src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy

move connected_real_lemma to the one place it is used

(*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
*)

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

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


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

lemma linear_scaleR: "linear (\<lambda>x. scaleR c x)"
  by (simp add: linear_def scaleR_add_right)

lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>(x::'a::real_vector). scaleR c x)"
  by (simp add: inj_on_def)

lemma linear_add_cmul:
assumes "linear f"
shows "f(a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
using linear_add[of f] linear_cmul[of f] assms by (simp) 

lemma mem_convex_2:
  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v=1"
  shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
  using assms convex_def[of S] by auto

lemma mem_convex_alt:
  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
apply (subst mem_convex_2) 
using assms apply (auto simp add: algebra_simps zero_le_divide_iff)
using add_divide_distrib[of u v "u+v"] by auto

lemma card_ge1: assumes "d ~= {}" "finite d" shows "card d >= 1"
by (metis Suc_eq_plus1 assms(1) assms(2) card_eq_0_iff fact_ge_one_nat fact_num_eq_if_nat one_le_mult_iff plus_nat.add_0)

lemma inj_on_image_mem_iff: "inj_on f B ==> (A <= B) ==> (f a : f`A) ==> (a : B) ==> (a : A)" 
by (blast dest: inj_onD)

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

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

lemma linear_injective_on_subspace_0:
assumes lf: "linear f" and "subspace S"
  shows "inj_on f S <-> (!x : S. f x = 0 --> x = 0)"
proof-
  have "inj_on f S <-> (!x : S. !y : S. f x = f y --> x = y)" by (simp add: inj_on_def)
  also have "... <-> (!x : S. !y : S. f x - f y = 0 --> x - y = 0)" by simp
  also have "... <-> (!x : S. !y : S. f (x - y) = 0 --> x - y = 0)"
    by (simp add: linear_sub[OF lf])
  also have "... <-> (! x : S. f x = 0 --> x = 0)" 
    using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
  finally show ?thesis .
qed

lemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"
  unfolding subspace_def by auto 

lemma span_eq[simp]: "(span s = s) <-> subspace s"
  unfolding span_def by (rule hull_eq, rule subspace_Inter)

lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
  by(auto simp add: inj_on_def euclidean_eq[where 'a='n])
  
lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
proof-
  have eq: "?S = basis ` d" by blast
  show ?thesis unfolding eq apply(rule finite_subset[OF _ range_basis_finite]) by auto
qed

lemma card_substdbasis: assumes "d \<subseteq> {..<DIM('n::euclidean_space)}"
  shows "card {basis i ::'n::euclidean_space | i. i : d} = card d" (is "card ?S = _")
proof-
  have eq: "?S = basis ` d" by blast
  show ?thesis unfolding eq using card_image[OF basis_inj_on[of d]] assms by auto
qed

lemma substdbasis_expansion_unique: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  shows "setsum (%i. f i *\<^sub>R basis i) d = (x::'a::euclidean_space)
      <-> (!i<DIM('a). (i:d --> f i = x$$i) & (i ~: d --> x $$ i = 0))"
proof- have *:"\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto
  have **:"finite d" apply(rule finite_subset[OF assms]) by fastsimp
  have ***:"\<And>i. (setsum (%i. f i *\<^sub>R ((basis i)::'a)) d) $$ i = (\<Sum>x\<in>d. if x = i then f x else 0)"
    unfolding euclidean_component_setsum euclidean_component_scaleR basis_component *
    apply(rule setsum_cong2) using assms by auto
  show ?thesis unfolding euclidean_eq[where 'a='a] *** setsum_delta[OF **] using assms by auto
qed

lemma independent_substdbasis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  shows "independent {basis i ::'a::euclidean_space |i. i : (d :: nat set)}" (is "independent ?A")
proof -
  have *: "{basis i |i. i < DIM('a)} = basis ` {..<DIM('a)}" by auto
  show ?thesis
    apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
    using independent_basis[where 'a='a] assms by (auto simp: *)
qed

lemma dim_cball: 
assumes "0<e"
shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
proof-
{ fix x :: "'n::euclidean_space" def y == "(e/norm x) *\<^sub>R x"
  hence "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto
  moreover have "x = (norm x/e) *\<^sub>R y" using y_def assms by simp
  moreover hence "x = (norm x/e) *\<^sub>R y"  by auto
  ultimately have "x : span (cball 0 e)"
     using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
} hence "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto 
from this show ?thesis using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
qed

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

lemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"
  apply(rule ccontr) by auto

lemma translate_inj_on: 
fixes A :: "('a::ab_group_add) set"
shows "inj_on (%x. a+x) A" unfolding inj_on_def by auto

lemma translation_assoc:
  fixes a b :: "'a::ab_group_add"
  shows "(\<lambda>x. b+x) ` ((\<lambda>x. a+x) ` S) = (\<lambda>x. (a+b)+x) ` S" by auto

lemma translation_invert:
  fixes a :: "'a::ab_group_add"
  assumes "(\<lambda>x. a+x) ` A = (\<lambda>x. a+x) ` B"
  shows "A=B"
proof-
  have "(%x. -a+x) ` ((%x. a+x) ` A) = (%x. -a+x) ` ((%x. a+x) ` B)" using assms by auto
  from this show ?thesis using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto 
qed

lemma translation_galois:
  fixes a :: "'a::ab_group_add"
  shows "T=((\<lambda>x. a+x) ` S) <-> S=((\<lambda>x. (-a)+x) ` T)"
  using translation_assoc[of "-a" a S] apply auto
  using translation_assoc[of a "-a" T] by auto

lemma translation_inverse_subset:
  assumes "((%x. -a+x) ` V) <= (S :: 'n::ab_group_add set)" 
  shows "V <= ((%x. a+x) ` S)"
proof-
{ fix x assume "x:V" hence "x-a : S" using assms by auto 
  hence "x : {a + v |v. v : S}" apply auto apply (rule exI[of _ "x-a"]) apply simp done 
  hence "x : ((%x. a+x) ` S)" by auto }
  from this show ?thesis by auto
qed

lemma basis_to_basis_subspace_isomorphism:
  assumes s: "subspace (S:: ('n::euclidean_space) set)"
  and t: "subspace (T :: ('m::euclidean_space) set)"
  and d: "dim S = dim T"
  and B: "B <= S" "independent B" "S <= span B" "card B = dim S"
  and C: "C <= T" "independent C" "T <= span C" "card C = dim T"
  shows "EX f. linear f & f ` B = C & f ` S = T & inj_on f S"
proof-
(* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
*)
  from B independent_bound have fB: "finite B" by blast
  from C independent_bound have fC: "finite C" by blast
  from B(4) C(4) card_le_inj[of B C] d obtain f where
    f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
  from linear_independent_extend[OF B(2)] obtain g where
    g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
  from inj_on_iff_eq_card[OF fB, of f] f(2)
  have "card (f ` B) = card B" by simp
  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
    by simp
  have "g ` B = f ` B" using g(2)
    by (auto simp add: image_iff)
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
  finally have gBC: "g ` B = C" .
  have gi: "inj_on g B" using f(2) g(2)
    by (auto simp add: inj_on_def)
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
    have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
    have "x=y" using g0[OF th1 th0] by simp }
  then have giS: "inj_on g S"
    unfolding inj_on_def by blast
  from span_subspace[OF B(1,3) s]
  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
  also have "\<dots> = span C" unfolding gBC ..
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
  finally have gS: "g ` S = T" .
  from g(1) gS giS gBC show ?thesis by blast
qed

lemma closure_linear_image:
fixes f :: "('m::euclidean_space) => ('n::real_normed_vector)"
assumes "linear f"
shows "f ` (closure S) <= closure (f ` S)"
using image_closure_subset[of S f "closure (f ` S)"] assms linear_conv_bounded_linear[of f] 
linear_continuous_on[of f "closure S"] closed_closure[of "f ` S"] closure_subset[of "f ` S"] by auto

lemma closure_injective_linear_image:
fixes f :: "('n::euclidean_space) => ('n::euclidean_space)"
assumes "linear f" "inj f"
shows "f ` (closure S) = closure (f ` S)"
proof-
obtain f' where f'_def: "linear f' & f o f' = id & f' o f = id" 
   using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
hence "f' ` closure (f ` S) <= closure (S)"
   using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto
hence "f ` f' ` closure (f ` S) <= f ` closure (S)" by auto
hence "closure (f ` S) <= f ` closure (S)" using image_compose[of f f' "closure (f ` S)"] f'_def by auto
from this show ?thesis using closure_linear_image[of f S] assms by auto 
qed

lemma closure_direct_sum:
shows "closure (S <*> T) = closure S <*> closure T"
  by (rule closure_Times)

lemma closure_scaleR:  (* TODO: generalize to real_normed_vector *)
fixes S :: "('n::euclidean_space) set"
shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
proof-
{ assume "c ~= 0" hence ?thesis using closure_injective_linear_image[of "(op *\<^sub>R c)" S]
      linear_scaleR injective_scaleR by auto 
}
moreover
{ assume zero: "c=0 & S ~= {}"
  hence "closure S ~= {}" using closure_subset by auto
  hence "op *\<^sub>R c ` (closure S) = {0}" using zero by auto
  moreover have "op *\<^sub>R 0 ` S = {0}" using zero by auto
  ultimately have ?thesis using zero by auto
}
moreover
{ assume "S={}" hence ?thesis by auto }
ultimately show ?thesis by blast
qed

lemma fst_linear: "linear fst" unfolding linear_def by (simp add: algebra_simps)

lemma snd_linear: "linear snd" unfolding linear_def by (simp add: algebra_simps)

lemma fst_snd_linear: "linear (%(x,y). x + y)" unfolding linear_def by (simp add: algebra_simps)

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 vector_choose_size: "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
  apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"]) using DIM_positive[where 'a='a] by auto

lemma setsum_delta_notmem: assumes "x\<notin>s"
  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
        "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
        "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
        "setsum (\<lambda>y. 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 "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
proof-
  have *:"\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" by auto
  show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
qed

lemma if_smult:"(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" by auto

lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
  (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
  using image_affinity_interval[of m 0 a b] by auto

lemma dist_triangle_eq:
  fixes x y z :: "'a::real_inner"
  shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
proof- have *:"x - y + (y - z) = x - z" by auto
  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
    by(auto simp add:norm_minus_commute) qed

lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto

lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
  unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto

lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
  unfolding norm_eq_sqrt_inner by simp

lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
  unfolding norm_eq_sqrt_inner by simp



subsection {* Affine set and affine hull.*}

definition
  affine :: "'a::real_vector set \<Rightarrow> bool" where
  "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"

lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> 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: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
  unfolding affine_def by auto 

lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
  unfolding affine_def by auto

lemma affine_affine_hull: "affine(affine hull s)"
  unfolding hull_def using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"]
  by auto

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

subsection {* Some explicit formulations (from Lars Schewe). *}

lemma affine: fixes V::"'a::real_vector set"
  shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> 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 \<in> V" "y \<in> V" "u + v = (1::real)"
    "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
  thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
    using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. 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:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
  def n \<equiv> "card s"
  have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> 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 \<Rightarrow> real"
      assume IA:"\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
               s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
        as:"Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
      have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
        assume " \<not> (\<exists>x\<in>s. u x \<noteq> 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\<in>s" "u x \<noteq> 1" by auto

      have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
      have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>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 \<noteq> 1` by auto
      have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
        case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
          assume "\<not> s - {x} \<noteq> {}" 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}" "\<lambda>y. (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\<noteq>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 \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
      hence "u x + (1 - u x) = 1 \<Longrightarrow> u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa) /\<^sub>R (1 - u x)) \<in> V"
        apply-apply(rule as(3)[rule_format]) 
        unfolding  RealVector.scaleR_right.setsum using x(1) as(6) by auto
      thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
         apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
         using `u x \<noteq> 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\<noteq>{}` `finite s`, auto)
qed

lemma affine_hull_explicit:
  "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
  apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq
  apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
  fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
    apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
next
  fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
  thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
next
  show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>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 "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
    then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
    fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
    then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
    have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
    have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
    show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
      apply(rule_tac x="sx \<union> sy" in exI)
      apply(rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>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] RealVector.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. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq apply (rule,rule)
  apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
  fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
  thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>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 \<subseteq> s" hence *:"s \<inter> t = t" by auto
  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
  thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>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) by auto

lemma affine_hull_finite_step:
  fixes y :: "'a::real_vector"
  shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
  "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
                (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
proof-
  show ?th1 by simp
  assume ?as 
  { assume ?lhs
    then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
    have ?rhs proof(cases "a\<in>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"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
    have *:"\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" by auto
    have ?lhs proof(cases "a\<in>s")
      case True thus ?thesis
        apply(rule_tac x="\<lambda>x. (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 **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
               "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
      from False show ?thesis
        apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
        unfolding setsum_clauses(2)[OF `?as`] and * using vu
        using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
        using setsum_cong2[of s u "\<lambda>x. 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 *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
proof-
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
    using affine_hull_finite[of "{a,b}"] by auto
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
    by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
  also have "\<dots> = ?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 *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
proof-
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
         "\<And>x y z. z = x - y \<longleftrightarrow> 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

lemma mem_affine:
  assumes "affine S" "x : S" "y : S" "u+v=1"
  shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
  using assms affine_def[of S] by auto

lemma mem_affine_3:
  assumes "affine S" "x : S" "y : S" "z : S" "u+v+w=1"
  shows "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : S"
proof-
have "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : affine hull {x, y, z}"
  using affine_hull_3[of x y z] assms by auto
moreover have " affine hull {x, y, z} <= affine hull S" 
  using hull_mono[of "{x, y, z}" "S"] assms by auto
moreover have "affine hull S = S" 
  using assms affine_hull_eq[of S] by auto
ultimately show ?thesis by auto 
qed

lemma mem_affine_3_minus:
  assumes "affine S" "x : S" "y : S" "z : S"
  shows "x + v *\<^sub>R (y-z) : S"
using mem_affine_3[of S x y z 1 v "-v"] assms by (simp add: algebra_simps)


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

lemma affine_hull_insert_subset_span:
  shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
  unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
  apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
  fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
  thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
    apply(rule_tac x="x - a" in exI)
    apply (rule conjI, simp)
    apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
    apply(rule_tac x="\<lambda>x. 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:
  assumes "a \<notin> s"
  shows "affine hull (insert a s) =
            {a + v | v . v \<in> span {x - a | x.  x \<in> 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 \<in> span {x - a |x. x \<in> s}"
  then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y" unfolding span_explicit by auto
  def f \<equiv> "(\<lambda>x. x + a) ` t"
  have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt 
    by(auto simp add: setsum_reindex[unfolded inj_on_def])
  have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
  show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
    apply(rule_tac x="insert a f" in exI)
    apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. 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 "\<lambda>x. x = a"]
    by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed

lemma affine_hull_span:
  assumes "a \<in> s"
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto

subsection{* Parallel Affine Sets *}

definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
where "affine_parallel S T = (? a. T = ((%x. a + x) ` S))"

lemma affine_parallel_expl_aux:
   fixes S T :: "'a::real_vector set"
   assumes "!x. (x : S <-> (a+x) : T)" 
   shows "T = ((%x. a + x) ` S)"
proof-
{ fix x assume "x : T" hence "(-a)+x : S" using assms by auto
  hence " x : ((%x. a + x) ` S)" using imageI[of "-a+x" S "(%x. a+x)"] by auto}
moreover have "T >= ((%x. a + x) ` S)" using assms by auto 
ultimately show ?thesis by auto
qed

lemma affine_parallel_expl: 
   "affine_parallel S T = (? a. !x. (x : S <-> (a+x) : T))" 
   unfolding affine_parallel_def using affine_parallel_expl_aux[of S _ T] by auto

lemma affine_parallel_reflex: "affine_parallel S S" unfolding affine_parallel_def apply (rule exI[of _ "0"]) by auto

lemma affine_parallel_commut:
assumes "affine_parallel A B" shows "affine_parallel B A" 
proof-
from assms obtain a where "B=((%x. a + x) ` A)" unfolding affine_parallel_def by auto 
from this show ?thesis using translation_galois[of B a A] unfolding affine_parallel_def by auto
qed

lemma affine_parallel_assoc:
assumes "affine_parallel A B" "affine_parallel B C"
shows "affine_parallel A C" 
proof-
from assms obtain ab where "B=((%x. ab + x) ` A)" unfolding affine_parallel_def by auto 
moreover 
from assms obtain bc where "C=((%x. bc + x) ` B)" unfolding affine_parallel_def by auto
ultimately show ?thesis using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto 
qed

lemma affine_translation_aux:
  fixes a :: "'a::real_vector"
  assumes "affine ((%x. a + x) ` S)" shows "affine S"
proof-
{ fix x y u v
  assume xy: "x : S" "y : S" "(u :: real)+v=1"
  hence "(a+x):((%x. a + x) ` S)" "(a+y):((%x. a + x) ` S)" by auto
  hence h1: "u *\<^sub>R  (a+x) + v *\<^sub>R (a+y) : ((%x. a + x) ` S)" using xy assms unfolding affine_def by auto
  have "u *\<^sub>R (a+x) + v *\<^sub>R (a+y) = (u+v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)" by (simp add:algebra_simps)
  also have "...= a + (u *\<^sub>R x + v *\<^sub>R y)" using `u+v=1` by auto
  ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) : ((%x. a + x) ` S)" using h1 by auto
  hence "u *\<^sub>R x + v *\<^sub>R y : S" by auto
} from this show ?thesis unfolding affine_def by auto
qed

lemma affine_translation:
  fixes a :: "'a::real_vector"
  shows "affine S <-> affine ((%x. a + x) ` S)"
proof-
have "affine S ==> affine ((%x. a + x) ` S)" using affine_translation_aux[of "-a" "((%x. a + x) ` S)"]  using translation_assoc[of "-a" a S] by auto
from this show ?thesis using affine_translation_aux by auto
qed

lemma parallel_is_affine:
fixes S T :: "'a::real_vector set"
assumes "affine S" "affine_parallel S T"
shows "affine T"
proof-
  from assms obtain a where "T=((%x. a + x) ` S)" unfolding affine_parallel_def by auto 
  from this show ?thesis using affine_translation assms by auto
qed

lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
  unfolding subspace_def affine_def by auto

subsection{* Subspace Parallel to an Affine Set *}

lemma subspace_affine:
  shows "subspace S <-> (affine S & 0 : S)"
proof-
have h0: "subspace S ==> (affine S & 0 : S)" using subspace_imp_affine[of S] subspace_0 by auto
{ assume assm: "affine S & 0 : S"
  { fix c :: real 
    fix x assume x_def: "x : S"
    have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
    moreover have "(1-c) *\<^sub>R 0 + c *\<^sub>R x : S" using affine_alt[of S] assm x_def by auto
    ultimately have "c *\<^sub>R x : S" by auto
  } hence h1: "!c. !x : S. c *\<^sub>R x : S" by auto
  { fix x y assume xy_def: "x : S" "y : S"
    def u == "(1 :: real)/2"
    have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)" by auto
    moreover have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y" by (simp add: algebra_simps)
    moreover have "(1-u) *\<^sub>R x + u *\<^sub>R y : S" using affine_alt[of S] assm xy_def by auto
    ultimately have "(1/2) *\<^sub>R (x+y) : S" using u_def by auto
    moreover have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" by auto
    ultimately have "(x+y) : S" using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
  } hence "!x : S. !y : S. (x+y) : S" by auto 
  hence "subspace S" using h1 assm unfolding subspace_def by auto
} from this show ?thesis using h0 by metis
qed

lemma affine_diffs_subspace:
  assumes "affine S" "a : S"
  shows "subspace ((%x. (-a)+x) ` S)"
proof-
have "affine ((%x. (-a)+x) ` S)" using  affine_translation assms by auto  
moreover have "0 : ((%x. (-a)+x) ` S)" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
ultimately show ?thesis using subspace_affine by auto 
qed

lemma parallel_subspace_explicit:
assumes "affine S" "a : S"
assumes "L == {y. ? x : S. (-a)+x=y}" 
shows "subspace L & affine_parallel S L" 
proof-
have par: "affine_parallel S L" unfolding affine_parallel_def using assms by auto
hence "affine L" using assms parallel_is_affine by auto  
moreover have "0 : L" using assms apply auto using exI[of "(%x. x:S & -a+x=0)" a] by auto
ultimately show ?thesis using subspace_affine par by auto 
qed

lemma parallel_subspace_aux:
assumes "subspace A" "subspace B" "affine_parallel A B"
shows "A>=B"
proof-
from assms obtain a where a_def: "!x. (x : A <-> (a+x) : B)" using affine_parallel_expl[of A B] by auto
hence "-a : A" using assms subspace_0[of B] by auto
hence "a : A" using assms subspace_neg[of A "-a"] by auto
from this show ?thesis using assms a_def unfolding subspace_def by auto
qed

lemma parallel_subspace:
assumes "subspace A" "subspace B" "affine_parallel A B"
shows "A=B"
proof-
have "A>=B" using assms parallel_subspace_aux by auto
moreover have "A<=B" using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
ultimately show ?thesis by auto  
qed

lemma affine_parallel_subspace:
assumes "affine S" "S ~= {}"
shows "?!L. subspace L & affine_parallel S L" 
proof-
have ex: "? L. subspace L & affine_parallel S L" using assms  parallel_subspace_explicit by auto 
{ fix L1 L2 assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"
  hence "affine_parallel L1 L2" using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
  hence "L1=L2" using ass parallel_subspace by auto
} from this show ?thesis using ex by auto
qed

subsection {* Cones. *}

definition
  cone :: "'a::real_vector set \<Rightarrow> bool" where
  "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> 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]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
  unfolding cone_def by auto

subsection {* Conic hull. *}

lemma cone_cone_hull: "cone (cone hull s)"
  unfolding hull_def by auto

lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
  apply(rule hull_eq)
  using cone_Inter unfolding subset_eq by auto

lemma mem_cone:
  assumes "cone S" "x : S" "c>=0"
  shows "c *\<^sub>R x : S"
  using assms cone_def[of S] by auto

lemma cone_contains_0:
assumes "cone S"
shows "(S ~= {}) <-> (0 : S)"
proof-
{ assume "S ~= {}" from this obtain a where "a:S" by auto
  hence "0 : S" using assms mem_cone[of S a 0] by auto
} from this show ?thesis by auto
qed

lemma cone_0: "cone {0}"
unfolding cone_def by auto

lemma cone_Union[intro]: "(!s:f. cone s) --> (cone (Union f))"
  unfolding cone_def by blast

lemma cone_iff:
assumes "S ~= {}"
shows "cone S <-> 0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
proof-
{ assume "cone S"
  { fix c assume "(c :: real)>0"
    { fix x assume "x : S" hence "x : (op *\<^sub>R c) ` S" unfolding image_def
        using `cone S` `c>0` mem_cone[of S x "1/c"]
        exI[of "(%t. t:S & x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"] by auto
    }
    moreover
    { fix x assume "x : (op *\<^sub>R c) ` S"
      (*from this obtain t where "t:S & x = c *\<^sub>R t" by auto*)
      hence "x:S" using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto
    }
    ultimately have "((op *\<^sub>R c) ` S) = S" by auto
  } hence "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)" using `cone S` cone_contains_0[of S] assms by auto
}
moreover
{ assume a: "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
  { fix x assume "x:S"
    fix c1 assume "(c1 :: real)>=0"
    hence "(c1=0) | (c1>0)" by auto
    hence "c1 *\<^sub>R x : S" using a `x:S` by auto
  }
 hence "cone S" unfolding cone_def by auto
} ultimately show ?thesis by blast
qed

lemma cone_hull_empty:
"cone hull {} = {}"
by (metis cone_empty cone_hull_eq)

lemma cone_hull_empty_iff:
shows "(S = {}) <-> (cone hull S = {})"
by (metis bot_least cone_hull_empty hull_subset xtrans(5))

lemma cone_hull_contains_0: 
shows "(S ~= {}) <-> (0 : cone hull S)"
using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] by auto

lemma mem_cone_hull:
  assumes "x : S" "c>=0"
  shows "c *\<^sub>R x : cone hull S"
by (metis assms cone_cone_hull hull_inc mem_cone)

lemma cone_hull_expl:
shows "cone hull S = {c *\<^sub>R x | c x. c>=0 & x : S}" (is "?lhs = ?rhs")
proof-
{ fix x assume "x : ?rhs"
  from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
  fix c assume c_def: "(c :: real)>=0"
  hence "c *\<^sub>R x = (c*cx) *\<^sub>R xx" using x_def by (simp add: algebra_simps)
  moreover have "(c*cx) >= 0" using c_def x_def using mult_nonneg_nonneg by auto
  ultimately have "c *\<^sub>R x : ?rhs" using x_def by auto
} hence "cone ?rhs" unfolding cone_def by auto
  hence "?rhs : Collect cone" unfolding mem_Collect_eq by auto
{ fix x assume "x : S" hence "1 *\<^sub>R x : ?rhs" apply auto apply(rule_tac x="1" in exI) by auto
  hence "x : ?rhs" by auto
} hence "S <= ?rhs" by auto
hence "?lhs <= ?rhs" using `?rhs : Collect cone` hull_minimal[of S "?rhs" "cone"] by auto
moreover
{ fix x assume "x : ?rhs"
  from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
  hence "xx : cone hull S" using hull_subset[of S] by auto
  hence "x : ?lhs" using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
} ultimately show ?thesis by auto
qed

lemma cone_closure:
fixes S :: "('m::euclidean_space) set"
assumes "cone S"
shows "cone (closure S)"
proof-
{ assume "S = {}" hence ?thesis by auto }
moreover
{ assume "S ~= {}" hence "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  hence "0:(closure S) & (!c. c>0 --> op *\<^sub>R c ` (closure S) = (closure S))"
     using closure_subset by (auto simp add: closure_scaleR)
  hence ?thesis using cone_iff[of "closure S"] by auto
}
ultimately show ?thesis by blast
qed

subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}

definition
  affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
  "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"

lemma affine_dependent_explicit:
  "affine_dependent p \<longleftrightarrow>
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
    (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>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 \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
  have "x\<notin>s" using as(1,4) by auto
  show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
    apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. 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\<notin>s`] and as using as by auto 
next
  fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
  have "s \<noteq> {v}" using as(3,6) by auto
  thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
    apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (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 \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
  (is "?lhs = ?rhs")
proof
  have *:"\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::'a))" by auto
  assume ?lhs
  then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
    unfolding affine_dependent_explicit by auto
  thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
    apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
    unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
next
  assume ?rhs
  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
  thus ?lhs unfolding affine_dependent_explicit using assms by auto
qed

subsection {* Connectedness of convex sets *}

lemma connected_real_lemma:
  fixes f :: "real \<Rightarrow> 'a::metric_space"
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
proof-
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
  have Sub: "\<exists>y. isUb UNIV ?S y"
    apply (rule exI[where x= b])
    using ab fb e12 by (auto simp add: isUb_def setle_def)
  from reals_complete[OF Se Sub] obtain l where
    l: "isLub UNIV ?S l"by blast
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
    by (metis linorder_linear)
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
    by (metis linorder_linear not_le)
    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
    have "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
    then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
    {assume le2: "f l \<in> e2"
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
      hence lap: "l - a > 0" using alb by arith
      from e2[rule_format, OF le2] obtain e where
        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
      from dst[OF alb e(1)] obtain d where
        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
      let ?d' = "min (d/2) ((l - a)/2)"
      have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
        by (simp add: min_max.less_infI2)
      then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
      moreover
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
      ultimately have False using e12 alb d' by auto}
    moreover
    {assume le1: "f l \<in> e1"
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
      hence blp: "b - l > 0" using alb by arith
      from e1[rule_format, OF le1] obtain e where
        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
      from dst[OF alb e(1)] obtain d where
        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
      have "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
      then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
      then obtain d' where d': "d' > 0" "d' < d" by metis
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
      with l d' have False
        by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
    ultimately show ?thesis using alb by metis
qed

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 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
    assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
    then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>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 \<le> x" "x \<le> 1" "0 < e"
      { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
          by (simp add: algebra_simps)
        assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
        hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
          unfolding * and scaleR_right_diff_distrib[THEN sym]
          unfolding less_divide_eq using n by auto  }
      hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>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 "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
      apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>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\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> 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: fixes a::"'a::euclidean_space"
  assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
  shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
  using assms unfolding convex_def by auto

lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> 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 \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
  shows "\<forall>y\<in>s. f x \<le> f y"
proof(rule ccontr)
  have "x\<in>s" using assms(1,3) by auto
  assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
  then obtain y where "y\<in>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 \<le> 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) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>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) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> 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 \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>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 \<le> u" "0 \<le> v" "u + v = 1"
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> 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 *\<^sub>R y + v *\<^sub>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 \<le> e" "dist x z \<le> e"
  fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> 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 *\<^sub>R y + v *\<^sub>R z) \<le> 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 t \<and> s \<subseteq> t}"]
  by auto

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

lemma bounded_convex_hull:
  fixes s :: "'a::real_normed_vector set"
  assumes "bounded s" shows "bounded(convex hull s)"
proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
  show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
    unfolding subset_hull[of convex, OF convex_cball]
    unfolding subset_eq mem_cball dist_norm using B by auto qed

lemma finite_imp_bounded_convex_hull:
  fixes s :: "'a::real_normed_vector set"
  shows "finite s \<Longrightarrow> bounded(convex hull s)"
  using bounded_convex_hull finite_imp_bounded 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 \<in> 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 \<in> 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 \<in> convex hull s"
  shows "(f x) \<in> convex hull (f ` s)"
using convex_hull_linear_image[OF assms(1)] assms(2) by auto

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

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

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

lemma convex_hull_insert:
  fixes s :: "'a::real_vector set"
  assumes "s \<noteq> {}"
  shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
                                    b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
 apply(rule,rule hull_minimal,rule) unfolding insert_iff prefer 3 apply rule proof-
 fix x assume x:"x = a \<or> x \<in> s"
 thus "x\<in>?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\<in>?hull"
  then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
  have "a\<in>convex hull insert a s" "b\<in>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\<in> 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) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
    from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
    from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
    have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
    have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
    proof(cases "u * v1 + v * v2 = 0")
      have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
      case True hence **:"u * v1 = 0" "v * v2 = 0"
        using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>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 "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
      also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
      case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> 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)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>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 zero_le_divide_iff
        by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
    qed note * = this
    have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
    have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
    have "u1 * u + u2 * v \<le> (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 "\<dots> \<le> 1" unfolding right_distrib[THEN sym] and as(3) using u1 u2 by auto
    finally 
    show "u *\<^sub>R x + v *\<^sub>R y \<in> ?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. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>