolding convex_def cone_def by blast
qed

lemma affine_dependent_biggerset: fixes s::"(real^'n) set"
assumes "finite s" "card s ≥ CARD('n) + 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 by auto
also have "… > CARD('n)" using assms(2)
unfolding card_Diff_singleton[OF assms(1) `a∈s`] by auto
finally show ?thesis apply(subst insert_Diff[OF `a∈s`, THEN sym])
apply(rule dependent_imp_affine_dependent)
apply(rule dependent_biggerset) by auto qed

lemma affine_dependent_biggerset_general:
assumes "finite (s::(real^'n) set)" "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 by auto
have "dim {x - a |x. x ∈ s - {a}} ≤ dim s"
apply(rule subset_le_dim) unfolding subset_eq
using `a∈s` by (auto simp add:span_superset span_sub)
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`, THEN sym])
apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed

subsection {* Caratheodory's theorem. *}

lemma convex_hull_caratheodory: fixes p::"(real^'n) set"
shows "convex hull p = {y. ∃s u. finite s ∧ s ⊆ p ∧ card s ≤ CARD('n) + 1 ∧
(∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ setsum (λv. u v *R v) s = y}"

unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
proof(rule,rule)
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
hence "∃n≤N. (∀k<n. ¬ ?P k) ∧ ?P n" apply(rule_tac ex_least_nat_le) by auto
then obtain n where "?P n" and smallest:"∀k<n. ¬ ?P k" by blast
then obtain s u where obt:"finite s" "card s = n" "s⊆p" "∀x∈s. 0 ≤ u x" "setsum u s = 1" "(∑v∈s. u v *R v) = y" by auto

have "card s ≤ CARD('n) + 1" proof(rule ccontr, simp only: not_le)
assume "CARD('n) + 1 < card s"
hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
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"
hence "setsum w (s - {v}) ≥ 0" apply(rule_tac setsum_nonneg) by auto
hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v∈s`]
using as[THEN bspec[where x=v]] and `v∈s` using `w v ≠ 0` by auto
thus False using wv(1) by auto
qed hence "i≠{}" unfolding i_def by auto

hence "t ≥ 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto
have t:"∀v∈s. u v + t * w v ≥ 0" proof
fix v assume "v∈s" hence 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 apply(rule_tac add_nonneg_nonneg)
using v apply simp apply(rule mult_nonneg_nonneg) using `t≥0` by auto next
case True hence "t ≤ u v / (- w v)" using `v∈s`
unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto
thus ?thesis unfolding real_0_le_add_iff
using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] 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
hence a:"a∈s" "u a + t * w a = 0" by auto
have *:"!!f. setsum f (s - {a}) = setsum f s - ((f a)::'a::ring)" unfolding setsum_diff1'[OF obt(1) `a∈s`] by auto
have "(∑v∈s. u v + t * w v) = 1"
unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] 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_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]]
by (simp add: vector_smult_lneg)
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 by (auto simp add: * scaleR_left_distrib)
thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
thus "∃s u. finite s ∧ s ⊆ p ∧ card s ≤ CARD('n) + 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:
"convex hull p = {x::real^'n. ∃s. finite s ∧ s ⊆ p ∧
card s ≤ CARD('n) + 1 ∧ x ∈ convex hull s}"

unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
fix x assume "x ∈ convex hull p"
then obtain s u where "finite s" "s ⊆ p" "card s ≤ CARD('n) + 1"
"∀x∈s. 0 ≤ u x" "setsum u s = 1" "(∑v∈s. u v *R v) = x"
unfolding convex_hull_caratheodory by auto
thus "∃s. finite s ∧ s ⊆ p ∧ card s ≤ CARD('n) + 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]] by auto
next
fix x assume "∃s. finite s ∧ s ⊆ p ∧ card s ≤ CARD('n) + 1 ∧ x ∈ convex hull s"
then obtain s where "finite s" "s ⊆ p" "card s ≤ CARD('n) + 1" "x ∈ convex hull s" by auto
thus "x ∈ convex hull p" using hull_mono[OF `s⊆p`] by auto
qed

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(10)
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≠{}" unfolding i_def 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` by auto
next fix y assume "y ∈ cball a (Min i)"
hence y:"norm (a - y) ≤ Min i" unfolding dist_norm[THEN sym] by auto
{ fix x assume "x∈t"
hence "Min i ≤ b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
hence "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_addf setsum_subtractf scaleR_left.setsum[THEN sym] 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) by auto
qed
qed

(* TODO: move *)
lemma compact_real_interval:
fixes a b :: real shows "compact {a..b}"

proof (rule bounded_closed_imp_compact)
have "∀y∈{a..b}. dist a y ≤ dist a b"
unfolding dist_real_def by auto
thus "bounded {a..b}" unfolding bounded_def by fast
show "closed {a..b}" by (rule closed_real_atLeastAtMost)
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_ext) unfolding image_iff mem_Collect_eq
apply rule apply auto
apply (rule_tac x=u in rev_bexI, simp)
apply (erule rev_bexI, erule rev_bexI, simp)
by auto
have "continuous_on ({0..1} × s × t)
(λz. (1 - fst z) *R fst (snd z) + fst z *R snd (snd z))"

unfolding continuous_on by (rule ballI) (intro tendsto_intros)
thus ?thesis unfolding *
apply (rule compact_continuous_image)
apply (intro compact_Times compact_real_interval assms)
done
qed

lemma compact_convex_hull: fixes s::"(real^'n) set"
assumes "compact s" shows "compact(convex hull s)"

proof(cases "s={}")
case True thus ?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 ("CARD('n) + 1"))
have *:"{x.∃sa. finite sa ∧ sa ⊆ s ∧ card sa ≤ 0 ∧ x ∈ convex hull sa} = {}"
using compact_empty by auto
case 0 thus ?case unfolding * by simp
next
case (Suc n)
show ?case proof(cases "n=0")
case True have "{x. ∃t. finite t ∧ t ⊆ s ∧ card t ≤ Suc n ∧ x ∈ convex hull t} = s"
unfolding expand_set_eq 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 thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
next
case False hence "card t = Suc 0" using t(3) `n=0` by auto
then obtain a where "t = {a}" unfolding card_Suc_eq by auto
thus ?thesis using t(2,4) by simp
qed
next
fix x assume "x∈s"
thus "∃t. finite t ∧ t ⊆ s ∧ card t ≤ Suc n ∧ x ∈ convex hull t"
apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto
qed thus ?thesis using assms by simp
next
case False have "{x. ∃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 expand_set_eq 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 mem_convex) 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"] by auto
ultimately show "∃t. finite t ∧ t ⊆ s ∧ card t ≤ Suc n ∧ x ∈ convex hull t"
apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
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
let ?P = "∃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)"

show ?P proof(cases "card t = Suc n")
case False hence "card t ≤ n" using t(3) by auto
thus ?P apply(rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) using `w∈s` and t
by(auto intro!: exI[where x=t])
next
case True then obtain a u where au:"t = insert a u" "a∉u" apply(drule_tac card_eq_SucD) by auto
show ?P proof(cases "u={}")
case True hence "x=a" using t(4)[unfolded au] by auto
show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
using t and `n≠0` unfolding au by(auto intro!: exI[where x="{a}"])
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 ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
by(auto intro!: exI[where x=u])
qed
qed
qed
thus ?thesis using compact_convex_combinations[OF assms Suc] by simp
qed
qed
qed

lemma finite_imp_compact_convex_hull:
fixes s :: "(real ^ _) set"
shows "finite s ==> compact(convex hull s)"

by (metis compact_convex_hull finite_imp_compact)

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 hence "0 < inner d d + (inner a d * 2 - inner b d * 2)"
apply(rule_tac add_pos_pos) using assms by auto
thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
by (simp add: algebra_simps inner_commute)
next
case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)"
apply(rule_tac add_pos_nonneg) using assms by auto
thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
by (simp add: algebra_simps inner_commute)
qed

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))"

proof(induct_tac rule: finite_induct[of s])
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
thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
next
case False show ?thesis using obt(3) proof(cases "u=0", case_tac[!] "v=0")
assume "u=0" "v≠0" hence "y = b" using obt by auto
thus ?thesis using False and obt(4) by auto
next
assume "u≠0" "v=0" hence "y = x" using obt by auto
thus ?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(rule ccontr)
assume "¬ x≠b" hence "y=b" unfolding obt(5)
using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
thus False using obt(4) and False by simp qed
hence *:"w *R (x - b) ≠ 0" using w(1) by auto
show ?thesis using dist_increases_online[OF *, of a y]
proof(erule_tac disjE)
assume "dist a y < dist a (y + w *R (x - b))"
hence "norm (y - a) < norm ((u + w) *R x + (v - w) *R b - a)"
unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
moreover have "(u + w) *R x + (v - w) *R b ∈ convex hull insert x s"
unfolding convex_hull_insert[OF `s≠{}`] 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) by auto
ultimately show ?thesis by auto
next
assume "dist a y < dist a (y - w *R (x - b))"
hence "norm (y - a) < norm ((u - w) *R x + (v + w) *R b - a)"
unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
moreover have "(u - w) *R x + (v + w) *R b ∈ convex hull insert x s"
unfolding convex_hull_insert[OF `s≠{}`] 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) by auto
ultimately show ?thesis by auto
qed
qed auto
qed
qed auto
qed (auto simp add: assms)

lemma simplex_furthest_le:
fixes s :: "(real ^ _) set"
assumes "finite s" "s ≠ {}"
shows "∃y∈s. ∀x∈(convex hull s). norm(x - a) ≤ norm(y - a)"

proof-
have "convex hull s ≠ {}" using hull_subset[of s convex] and assms(2) by auto
then obtain x where x:"x∈convex hull s" "∀y∈convex hull s. norm (y - a) ≤ norm (x - a)"
using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
unfolding dist_commute[of a] unfolding dist_norm by auto
thus ?thesis proof(cases "x∈s")
case False then obtain y where "y∈convex hull s" "norm (x - a) < norm (y - a)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
qed auto
qed

lemma simplex_furthest_le_exists:
fixes s :: "(real ^ _) set"
shows "finite s ==> (∀x∈(convex hull s). ∃y∈s. norm(x - a) ≤ norm(y - a))"

using simplex_furthest_le[of s] by (cases "s={}")auto

lemma simplex_extremal_le:
fixes s :: "(real ^ _) set"
assumes "finite s" "s ≠ {}"
shows "∃u∈s. ∃v∈s. ∀x∈convex hull s. ∀y ∈ convex hull s. norm(x - y) ≤ norm(u - v)"

proof-
have "convex hull s ≠ {}" using hull_subset[of s convex] and assms(2) by auto
then obtain u v where obt:"u∈convex hull s" "v∈convex hull s"
"∀x∈convex hull s. ∀y∈convex hull s. norm (x - y) ≤ norm (u - v)"

using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
thus ?thesis proof(cases "u∉s ∨ v∉s", erule_tac disjE)
assume "u∉s" then obtain y where "y∈convex hull s" "norm (u - v) < norm (y - v)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
next
assume "v∉s" then obtain y where "y∈convex hull s" "norm (v - u) < norm (y - u)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
by (auto simp add: norm_minus_commute)
qed auto
qed

lemma simplex_extremal_le_exists:
fixes s :: "(real ^ _) set"
shows "finite s ==> x ∈ convex hull s ==> y ∈ convex hull s
==> (∃u∈s. ∃v∈s. norm(x - y) ≤ norm(u - v))"

using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto

subsection {* Closest point of a convex set is unique, with a continuous projection. *}

definition
closest_point :: "'a::{real_inner,heine_borel} set => 'a => 'a" where
"closest_point s a = (SOME x. x ∈ s ∧ (∀y∈s. dist a x ≤ dist a y))"


lemma closest_point_exists:
assumes "closed s" "s ≠ {}"
shows "closest_point s a ∈ s" "∀y∈s. dist a (closest_point s a) ≤ dist a y"

unfolding closest_point_def apply(rule_tac[!] someI2_ex)
using distance_attains_inf[OF assms(1,2), of a] by auto

lemma closest_point_in_set:
"closed s ==> s ≠ {} ==> (closest_point s a) ∈ s"

by(meson closest_point_exists)

lemma closest_point_le:
"closed s ==> x ∈ s ==> dist a (closest_point s a) ≤ dist a x"

using closest_point_exists[of s] by auto

lemma closest_point_self:
assumes "x ∈ s" shows "closest_point s x = x"

unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x])
using assms by auto

lemma closest_point_refl:
"closed s ==> s ≠ {} ==> (closest_point s x = x <-> x ∈ s)"

using closest_point_in_set[of s x] closest_point_self[of x s] by auto

(* TODO: move *)
lemma norm_lt: "norm x < norm y <-> inner x x < inner y y"
unfolding norm_eq_sqrt_inner by simp

(* TODO: move *)
lemma norm_le: "norm x ≤ norm y <-> inner x x ≤ inner y y"
unfolding norm_eq_sqrt_inner by simp

lemma closer_points_lemma:
assumes "inner y z > 0"
shows "∃u>0. ∀v>0. v ≤ u --> norm(v *R z - y) < norm y"

proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
fix v assume "0<v" "v ≤ inner y z / inner z z"
thus "norm (v *R z - y) < norm y" unfolding norm_lt using z and assms
by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
qed(rule divide_pos_pos, auto) qed

lemma closer_point_lemma:
assumes "inner (y - x) (z - x) > 0"
shows "∃u>0. u ≤ 1 ∧ dist (x + u *R (z - x)) y < dist x y"

proof- obtain u where "u>0" and u:"∀v>0. v ≤ u --> norm (v *R (z - x) - (y - x)) < norm (y - x)"
using closer_points_lemma[OF assms] by auto
show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed

lemma any_closest_point_dot:
assumes "convex s" "closed s" "x ∈ s" "y ∈ s" "∀z∈s. dist a x ≤ dist a z"
shows "inner (a - x) (y - x) ≤ 0"

proof(rule ccontr) assume "¬ inner (a - x) (y - x) ≤ 0"
then obtain u where u:"u>0" "u≤1" "dist (x + u *R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
let ?z = "(1 - u) *R x + u *R y" have "?z ∈ s" using mem_convex[OF assms(1,3,4), of u] using u by auto
thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed

lemma any_closest_point_unique:
fixes x :: "'a::real_inner"
assumes "convex s" "closed s" "x ∈ s" "y ∈ s"
"∀z∈s. dist a x ≤ dist a z" "∀z∈s. dist a y ≤ dist a z"
shows "x = y"
using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
unfolding norm_pths(1) and norm_le_square
by (auto simp add: algebra_simps)

lemma closest_point_unique:
assumes "convex s" "closed s" "x ∈ s" "∀z∈s. dist a x ≤ dist a z"
shows "x = closest_point s a"

using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"]
using closest_point_exists[OF assms(2)] and assms(3) by auto

lemma closest_point_dot:
assumes "convex s" "closed s" "x ∈ s"
shows "inner (a - closest_point s a) (x - closest_point s a) ≤ 0"

apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
using closest_point_exists[OF assms(2)] and assms(3) by auto

lemma closest_point_lt:
assumes "convex s" "closed s" "x ∈ s" "x ≠ closest_point s a"
shows "dist a (closest_point s a) < dist a x"

apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
apply(rule closest_point_unique[OF assms(1-3), of a])
using closest_point_le[OF assms(2), of _ a] by fastsimp

lemma closest_point_lipschitz:
assumes "convex s" "closed s" "s ≠ {}"
shows "dist (closest_point s x) (closest_point s y) ≤ dist x y"

proof-
have "inner (x - closest_point s x) (closest_point s y - closest_point s x) ≤ 0"
"inner (y - closest_point s y) (closest_point s x - closest_point s y) ≤ 0"

apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
using closest_point_exists[OF assms(2-3)] by auto
thus ?thesis unfolding dist_norm and norm_le
using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
by (simp add: inner_add inner_diff inner_commute) qed

lemma continuous_at_closest_point:
assumes "convex s" "closed s" "s ≠ {}"
shows "continuous (at x) (closest_point s)"

unfolding continuous_at_eps_delta
using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto

lemma continuous_on_closest_point:
assumes "convex s" "closed s" "s ≠ {}"
shows "continuous_on t (closest_point s)"

by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])

subsection {* Various point-to-set separating/supporting hyperplane theorems. *}

lemma supporting_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex s" "closed s" "s ≠ {}" "z ∉ s"
shows "∃a b. ∃y∈s. inner a z < b ∧ (inner a y = b) ∧ (∀x∈s. inner a x ≥ b)"

proof-
from distance_attains_inf[OF assms(2-3)] obtain y where "y∈s" and y:"∀x∈s. dist z y ≤ dist z x" by auto
show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
apply rule defer apply rule defer apply(rule, rule ccontr) using `y∈s` proof-
show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y∈s` `z∉s` by auto
next
fix x assume "x∈s" have *:"∀u. 0 ≤ u ∧ u ≤ 1 --> dist z y ≤ dist z ((1 - u) *R y + u *R x)"
using assms(1)[unfolded convex_alt] and y and `x∈s` and `y∈s` by auto
assume "¬ inner (y - z) y ≤ inner (y - z) x" then obtain v where
"v>0" "v≤1" "dist (y + v *R (x - y)) z < dist y z"
using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
qed auto
qed

lemma separating_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex s" "closed s" "z ∉ s"
shows "∃a b. inner a z < b ∧ (∀x∈s. inner a x > b)"

proof(cases "s={}")
case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
using less_le_trans[OF _ inner_ge_zero[of z]] by auto
next
case False obtain y where "y∈s" and y:"∀x∈s. dist z y ≤ dist z x"
using distance_attains_inf[OF assms(2) False] by auto
show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))² / 2" in exI)
apply rule defer apply rule proof-
fix x assume "x∈s"
have "¬ 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
assume "∃u>0. u ≤ 1 ∧ dist (y + u *R (x - y)) z < dist y z"
then obtain u where "u>0" "u≤1" "dist (y + u *R (x - y)) z < dist y z" by auto
thus False using y[THEN bspec[where x="y + u *R (x - y)"]]
using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
using `x∈s` `y∈s` by (auto simp add: dist_commute algebra_simps) qed
moreover have "0 < norm (y - z) ^ 2" using `y∈s` `z∉s` by auto
hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
ultimately show "inner (y - z) z + (norm (y - z))² / 2 < inner (y - z) x"
unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
qed(insert `y∈s` `z∉s`, auto)
qed

lemma separating_hyperplane_closed_0:
assumes "convex (s::(real^'n) set)" "closed s" "0 ∉ s"
shows "∃a b. a ≠ 0 ∧ 0 < b ∧ (∀x∈s. inner a x > b)"

proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
case True have "norm ((basis a)::real^'n) = 1"
using norm_basis and dimindex_ge_1 by auto
thus ?thesis apply(rule_tac x="basis a" in exI, rule_tac x=1 in exI) using True by auto
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed

subsection {* Now set-to-set for closed/compact sets. *}

lemma separating_hyperplane_closed_compact:
assumes "convex (s::(real^'n) set)" "closed s" "convex t" "compact t" "t ≠ {}" "s ∩ t = {}"
shows "∃a b. (∀x∈s. inner a x < b) ∧ (∀x∈t. inner a x > b)"

proof(cases "s={}")
case True
obtain b where b:"b>0" "∀x∈t. norm x ≤ b" using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
obtain z::"real^'n" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
hence "z∉t" using b(2)[THEN bspec[where x=z]] by auto
then obtain a b where ab:"inner a z < b" "∀x∈t. b < inner a x"
using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
thus ?thesis using True by auto
next
case False then obtain y where "y∈s" by auto
obtain a b where "0 < b" "∀x∈{x - y |x y. x ∈ s ∧ y ∈ t}. b < inner a x"
using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
hence ab:"∀x∈s. ∀y∈t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
def k "Sup ((λx. inner a x) ` t)"
show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
from ab have "((λx. inner a x) ` t) *<= (inner a y - b)"
apply(erule_tac x=y in ballE) apply(rule setleI) using `y∈s` by auto
hence k:"isLub UNIV ((λx. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
fix x assume "x∈t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
next
fix x assume "x∈s"
hence "k ≤ inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
using ab[THEN bspec[where x=x]] by auto
thus "k + b / 2 < inner a x" using `0 < b` by auto
qed
qed

lemma separating_hyperplane_compact_closed:
fixes s :: "(real ^ _) set"
assumes "convex s" "compact s" "s ≠ {}" "convex t" "closed t" "s ∩ t = {}"
shows "∃a b. (∀x∈s. inner a x < b) ∧ (∀x∈t. inner a x > b)"

proof- obtain a b where "(∀x∈t. inner a x < b) ∧ (∀x∈s. b < inner a x)"
using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed

subsection {* General case without assuming closure and getting non-strict separation. *}

lemma separating_hyperplane_set_0:
assumes "convex s" "(0::real^'n) ∉ s"
shows "∃a. a ≠ 0 ∧ (∀x∈s. 0 ≤ inner a x)"

proof- let ?k = "λc. {x::real^'n. 0 ≤ inner c x}"
have "frontier (cball 0 1) ∩ (\<Inter> (?k ` s)) ≠ {}"
apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
defer apply(rule,rule,erule conjE) proof-
fix f assume as:"f ⊆ ?k ` s" "finite f"
obtain c where c:"f = ?k ` c" "c⊆s" "finite c" using finite_subset_image[OF as(2,1)] by auto
then obtain a b where ab:"a ≠ 0" "0 < b" "∀x∈convex hull c. b < inner a x"
using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
hence "∃x. norm x = 1 ∧ (∀y∈c. 0 ≤ inner y x)" apply(rule_tac x="inverse(norm a) *R a" in exI)
using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
by(auto simp add: inner_commute elim!: ballE)
thus "frontier (cball 0 1) ∩ \<Inter>f ≠ {}" unfolding c(1) frontier_cball dist_norm by auto
qed(insert closed_halfspace_ge, auto)
then obtain x where "norm x = 1" "∀y∈s. x∈?k y" unfolding frontier_cball dist_norm by auto
thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed

lemma separating_hyperplane_sets:
assumes "convex s" "convex (t::(real^'n) set)" "s ≠ {}" "t ≠ {}" "s ∩ t = {}"
shows "∃a b. a ≠ 0 ∧ (∀x∈s. inner a x ≤ b) ∧ (∀x∈t. inner a x ≥ b)"

proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
obtain a where "a≠0" "∀x∈{x - y |x y. x ∈ t ∧ y ∈ s}. 0 ≤ inner a x"
using assms(3-5) by auto
hence "∀x∈t. ∀y∈s. inner a y ≤ inner a x"
by (force simp add: inner_diff)
thus ?thesis
apply(rule_tac x=a in exI, rule_tac x="Sup ((λx. inner a x) ` s)" in exI) using `a≠0`
apply auto
apply (rule Sup[THEN isLubD2])
prefer 4
apply (rule Sup_least)
using assms(3-5) apply (auto simp add: setle_def)
apply metis
done
qed

subsection {* More convexity generalities. *}

lemma convex_closure:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "convex(closure s)"

unfolding convex_def Ball_def closure_sequential
apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
apply(rule_tac x="λn. u *R xb n + v *R xc n" in exI) apply(rule,rule)
apply(rule assms[unfolded convex_def, rule_format]) prefer 6
apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto

lemma convex_interior:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "convex(interior s)"

unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
fix x y u assume u:"0 ≤ u" "u ≤ (1::real)"
fix e d assume ed:"ball x e ⊆ s" "ball y d ⊆ s" "0<d" "0<e"
show "∃e>0. ball ((1 - u) *R x + u *R y) e ⊆ s" apply(rule_tac x="min d e" in exI)
apply rule unfolding subset_eq defer apply rule proof-
fix z assume "z ∈ ball ((1 - u) *R x + u *R y) (min d e)"
hence "(1- u) *R (z - u *R (y - x)) + u *R (z + (1 - u) *R (y - x)) ∈ s"
apply(rule_tac assms[unfolded convex_alt, rule_format])
using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
thus "z ∈ s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed

lemma convex_hull_eq_empty[simp]: "convex hull s = {} <-> s = {}"
using hull_subset[of s convex] convex_hull_empty by auto

subsection {* Moving and scaling convex hulls. *}

lemma convex_hull_translation_lemma:
"convex hull ((λx. a + x) ` s) ⊆ (λx. a + x) ` (convex hull s)"

by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono mem_def)

lemma convex_hull_bilemma: fixes neg
assumes "(∀s a. (convex hull (up a s)) ⊆ up a (convex hull s))"
shows "(∀s. up a (up (neg a) s) = s) ∧ (∀s. up (neg a) (up a s) = s) ∧ (∀s t a. s ⊆ t --> up a s ⊆ up a t)
==> ∀s. (convex hull (up a s)) = up a (convex hull s)"

using assms by(metis subset_antisym)

lemma convex_hull_translation:
"convex hull ((λx. a + x) ` s) = (λx. a + x) ` (convex hull s)"

apply(rule convex_hull_bilemma[rule_format, of _ _ "λa. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto

lemma convex_hull_scaling_lemma:
"(convex hull ((λx. c *R x) ` s)) ⊆ (λx. c *R x) ` (convex hull s)"

by (metis convex_convex_hull convex_scaling hull_subset mem_def subset_hull subset_image_iff)

lemma convex_hull_scaling:
"convex hull ((λx. c *R x) ` s) = (λx. c *R x) ` (convex hull s)"

apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)

lemma convex_hull_affinity:
"convex hull ((λx. a + c *R x) ` s) = (λx. a + c *R x) ` (convex hull s)"

by(simp only: image_image[THEN sym] convex_hull_scaling convex_hull_translation)

subsection {* Convex set as intersection of halfspaces. *}

lemma convex_halfspace_intersection:
fixes s :: "(real ^ _) set"
assumes "closed s" "convex s"
shows "s = \<Inter> {h. s ⊆ h ∧ (∃a b. h = {x. inner a x ≤ b})}"

apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof-
fix x assume "∀xa. s ⊆ xa ∧ (∃a b. xa = {x. inner a x ≤ b}) --> x ∈ xa"
hence "∀a b. s ⊆ {x. inner a x ≤ b} --> x ∈ {x. inner a x ≤ b}" by blast
thus "x∈s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
qed auto

subsection {* Radon's theorem (from Lars Schewe). *}

lemma radon_ex_lemma:
assumes "finite c" "affine_dependent c"
shows "∃u. setsum u c = 0 ∧ (∃v∈c. u v ≠ 0) ∧ setsum (λv. u v *R v) c = 0"

proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
thus ?thesis apply(rule_tac x="λv. if v∈s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
and setsum_restrict_set[OF assms(1), THEN sym]
by(auto simp add: Int_absorb1) qed

lemma radon_s_lemma:
assumes "finite s" "setsum f s = (0::real)"
shows "setsum f {x∈s. 0 < f x} = - setsum f {x∈s. f x < 0}"

proof- have *:"!!x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" by auto
show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
using assms(2) by assumption qed

lemma radon_v_lemma:
assumes "finite s" "setsum f s = 0" "∀x. g x = (0::real) --> f x = (0::real^_)"
shows "(setsum f {x∈s. 0 < g x}) = - setsum f {x∈s. g x < 0}"

proof-
have *:"!!x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" using assms(3) by auto
show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
using assms(2) by assumption qed

lemma radon_partition:
assumes "finite c" "affine_dependent c"
shows "∃m p. m ∩ p = {} ∧ m ∪ p = c ∧ (convex hull m) ∩ (convex hull p) ≠ {}"
proof-
obtain u v where uv:"setsum u c = 0" "v∈c" "u v ≠ 0" "(∑v∈c. u v *R v) = 0" using radon_ex_lemma[OF assms] by auto
have fin:"finite {x ∈ c. 0 < u x}" "finite {x ∈ c. 0 > u x}" using assms(1) by auto
def z "(inverse (setsum u {x∈c. u x > 0})) *R setsum (λx. u x *R x) {x∈c. u x > 0}"
have "setsum u {x ∈ c. 0 < u x} ≠ 0" proof(cases "u v ≥ 0")
case False hence "u v < 0" by auto
thus ?thesis proof(cases "∃w∈{x ∈ c. 0 < u x}. u w > 0")
case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
next
case False hence "setsum u c ≤ setsum (λx. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)

hence *:"setsum u {x∈c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
moreover have "setsum u ({x ∈ c. 0 < u x} ∪ {x ∈ c. u x < 0}) = setsum u c"
"(∑x∈{x ∈ c. 0 < u x} ∪ {x ∈ c. u x < 0}. u x *R x) = (∑x∈c. u x *R x)"

using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
hence "setsum u {x ∈ c. 0 < u x} = - setsum u {x ∈ c. 0 > u x}"
"(∑x∈{x ∈ c. 0 < u x}. u x *R x) = - (∑x∈{x ∈ c. 0 > u x}. u x *R x)"

unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add: setsum_Un_zero[OF fin, THEN sym])
moreover have "∀x∈{v ∈ c. u v < 0}. 0 ≤ inverse (setsum u {x ∈ c. 0 < u x}) * - u x"
apply (rule) apply (rule mult_nonneg_nonneg) using * by auto

ultimately have "z ∈ convex hull {v ∈ c. u v ≤ 0}" unfolding convex_hull_explicit mem_Collect_eq
apply(rule_tac x="{v ∈ c. u v < 0}" in exI, rule_tac x="λy. inverse (setsum u {x∈c. u x > 0}) * - u y" in exI)
using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
moreover have "∀x∈{v ∈ c. 0 < u v}. 0 ≤ inverse (setsum u {x ∈ c. 0 < u x}) * u x"
apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
hence "z ∈ convex hull {v ∈ c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
apply(rule_tac x="{v ∈ c. 0 < u v}" in exI, rule_tac x="λy. inverse (setsum u {x∈c. u x > 0}) * u y" in exI)
using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
ultimately show ?thesis apply(rule_tac x="{v∈c. u v ≤ 0}" in exI, rule_tac x="{v∈c. u v > 0}" in exI) by auto
qed

lemma radon: assumes "affine_dependent c"
obtains m p where "m⊆c" "p⊆c" "m ∩ p = {}" "(convex hull m) ∩ (convex hull p) ≠ {}"

proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
hence *:"finite s" "affine_dependent s" and s:"s ⊆ c" unfolding affine_dependent_explicit by auto
from radon_partition[OF *] guess m .. then guess p ..
thus ?thesis apply(rule_tac that[of p m]) using s by auto qed

subsection {* Helly's theorem. *}

lemma helly_induct: fixes f::"(real^'n) set set"
assumes "card f = n" "n ≥ CARD('n) + 1"
"∀s∈f. convex s" "∀t⊆f. card t = CARD('n) + 1 --> \<Inter> t ≠ {}"
shows "\<Inter> f ≠ {}"

using assms proof(induct n arbitrary: f)
case (Suc n)
have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
show "\<Inter> f ≠ {}" apply(cases "n = CARD('n)") apply(rule Suc(5)[rule_format])
unfolding `card f = Suc n` proof-
assume ng:"n ≠ CARD('n)" hence "∃X. ∀s∈f. X s ∈ \<Inter>(f - {s})" apply(rule_tac bchoice) unfolding ex_in_conv
apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
then obtain X where X:"∀s∈f. X s ∈ \<Inter>(f - {s})" by auto
show ?thesis proof(cases "inj_on X f")
case False then obtain s t where st:"s≠t" "s∈f" "t∈f" "X s = X t" unfolding inj_on_def by auto
hence *:"\<Inter> f = \<Inter> (f - {s}) ∩ \<Inter> (f - {t})" by auto
show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
apply(rule, rule X[rule_format]) using X st by auto
next case True then obtain m p where mp:"m ∩ p = {}" "m ∪ p = X ` f" "convex hull m ∩ convex hull p ≠ {}"
using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
have "m ⊆ X ` f" "p ⊆ X ` f" using mp(2) by auto
then obtain g h where gh:"m = X ` g" "p = X ` h" "g ⊆ f" "h ⊆ f" unfolding subset_image_iff by auto
hence "f ∪ (g ∪ h) = f" by auto
hence f:"f = g ∪ h" using inj_on_Un_image_eq_iff[of X f "g ∪ h"] and True
unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
have *:"g ∩ h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
have "convex hull (X ` h) ⊆ \<Inter> g" "convex hull (X ` g) ⊆ \<Inter> h"
apply(rule_tac [!] hull_minimal) using Suc gh(3-4) unfolding mem_def unfolding subset_eq
apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
fix x assume "x∈X ` g" then guess y unfolding image_iff ..
thus "x∈\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
fix x assume "x∈X ` h" then guess y unfolding image_iff ..
thus "x∈\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
qed(auto)
thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
qed(insert dimindex_ge_1, auto) qed(auto)

lemma helly: fixes f::"(real^'n) set set"
assumes "card f ≥ CARD('n) + 1" "∀s∈f. convex s"
"∀t⊆f. card t = CARD('n) + 1 --> \<Inter> t ≠ {}"
shows "\<Inter> f ≠{}"

apply(rule helly_induct) using assms by auto

subsection {* Convex hull is "preserved" by a linear function. *}

lemma convex_hull_linear_image:
assumes "bounded_linear f"
shows "f ` (convex hull s) = convex hull (f ` s)"

apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3
apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
proof-
interpret f: bounded_linear f by fact
show "convex {x. f x ∈ convex hull f ` s}"
unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
interpret f: bounded_linear f by fact
show "convex {x. x ∈ f ` (convex hull s)}" using convex_convex_hull[unfolded convex_def, of s]
unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
qed auto

lemma in_convex_hull_linear_image:
assumes "bounded_linear f" "x ∈ convex hull s"
shows "(f x) ∈ convex hull (f ` s)"

using convex_hull_linear_image[OF assms(1)] assms(2) by auto

subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}

lemma compact_frontier_line_lemma:
fixes s :: "(real ^ _) set"
assumes "compact s" "0 ∈ s" "x ≠ 0"
obtains u where "0 ≤ u" "(u *R x) ∈ frontier s" "∀v>u. (v *R x) ∉ s"

proof-
obtain b where b:"b>0" "∀x∈s. norm x ≤ b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
let ?A = "{y. ∃u. 0 ≤ u ∧ u ≤ b / norm(x) ∧ (y = u *R x)}"
have A:"?A = (λu. u *R x) ` {0 .. b / norm x}"
by auto
have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
apply(rule, rule continuous_vmul)
apply(rule continuous_at_id) by(rule compact_real_interval)
moreover have "{y. ∃u≥0. u ≤ b / norm x ∧ y = u *R x} ∩ s ≠ {}" apply(rule not_disjointI[OF _ assms(2)])
unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
ultimately obtain u y where obt: "u≥0" "u ≤ b / norm x" "y = u *R x"
"y∈?A" "y∈s" "∀z∈?A ∩ s. dist 0 z ≤ dist 0 y"
using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto

have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
{ fix v assume as:"v > u" "v *R x ∈ s"
hence "v ≤ b / norm x" using b(2)[rule_format, OF as(2)]
using `u≥0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
hence "norm (v *R x) ≤ norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer
apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI)
using as(1) `u≥0` by(auto simp add:field_simps)
hence False unfolding obt(3) using `u≥0` `norm x > 0` `v>u` by(auto simp add:field_simps)
} note u_max = this

have "u *R x ∈ frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *R x" in bexI) unfolding obt(3)[THEN sym]
prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *R x" in exI) apply(rule, rule) proof-
fix e assume "0 < e" and as:"(u + e / 2 / norm x) *R x ∈ s"
hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
thus False using u_max[OF _ as] by auto
qed(insert `y∈s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
thus ?thesis by(metis that[of u] u_max obt(1))
qed

lemma starlike_compact_projective:
assumes "compact s" "cball (0::real^'n) 1 ⊆ s "
"∀x∈s. ∀u. 0 ≤ u ∧ u < 1 --> (u *R x) ∈ (s - frontier s )"
shows "s homeomorphic (cball (0::real^'n) 1)"

proof-
have fs:"frontier s ⊆ s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
def pi "λx::real^'n. inverse (norm x) *R x"
have "0 ∉ frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
have injpi:"!!x y. pi x = pi y ∧ norm x = norm y <-> x = y" unfolding pi_def by auto

have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
apply rule unfolding pi_def
apply (rule continuous_mul)
apply (rule continuous_at_inv[unfolded o_def])
apply (rule continuous_at_norm)
apply simp
apply (rule continuous_at_id)
done
def sphere "{x::real^'n. norm x = 1}"
have pi:"!!x. x ≠ 0 ==> pi x ∈ sphere" "!!x u. u>0 ==> pi (u *R x) = pi x" unfolding pi_def sphere_def by auto

have "0∈s" using assms(2) and centre_in_cball[of 0 1] by auto
have front_smul:"∀x∈frontier s. ∀u≥0. u *R x ∈ s <-> u ≤ 1" proof(rule,rule,rule)
fix x u assume x:"x∈frontier s" and "(0::real)≤u"
hence "x≠0" using `0∉frontier s` by auto
obtain v where v:"0 ≤ v" "v *R x ∈ frontier s" "∀w>v. w *R x ∉ s"
using compact_frontier_line_lemma[OF assms(1) `0∈s` `x≠0`] by auto
have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
assume "v>1" thus False using assms(3)[THEN bspec[where x="v *R x"], THEN spec[where x="inverse v"]]
using v and x and fs unfolding inverse_less_1_iff by auto qed
show "u *R x ∈ s <-> u ≤ 1" apply rule using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
assume "u≤1" thus "u *R x ∈ s" apply(cases "u=1")
using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0≤u` and x and fs by auto qed auto qed

have "∃surf. homeomorphism (frontier s) sphere pi surf"
apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_ext,rule)
unfolding inj_on_def prefer 3 apply(rule,rule,rule)
proof- fix x assume "x∈pi ` frontier s" then obtain y where "y∈frontier s" "x = pi y" by auto
thus "x ∈ sphere" using pi(1)[of y] and `0 ∉ frontier s` by auto
next fix x assume "x∈sphere" hence "norm x = 1" "x≠0" unfolding sphere_def by auto
then obtain u where "0 ≤ u" "u *R x ∈ frontier s" "∀v>u. v *R x ∉ s"
using compact_frontier_line_lemma[OF assms(1) `0∈s`, of x] by auto
thus "x ∈ pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *R x" in bexI) using `norm x = 1` `0∉frontier s` by auto
next fix x y assume as:"x ∈ frontier s" "y ∈ frontier s" "pi x = pi y"
hence xys:"x∈s" "y∈s" using fs by auto
from as(1,2) have nor:"norm x ≠ 0" "norm y ≠ 0" using `0∉frontier s` by auto
from nor have x:"x = norm x *R ((inverse (norm y)) *R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto
from nor have y:"y = norm y *R ((inverse (norm x)) *R x)" unfolding as(3)[unfolded pi_def] by auto
have "0 ≤ norm y * inverse (norm x)" "0 ≤ norm x * inverse (norm y)"
unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
qed(insert `0 ∉ frontier s`, auto)
then obtain surf where surf:"∀x∈frontier s. surf (pi x) = x" "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
"∀y∈sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf"
unfolding homeomorphism_def by auto

have cont_surfpi:"continuous_on (UNIV - {0}) (surf o pi)" apply(rule continuous_on_compose, rule contpi)
apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto

{ fix x assume as:"x ∈ cball (0::real^'n) 1"
have "norm x *R surf (pi x) ∈ s" proof(cases "x=0 ∨ norm x = 1")
case False hence "pi x ∈ sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
apply(rule_tac fs[unfolded subset_eq, rule_format])
unfolding surf(5)[THEN sym] by auto
next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
unfolding surf(5)[unfolded sphere_def, THEN sym] using `0∈s` by auto qed } note hom = this

{ fix x assume "x∈s"
hence "x ∈ (λx. norm x *R surf (pi x)) ` cball 0 1" proof(cases "x=0")
case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
next let ?a = "inverse (norm (surf (pi x)))"
case False hence invn:"inverse (norm x) ≠ 0" by auto
from False have pix:"pi x∈sphere" using pi(1) by auto
hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
hence **:"norm x *R (?a *R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
hence *:"?a * norm x > 0" and"?a > 0" "?a ≠ 0" using surf(5) `0∉frontier s` apply -
apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
have "norm (surf (pi x)) ≠ 0" using ** False by auto
hence "norm x = norm ((?a * norm x) *R surf (pi x))"
unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *R surf (pi x))"
unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
moreover have "surf (pi x) ∈ frontier s" using surf(5) pix by auto
hence "dist 0 (inverse (norm (surf (pi x))) *R x) ≤ 1" unfolding dist_norm
using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
using False `x∈s` by(auto simp add:field_simps)
ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *R x" in bexI)
apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
unfolding pi(2)[OF `?a > 0`] by auto
qed } note hom2 = this

show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="λx. norm x *R surf (pi x)"])
apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
fix x::"real^'n" assume as:"x ∈ cball 0 1"
thus "continuous (at x) (λx. norm x *R surf (pi x))" proof(cases "x=0")
case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
next guess a using UNIV_witness[where 'a = 'n] ..
obtain B where B:"∀x∈s. norm x ≤ B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis a" in ballE) defer apply(erule_tac x="basis a" in ballE)
unfolding Ball_def mem_cball dist_norm by (auto simp add: norm_basis[unfolded One_nat_def])
case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
fix e and x::"real^'n" assume as:"norm x < e / B" "0 < norm x" "0<e"
hence "surf (pi x) ∈ frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
hence "norm (surf (pi x)) ≤ B" using B fs by auto
hence "norm x * norm (surf (pi x)) ≤ norm x * B" using as(2) by auto
also have "… < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
also have "… = e" using `B>0` by auto
finally show "norm x * norm (surf (pi x)) < e" by assumption
qed(insert `B>0`, auto) qed
next { fix x assume as:"surf (pi x) = 0"
have "x = 0" proof(rule ccontr)
assume "x≠0" hence "pi x ∈ sphere" using pi(1) by auto
hence "surf (pi x) ∈ frontier s" using surf(5) by auto
thus False using `0∉frontier s` unfolding as by simp qed
} note surf_0 = this
show "inj_on (λx. norm x *R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
fix x y assume as:"x ∈ cball 0 1" "y ∈ cball 0 1" "norm x *R surf (pi x) = norm y *R surf (pi y)"
thus "x=y" proof(cases "x=0 ∨ y=0")
case True thus ?thesis using as by(auto elim: surf_0) next
case False
hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
moreover have "pi x ∈ sphere" "pi y ∈ sphere" using pi(1) False by auto
ultimately have *:"pi x = pi y" using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]] by auto
moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
ultimately show ?thesis using injpi by auto qed qed
qed auto qed

lemma homeomorphic_convex_compact_lemma: fixes s::"(real^'n) set"
assumes "convex s" "compact s" "cball 0 1 ⊆ s"
shows "s homeomorphic (cball (0::real^'n) 1)"

apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
fix x u assume as:"x ∈ s" "0 ≤ u" "u < (1::real)"
hence "u *R x ∈ interior s" unfolding interior_def mem_Collect_eq
apply(rule_tac x="ball (u *R x) (1 - u)" in exI) apply(rule, rule open_ball)
unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
fix y assume "dist (u *R x) y < 1 - u"
hence "inverse (1 - u) *R (y - u *R x) ∈ s"
using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
apply (rule mult_left_le_imp_le[of "1 - u"])
unfolding mult_assoc[symmetric] using `u<1` by auto
thus "y ∈ s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *R (y - u *R x)" x "1 - u" u]
using as unfolding scaleR_scaleR by auto qed auto
thus "u *R x ∈ s - frontier s" using frontier_def and interior_subset by auto qed

lemma homeomorphic_convex_compact_cball: fixes e::real and s::"(real^'n) set"
assumes "convex s" "compact s" "interior s ≠ {}" "0 < e"
shows "s homeomorphic (cball (b::real^'n) e)"

proof- obtain a where "a∈interior s" using assms(3) by auto
then obtain d where "d>0" and d:"cball a d ⊆ s" unfolding mem_interior_cball by auto
let ?d = "inverse d" and ?n = "0::real^'n"
have "cball ?n 1 ⊆ (λx. inverse d *R (x - a)) ` s"
apply(rule, rule_tac x="d *R x + a" in image_eqI) defer
apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
by(auto simp add: mult_right_le_one_le)
hence "(λx. inverse d *R (x - a)) ` s homeomorphic cball ?n 1"
using homeomorphic_convex_compact_lemma[of "(λx. ?d *R -a + ?d *R x) ` s", OF convex_affinity compact_affinity]
using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *R -a"]])
using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed

lemma homeomorphic_convex_compact: fixes s::"(real^'n) set" and t::"(real^'n) set"
assumes "convex s" "compact s" "interior s ≠ {}"
"convex t" "compact t" "interior t ≠ {}"
shows "s homeomorphic t"

using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)

subsection {* Epigraphs of convex functions. *}

definition "epigraph s (f::_ => real) = {xy. fst xy ∈ s ∧ f (fst xy) ≤ snd xy}"

lemma mem_epigraph: "(x, y) ∈ epigraph s f <-> x ∈ s ∧ f x ≤ y" unfolding epigraph_def by auto

(** This might break sooner or later. In fact it did already once. **)
lemma convex_epigraph:
"convex(epigraph s f) <-> convex_on s f ∧ convex s"

unfolding convex_def convex_on_def
unfolding Ball_def split_paired_All epigraph_def
unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)

lemma convex_epigraphI:
"convex_on s f ==> convex s ==> convex(epigraph s f)"

unfolding convex_epigraph by auto

lemma convex_epigraph_convex:
"convex s ==> convex_on s f <-> convex(epigraph s f)"

by(simp add: convex_epigraph)

subsection {* Use this to derive general bound property of convex function. *}

lemma convex_on:
assumes "convex s"
shows "convex_on s f <-> (∀k u x. (∀i∈{1..k::nat}. 0 ≤ u i ∧ x i ∈ s) ∧ setsum u {1..k} = 1 -->
f (setsum (λi. u i *R x i) {1..k} ) ≤ setsum (λi. u i * f(x i)) {1..k} ) "

unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
apply safe
apply (drule_tac x=k in spec)
apply (drule_tac x=u in spec)
apply (drule_tac x="λi. (x i, f (x i))" in spec)
apply simp
using assms[unfolded convex] apply simp
apply(rule_tac y="∑i = 1..k. u i * f (fst (x i))" in order_trans)
defer apply(rule setsum_mono) apply(erule_tac x=i in allE) unfolding real_scaleR_def
apply(rule mult_left_mono)using assms[unfolded convex] by auto


subsection {* Convexity of general and special intervals. *}

lemma is_interval_convex:
fixes s :: "(real ^ _) set"
assumes "is_interval s" shows "convex s"

unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
fix x y u v assume as:"x ∈ s" "y ∈ s" "0 ≤ u" "0 ≤ v" "u + v = (1::real)"
hence *:"u = 1 - v" "1 - v ≥ 0" and **:"v = 1 - u" "1 - u ≥ 0" by auto
{ fix a b assume "¬ b ≤ u * a + v * b"
hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
hence "a < b" unfolding * using as(4) *(2) apply(rule_tac mult_left_less_imp_less[of "1 - v"]) by(auto simp add: field_simps)
hence "a ≤ u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
} moreover
{ fix a b assume "¬ u * a + v * b ≤ a"
hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
hence "u * a + v * b ≤ b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
ultimately show "u *R x + v *R y ∈ s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
using as(3-) dimindex_ge_1 by auto qed

lemma is_interval_connected:
fixes s :: "(real ^ _) set"
shows "is_interval s ==> connected s"

using is_interval_convex convex_connected by auto

lemma convex_interval: "convex {a .. b}" "convex {a<..<b::real^'n}"
apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto

(* FIXME: rewrite these lemmas without using vec1
subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}

lemma is_interval_1:
"is_interval s <-> (∀a∈s. ∀b∈s. ∀ x. dest_vec1 a ≤ dest_vec1 x ∧ dest_vec1 x ≤ dest_vec1 b --> x ∈ s)"
unfolding is_interval_def forall_1 by auto

lemma is_interval_connected_1: "is_interval s <-> connected (s::(real^1) set)"
apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
fix a b x assume as:"connected s" "a ∈ s" "b ∈ s" "dest_vec1 a ≤ dest_vec1 x" "dest_vec1 x ≤ dest_vec1 b" "x∉s"
hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
{ fix y assume "y ∈ s" have "y ∈ ?halfr ∪ ?halfl" apply(rule ccontr)
using as(6) `y∈s` by (auto simp add: inner_vector_def) }
moreover have "a∈?halfl" "b∈?halfr" using * by (auto simp add: inner_vector_def)
hence "?halfl ∩ s ≠ {}" "?halfr ∩ s ≠ {}" using as(2-3) by auto
ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI)
apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
by(auto simp add: field_simps) qed

lemma is_interval_convex_1:
"is_interval s <-> convex (s::(real^1) set)"
by(metis is_interval_convex convex_connected is_interval_connected_1)

lemma convex_connected_1:
"connected s <-> convex (s::(real^1) set)"
by(metis is_interval_convex convex_connected is_interval_connected_1)
*)

subsection {* Another intermediate value theorem formulation. *}

lemma ivt_increasing_component_on_1: fixes f::"real => real^'n"
assumes "a ≤ b" "continuous_on {a .. b} f" "(f a)$k ≤ y" "y ≤ (f b)$k"
shows "∃x∈{a..b}. (f x)$k = y"

proof- have "f a ∈ f ` {a..b}" "f b ∈ f ` {a..b}" apply(rule_tac[!] imageI)
using assms(1) by(auto simp add: vector_le_def)
thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
using assms by(auto intro!: imageI) qed

lemma ivt_increasing_component_1: fixes f::"real => real^'n"
shows "a ≤ b ==> ∀x∈{a .. b}. continuous (at x) f
==> f a$k ≤ y ==> y ≤ f b$k ==> ∃x∈{a..b}. (f x)$k = y"

by(rule ivt_increasing_component_on_1)
(auto simp add: continuous_at_imp_continuous_on)


lemma ivt_decreasing_component_on_1: fixes f::"real => real^'n"
assumes "a ≤ b" "continuous_on {a .. b} f" "(f b)$k ≤ y" "y ≤ (f a)$k"
shows "∃x∈{a..b}. (f x)$k = y"

apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
by auto

lemma ivt_decreasing_component_1: fixes f::"real => real^'n"
shows "a ≤ b ==> ∀x∈{a .. b}. continuous (at x) f
==> f b$k ≤ y ==> y ≤ f a$k ==> ∃x∈{a..b}. (f x)$k = y"

by(rule ivt_decreasing_component_on_1)
(auto simp: continuous_at_imp_continuous_on)


subsection {* A bound within a convex hull, and so an interval. *}

lemma convex_on_convex_hull_bound:
assumes "convex_on (convex hull s) f" "∀x∈s. f x ≤ b"
shows "∀x∈ convex hull s. f x ≤ b"
proof
fix x assume "x∈convex hull s"
then obtain k u v where obt:"∀i∈{1..k::nat}. 0 ≤ u i ∧ v i ∈ s" "setsum u {1..k} = 1" "(∑i = 1..k. u i *R v i) = x"
unfolding convex_hull_indexed mem_Collect_eq by auto
have "(∑i = 1..k. u i * f (v i)) ≤ b" using setsum_mono[of "{1..k}" "λi. u i * f (v i)" "λi. u i * b"]
unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
using assms(2) obt(1) by auto
thus "f x ≤ b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed

lemma unit_interval_convex_hull:
"{0::real^'n .. 1} = convex hull {x. ∀i. (x$i = 0) ∨ (x$i = 1)}" (is "?int = convex hull ?points")

proof- have 01:"{0,1} ⊆ convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
{ fix n x assume "x∈{0::real^'n .. 1}" "n ≤ CARD('n)" "card {i. x$i ≠ 0} ≤ n"
hence "x∈convex hull ?points" proof(induct n arbitrary: x)
case 0 hence "x = 0" apply(subst Cart_eq) apply rule by auto
thus "x∈convex hull ?points" using 01 by auto
next
case (Suc n) show "x∈convex hull ?points" proof(cases "{i. x$i ≠ 0} = {}")
case True hence "x = 0" unfolding Cart_eq by auto
thus "x∈convex hull ?points" using 01 by auto
next
case False def xi "Min ((λi. x$i) ` {i. x$i ≠ 0})"
have "xi ∈ (λi. x$i) ` {i. x$i ≠ 0}" unfolding xi_def apply(rule Min_in) using False by auto
then obtain i where i':"x$i = xi" "x$i ≠ 0" by auto
have i:"!!j. x$j > 0 ==> x$i ≤ x$j"
unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
defer apply(rule_tac x=j in bexI) using i' by auto
have i01:"x$i ≤ 1" "x$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i] using i'(2) `x$i ≠ 0`
by auto
show ?thesis proof(cases "x$i=1")
case True have "∀j∈{i. x$i ≠ 0}. x$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq proof-
fix j assume "x $ j ≠ 0" "x $ j ≠ 1"
hence j:"x$j ∈ {0<..<1}" using Suc(2) by(auto simp add: vector_le_def elim!:allE[where x=j])
hence "x$j ∈ op $ x ` {i. x $ i ≠ 0}" by auto
hence "x$j ≥ x$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
thus False using True Suc(2) j by(auto simp add: vector_le_def elim!:ballE[where x=j]) qed
thus "x∈convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
by auto
next let ?y = "λj. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
case False hence *:"x = x$i *R (χ j. if x$j = 0 then 0 else 1) + (1 - x$i) *R (χ j. ?y j)" unfolding Cart_eq
by(auto simp add: field_simps)
{ fix j have "x$j ≠ 0 ==> 0 ≤ (x $ j - x $ i) / (1 - x $ i)" "(x $ j - x $ i) / (1 - x $ i) ≤ 1"
apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps)
hence "0 ≤ ?y j ∧ ?y j ≤ 1" by auto }
moreover have "i∈{j. x$j ≠ 0} - {j. ((χ j. ?y j)::real^'n) $ j ≠ 0}" using i01 by auto
hence "{j. x$j ≠ 0} ≠ {j. ((χ j. ?y j)::real^'n) $ j ≠ 0}" by auto
hence **:"{j. ((χ j. ?y j)::real^'n) $ j ≠ 0} ⊂ {j. x$j ≠ 0}" apply - apply rule by auto
have "card {j. ((χ j. ?y j)::real^'n) $ j ≠ 0} ≤ n" using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
unfolding mem_interval using i01 Suc(3) by auto
qed qed qed } note * = this
show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
apply(rule_tac n2="CARD('n)" in *) prefer 3 apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
by(auto simp add: vector_le_def mem_def[of _ convex]) qed

subsection {* And this is a finite set of vertices. *}

lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. 1::real^'n} = convex hull s"
apply(rule that[of "{x::real^'n. ∀i. x$i=0 ∨ x$i=1}"])
apply(rule finite_subset[of _ "(λs. (χ i. if i∈s then 1::real else 0)::real^'n) ` UNIV"])
prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
fix x::"real^'n" assume as:"∀i. x $ i = 0 ∨ x $ i = 1"
show "x ∈ (λs. χ i. if i ∈ s then 1 else 0) ` UNIV" apply(rule image_eqI[where x="{i. x$i = 1}"])
unfolding Cart_eq using as by auto qed auto

subsection {* Hence any cube (could do any nonempty interval). *}

lemma cube_convex_hull:
assumes "0 < d" obtains s::"(real^'n) set" where "finite s" "{x - (χ i. d) .. x + (χ i. d)} = convex hull s"
proof-
let ?d = "(χ i. d)::real^'n"
have *:"{x - ?d .. x + ?d} = (λy. x - ?d + (2 * d) *R y) ` {0 .. 1}" apply(rule set_ext, rule)
unfolding image_iff defer apply(erule bexE) proof-
fix y assume as:"y∈{x - ?d .. x + ?d}"
{ fix i::'n have "x $ i ≤ d + y $ i" "y $ i ≤ d + x $ i" using as[unfolded mem_interval, THEN spec[where x=i]]
by auto
hence "1 ≥ inverse d * (x $ i - y $ i)" "1 ≥ inverse d * (y $ i - x $ i)"
apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
using assms by(auto simp add: field_simps)
hence "inverse d * (x $ i * 2) ≤ 2 + inverse d * (y $ i * 2)"
"inverse d * (y $ i * 2) ≤ 2 + inverse d * (x $ i * 2)"
by(auto simp add:field_simps) }
hence "inverse (2 * d) *R (y - (x - ?d)) ∈ {0..1}" unfolding mem_interval using assms
by(auto simp add: Cart_eq field_simps)
thus "∃z∈{0..1}. y = x - ?d + (2 * d) *R z" apply- apply(rule_tac x="inverse (2 * d) *R (y - (x - ?d))" in bexI)
using assms by(auto simp add: Cart_eq vector_le_def)
next
fix y z assume as:"z∈{0..1}" "y = x - ?d + (2*d) *R z"
have "!!i. 0 ≤ d * z $ i ∧ d * z $ i ≤ d" using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
using assms by(auto simp add: Cart_eq)
thus "y ∈ {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
apply(erule_tac x=i in allE) using assms by(auto simp add: Cart_eq) qed
obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
thus ?thesis apply(rule_tac that[of "(λy. x - ?d + (2 * d) *R y)` s"]) unfolding * and convex_hull_affinity by auto qed

subsection {* Bounded convex function on open set is continuous. *}

lemma convex_on_bounded_continuous:
fixes s :: "('a::real_normed_vector) set"
assumes "open s" "convex_on s f" "∀x∈s. abs(f x) ≤ b"
shows "continuous_on s f"

apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
fix x e assume "x∈s" "(0::real) < e"
def B "abs b + 1"
have B:"0 < B" "!!x. x∈s ==> abs (f x) ≤ B"
unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
obtain k where "k>0"and k:"cball x k ⊆ s" using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]] using `x∈s` by auto
show "∃d>0. ∀x'. norm (x' - x) < d --> ¦f x' - f x¦ < e"
apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)"
show "¦f y - f x¦ < e" proof(cases "y=x")
case False def t "k / norm (y - x)"
have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
have "y∈s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute)
{ def w "x + t *R (y - x)"
have "w∈s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
unfolding t_def using `k>0` by auto
have "(1 / t) *R x + - x + ((t - 1) / t) *R x = (1 / t - 1 + (t - 1) / t) *R x" by (auto simp add: algebra_simps)
also have "… = 0" using `t>0` by(auto simp add:field_simps)
finally have w:"(1 / t) *R w + ((t - 1) / t) *R x = y" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
have "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps)
hence "(f w - f x) / t < e"
using B(2)[OF `w∈s`] and B(2)[OF `x∈s`] using `t>0` by(auto simp add:field_simps)
hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
using `0<t` `2<t` and `x∈s` `w∈s` by(auto simp add:field_simps) }
moreover
{ def w "x - t *R (y - x)"
have "w∈s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
unfolding t_def using `k>0` by auto
have "(1 / (1 + t)) *R x + (t / (1 + t)) *R x = (1 / (1 + t) + t / (1 + t)) *R x" by (auto simp add: algebra_simps)
also have "…=x" using `t>0` by (auto simp add:field_simps)
finally have w:"(1 / (1+t)) *R w + (t / (1 + t)) *R y = x" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
have "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps)
hence *:"(f w - f y) / t < e" using B(2)[OF `w∈s`] and B(2)[OF `y∈s`] using `t>0` by(auto simp add:field_simps)
have "f x ≤ 1 / (1 + t) * f w + (t / (1 + t)) * f y"
using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
using `0<t` `2<t` and `y∈s` `w∈s` by (auto simp add:field_simps)
also have "… = (f w + t * f y) / (1 + t)" using `t>0` unfolding divide_inverse by (auto simp add:field_simps)
also have "… < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
finally have "f x - f y < e" by auto }
ultimately show ?thesis by auto
qed(insert `0<e`, auto)
qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed

subsection {* Upper bound on a ball implies upper and lower bounds. *}

lemma scaleR_2:
fixes x :: "'a::real_vector"
shows "scaleR 2 x = x + x"

unfolding one_add_one_is_two [symmetric] scaleR_left_distrib by simp

lemma convex_bounds_lemma:
fixes x :: "'a::real_normed_vector"
assumes "convex_on (cball x e) f" "∀y ∈ cball x e. f y ≤ b"
shows "∀y ∈ cball x e. abs(f y) ≤ b + 2 * abs(f x)"

apply(rule) proof(cases "0 ≤ e") case True
fix y assume y:"y∈cball x e" def z "2 *R x - y"
have *:"x - (2 *R x - y) = y - x" by (simp add: scaleR_2)
have z:"z∈cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
have "(1 / 2) *R y + (1 / 2) *R z = x" unfolding z_def by (auto simp add: algebra_simps)
thus "¦f y¦ ≤ b + 2 * ¦f x¦" using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
next case False fix y assume "y∈cball x e"
hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
thus "¦f y¦ ≤ b + 2 * ¦f x¦" using zero_le_dist[of x y] by auto qed

subsection {* Hence a convex function on an open set is continuous. *}

lemma convex_on_continuous:
assumes "open (s::(real^'n) set)" "convex_on s f"
shows "continuous_on s f"

unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
note dimge1 = dimindex_ge_1[where 'a='n]
fix x assume "x∈s"
then obtain e where e:"cball x e ⊆ s" "e>0" using assms(1) unfolding open_contains_cball by auto
def d "e / real CARD('n)"
have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto)
let ?d = "(χ i. d)::real^'n"
obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
have "x∈{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by auto
hence "c≠{}" using c by auto
def k "Max (f ` c)"
have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof
fix z assume z:"z∈{x - ?d..x + ?d}"
have e:"e = setsum (λi. d) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
by (metis eq_divide_imp times_divide_eq_left real_dimindex_gt_0 real_eq_of_nat less_le mult_commute)
show "dist x z ≤ e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
hence k:"∀y∈{x - ?d..x + ?d}. f y ≤ k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
have "d ≤ e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 using real_dimindex_ge_1 by auto
hence dsube:"cball x d ⊆ cball x e" unfolding subset_eq Ball_def mem_cball by auto
have conv:"convex_on (cball x d) f" apply(rule convex_on_subset, rule convex_on_subset[OF assms(2)]) apply(rule e(1)) using dsube by auto
hence "∀y∈cball x d. abs (f y) ≤ k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
fix y assume y:"y∈cball x d"
{ fix i::'n have "x $ i - d ≤ y $ i" "y $ i ≤ x $ i + d"
using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by auto }
thus "f y ≤ k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
by auto qed
hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
apply force
done
thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
using `d>0` by auto
qed

subsection {* Line segments, Starlike Sets, etc.*}

(* Use the same overloading tricks as for intervals, so that
segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)


definition
midpoint :: "'a::real_vector => 'a => 'a" where
"midpoint a b = (inverse (2::real)) *R (a + b)"


definition
open_segment :: "'a::real_vector => 'a => 'a set" where
"open_segment a b = {(1 - u) *R a + u *R b | u::real. 0 < u ∧ u < 1}"


definition
closed_segment :: "'a::real_vector => 'a => 'a set" where
"closed_segment a b = {(1 - u) *R a + u *R b | u::real. 0 ≤ u ∧ u ≤ 1}"


definition "between = (λ (a,b). closed_segment a b)"

lemmas segment = open_segment_def closed_segment_def

definition "starlike s <-> (∃a∈s. ∀x∈s. closed_segment a x ⊆ s)"

lemma midpoint_refl: "midpoint x x = x"
unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto

lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)

lemma midpoint_eq_iff: "midpoint a b = c <-> a + b = c + c"
proof -
have "midpoint a b = c <-> scaleR 2 (midpoint a b) = scaleR 2 c"
by simp
thus ?thesis
unfolding midpoint_def scaleR_2 [symmetric] by simp
qed

lemma dist_midpoint:
fixes a b :: "'a::real_normed_vector" shows
"dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
"dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
"dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
"dist (midpoint a b) b = (dist a b) / 2" (is ?t4)

proof-
have *: "!!x y::'a. 2 *R x = - y ==> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
have **:"!!x y::'a. 2 *R x = y ==> norm x = (norm y) / 2" by auto
note scaleR_right_distrib [simp]
show ?t1 unfolding midpoint_def dist_norm apply (rule **)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
show ?t2 unfolding midpoint_def dist_norm apply (rule *)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
show ?t3 unfolding midpoint_def dist_norm apply (rule *)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
show ?t4 unfolding midpoint_def dist_norm apply (rule **)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
qed

lemma midpoint_eq_endpoint:
"midpoint a b = a <-> a = b"
"midpoint a b = b <-> a = b"

unfolding midpoint_eq_iff by auto

lemma convex_contains_segment:
"convex s <-> (∀a∈s. ∀b∈s. closed_segment a b ⊆ s)"

unfolding convex_alt closed_segment_def by auto

lemma convex_imp_starlike:
"convex s ==> s ≠ {} ==> starlike s"

unfolding convex_contains_segment starlike_def by auto

lemma segment_convex_hull:
"closed_segment a b = convex hull {a,b}"
proof-
have *:"!!x. {x} ≠ {}" by auto
have **:"!!u v. u + v = 1 <-> u = 1 - (v::real)" by auto
show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_ext)
unfolding mem_Collect_eq apply(rule,erule exE)
apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed

lemma convex_segment: "convex (closed_segment a b)"
unfolding segment_convex_hull by(rule convex_convex_hull)

lemma ends_in_segment: "a ∈ closed_segment a b" "b ∈ closed_segment a b"
unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto

lemma segment_furthest_le:
fixes a b x y :: "real ^ 'n"
assumes "x ∈ closed_segment a b" shows "norm(y - x) ≤ norm(y - a) ∨ norm(y - x) ≤ norm(y - b)"
proof-
obtain z where "z∈{a, b}" "norm (x - y) ≤ norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
using assms[unfolded segment_convex_hull] by auto
thus ?thesis by(auto simp add:norm_minus_commute) qed

lemma segment_bound:
fixes x a b :: "real ^ 'n"
assumes "x ∈ closed_segment a b"
shows "norm(x - a) ≤ norm(b - a)" "norm(x - b) ≤ norm(b - a)"

using segment_furthest_le[OF assms, of a]
using segment_furthest_le[OF assms, of b]
by (auto simp add:norm_minus_commute)

lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)

lemma between_mem_segment: "between (a,b) x <-> x ∈ closed_segment a b"
unfolding between_def mem_def by auto

lemma between:"between (a,b) (x::real^'n) <-> dist a b = (dist a x) + (dist x b)"
proof(cases "a = b")
case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
by(auto simp add:segment_refl dist_commute) next
case False hence Fal:"norm (a - b) ≠ 0" and Fal2: "norm (a - b) > 0" by auto
have *:"!!u. a - ((1 - u) *R a + u *R b) = u *R (a - b)" by (auto simp add: algebra_simps)
show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
fix u assume as:"x = (1 - u) *R a + u *R b" "0 ≤ u" "u ≤ 1"
hence *:"a - x = u *R (a - b)" "x - b = (1 - u) *R (a - b)"
unfolding as(1) by(auto simp add:algebra_simps)
show "norm (a - x) *R (x - b) = norm (x - b) *R (a - x)"
unfolding norm_minus_commute[of x a] * Cart_eq using as(2,3)
by(auto simp add: field_simps)
next assume as:"dist a b = dist a x + dist x b"
have "norm (a - x) / norm (a - b) ≤ 1" unfolding divide_le_eq_1_pos[OF Fal2] unfolding as[unfolded dist_norm] norm_ge_zero by auto
thus "∃u. x = (1 - u) *R a + u *R b ∧ 0 ≤ u ∧ u ≤ 1" apply(rule_tac x="dist a x / dist a b" in exI)
unfolding dist_norm Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
fix i::'n have "((1 - norm (a - x) / norm (a - b)) *R a + (norm (a - x) / norm (a - b)) *R b) $ i =
((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"

using Fal by(auto simp add: field_simps)
also have "… = x$i" apply(rule divide_eq_imp[OF Fal])
unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
by(auto simp add:field_simps)
finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *R a + (norm (a - x) / norm (a - b)) *R b) $ i" by auto
qed(insert Fal2, auto) qed qed

lemma between_midpoint: fixes a::"real^'n" shows
"between (a,b) (midpoint a b)" (is ?t1)
"between (b,a) (midpoint a b)" (is ?t2)

proof- have *:"!!x y z. x = (1/2::real) *R z ==> y = (1/2) *R z ==> norm z = norm x + norm y" by auto
show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
by(auto simp add:field_simps Cart_eq) qed

lemma between_mem_convex_hull:
"between (a,b) x <-> x ∈ convex hull {a,b}"

unfolding between_mem_segment segment_convex_hull ..

subsection {* Shrinking towards the interior of a convex set. *}

lemma mem_interior_convex_shrink:
fixes s :: "(real ^ _) set"
assumes "convex s" "c ∈ interior s" "x ∈ s" "0 < e" "e ≤ 1"
shows "x - e *R (x - c) ∈ interior s"

proof- obtain d where "d>0" and d:"ball c d ⊆ s" using assms(2) unfolding mem_interior by auto
show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
fix y assume as:"dist (x - e *R (x - c)) y < e * d"
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 "dist c ((1 / e) *R y - ((1 - e) / e) *R x) = abs(1/e) * norm (e *R c - y + (1 - e) *R x)"
unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule norm_eqI) using `e>0`
by(auto simp add: Cart_eq field_simps)
also have "… = abs(1/e) * norm (x - e *R (x - c) - y)" by(auto intro!:norm_eqI 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 show "y ∈ s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed

lemma mem_interior_closure_convex_shrink:
fixes s :: "(real ^ _) set"
assumes "convex s" "c ∈ interior s" "x ∈ closure s" "0 < e" "e ≤ 1"
shows "x - e *R (x - c) ∈ interior s"

proof- obtain d where "d>0" and d:"ball c d ⊆ s" using assms(2) unfolding mem_interior by auto
have "∃y∈s. norm (y - x) * (1 - e) < e * d" proof(cases "x∈s")
case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
show ?thesis proof(cases "e=1")
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
thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
using `e≤1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
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
thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto 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 "z∈interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
by(auto simp add:field_simps norm_minus_commute)
thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink)
using assms(1,4-5) `y∈s` by auto qed

subsection {* Some obvious but surprisingly hard simplex lemmas. *}

lemma simplex:
assumes "finite s" "0 ∉ s"
shows "convex hull (insert 0 s) = { y. (∃u. (∀x∈s. 0 ≤ u x) ∧ setsum u s ≤ 1 ∧ setsum (λx. u x *R x) s = y)}"

unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_ext, rule) unfolding mem_Collect_eq
apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
apply(rule_tac x=u in exI) defer apply(rule_tac x="λx. if x = 0 then 1 - setsum u s else u x" in exI) using assms(2)
unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto

lemma std_simplex:
"convex hull (insert 0 { basis i | i. i∈UNIV}) =
{x::real^'n . (∀i. 0 ≤ x$i) ∧ setsum (λi. x$i) UNIV ≤ 1 }"
(is "convex hull (insert 0 ?p) = ?s")

proof- let ?D = "UNIV::'n set"
have "0∉?p" by(auto simp add: basis_nonzero)
have "{(basis i)::real^'n |i. i ∈ ?D} = basis ` ?D" by auto
note sumbas = this setsum_reindex[OF basis_inj, unfolded o_def]
show ?thesis unfolding simplex[OF finite_stdbasis `0∉?p`] apply(rule set_ext) unfolding mem_Collect_eq apply rule
apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
fix x::"real^'n" and u assume as: "∀x∈{basis i |i. i ∈?D}. 0 ≤ u x" "setsum u {basis i |i. i ∈ ?D} ≤ 1" "(∑x∈{basis i |i. i ∈?D}. u x *R x) = x"
have *:"∀i. u (basis i) = x$i" using as(3) unfolding sumbas and basis_expansion_unique [where 'a=real, unfolded smult_conv_scaleR] by auto
hence **:"setsum u {basis i |i. i ∈ ?D} = setsum (op $ x) ?D" unfolding sumbas by(rule_tac setsum_cong, auto)
show " (∀i. 0 ≤ x $ i) ∧ setsum (op $ x) ?D ≤ 1" apply - proof(rule,rule)
fix i::'n show "0 ≤ x$i" unfolding *[rule_format,of i,THEN sym] apply(rule_tac as(1)[rule_format]) by auto
qed(insert as(2)[unfolded **], auto)
next fix x::"real^'n" assume as:"∀i. 0 ≤ x $ i" "setsum (op $ x) ?D ≤ 1"
show "∃u. (∀x∈{basis i |i. i ∈ ?D}. 0 ≤ u x) ∧ setsum u {basis i |i. i ∈ ?D} ≤ 1 ∧ (∑x∈{basis i |i. i ∈ ?D}. u x *R x) = x"
apply(rule_tac x="λy. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE) using as(1) apply(erule_tac x=i in allE)
unfolding sumbas using as(2) and basis_expansion_unique [where 'a=real, unfolded smult_conv_scaleR] by(auto simp add:inner_basis) qed qed

lemma interior_std_simplex:
"interior (convex hull (insert 0 { basis i| i. i∈UNIV})) =
{x::real^'n. (∀i. 0 < x$i) ∧ setsum (λi. x$i) UNIV < 1 }"

apply(rule set_ext) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
fix x::"real^'n" and e assume "0<e" and as:"∀xa. dist x xa < e --> (∀x. 0 ≤ xa $ x) ∧ setsum (op $ xa) UNIV ≤ 1"
show "(∀xa. 0 < x $ xa) ∧ setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *R basis i"]] and `e>0`
unfolding dist_norm by(auto simp add: norm_basis elim:allE[where x=i])
next guess a using UNIV_witness[where 'a='n] ..
have **:"dist x (x + (e / 2) *R basis a) < e" using `e>0` and norm_basis[of a]
unfolding dist_norm by(auto intro!: mult_strict_left_mono_comm)
have "!!i. (x + (e / 2) *R basis a) $ i = x$i + (if i = a then e/2 else 0)" by auto
hence *:"setsum (op $ (x + (e / 2) *R basis a)) UNIV = setsum (λi. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto)
have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *R basis a)) UNIV" unfolding * setsum_addf
using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
also have "… ≤ 1" using ** apply(drule_tac as[rule_format]) by auto
finally show "setsum (op $ x) UNIV < 1" by auto qed
next
fix x::"real^'n" assume as:"∀i. 0 < x $ i" "setsum (op $ x) UNIV < 1"
guess a using UNIV_witness[where 'a='b] ..
let ?d = "(1 - setsum (op $ x) UNIV) / real (CARD('n))"
have "Min ((op $ x) ` UNIV) > 0" apply(rule Min_grI) using as(1) dimindex_ge_1 by auto
moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) using dimindex_ge_1 by(auto simp add: Suc_le_eq)
ultimately show "∃e>0. ∀y. dist x y < e --> (∀i. 0 ≤ y $ i) ∧ setsum (op $ y) UNIV ≤ 1"
apply(rule_tac x="min (Min ((op $ x) ` UNIV)) ?D" in exI) apply rule defer apply(rule,rule) proof-
fix y assume y:"dist x y < min (Min (op $ x ` UNIV)) ?d"
have "setsum (op $ y) UNIV ≤ setsum (λi. x$i + ?d) UNIV" proof(rule setsum_mono)
fix i::'n have "abs (y$i - x$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
thus "y $ i ≤ x $ i + ?d" by auto qed
also have "… ≤ 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat using dimindex_ge_1 by(auto simp add: Suc_le_eq)
finally show "(∀i. 0 ≤ y $ i) ∧ setsum (op $ y) UNIV ≤ 1" apply- proof(rule,rule)
fix i::'n have "norm (x - y) < x$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
by auto
thus "0 ≤ y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
qed auto qed auto qed

lemma interior_std_simplex_nonempty: obtains a::"real^'n" where
"a ∈ interior(convex hull (insert 0 {basis i | i . i ∈ UNIV}))"
proof-
let ?D = "UNIV::'n set" let ?a = "setsum (λb::real^'n. inverse (2 * real CARD('n)) *R b) {(basis i) | i. i ∈ ?D}"
have *:"{basis i :: real ^ 'n | i. i ∈ ?D} = basis ` ?D" by auto
{ fix i have "?a $ i = inverse (2 * real CARD('n))"
unfolding setsum_component vector_smult_component and * and setsum_reindex[OF basis_inj] and o_def
apply(rule trans[of _ "setsum (λj. if i = j then inverse (2 * real CARD('n)) else 0) ?D"]) apply(rule setsum_cong2)
unfolding setsum_delta'[OF finite_UNIV[where 'a='n]] and real_dimindex_ge_1[where 'n='n] by(auto simp add: basis_component[of i]) }
note ** = this
show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof(rule,rule)
fix i::'n show "0 < ?a $ i" unfolding ** using dimindex_ge_1 by(auto simp add: Suc_le_eq) next
have "setsum (op $ ?a) ?D = setsum (λi. inverse (2 * real CARD('n))) ?D" by(rule setsum_cong2, rule **)
also have "… < 1" unfolding setsum_constant card_enum real_eq_of_nat divide_inverse[THEN sym] by (auto simp add:field_simps)
finally show "setsum (op $ ?a) ?D < 1" by auto qed qed

end