# Theory Convex_Euclidean_Space

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theory Convex_Euclidean_Space
imports Topology_Euclidean_Space Convex Set_Algebras
`(*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy    Author:     Robert Himmelmann, TU Muenchen    Author:     Bogdan Grechuk, University of Edinburgh*)header {* Convex sets, functions and related things. *}theory Convex_Euclidean_Spaceimports  Topology_Euclidean_Space  "~~/src/HOL/Library/Convex"  "~~/src/HOL/Library/Set_Algebras"begin(* ------------------------------------------------------------------------- *)(* To be moved elsewhere                                                     *)(* ------------------------------------------------------------------------- *)lemma linear_scaleR: "linear (λx. scaleR c x)"  by (simp add: linear_def scaleR_add_right)lemma injective_scaleR: "c ≠ 0 ==> inj (λ(x::'a::real_vector). scaleR c x)"  by (simp add: inj_on_def)lemma linear_add_cmul:  assumes "linear f"  shows "f(a *⇩R x + b *⇩R y) = a *⇩R f x +  b *⇩R f y"  using linear_add[of f] linear_cmul[of f] assms by simplemma mem_convex_2:  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v=1"  shows "(u *⇩R x + v *⇩R y) : S"  using assms convex_def[of S] by autolemma mem_convex_alt:  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"  shows "((u/(u+v)) *⇩R x + (v/(u+v)) *⇩R y) : S"  apply (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"] apply auto  donelemma 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 blastqedlemma 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  then have "span S = span B"    using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto  then have "independent (f ` B)"    using independent_injective_on_span_image[of B f] B_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  then show ?thesis using dim_image_le[of f S] assms by autoqedlemma 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 .qedlemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"  unfolding subspace_def by autolemma span_eq[simp]: "(span s = s) <-> subspace s"  unfolding span_def by (rule hull_eq, rule subspace_Inter)lemma substdbasis_expansion_unique:  assumes d: "d ⊆ Basis"  shows "(∑i∈d. f i *⇩R i) = (x::'a::euclidean_space)      <-> (∀i∈Basis. (i ∈ d --> f i = x • i) ∧ (i ∉ d --> x • i = 0))"proof -  have *: "!!x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto  have **: "finite d" by (auto intro: finite_subset[OF assms])  have ***: "!!i. i ∈ Basis ==> (∑i∈d. f i *⇩R i) • i = (∑x∈d. if x = i then f x else 0)"    using d    by (auto intro!: setsum_cong simp: inner_Basis inner_setsum_left)  show ?thesis    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum_delta[OF **] ***)qedlemma independent_substdbasis: "d ⊆ Basis ==> independent d"  by (rule independent_mono[OF independent_Basis])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) *⇩R x"    then have "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto    moreover have *: "x = (norm x/e) *⇩R y" using y_def assms by simp    moreover from * have "x = (norm x/e) *⇩R y" by auto    ultimately have "x : span (cball 0 e)"      using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto  } then have "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto  then show ?thesis    using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)qedlemma 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 autolemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"  by (rule ccontr) autolemma translate_inj_on:  fixes A :: "('a::ab_group_add) set"  shows "inj_on (%x. a+x) A"  unfolding inj_on_def by autolemma translation_assoc:  fixes a b :: "'a::ab_group_add"  shows "(λx. b+x) ` ((λx. a+x) ` S) = (λx. (a+b)+x) ` S"  by autolemma translation_invert:  fixes a :: "'a::ab_group_add"  assumes "(λx. a+x) ` A = (λx. 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  then show ?thesis    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by autoqedlemma translation_galois:  fixes a :: "'a::ab_group_add"  shows "T=((λx. a+x) ` S) <-> S=((λx. (-a)+x) ` T)"  using translation_assoc[of "-a" a S] apply auto  using translation_assoc[of a "-a" T] apply auto  donelemma 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"    then have "x-a : S" using assms by auto    then have "x : {a + v |v. v : S}"      apply auto      apply (rule exI[of _ "x-a"])      apply simp      done    then have "x : ((%x. a+x) ` S)" by auto  } then show ?thesis by autoqedlemma 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 ⊆ C" "inj_on f B" using `finite B` `finite C` by auto  from linear_independent_extend[OF B(2)] obtain g where    g: "linear g" "∀x∈ B. g x = f x" by blast  from inj_on_iff_eq_card[OF fB, of f] f(2)  have "card (f ` B) = card B" by simp  with B(4) C(4) have ceq: "card (f ` B) = card C" using d    by simp  have "g ` B = f ` B" using g(2)    by (auto simp add: image_iff)  also have "… = C" using card_subset_eq[OF fC f(1) ceq] .  finally have gBC: "g ` B = C" .  have gi: "inj_on g B" using f(2) g(2)    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 ∈ S" and y: "y ∈ S" and gxy: "g x = g y"    from B(3) x y have x': "x ∈ span B" and y': "y ∈ span B" by blast+    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])    have th1: "x - y ∈ span B" using x' y' by (metis span_sub)    have "x=y" using g0[OF th1 th0] by simp  } then have giS: "inj_on g S" unfolding inj_on_def by blast  from span_subspace[OF B(1,3) s]  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])  also have "… = span C" unfolding gBC ..  also have "… = T" using span_subspace[OF C(1,3) t] .  finally have gS: "g ` S = T" .  from g(1) gS giS gBC show ?thesis by blastqedlemma closure_bounded_linear_image:  assumes f: "bounded_linear f"  shows "f ` (closure S) ⊆ closure (f ` S)"  using linear_continuous_on [OF f] closed_closure closure_subset  by (rule image_closure_subset)lemma closure_linear_image:  fixes f :: "('m::euclidean_space) => ('n::real_normed_vector)"  assumes "linear f"  shows "f ` (closure S) <= closure (f ` S)"  using assms unfolding linear_conv_bounded_linear  by (rule closure_bounded_linear_image)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  then have "f' ` closure (f ` S) <= closure (S)"    using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto  then have "f ` f' ` closure (f ` S) <= f ` closure (S)" by auto  then have "closure (f ` S) <= f ` closure (S)"    using image_compose[of f f' "closure (f ` S)"] f'_def by auto  then show ?thesis using closure_linear_image[of f S] assms by autoqedlemma closure_direct_sum:  shows "closure (S <*> T) = closure S <*> closure T"  by (rule closure_Times)lemma closure_scaleR:  fixes S :: "('a::real_normed_vector) set"  shows "(op *⇩R c) ` (closure S) = closure ((op *⇩R c) ` S)"proof  show "(op *⇩R c) ` (closure S) ⊆ closure ((op *⇩R c) ` S)"    using bounded_linear_scaleR_right by (rule closure_bounded_linear_image)  show "closure ((op *⇩R c) ` S) ⊆ (op *⇩R c) ` (closure S)"    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)qedlemma 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 [symmetric] scaleR_left_distrib by simplemma vector_choose_size:  "0 <= c ==> ∃(x::'a::euclidean_space). norm x = c"  apply (rule exI[where x="c *⇩R (SOME i. i ∈ Basis)"])  apply (auto simp: SOME_Basis)  donelemma setsum_delta_notmem:  assumes "x ∉ s"  shows "setsum (λy. if (y = x) then P x else Q y) s = setsum Q s"    and "setsum (λy. if (x = y) then P x else Q y) s = setsum Q s"    and "setsum (λy. if (y = x) then P y else Q y) s = setsum Q s"    and "setsum (λy. if (x = y) then P y else Q y) s = setsum Q s"  apply (rule_tac [!] setsum_cong2)  using assms apply auto  donelemma 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 autoqedlemma if_smult:"(if P then x else (y::real)) *⇩R v = (if P then x *⇩R v else y *⇩R v)" by autolemma image_smult_interval:  "(λx. m *⇩R (x::'a::ordered_euclidean_space)) ` {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 autolemma dist_triangle_eq:  fixes x y z :: "'a::real_inner"  shows "dist x z = dist x y + dist y z <-> norm (x - y) *⇩R (y - z) = norm (y - z) *⇩R (x - y)"proof -  have *: "x - y + (y - z) = x - z" by auto  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]    by (auto simp add:norm_minus_commute)qedlemma norm_minus_eqI:"x = - y ==> norm x = norm y" by autolemma 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 autolemma norm_lt: "norm x < norm y <-> inner x x < inner y y"  unfolding norm_eq_sqrt_inner by simplemma norm_le: "norm x ≤ norm y <-> inner x x ≤ inner y y"  unfolding norm_eq_sqrt_inner by simpsubsection {* 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 autolemma 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 autolemma affine_Inter: "(∀s∈f. affine s) ==> affine (\<Inter> f)"  unfolding affine_def by autolemma affine_Int: "affine s ==> affine t ==> affine (s ∩ t)"  unfolding affine_def by autolemma affine_affine_hull: "affine(affine hull s)"  unfolding hull_def  using affine_Inter[of "{t. affine t ∧ s ⊆ t}"] by autolemma affine_hull_eq[simp]: "(affine hull s = s) <-> affine s"  by (metis affine_affine_hull hull_same)subsubsection {* Some explicit formulations (from Lars Schewe) *}lemma affine:  fixes V::"'a::real_vector set"  shows "affine V <->    (∀s u. finite s ∧ s ≠ {} ∧ s ⊆ V ∧ setsum u s = 1 --> (setsum (λx. (u x) *⇩R x)) s ∈ V)"  unfolding affine_def  apply rule  apply(rule, rule, rule)  apply(erule conjE)+  defer  apply (rule, rule, rule, rule, rule)proof -  fix x y u v  assume as: "x ∈ V" "y ∈ V" "u + v = (1::real)"    "∀s u. finite s ∧ s ≠ {} ∧ s ⊆ V ∧ setsum u s = 1 --> (∑x∈s. u x *⇩R x) ∈ V"  then show "u *⇩R x + v *⇩R y ∈ V"    apply (cases "x = y")    using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="λw. if w = x then u else v"]]      and as(1-3)    by (auto simp add: scaleR_left_distrib[symmetric])next  fix s u  assume as: "∀x∈V. ∀y∈V. ∀u v. u + v = 1 --> u *⇩R x + v *⇩R y ∈ V"    "finite s" "s ≠ {}" "s ⊆ V" "setsum u s = (1::real)"  def n ≡ "card s"  have "card s = 0 ∨ card s = 1 ∨ card s = 2 ∨ card s > 2" by auto  then show "(∑x∈s. u x *⇩R x) ∈ V"  proof (auto simp only: disjE)    assume "card s = 2"    then have "card s = Suc (Suc 0)" by auto    then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto    then show ?thesis      using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)      by (auto simp add: setsum_clauses(2))  next    assume "card s > 2"    then show ?thesis using as and n_def    proof (induct n arbitrary: u s)      case 0      then show ?case by auto    next      case (Suc n)      fix s :: "'a set" and u :: "'a => real"      assume IA:        "!!u s.  [|2 < card s; ∀x∈V. ∀y∈V. ∀u v. u + v = 1 --> u *⇩R x + v *⇩R y ∈ V; finite s;          s ≠ {}; s ⊆ V; setsum u s = 1; n = card s |] ==> (∑x∈s. u x *⇩R x) ∈ V"        and as:          "Suc n = card s" "2 < card s" "∀x∈V. ∀y∈V. ∀u v. u + v = 1 --> u *⇩R x + v *⇩R y ∈ V"           "finite s" "s ≠ {}" "s ⊆ V" "setsum u s = 1"      have "∃x∈s. u x ≠ 1"      proof (rule ccontr)        assume "¬ ?thesis"        then have "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto        then show False          using as(7) and `card s > 2`          by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)      qed      then obtain x where x:"x∈s" "u x ≠ 1" by auto      have c: "card (s - {x}) = card s - 1"        apply (rule card_Diff_singleton) using `x∈s` as(4) 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)[symmetric] as(7)] by auto      have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"        unfolding ** using `u x ≠ 1` by auto      have "(∑xa∈s - {x}. (inverse (1 - u x) * u xa) *⇩R xa) ∈ V"      proof (cases "card (s - {x}) > 2")        case True        then have "s - {x} ≠ {}" "card (s - {x}) = n"          unfolding c and as(1)[symmetric]        proof (rule_tac ccontr)          assume "¬ s - {x} ≠ {}"          then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp          then show False using True by auto        qed auto        then show ?thesis          apply (rule_tac IA[of "s - {x}" "λy. (inverse (1 - u x) * u y)"])          unfolding setsum_right_distrib[symmetric] using as and *** and True          apply auto          done      next        case False        then have "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto        then obtain a b where "(s - {x}) = {a, b}" "a≠b" unfolding card_Suc_eq by auto        then show ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]          using *** *(2) and `s ⊆ V`          unfolding setsum_right_distrib by (auto simp add: setsum_clauses(2))      qed      then have "u x + (1 - u x) = 1 ==>          u x *⇩R x + (1 - u x) *⇩R ((∑xa∈s - {x}. u xa *⇩R xa) /⇩R (1 - u x)) ∈ V"        apply -        apply (rule as(3)[rule_format])        unfolding  RealVector.scaleR_right.setsum        using x(1) as(6) apply auto        done      then show "(∑x∈s. u x *⇩R x) ∈ V"        unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]        apply (subst *)        unfolding setsum_clauses(2)[OF *(2)]        using `u x ≠ 1` apply auto        done    qed  next    assume "card s = 1"    then obtain a where "s={a}" by (auto simp add: card_Suc_eq)    then show ?thesis using as(4,5) by simp  qed (insert `s≠{}` `finite s`, auto)qedlemma affine_hull_explicit:  "affine hull p = {y. ∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ setsum (λv. (u v) *⇩R v) s = y}"  apply (rule hull_unique)  apply (subst subset_eq)  prefer 3  apply rule  unfolding mem_Collect_eq  apply (erule exE)+  apply (erule conjE)+  prefer 2  apply ruleproof -  fix x  assume "x∈p"  then show "∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"    apply (rule_tac x="{x}" in exI, rule_tac x="λx. 1" in exI)    apply auto    donenext  fix t x s u  assume as: "p ⊆ t" "affine t" "finite s" "s ≠ {}" "s ⊆ p" "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"  then show "x ∈ t"    using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by autonext  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, symmetric]      unfolding scaleR_scaleR[symmetric] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[symmetric]      unfolding x y      using x(1-3) y(1-3) uv apply simp      done  qedqedlemma affine_hull_finite:  assumes "finite s"  shows "affine hull s = {y. ∃u. setsum u s = 1 ∧ setsum (λv. u v *⇩R v) s = y}"  unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq apply (rule,rule)  apply(erule exE)+  apply(erule conjE)+  defer  apply (erule exE)  apply (erule conjE)proof -  fix x u  assume "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"  then show "∃sa u. finite sa ∧      ¬ (∀x. (x ∈ sa) = (x ∈ {})) ∧ sa ⊆ s ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = x"    apply (rule_tac x=s in exI, rule_tac x=u in exI)    using assms apply auto    donenext  fix x t u  assume "t ⊆ s"  then have *: "s ∩ t = t" by auto  assume "finite t" "¬ (∀x. (x ∈ t) = (x ∈ {}))" "setsum u t = 1" "(∑v∈t. u v *⇩R v) = x"  then show "∃u. setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"    apply (rule_tac x="λx. if x∈t then u x else 0" in exI)    unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, symmetric] and *    apply auto    doneqedsubsubsection {* Stepping theorems and hence small special cases *}lemma affine_hull_empty[simp]: "affine hull {} = {}"  by (rule hull_unique) autolemma 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      then have *: "insert a s = s" by auto      show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto    next      case False      then show ?thesis        apply (rule_tac x="u a" in exI)        using u and `?as` apply auto        done    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      then show ?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        apply (auto simp add: setsum_delta[OF `?as`])        done    next      case False      then have **:        "!!x. x ∈ s ==> u x = (if x = a then v else u x)"        "!!x. x ∈ s ==> u x *⇩R x = (if x = a then v *⇩R x else u x *⇩R x)" by auto      from False show ?thesis        apply (rule_tac x="λx. if x=a then v else u x" in exI)        unfolding setsum_clauses(2)[OF `?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)]        apply auto        done    qed  }  ultimately show "?lhs = ?rhs" by blastqedlemma 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 autoqedlemma 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) apply force    doneqedlemma mem_affine:  assumes "affine S" "x : S" "y : S" "u + v = 1"  shows "(u *⇩R x + v *⇩R y) : S"  using assms affine_def[of S] by autolemma mem_affine_3:  assumes "affine S" "x : S" "y : S" "z : S" "u + v + w = 1"  shows "(u *⇩R x + v *⇩R y + w *⇩R z) : S"proof -  have "(u *⇩R x + v *⇩R y + w *⇩R z) : affine hull {x, y, z}"    using affine_hull_3[of x y z] assms by auto  moreover  have "affine hull {x, y, z} <= affine hull S"    using hull_mono[of "{x, y, z}" "S"] assms by auto  moreover  have "affine hull S = S" using assms affine_hull_eq[of S] by auto  ultimately show ?thesis by autoqedlemma mem_affine_3_minus:  assumes "affine S" "x : S" "y : S" "z : S"  shows "x + v *⇩R (y-z) : S"  using mem_affine_3[of S x y z 1 v "-v"] assms by (simp add: algebra_simps)subsubsection {* Some relations between affine hull and subspaces *}lemma affine_hull_insert_subset_span:  "affine hull (insert a s) ⊆ {a + v| v . v ∈ span {x - a | x . x ∈ s}}"  unfolding subset_eq Ball_def  unfolding affine_hull_explicit span_explicit mem_Collect_eq  apply (rule, rule)  apply (erule exE)+  apply (erule conjE)+proof -  fix x t u  assume as: "finite t" "t ≠ {}" "t ⊆ insert a s" "setsum u t = 1" "(∑v∈t. u v *⇩R v) = x"  have "(λx. x - a) ` (t - {a}) ⊆ {x - a |x. x ∈ s}" using as(3) by auto  then show "∃v. x = a + v ∧ (∃S u. finite S ∧ S ⊆ {x - a |x. x ∈ s} ∧ (∑v∈S. u v *⇩R v) = v)"    apply (rule_tac x="x - a" in exI)    apply (rule conjI, simp)    apply (rule_tac x="(λx. x - a) ` (t - {a})" in exI)    apply (rule_tac x="λx. u (x + a)" in exI)    apply (rule conjI) using as(1) apply simp    apply (erule conjI)    using as(1)    apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib      setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)    unfolding as    apply simp    doneqedlemma affine_hull_insert_span:  assumes "a ∉ s"  shows "affine hull (insert a s) = {a + v | v . v ∈ span {x - a | x.  x ∈ s}}"  apply (rule, rule affine_hull_insert_subset_span)  unfolding subset_eq Ball_def  unfolding affine_hull_explicit and mem_Collect_eqproof (rule, rule, erule exE, erule conjE)  fix y v  assume "y = a + v" "v ∈ span {x - a |x. x ∈ s}"  then obtain t u where obt:"finite t" "t ⊆ {x - a |x. x ∈ s}" "a + (∑v∈t. u v *⇩R v) = y"    unfolding span_explicit by auto  def f ≡ "(λx. x + a) ` t"  have f:"finite f" "f ⊆ s" "(∑v∈f. u (v - a) *⇩R (v - a)) = y - a"    unfolding f_def using obt by (auto simp add: setsum_reindex[unfolded inj_on_def])  have *: "f ∩ {a} = {}" "f ∩ - {a} = f" using f(2) assms by auto  show "∃sa u. finite sa ∧ sa ≠ {} ∧ sa ⊆ insert a s ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = y"    apply (rule_tac x = "insert a f" in exI)    apply (rule_tac x = "λx. if x=a then 1 - setsum (λx. u (x - a)) f else u (x - a)" in exI)    using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult    unfolding setsum_cases[OF f(1), of "λx. x = a"]    apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)    doneqedlemma affine_hull_span:  assumes "a ∈ s"  shows "affine hull s = {a + v | v. v ∈ span {x - a | x. x ∈ s - {a}}}"  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by autosubsubsection {* 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"    then have "(-a)+x : S" using assms by auto    then have "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 autoqedlemma 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 autolemma affine_parallel_reflex: "affine_parallel S S"  unfolding affine_parallel_def apply (rule exI[of _ "0"]) by autolemma 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  then show ?thesis    using translation_galois[of B a A] unfolding affine_parallel_def by autoqedlemma 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 autoqedlemma 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"    then have "(a+x):((%x. a + x) ` S)" "(a+y):((%x. a + x) ` S)" by auto    then have h1: "u *⇩R  (a+x) + v *⇩R (a+y) : ((%x. a + x) ` S)"      using xy assms unfolding affine_def by auto    have "u *⇩R (a+x) + v *⇩R (a+y) = (u+v) *⇩R a + (u *⇩R x + v *⇩R y)"      by (simp add: algebra_simps)    also have "...= a + (u *⇩R x + v *⇩R y)" using `u+v=1` by auto    ultimately have "a + (u *⇩R x + v *⇩R y) : ((%x. a + x) ` S)" using h1 by auto    then have "u *⇩R x + v *⇩R y : S" by auto  }  then show ?thesis unfolding affine_def by autoqedlemma 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  then show ?thesis using affine_translation_aux by autoqedlemma 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  then show ?thesis using affine_translation assms by autoqedlemma subspace_imp_affine: "subspace s ==> affine s"  unfolding subspace_def affine_def by autosubsubsection {* Subspace parallel to an affine set *}lemma subspace_affine: "subspace S <-> (affine S & 0 : S)"proof -  have h0: "subspace S ==> (affine S & 0 : S)"    using subspace_imp_affine[of S] subspace_0 by auto  { assume assm: "affine S & 0 : S"    { fix c :: real      fix x assume x_def: "x : S"      have "c *⇩R x = (1-c) *⇩R 0 + c *⇩R x" by auto      moreover      have "(1-c) *⇩R 0 + c *⇩R x : S" using affine_alt[of S] assm x_def by auto      ultimately have "c *⇩R x : S" by auto    }    then have h1: "!c. !x : S. c *⇩R x : S" by auto    { fix x y assume xy_def: "x : S" "y : S"      def u == "(1 :: real)/2"      have "(1/2) *⇩R (x+y) = (1/2) *⇩R (x+y)" by auto      moreover      have "(1/2) *⇩R (x+y)=(1/2) *⇩R x + (1-(1/2)) *⇩R y" by (simp add: algebra_simps)      moreover      have "(1-u) *⇩R x + u *⇩R y : S" using affine_alt[of S] assm xy_def by auto      ultimately      have "(1/2) *⇩R (x+y) : S" using u_def by auto      moreover      have "(x+y) = 2 *⇩R ((1/2) *⇩R (x+y))" by auto      ultimately      have "(x+y) : S" using h1[rule_format, of "(1/2) *⇩R (x+y)" "2"] by auto    }    then have "!x : S. !y : S. (x+y) : S" by auto    then have "subspace S" using h1 assm unfolding subspace_def by auto  }  then show ?thesis using h0 by metisqedlemma 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 autoqedlemma 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  then have "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] apply auto    done  ultimately show ?thesis using subspace_affine par by autoqedlemma 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  then have "-a : A" using assms subspace_0[of B] by auto  then have "a : A" using assms subspace_neg[of A "-a"] by auto  then show ?thesis using assms a_def unfolding subspace_def by autoqedlemma parallel_subspace:  assumes "subspace A" "subspace B" "affine_parallel A B"  shows "A = B"proof  show "A >= B"    using assms parallel_subspace_aux by auto  show "A <= B"    using assms parallel_subspace_aux[of B A] affine_parallel_commut by autoqedlemma affine_parallel_subspace:  assumes "affine S" "S ~= {}"  shows "?!L. subspace L & affine_parallel S L"proof -  have ex: "? L. subspace L & affine_parallel S L"    using assms parallel_subspace_explicit by auto  { fix L1 L2    assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"    then have "affine_parallel L1 L2"      using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto    then have "L1 = L2"      using ass parallel_subspace by auto  }  then show ?thesis using ex by autoqedsubsection {* 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 autolemma cone_univ[intro, simp]: "cone UNIV"  unfolding cone_def by autolemma cone_Inter[intro]: "(∀s∈f. cone s) ==> cone(\<Inter> f)"  unfolding cone_def by autosubsubsection {* Conic hull *}lemma cone_cone_hull: "cone (cone hull s)"  unfolding hull_def by autolemma cone_hull_eq: "(cone hull s = s) <-> cone s"  apply (rule hull_eq)  using cone_Inter unfolding subset_eq apply auto  donelemma mem_cone:  assumes "cone S" "x : S" "c>=0"  shows "c *⇩R x : S"  using assms cone_def[of S] by autolemma cone_contains_0:  assumes "cone S"  shows "(S ~= {}) <-> (0 : S)"proof -  { assume "S ~= {}" then obtain a where "a:S" by auto    then have "0 : S" using assms mem_cone[of S a 0] by auto  }  then show ?thesis by autoqedlemma cone_0: "cone {0}"  unfolding cone_def by autolemma cone_Union[intro]: "(!s:f. cone s) --> (cone (Union f))"  unfolding cone_def by blastlemma cone_iff:  assumes "S ~= {}"  shows "cone S <-> 0:S & (!c. c>0 --> ((op *⇩R c) ` S) = S)"proof -  { assume "cone S"    { fix c      assume "(c :: real) > 0"      { fix x        assume "x : S"        then have "x : (op *⇩R c) ` S"          unfolding image_def          using `cone S` `c>0` mem_cone[of S x "1/c"]            exI[of "(%t. t:S & x = c *⇩R t)" "(1 / c) *⇩R x"] apply auto          done      }      moreover      { fix x assume "x : (op *⇩R c) ` S"        (*from this obtain t where "t:S & x = c *⇩R t" by auto*)        then have "x:S"          using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto      }      ultimately have "((op *⇩R c) ` S) = S" by auto    }    then have "0:S & (!c. c>0 --> ((op *⇩R c) ` S) = S)"      using `cone S` cone_contains_0[of S] assms by auto  }  moreover  { assume a: "0:S & (!c. c>0 --> ((op *⇩R c) ` S) = S)"    { fix x assume "x:S"      fix c1      assume "(c1 :: real) >= 0"      then have "(c1=0) | (c1>0)" by auto      then have "c1 *⇩R x : S" using a `x:S` by auto    }    then have "cone S" unfolding cone_def by auto  }  ultimately show ?thesis by blastqedlemma cone_hull_empty: "cone hull {} = {}"  by (metis cone_empty cone_hull_eq)lemma cone_hull_empty_iff: "(S = {}) <-> (cone hull S = {})"  by (metis bot_least cone_hull_empty hull_subset xtrans(5))lemma cone_hull_contains_0: "(S ~= {}) <-> (0 : cone hull S)"  using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]  by autolemma mem_cone_hull:  assumes "x : S" "c>=0"  shows "c *⇩R x : cone hull S"  by (metis assms cone_cone_hull hull_inc mem_cone)lemma cone_hull_expl: "cone hull S = {c *⇩R x | c x. c>=0 & x : S}" (is "?lhs = ?rhs")proof -  { fix x    assume "x : ?rhs"    then obtain cx xx where x_def: "x= cx *⇩R xx & (cx :: real)>=0 & xx : S"      by auto    fix c    assume c_def: "(c :: real) >= 0"    then have "c *⇩R x = (c*cx) *⇩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 *⇩R x : ?rhs" using x_def by auto  } then have "cone ?rhs" unfolding cone_def by auto  then have "?rhs : Collect cone" unfolding mem_Collect_eq by auto  { fix x    assume "x : S"    then have "1 *⇩R x : ?rhs"      apply auto      apply (rule_tac x="1" in exI)      apply auto      done    then have "x : ?rhs" by auto  } then have "S <= ?rhs" by auto  then have "?lhs <= ?rhs"    using `?rhs : Collect cone` hull_minimal[of S "?rhs" "cone"] by auto  moreover  { fix x    assume "x : ?rhs"    then obtain cx xx where x_def: "x= cx *⇩R xx & (cx :: real)>=0 & xx : S" by auto    then have "xx : cone hull S" using hull_subset[of S] by auto    then have "x : ?lhs"      using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto  }  ultimately show ?thesis by autoqedlemma cone_closure:  fixes S :: "('a::real_normed_vector) set"  assumes "cone S"  shows "cone (closure S)"proof (cases "S = {}")  case True  then show ?thesis by autonext  case False  then have "0:S & (!c. c>0 --> op *⇩R c ` S = S)"    using cone_iff[of S] assms by auto  then have "0:(closure S) & (!c. c>0 --> op *⇩R c ` (closure S) = (closure S))"    using closure_subset by (auto simp add: closure_scaleR)  then show ?thesis using cone_iff[of "closure S"] by autoqedsubsection {* Affine dependence and consequential theorems (from Lars Schewe) *}definition affine_dependent :: "'a::real_vector set => bool"  where "affine_dependent s <-> (∃x∈s. x ∈ (affine hull (s - {x})))"lemma affine_dependent_explicit:  "affine_dependent p <->    (∃s u. finite s ∧ s ⊆ p ∧ setsum u s = 0 ∧    (∃v∈s. u v ≠ 0) ∧ setsum (λv. u v *⇩R v) s = 0)"  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq  apply rule  apply (erule bexE, erule exE, erule exE)  apply (erule conjE)+  defer  apply (erule exE, erule exE)  apply (erule conjE)+  apply (erule bexE)proof -  fix x s u  assume as: "x ∈ p" "finite s" "s ≠ {}" "s ⊆ p - {x}" "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"  have "x∉s" using as(1,4) by auto  show "∃s u. finite s ∧ s ⊆ p ∧ setsum u s = 0 ∧ (∃v∈s. u v ≠ 0) ∧ (∑v∈s. u v *⇩R v) = 0"    apply (rule_tac x="insert x s" in exI, rule_tac x="λv. if v = x then - 1 else u v" in exI)    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x∉s`] and as    using as apply auto    donenext  fix s u v  assume as:"finite s" "s ⊆ p" "setsum u s = 0" "(∑v∈s. u v *⇩R v) = 0" "v ∈ s" "u v ≠ 0"  have "s ≠ {v}" using as(3,6) by auto  then show "∃x∈p. ∃s u. finite s ∧ s ≠ {} ∧ s ⊆ p - {x} ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"    apply (rule_tac x=v in bexI, rule_tac x="s - {v}" in exI,      rule_tac x="λx. - (1 / u v) * u x" in exI)    unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]    unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]    using as apply auto    doneqedlemma affine_dependent_explicit_finite:  fixes s :: "'a::real_vector set"  assumes "finite s"  shows "affine_dependent s <-> (∃u. setsum u s = 0 ∧ (∃v∈s. u v ≠ 0) ∧ setsum (λv. u v *⇩R v) s = 0)"  (is "?lhs = ?rhs")proof  have *: "!!vt u v. (if vt then u v else 0) *⇩R v = (if vt then (u v) *⇩R v else (0::'a))"    by auto  assume ?lhs  then obtain t u v where      "finite t" "t ⊆ s" "setsum u t = 0" "v∈t" "u v ≠ 0"  "(∑v∈t. u v *⇩R v) = 0"    unfolding affine_dependent_explicit by auto  then show ?rhs    apply (rule_tac x="λx. if x∈t then u x else 0" in exI)    apply auto unfolding * and setsum_restrict_set[OF assms, symmetric]    unfolding Int_absorb1[OF `t⊆s`]    apply auto    donenext  assume ?rhs  then obtain u v where "setsum u s = 0"  "v∈s" "u v ≠ 0" "(∑v∈s. u v *⇩R v) = 0" by auto  then show ?lhs unfolding affine_dependent_explicit    using assms by autoqedsubsection {* Connectedness of convex sets *}lemma connected_real_lemma:  fixes f :: "real => 'a::metric_space"  assumes ab: "a ≤ b" and fa: "f a ∈ e1" and fb: "f b ∈ e2"    and dst: "!!e x. a <= x ==> x <= b ==> 0 < e ==> ∃d > 0. ∀y. abs(y - x) < d --> dist(f y) (f x) < e"    and e1: "∀y ∈ e1. ∃e > 0. ∀y'. dist y' y < e --> y' ∈ e1"    and e2: "∀y ∈ e2. ∃e > 0. ∀y'. dist y' y < e --> y' ∈ e2"    and e12: "~(∃x ≥ a. x <= b ∧ f x ∈ e1 ∧ f x ∈ e2)"  shows "∃x ≥ a. x <= b ∧ f x ∉ e1 ∧ f x ∉ e2" (is "∃ x. ?P x")proof -  let ?S = "{c. ∀x ≥ a. x <= c --> f x ∈ e1}"  have Se: " ∃x. x ∈ ?S"    apply (rule exI[where x=a])    apply (auto simp add: fa)    done  have Sub: "∃y. isUb UNIV ?S y"    apply (rule exI[where x= b])    using ab fb e12 apply (auto simp add: isUb_def setle_def)    done  from reals_complete[OF Se Sub] obtain l where    l: "isLub UNIV ?S l"by blast  have alb: "a ≤ l" "l ≤ b" using l ab fa fb e12    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)    apply (metis linorder_linear)    done  have ale1: "∀z ≥ a. z < l --> f z ∈ e1" using l    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)    apply (metis linorder_linear not_le)    done  have th1: "!!z x e d :: real. z <= x + e ==> e < d ==> z < x ∨ abs(z - x) < d" by arith  have th2: "!!e x:: real. 0 < e ==> ~(x + e <= x)" by arith  have "!!d::real. 0 < d ==> 0 < d/2 ∧ d/2 < d" by simp  then have th3: "!!d::real. d > 0 ==> ∃e > 0. e < d" by blast  { assume le2: "f l ∈ e2"    from le2 fa fb e12 alb have la: "l ≠ a" by metis    then have lap: "l - a > 0" using alb by arith    from e2[rule_format, OF le2] obtain e where      e: "e > 0" "∀y. dist y (f l) < e --> y ∈ e2" by metis    from dst[OF alb e(1)] obtain d where      d: "d > 0" "∀y. ¦y - l¦ < d --> dist (f y) (f l) < e" by metis    let ?d' = "min (d/2) ((l - a)/2)"    have "?d' < d ∧ 0 < ?d' ∧ ?d' < l - a" using lap d(1)      by (simp add: min_max.less_infI2)    then have "∃d'. d' < d ∧ d' >0 ∧ l - d' > a" by auto    then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis    from d e have th0: "∀y. ¦y - l¦ < d --> f y ∈ e2" by metis    from th0[rule_format, of "l - d'"] d' have "f (l - d') ∈ e2" by auto    moreover    have "f (l - d') ∈ 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 ∈ e1"    from le1 fa fb e12 alb have lb: "l ≠ b" by metis    then have blp: "b - l > 0" using alb by arith    from e1[rule_format, OF le1] obtain e where      e: "e > 0" "∀y. dist y (f l) < e --> y ∈ e1" by metis    from dst[OF alb e(1)] obtain d where      d: "d > 0" "∀y. ¦y - l¦ < d --> dist (f y) (f l) < e" by metis    have "!!d::real. 0 < d ==> d/2 < d ∧ 0 < d/2" by simp    then have "∃d'. d' < d ∧ d' >0" using d(1) by blast    then obtain d' where d': "d' > 0" "d' < d" by metis    from d e have th0: "∀y. ¦y - l¦ < d --> f y ∈ e1" by auto    then have "∀y. l ≤ y ∧ y ≤ l + d' --> f y ∈ e1" using d' by auto    with ale1 have "∀y. a ≤ y ∧ y ≤ l + d' --> f y ∈ 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 metisqedlemma 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    then have 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[symmetric]          unfolding less_divide_eq using n by auto      }      then have "∃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 apply simp        done    } 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) apply auto      done    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    then have 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    }  then show ?thesis unfolding connected_def by autoqedtext {* One rather trivial consequence. *}lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"  by(simp add: convex_connected convex_UNIV)text {* Balls, being convex, are connected. *}lemma convex_box: fixes a::"'a::euclidean_space"  assumes "!!i. i∈Basis ==> convex {x. P i x}"  shows "convex {x. ∀i∈Basis. P i (x•i)}"  using assms unfolding convex_def  by (auto simp: inner_add_left)lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (∀i∈Basis. 0 ≤ x•i)}"  by (rule convex_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[symmetric])  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[symmetric]    using u unfolding pos_less_divide_eq[OF xy] by auto  then have "f x ≤ f ((1 - u) *⇩R x + u *⇩R y)" using assms(4) by auto  ultimately show False    using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by autoqedlemma 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  then show "dist x (u *⇩R y + v *⇩R z) < e"    using convex_bound_lt[OF yz uv] by autoqedlemma 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  then show "dist x (u *⇩R y + v *⇩R z) ≤ e"    using convex_bound_le[OF yz uv] by autoqedlemma connected_ball:  fixes x :: "'a::real_normed_vector"  shows "connected (ball x e)"  using convex_connected convex_ball by autolemma connected_cball:  fixes x :: "'a::real_normed_vector"  shows "connected(cball x e)"  using convex_connected convex_cball by autosubsection {* Convex hull *}lemma convex_convex_hull: "convex(convex hull s)"  unfolding hull_def using convex_Inter[of "{t. convex t ∧ s ⊆ t}"]  by autolemma convex_hull_eq: "convex hull s = s <-> convex s"  by (metis convex_convex_hull hull_same)lemma bounded_convex_hull:  fixes s :: "'a::real_normed_vector set"  assumes "bounded s" shows "bounded(convex hull s)"proof -  from assms obtain B where B: "∀x∈s. norm x ≤ B"    unfolding bounded_iff by auto  show ?thesis    apply (rule bounded_subset[OF bounded_cball, of _ 0 B])    unfolding subset_hull[of convex, OF convex_cball]    unfolding subset_eq mem_cball dist_norm using B apply auto    doneqedlemma 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 autosubsubsection {* 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 assumptionproof -  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 autolemma 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 autosubsubsection {* Stepping theorems for convex hulls of finite sets *}lemma convex_hull_empty[simp]: "convex hull {} = {}"  by (rule hull_unique) autolemma convex_hull_singleton[simp]: "convex hull {a} = {a}"  by (rule hull_unique) autolemma 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 insert_iff  prefer 3  apply ruleproof -  fix x  assume x: "x = a ∨ x ∈ s"  then show "x ∈ ?hull"    apply rule    unfolding mem_Collect_eq    apply (rule_tac x=1 in exI)    defer    apply (rule_tac x=0 in exI)    using assms hull_subset[of s convex]    apply auto    donenext  fix x  assume "x ∈ ?hull"  then obtain u v b where obt: "u≥0" "v≥0" "u + v = 1" "b ∈ convex hull s" "x = u *⇩R a + v *⇩R b"    by auto  have "a ∈ convex hull insert a s" "b∈convex hull insert a s"    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4)    by auto  then show "x ∈ convex hull insert a s"    unfolding obt(5)    using 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 apply auto    donenext  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")      case True      have *: "!!(x::'a) s1 s2. x - s1 *⇩R x - s2 *⇩R x = ((1::real) - (s1 + s2)) *⇩R x"        by (auto simp add: algebra_simps)      from True have ***: "u * v1 = 0" "v * v2 = 0"        using mult_nonneg_nonneg[OF `u≥0` `v1≥0`] mult_nonneg_nonneg[OF `v≥0` `v2≥0`] by arith+      then have "u * u1 + v * u2 = 1"        using as(3) obt1(3) obt2(3) by auto      then show ?thesis        unfolding obt1(5) obt2(5) *        using assms hull_subset[of s convex]        by (auto simp add: *** scaleR_right_distrib)    next      case False      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"        using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)      also have "… = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"        using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)      also have "… = u * v1 + v * v2"        by simp      finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto      have "0 ≤ u * v1 + v * v2" "0 ≤ u * v1" "0 ≤ u * v1 + v * v2" "0 ≤ v * v2"        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) apply auto        done      then show ?thesis        unfolding obt1(5) obt2(5)        unfolding * and **        using False        apply (rule_tac x = "((u * v1) / (u * v1 + v * v2)) *⇩R b1 + ((v * v2) / (u * v1 + v * v2)) *⇩R b2" in bexI)        defer        apply (rule convex_convex_hull[of s, unfolded convex_def, rule_format])        using obt1(4) obt2(4)        unfolding add_divide_distrib[symmetric] and zero_le_divide_iff        apply (auto simp add: scaleR_left_distrib scaleR_right_distrib)        done    qed    have u1: "u1 ≤ 1"      unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto    have u2: "u2 ≤ 1"      unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto    have "u1 * u + u2 * v ≤ (max u1 u2) * u + (max u1 u2) * v"      apply (rule add_mono)      apply (rule_tac [!] mult_right_mono)      using as(1,2) obt1(1,2) obt2(1,2)      apply auto      done    also have "… ≤ 1"      unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto    finally show "u *⇩R x + v *⇩R y ∈ ?hull"      unfolding mem_Collect_eq      apply (rule_tac x="u * u1 + v * u2" in exI)      apply (rule conjI)      defer      apply (rule_tac x="1 - u * u1 - v * u2" in exI)      unfolding Bex_def      using as(1,2) obt1(1,2) obt2(1,2) **      apply (auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)      done  qedqedsubsubsection {* Explicit expression for convex hull *}lemma convex_hull_indexed:  fixes s :: "'a::real_vector set"  shows "convex hull s =    {y. ∃k u x. (∀i∈{1::nat .. k}. 0 ≤ u i ∧ x i ∈ s) ∧        (setsum u {1..k} = 1) ∧        (setsum (λi. u i *⇩R x i) {1..k} = y)}" (is "?xyz = ?hull")  apply (rule hull_unique)  apply rule  defer  apply (subst convex_def)  apply (rule, rule, rule, rule, rule, rule, rule)proof -  fix x  assume "x∈s"  then show "x ∈ ?hull"    unfolding mem_Collect_eq    apply (rule_tac x=1 in exI, rule_tac x="λx. 1" in exI)    apply auto    donenext  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) [symmetric]      apply (rule as(2)[unfolded convex, rule_format])      using assm(1,2) as(1) apply auto      done  qednext  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)    apply (auto simp add: not_le)    done  have inj: "inj_on (λi. i + k1) {1..k2}"    unfolding inj_on_def by auto  show "u *⇩R x + v *⇩R y ∈ ?hull"    apply rule    apply (rule_tac x="k1 + k2" in exI)    apply (rule_tac x="λi. if i ∈ {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)    apply (rule_tac x="λi. if i ∈ {1..k1} then x1 i else x2 (i - k1)" in exI)    apply (rule, rule)    defer    apply rule    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and      setsum_reindex[OF inj] and o_def Collect_mem_eq    unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]  proof -    fix i    assume i: "i ∈ {1..k1+k2}"    show "0 ≤ (if i ∈ {1..k1} then u * u1 i else v * u2 (i - k1)) ∧      (if i ∈ {1..k1} then x1 i else x2 (i - k1)) ∈ s"    proof (cases "i∈{1..k1}")      case True      then show ?thesis        using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto    next      case False      def j ≡ "i - k1"      from i False have "j ∈ {1..k2}" unfolding j_def by auto      then show ?thesis        unfolding j_def[symmetric]        using False        using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]]        apply auto        done    qed  qed (auto simp add: not_le x(2,3) y(2,3) uv(3))qedlemma convex_hull_finite:  fixes s :: "'a::real_vector set"  assumes "finite s"  shows "convex hull s = {y. ∃u. (∀x∈s. 0 ≤ u x) ∧    setsum u s = 1 ∧ setsum (λx. u x *⇩R x) s = y}" (is "?HULL = ?set")proof (rule hull_unique, auto simp add: convex_def[of ?set])  fix x  assume "x ∈ s"  then show "∃u. (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ (∑x∈s. u x *⇩R x) = x"    apply (rule_tac x="λy. if x=y then 1 else 0" in exI)    apply auto    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms]    apply auto    donenext  fix u v :: real  assume uv: "0 ≤ u" "0 ≤ v" "u + v = 1"  fix ux assume ux: "∀x∈s. 0 ≤ ux x" "setsum ux s = (1::real)"  fix uy assume uy: "∀x∈s. 0 ≤ uy x" "setsum uy s = (1::real)"  { fix x    assume "x∈s"    then have "0 ≤ u * ux x + v * uy x"      using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)      apply auto      apply (metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))      done  }  moreover  have "(∑x∈s. u * ux x + v * uy x) = 1"    unfolding setsum_addf and setsum_right_distrib[symmetric] and ux(2) uy(2) using uv(3) by auto  moreover  have "(∑x∈s. (u * ux x + v * uy x) *⇩R x) = u *⇩R (∑x∈s. ux x *⇩R x) + v *⇩R (∑x∈s. uy x *⇩R x)"    unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]    by auto  ultimately  show "∃uc. (∀x∈s. 0 ≤ uc x) ∧ setsum uc s = 1 ∧      (∑x∈s. uc x *⇩R x) = u *⇩R (∑x∈s. ux x *⇩R x) + v *⇩R (∑x∈s. uy x *⇩R x)"    apply (rule_tac x="λx. u * ux x + v * uy x" in exI)    apply auto    donenext  fix t  assume t: "s ⊆ t" "convex t"  fix u  assume u: "∀x∈s. 0 ≤ u x" "setsum u s = (1::real)"  then show "(∑x∈s. u x *⇩R x) ∈ t"    using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]    using assms and t(1) by autoqedsubsubsection {* 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)  donelemma convex_hull_explicit:  fixes p :: "'a::real_vector set"  shows "convex hull p = {y. ∃s u. finite s ∧ s ⊆ p ∧    (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ setsum (λv. u v *⇩R v) s = y}" (is "?lhs = ?rhs")proof -  { fix x assume "x∈?lhs"    then obtain k u y where        obt: "∀i∈{1::nat..k}. 0 ≤ u i ∧ y i ∈ p" "setsum u {1..k} = 1" "(∑i = 1..k. u i *⇩R y i) = x"      unfolding convex_hull_indexed by auto    have fin: "finite {1..k}" by auto    have fin': "!!v. finite {i ∈ {1..k}. y i = v}" by auto    { fix j      assume "j∈{1..k}"      then have "y j ∈ p" "0 ≤ setsum u {i. Suc 0 ≤ i ∧ i ≤ k ∧ y i = y j}"        using obt(1)[THEN bspec[where x=j]] and obt(2)        apply simp        apply (rule setsum_nonneg)        using obt(1)        apply auto        done    }    moreover    have "(∑v∈y ` {1..k}. setsum u {i ∈ {1..k}. y i = v}) = 1"      unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto    moreover have "(∑v∈y ` {1..k}. setsum u {i ∈ {1..k}. y i = v} *⇩R v) = x"      using setsum_image_gen[OF fin, of "λi. u i *⇩R y i" y, symmetric]      unfolding scaleR_left.setsum using obt(3) by auto    ultimately    have "∃s u. finite s ∧ s ⊆ p ∧ (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = x"      apply (rule_tac x="y ` {1..k}" in exI)      apply (rule_tac x="λv. setsum u {i∈{1..k}. y i = v}" in exI)      apply auto      done    then have "x∈?rhs" by auto  }  moreover  { fix y assume "y∈?rhs"    then obtain s u where      obt: "finite s" "s ⊆ p" "∀x∈s. 0 ≤ u x" "setsum u s = 1" "(∑v∈s. u v *⇩R v) = y" by auto    obtain f where f: "inj_on f {1..card s}" "f ` {1..card s} = s"      using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto    { fix i :: nat      assume "i∈{1..card s}"      then have "f i ∈ s"        apply (subst f(2)[symmetric])        apply auto        done      then have "0 ≤ u (f i)" "f i ∈ p" using obt(2,3) by auto    }    moreover have *:"finite {1..card s}" by auto    { fix y      assume "y∈s"      then obtain i where "i∈{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"]        by auto      then have "{x. Suc 0 ≤ x ∧ x ≤ card s ∧ f x = y} = {i}"        apply auto        using f(1)[unfolded inj_on_def]        apply(erule_tac x=x in ballE)        apply auto        done      then have "card {x. Suc 0 ≤ x ∧ x ≤ card s ∧ f x = y} = 1" by auto      then have "(∑x∈{x ∈ {1..card s}. f x = y}. u (f x)) = u y"          "(∑x∈{x ∈ {1..card s}. f x = y}. u (f x) *⇩R f x) = u y *⇩R y"        by (auto simp add: setsum_constant_scaleR)    }    then have "(∑x = 1..card s. u (f x)) = 1" "(∑i = 1..card s. u (f i) *⇩R f i) = y"      unfolding setsum_image_gen[OF *(1), of "λx. u (f x) *⇩R f x" f] and setsum_image_gen[OF *(1), of "λx. u (f x)" f]      unfolding f using setsum_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)      apply fastforce      done    then have "y ∈ ?lhs" unfolding convex_hull_indexed by auto  }  ultimately show ?thesis unfolding set_eq_iff by blastqedsubsubsection {* A stepping theorem for that expansion *}lemma convex_hull_finite_step:  fixes s :: "'a::real_vector set"  assumes "finite s"  shows "(∃u. (∀x∈insert a s. 0 ≤ u x) ∧ setsum u (insert a s) = w ∧ setsum (λx. u x *⇩R x) (insert a s) = y)     <-> (∃v≥0. ∃u. (∀x∈s. 0 ≤ u x) ∧ setsum u s = w - v ∧ setsum (λx. u x *⇩R x) s = y - v *⇩R a)" (is "?lhs = ?rhs")proof (rule, case_tac[!] "a∈s")  assume "a∈s"  then have *:" insert a s = s" by auto  assume ?lhs  then show ?rhs    unfolding *    apply (rule_tac x=0 in exI)    apply auto    donenext  assume ?lhs  then obtain u where u: "∀x∈insert a s. 0 ≤ u x" "setsum u (insert a s) = w" "(∑x∈insert a s. u x *⇩R x) = y"    by auto  assume "a ∉ s"  then show ?rhs    apply (rule_tac x="u a" in exI)    using u(1)[THEN bspec[where x=a]]    apply simp    apply (rule_tac x=u in exI)    using u[unfolded setsum_clauses(2)[OF assms]] and `a∉s`    apply auto    donenext  assume "a ∈ s"  then have *: "insert a s = s" by auto  have fin: "finite (insert a s)" using assms by auto  assume ?rhs  then obtain v u where uv: "v≥0" "∀x∈s. 0 ≤ u x" "setsum u s = w - v" "(∑x∈s. u x *⇩R x) = y - v *⇩R a"    by auto  show ?lhs    apply (rule_tac x = "λx. (if a = x then v else 0) + u x" in exI)    unfolding scaleR_left_distrib and setsum_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`    apply auto    donenext  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`    apply auto    done  ultimately show ?lhs    apply (rule_tac x="λx. if a = x then v else u x" in exI)    unfolding setsum_clauses(2)[OF assms]    apply auto    doneqedsubsubsection {* 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 autoshow ?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 qedlemma convex_hull_2_alt: "convex hull {a,b} = {a + u *⇩R (b - a) | u.  0 ≤ u ∧ u ≤ 1}"  unfolding convex_hull_2proof(rule Collect_cong) have *:"!!x y ::real. x + y = 1 <-> x = 1 - y" by auto  fix x show "(∃v u. x = v *⇩R a + u *⇩R b ∧ 0 ≤ v ∧ 0 ≤ u ∧ v + u = 1) = (∃u. x = a + u *⇩R (b - a) ∧ 0 ≤ u ∧ u ≤ 1)"    unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qedlemma convex_hull_3:  "convex hull {a,b,c} = { u *⇩R a + v *⇩R b + w *⇩R c | u v w. 0 ≤ u ∧ 0 ≤ v ∧ 0 ≤ w ∧ u + v + w = 1}"proof-  have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto  have *:"!!x y z ::real. x + y + z = 1 <-> x = 1 - y - z"    by (auto simp add: field_simps)  show ?thesis unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *    unfolding convex_hull_finite_step[OF fin(3)] apply(rule Collect_cong) 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 qedlemma 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) qedsubsection {* Relations among closure notions and corresponding hulls *}lemma affine_imp_convex: "affine s ==> convex s"  unfolding affine_def convex_def by autolemma subspace_imp_convex: "subspace s ==> convex s"  using subspace_imp_affine affine_imp_convex by autolemma affine_hull_subset_span: "(affine hull s) ⊆ (span s)"by (metis hull_minimal span_inc subspace_imp_affine subspace_span)lemma convex_hull_subset_span: "(convex hull s) ⊆ (span s)"by (metis hull_minimal span_inc subspace_imp_convex subspace_span)lemma convex_hull_subset_affine_hull: "(convex hull s) ⊆ (affine hull s)"by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)lemma affine_dependent_imp_dependent:  shows "affine_dependent s ==> dependent s"  unfolding affine_dependent_def dependent_def  using affine_hull_subset_span by autolemma dependent_imp_affine_dependent:  assumes "dependent {x - a| x . x ∈ s}" "a ∉ s"  shows "affine_dependent (insert a s)"proof-  from assms(1)[unfolded dependent_explicit] obtain S u v    where obt:"finite S" "S ⊆ {x - a |x. x ∈ s}" "v∈S" "u v  ≠ 0" "(∑v∈S. u v *⇩R v) = 0" by auto  def t ≡ "(λx. x + a) ` S"  have inj:"inj_on (λx. x + a) S" unfolding inj_on_def by auto  have "0∉S" using obt(2) assms(2) unfolding subset_eq by auto  have fin:"finite t" and  "t⊆s" unfolding t_def using obt(1,2) by auto  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: *)  ultimately show ?thesis unfolding affine_dependent_explicit    apply(rule_tac x="insert a t" in exI) by autoqedlemma 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 blastqedlemma affine_dependent_biggerset: fixes s::"('a::euclidean_space) set"  assumes "finite s" "card s ≥ DIM('a) + 2"  shows "affine_dependent s"proof-  have "s≠{}" using assms by auto then obtain a where "a∈s" by auto  have *:"{x - a |x. x ∈ s - {a}} = (λx. x - a) ` (s - {a})" by auto  have "card {x - a |x. x ∈ s - {a}} = card (s - {a})" unfolding *    apply(rule card_image) unfolding inj_on_def by auto  also have "… > DIM('a)" using assms(2)    unfolding card_Diff_singleton[OF assms(1) `a∈s`] by auto  finally show ?thesis apply(subst insert_Diff[OF `a∈s`, symmetric])    apply(rule dependent_imp_affine_dependent)    apply(rule dependent_biggerset) by auto qedlemma affine_dependent_biggerset_general:  assumes "finite (s::('a::euclidean_space) 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`, symmetric])    apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qedsubsection {* Caratheodory's theorem. *}lemma convex_hull_caratheodory: fixes p::"('a::euclidean_space) set"  shows "convex hull p = {y. ∃s u. finite s ∧ s ⊆ p ∧ card s ≤ DIM('a) + 1 ∧  (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ setsum (λv. u v *⇩R v) s = y}"  unfolding convex_hull_explicit set_eq_iff mem_Collect_eqproof(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 ≤ DIM('a) + 1" proof(rule ccontr, simp only: not_le)    assume "DIM('a) + 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[symmetric]]] by auto      qed qed    obtain a where "a∈s" and "t = (λv. (u v) / (- w v)) a" and "w a < 0"      using Min_in[OF _ `i≠{}`] and obt(1) unfolding i_def t_def by auto    hence a:"a∈s" "u a + t * w a = 0" by auto    have *:"!!f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"      unfolding setsum_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[symmetric] obt(5) by auto    moreover have "(∑v∈s. u v *⇩R v + (t * w v) *⇩R v) - (u a *⇩R a + (t * w a) *⇩R a) = y"      unfolding setsum_addf obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp    ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)      apply(rule_tac x="λv. u v + t * w v" in exI) using obt(1-3) and t and a      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 ≤ DIM('a) + 1    ∧ (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ∧ (∑v∈s. u v *⇩R v) = y" using obt by autoqed autolemma caratheodory: "convex hull p = {x::'a::euclidean_space. ∃s. finite s ∧ s ⊆ p ∧      card s ≤ DIM('a) + 1 ∧ x ∈ convex hull s}"  unfolding set_eq_iff 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 ≤ DIM('a) + 1"     "∀x∈s. 0 ≤ u x" "setsum u s = 1" "(∑v∈s. u v *⇩R v) = x"unfolding convex_hull_caratheodory by auto  thus "∃s. finite s ∧ s ⊆ p ∧ card s ≤ DIM('a) + 1 ∧ x ∈ convex hull s"    apply(rule_tac x=s in exI) using hull_subset[of s convex]  using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by autonext  fix x assume "∃s. finite s ∧ s ⊆ p ∧ card s ≤ DIM('a) + 1 ∧ x ∈ convex hull s"  then obtain s where "finite s" "s ⊆ p" "card s ≤ DIM('a) + 1" "x ∈ convex hull s" by auto  thus "x ∈ convex hull p" using hull_mono[OF `s⊆p`] by autoqedsubsection {* Some Properties of Affine Dependent Sets *}lemma affine_independent_empty: "~(affine_dependent {})"  by (simp add: affine_dependent_def)lemma affine_independent_sing:shows "~(affine_dependent {a})" by (simp add: affine_dependent_def)lemma affine_hull_translation:"affine hull ((%x. a + x) `  S) = (%x. a + x) ` (affine hull S)"proof-have "affine ((%x. a + x) ` (affine hull S))" using affine_translation affine_affine_hull by automoreover have "(%x. a + x) `  S <= (%x. a + x) ` (affine hull S)" using hull_subset[of S] by autoultimately have h1: "affine hull ((%x. a + x) `  S) <= (%x. a + x) ` (affine hull S)" by (metis hull_minimal)have "affine((%x. -a + x) ` (affine hull ((%x. a + x) `  S)))"  using affine_translation affine_affine_hull by automoreover have "(%x. -a + x) ` (%x. a + x) `  S <= (%x. -a + x) ` (affine hull ((%x. a + x) `  S))" using hull_subset[of "(%x. a + x) `  S"] by automoreover have "S=(%x. -a + x) ` (%x. a + x) `  S" using  translation_assoc[of "-a" a] by autoultimately have "(%x. -a + x) ` (affine hull ((%x. a + x) `  S)) >= (affine hull S)" by (metis hull_minimal)hence "affine hull ((%x. a + x) `  S) >= (%x. a + x) ` (affine hull S)" by autofrom this show ?thesis using h1 by autoqedlemma affine_dependent_translation:  assumes "affine_dependent S"  shows "affine_dependent ((%x. a + x) ` S)"proof-obtain x where x_def: "x : S & x : affine hull (S - {x})" using assms affine_dependent_def by autohave "op + a ` (S - {x}) = op + a ` S - {a + x}" by autohence "a+x : affine hull ((%x. a + x) ` S - {a+x})" using  affine_hull_translation[of a "S-{x}"] x_def by automoreover have "a+x : (%x. a + x) ` S" using x_def by autoultimately show ?thesis unfolding affine_dependent_def by autoqedlemma affine_dependent_translation_eq:  "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"proof-{ assume "affine_dependent ((%x. a + x) ` S)"  hence "affine_dependent S" using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] by auto} from this show ?thesis using affine_dependent_translation by autoqedlemma affine_hull_0_dependent:  assumes "0 : affine hull S"  shows "dependent S"proof-obtain s u where s_u_def: "finite s & s ~= {} & s <= S & setsum u s = 1 & (SUM v:s. u v *⇩R v) = 0" using assms affine_hull_explicit[of S] by autohence "EX v:s. u v ~= 0" using setsum_not_0[of "u" "s"] by autohence "finite s & s <= S & (EX v:s. u v ~= 0 & (SUM v:s. u v *⇩R v) = 0)" using s_u_def by autofrom this show ?thesis unfolding dependent_explicit[of S] by autoqedlemma affine_dependent_imp_dependent2:  assumes "affine_dependent (insert 0 S)"  shows "dependent S"proof-obtain x where x_def: "x:insert 0 S & x : affine hull (insert 0 S - {x})" using affine_dependent_def[of "(insert 0 S)"] assms by blasthence "x : span (insert 0 S - {x})" using affine_hull_subset_span by automoreover have "span (insert 0 S - {x}) = span (S - {x})" using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by autoultimately have "x : span (S - {x})" by autohence "(x~=0) ==> dependent S" using x_def dependent_def by automoreover{ assume "x=0" hence "0 : affine hull S" using x_def hull_mono[of "S - {0}" S] by auto               hence "dependent S" using affine_hull_0_dependent by auto} ultimately show ?thesis by autoqedlemma affine_dependent_iff_dependent:  assumes "a ~: S"  shows "affine_dependent (insert a S) <-> dependent ((%x. -a + x) ` S)"proof-have "(op + (- a) ` S)={x - a| x . x : S}" by autofrom this show ?thesis using affine_dependent_translation_eq[of "(insert a S)" "-a"]      affine_dependent_imp_dependent2 assms      dependent_imp_affine_dependent[of a S] by autoqedlemma affine_dependent_iff_dependent2:  assumes "a : S"  shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"proof-have "insert a (S - {a})=S" using assms by autofrom this show ?thesis using assms affine_dependent_iff_dependent[of a "S-{a}"] by autoqedlemma affine_hull_insert_span_gen:  shows "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)"proof-have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)" by auto{ assume "a ~: s" hence ?thesis using affine_hull_insert_span[of a s] h1 by auto}moreover{ assume a1: "a : s"  have "EX x. x:s & -a+x=0" apply (rule exI[of _ a]) using a1 by auto  hence "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s" by auto  hence "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)"    using span_insert_0[of "op + (- a) ` (s - {a})"] by auto  moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))" by auto  moreover have "insert a (s - {a})=(insert a s)" using assms by auto  ultimately have ?thesis using assms affine_hull_insert_span[of "a" "s-{a}"] by auto}ultimately show ?thesis by autoqedlemma affine_hull_span2:  assumes "a : s"  shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` (s-{a}))"  using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]] by autolemma affine_hull_span_gen:  assumes "a : affine hull s"  shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` s)"proof-have "affine hull (insert a s) = affine hull s" using hull_redundant[of a affine s] assms by autofrom this show ?thesis using affine_hull_insert_span_gen[of a "s"] by autoqedlemma affine_hull_span_0:  assumes "0 : affine hull S"  shows "affine hull S = span S"using affine_hull_span_gen[of "0" S] assms by autolemma extend_to_affine_basis:fixes S V :: "('n::euclidean_space) set"assumes "~(affine_dependent S)" "S <= V" "S~={}"shows "? T. ~(affine_dependent T) & S<=T & T<=V & affine hull T = affine hull V"proof-obtain a where a_def: "a : S" using assms by autohence h0: "independent  ((%x. -a + x) ` (S-{a}))" using affine_dependent_iff_dependent2 assms by autofrom this obtain B   where B_def: "(%x. -a+x) ` (S - {a}) <= B & B <= (%x. -a+x) ` V & independent B & (%x. -a+x) ` V <= span B"   using maximal_independent_subset_extend[of "(%x. -a+x) ` (S-{a})" "(%x. -a + x) ` V"] assms by blastdef T == "(%x. a+x) ` (insert 0 B)" hence "T=insert a ((%x. a+x) ` B)" by autohence "affine hull T = (%x. a+x) ` span B" using affine_hull_insert_span_gen[of a "((%x. a+x) ` B)"] translation_assoc[of "-a" a B] by autohence "V <= affine hull T" using B_def assms translation_inverse_subset[of a V "span B"] by automoreover have "T<=V" using T_def B_def a_def assms by autoultimately have "affine hull T = affine hull V"    by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)moreover have "S<=T" using T_def B_def translation_inverse_subset[of a "S-{a}" B] by automoreover have "~(affine_dependent T)" using T_def affine_dependent_translation_eq[of "insert 0 B"] affine_dependent_imp_dependent2 B_def by autoultimately show ?thesis using `T<=V` by autoqedlemma affine_basis_exists:fixes V :: "('n::euclidean_space) set"shows "? B. B <= V & ~(affine_dependent B) & affine hull V = affine hull B"proof-{ assume empt: "V={}" have "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)" using empt affine_independent_empty by auto}moreover{ assume nonempt: "V~={}" obtain x where "x:V" using nonempt by auto  hence "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)"  using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}:: ('n::euclidean_space) set" V] by auto}ultimately show ?thesis by autoqedsubsection {* Affine Dimension of a Set *}definition "aff_dim V = (SOME d :: int. ? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1))"lemma aff_dim_basis_exists:  fixes V :: "('n::euclidean_space) set"  shows "? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"proof-obtain B where B_def: "~(affine_dependent B) & (affine hull B=affine hull V)" using affine_basis_exists[of V] by autofrom this show ?thesis unfolding aff_dim_def some_eq_ex[of "%d. ? (B :: ('n::euclidean_space) set). (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1)"] apply auto apply (rule exI[of _ "int (card B)-(1 :: int)"]) apply (rule exI[of _ "B"]) by autoqedlemma affine_hull_nonempty: "(S ~= {}) <-> affine hull S ~= {}"proof-have "(S = {}) ==> affine hull S = {}"using affine_hull_empty by automoreover have "affine hull S = {} ==> S = {}" unfolding hull_def by autoultimately show "(S ~= {}) <-> affine hull S ~= {}" by blastqedlemma aff_dim_parallel_subspace_aux:fixes B :: "('n::euclidean_space) set"assumes "~(affine_dependent B)" "a:B"shows "finite B & ((card B) - 1 = dim (span ((%x. -a+x) ` (B-{a}))))"proof-have "independent ((%x. -a + x) ` (B-{a}))" using affine_dependent_iff_dependent2 assms by autohence fin: "dim (span ((%x. -a+x) ` (B-{a}))) = card ((%x. -a + x) ` (B-{a}))" "finite ((%x. -a + x) ` (B - {a}))"  using indep_card_eq_dim_span[of "(%x. -a+x) ` (B-{a})"] by auto{ assume emp: "(%x. -a + x) ` (B - {a}) = {}"  have "B=insert a ((%x. a + x) ` (%x. -a + x) ` (B - {a}))" using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto  hence "B={a}" using emp by auto  hence ?thesis using assms fin by auto}moreover{ assume "(%x. -a + x) ` (B - {a}) ~= {}"  hence "card ((%x. -a + x) ` (B - {a}))>0" using fin by auto  moreover have h1: "card ((%x. -a + x) ` (B-{a})) = card (B-{a})"     apply (rule card_image) using translate_inj_on by auto  ultimately have "card (B-{a})>0" by auto  hence "finite(B-{a})" using card_gt_0_iff[of "(B - {a})"] by auto  moreover hence "(card (B-{a})= (card B) - 1)" using card_Diff_singleton assms by auto  ultimately have ?thesis using fin h1 by auto} ultimately show ?thesis by autoqedlemma aff_dim_parallel_subspace:fixes V L :: "('n::euclidean_space) set"assumes "V ~= {}"assumes "subspace L" "affine_parallel (affine hull V) L"shows "aff_dim V=int(dim L)"proof-obtain B where B_def: "affine hull B = affine hull V & ~ affine_dependent B & int (card B) = aff_dim V + 1" using aff_dim_basis_exists by autohence "B~={}" using assms B_def  affine_hull_nonempty[of V] affine_hull_nonempty[of B] by autofrom this obtain a where a_def: "a : B" by autodef Lb == "span ((%x. -a+x) ` (B-{a}))"  moreover have "affine_parallel (affine hull B) Lb"     using Lb_def B_def assms affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"] unfolding affine_parallel_def by auto  moreover have "subspace Lb" using Lb_def subspace_span by auto  moreover have "affine hull B ~= {}" using assms B_def affine_hull_nonempty[of V] by auto  ultimately have "L=Lb" using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B_def by auto  hence "dim L=dim Lb" by auto  moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def B_def by auto(*  hence "card B=dim Lb+1" using `B~={}` card_gt_0_iff[of B] by auto *)  ultimately show ?thesis using B_def `B~={}` card_gt_0_iff[of B] by autoqedlemma aff_independent_finite:fixes B :: "('n::euclidean_space) set"assumes "~(affine_dependent B)"shows "finite B"proof-{ assume "B~={}" from this obtain a where "a:B" by auto  hence ?thesis using aff_dim_parallel_subspace_aux assms by auto} from this show ?thesis by autoqedlemma independent_finite:fixes B :: "('n::euclidean_space) set"assumes "independent B"shows "finite B"using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms by autolemma subspace_dim_equal:assumes "subspace (S :: ('n::euclidean_space) set)" "subspace T" "S <= T" "dim S >= dim T"shows "S=T"proof-obtain B where B_def: "B <= S & independent B & S <= span B & (card B = dim S)" using basis_exists[of S] by autohence "span B <= S" using span_mono[of B S] span_eq[of S] assms by metishence "span B = S" using B_def by autohave "dim S = dim T" using assms dim_subset[of S T] by autohence "T <= span B" using card_eq_dim[of B T] B_def independent_finite assms by autofrom this show ?thesis using assms `span B=S` by autoqedlemma span_substd_basis:  assumes d: "d ⊆ Basis"  shows "span d = {x. ∀i∈Basis. i ∉ d --> x•i = 0}" (is "_ = ?B")proof-have "d <= ?B" using d by (auto simp: inner_Basis)moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .ultimately have "span d <= ?B" using span_mono[of d "?B"] span_eq[of "?B"] by blastmoreover have "card d <= dim (span d)" using independent_card_le_dim[of d "span d"]   independent_substdbasis[OF assms] span_inc[of d] by automoreover hence "dim ?B <= dim (span d)" using dim_substandard[OF assms] by autoultimately show ?thesis using s subspace_dim_equal[of "span d" "?B"]  subspace_span[of d] by autoqedlemma basis_to_substdbasis_subspace_isomorphism:fixes B :: "('a::euclidean_space) set"assumes "independent B"shows "EX f (d::'a set). card d = card B ∧ linear f ∧ f ` B = d ∧       f ` span B = {x. ∀i∈Basis. i ∉ d --> x • i = 0} ∧ inj_on f (span B) ∧ d ⊆ Basis"proof-  have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto  have "dim B ≤ card (Basis :: 'a set)" using dim_subset_UNIV[of B] by simp  from ex_card[OF this] obtain d :: "'a set" where d: "d ⊆ Basis" and t: "card d = dim B" by auto  let ?t = "{x::'a::euclidean_space. ∀i∈Basis. i ~: d --> x•i = 0}"  have "EX f. linear f & f ` B = d & f ` span B = ?t & inj_on f (span B)"    apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "d"])    apply(rule subspace_span) apply(rule subspace_substandard) defer    apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)    unfolding span_substd_basis[OF d, symmetric]     apply(rule span_inc)    apply(rule independent_substdbasis[OF d]) apply(rule,assumption)    unfolding t[symmetric] span_substd_basis[OF d] dim_substandard[OF d] by auto  with t `card B = dim B` d show ?thesis by autoqedlemma aff_dim_empty:fixes S :: "('n::euclidean_space) set"shows "S = {} <-> aff_dim S = -1"proof-obtain B where "affine hull B = affine hull S & ~ affine_dependent B & int (card B) = aff_dim S + 1" using aff_dim_basis_exists by automoreover hence "S={} <-> B={}" using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by autoultimately show ?thesis using aff_independent_finite[of B] card_gt_0_iff[of B] by autoqedlemma aff_dim_affine_hull:shows "aff_dim (affine hull S)=aff_dim S"unfolding aff_dim_def using hull_hull[of _ S] by autolemma aff_dim_affine_hull2:assumes "affine hull S=affine hull T"shows "aff_dim S=aff_dim T" unfolding aff_dim_def using assms by autolemma aff_dim_unique:fixes B V :: "('n::euclidean_space) set"assumes "(affine hull B=affine hull V) & ~(affine_dependent B)"shows "of_nat(card B) = aff_dim V+1"proof-{ assume "B={}" hence "V={}" using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms by auto  hence "aff_dim V = (-1::int)"  using aff_dim_empty by auto  hence ?thesis using `B={}` by auto}moreover{ assume "B~={}" from this obtain a where a_def: "a:B" by auto  def Lb == "span ((%x. -a+x) ` (B-{a}))"  have "affine_parallel (affine hull B) Lb"     using Lb_def affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"]     unfolding affine_parallel_def by auto  moreover have "subspace Lb" using Lb_def subspace_span by auto  ultimately have "aff_dim B=int(dim Lb)" using aff_dim_parallel_subspace[of B Lb] `B~={}` by auto  moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def assms by auto  ultimately have "(of_nat(card B) = aff_dim B+1)" using  `B~={}` card_gt_0_iff[of B] by auto  hence ?thesis using aff_dim_affine_hull2 assms by auto} ultimately show ?thesis by blastqedlemma aff_dim_affine_independent:fixes B :: "('n::euclidean_space) set"assumes "~(affine_dependent B)"shows "of_nat(card B) = aff_dim B+1"  using aff_dim_unique[of B B] assms by autolemma aff_dim_sing:fixes a :: "'n::euclidean_space"shows "aff_dim {a}=0"  using aff_dim_affine_independent[of "{a}"] affine_independent_sing by autolemma aff_dim_inner_basis_exists:  fixes V :: "('n::euclidean_space) set"  shows "? B. B<=V & (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"proof-obtain B where B_def: "~(affine_dependent B) & B<=V & (affine hull B=affine hull V)" using affine_basis_exists[of V] by automoreover hence "of_nat(card B) = aff_dim V+1" using aff_dim_unique by autoultimately show ?thesis by autoqedlemma aff_dim_le_card:fixes V :: "('n::euclidean_space) set"assumes "finite V"shows "aff_dim V <= of_nat(card V) - 1" proof- obtain B where B_def: "B<=V & (of_nat(card B) = aff_dim V+1)" using aff_dim_inner_basis_exists[of V] by auto moreover hence "card B <= card V" using assms card_mono by auto ultimately show ?thesis by autoqedlemma aff_dim_parallel_eq:fixes S T :: "('n::euclidean_space) set"assumes "affine_parallel (affine hull S) (affine hull T)"shows "aff_dim S=aff_dim T"proof-{ assume "T~={}" "S~={}"  from this obtain L where L_def: "subspace L & affine_parallel (affine hull T) L"       using affine_parallel_subspace[of "affine hull T"] affine_affine_hull[of T] affine_hull_nonempty by auto  hence "aff_dim T = int(dim L)" using aff_dim_parallel_subspace `T~={}` by auto  moreover have "subspace L & affine_parallel (affine hull S) L"     using L_def affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto  moreover hence "aff_dim S = int(dim L)" using aff_dim_parallel_subspace `S~={}` by auto  ultimately have ?thesis by auto}moreover{ assume "S={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto  hence ?thesis using aff_dim_empty by auto}moreover{ assume "T={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto  hence ?thesis using aff_dim_empty by auto}ultimately show ?thesis by blastqedlemma aff_dim_translation_eq:fixes a :: "'n::euclidean_space"shows "aff_dim ((%x. a + x) ` S)=aff_dim S"proof-have "affine_parallel (affine hull S) (affine hull ((%x. a + x) ` S))" unfolding affine_parallel_def apply (rule exI[of _ "a"]) using affine_hull_translation[of a S] by autofrom this show ?thesis using  aff_dim_parallel_eq[of S "(%x. a + x) ` S"] by autoqedlemma aff_dim_affine:fixes S L :: "('n::euclidean_space) set"assumes "S ~= {}" "affine S"assumes "subspace L" "affine_parallel S L"shows "aff_dim S=int(dim L)"proof-have 1: "(affine hull S) = S" using assms affine_hull_eq[of S] by autohence "affine_parallel (affine hull S) L" using assms by (simp add:1)from this show ?thesis using assms aff_dim_parallel_subspace[of S L] by blastqedlemma dim_affine_hull:fixes S :: "('n::euclidean_space) set"shows "dim (affine hull S)=dim S"proof-have "dim (affine hull S)>=dim S" using dim_subset by automoreover have "dim(span S) >= dim (affine hull S)" using dim_subset affine_hull_subset_span by automoreover have "dim(span S)=dim S" using dim_span by autoultimately show ?thesis by autoqedlemma aff_dim_subspace:fixes S :: "('n::euclidean_space) set"assumes "S ~= {}" "subspace S"shows "aff_dim S=int(dim S)" using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] by autolemma aff_dim_zero:fixes S :: "('n::euclidean_space) set"assumes "0 : affine hull S"shows "aff_dim S=int(dim S)"proof-have "subspace(affine hull S)" using subspace_affine[of "affine hull S"] affine_affine_hull assms by autohence "aff_dim (affine hull S) =int(dim (affine hull S))" using assms aff_dim_subspace[of "affine hull S"] by autofrom this show ?thesis using aff_dim_affine_hull[of S] dim_affine_hull[of S] by autoqedlemma aff_dim_univ: "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))"  using aff_dim_subspace[of "(UNIV :: ('n::euclidean_space) set)"]    dim_UNIV[where 'a="'n::euclidean_space"] by autolemma aff_dim_geq:  fixes V :: "('n::euclidean_space) set"  shows "aff_dim V >= -1"proof-obtain B where B_def: "affine hull B = affine hull V & ~ affine_dependent B & int (card B) = aff_dim V + 1" using aff_dim_basis_exists by autofrom this show ?thesis by autoqedlemma independent_card_le_aff_dim:  assumes "(B::('n::euclidean_space) set) <= V"  assumes "~(affine_dependent B)"  shows "int(card B) <= aff_dim V+1"proof-{ assume "B~={}"  from this obtain T where T_def: "~(affine_dependent T) & B<=T & T<=V & affine hull T = affine hull V"  using assms extend_to_affine_basis[of B V] by auto  hence "of_nat(card T) = aff_dim V+1" using aff_dim_unique by auto  hence ?thesis using T_def card_mono[of T B] aff_independent_finite[of T] by auto}moreover{ assume "B={}"  moreover have "-1<= aff_dim V" using aff_dim_geq by auto  ultimately have ?thesis by auto}  ultimately show ?thesis by blastqedlemma aff_dim_subset:  fixes S T :: "('n::euclidean_space) set"  assumes "S <= T"  shows "aff_dim S <= aff_dim T"proof-obtain B where B_def: "~(affine_dependent B) & B<=S & (affine hull B=affine hull S) & of_nat(card B) = aff_dim S+1" using aff_dim_inner_basis_exists[of S] by automoreover hence "int (card B) <= aff_dim T + 1" using assms independent_card_le_aff_dim[of B T] by autoultimately show ?thesis by autoqedlemma aff_dim_subset_univ:fixes S :: "('n::euclidean_space) set"shows "aff_dim S <= int(DIM('n))"proof -  have "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))" using aff_dim_univ by auto  from this show "aff_dim (S:: ('n::euclidean_space) set) <= int(DIM('n))" using assms aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by autoqedlemma affine_dim_equal:assumes "affine (S :: ('n::euclidean_space) set)" "affine T" "S ~= {}" "S <= T" "aff_dim S = aff_dim T"shows "S=T"proof-obtain a where "a : S" using assms by autohence "a : T" using assms by autodef LS == "{y. ? x : S. (-a)+x=y}"hence ls: "subspace LS & affine_parallel S LS" using assms parallel_subspace_explicit[of S a LS] `a : S` by autohence h1: "int(dim LS) = aff_dim S" using assms aff_dim_affine[of S LS] by autohave "T ~= {}" using assms by autodef LT == "{y. ? x : T. (-a)+x=y}"hence lt: "subspace LT & affine_parallel T LT" using assms parallel_subspace_explicit[of T a LT] `a : T` by autohence "int(dim LT) = aff_dim T" using assms aff_dim_affine[of T LT] `T ~= {}` by autohence "dim LS = dim LT" using h1 assms by automoreover have "LS <= LT" using LS_def LT_def assms by autoultimately have "LS=LT" using subspace_dim_equal[of LS LT] ls lt by automoreover have "S = {x. ? y : LS. a+y=x}" using LS_def by automoreover have "T = {x. ? y : LT. a+y=x}" using LT_def by autoultimately show ?thesis by autoqedlemma affine_hull_univ:fixes S :: "('n::euclidean_space) set"assumes "aff_dim S = int(DIM('n))"shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"proof-have "S ~= {}" using assms aff_dim_empty[of S] by autohave h0: "S <= affine hull S" using hull_subset[of S _] by autohave h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S" using aff_dim_univ assms by autohence h2: "aff_dim (affine hull S) <= aff_dim (UNIV :: ('n::euclidean_space) set)" using aff_dim_subset_univ[of "affine hull S"] assms h0 by autohave h3: "aff_dim S <= aff_dim (affine hull S)" using h0 aff_dim_subset[of S "affine hull S"] assms by autohence h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" using h0 h1 h2 by autofrom this show ?thesis using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"] affine_affine_hull[of S] affine_UNIV assms h4 h0 `S ~= {}` by autoqedlemma aff_dim_convex_hull:fixes S :: "('n::euclidean_space) set"shows "aff_dim (convex hull S)=aff_dim S"  using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]  hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]  aff_dim_subset[of "convex hull S" "affine hull S"] by autolemma aff_dim_cball:fixes a :: "'n::euclidean_space"assumes "0<e"shows "aff_dim (cball a e) = int (DIM('n))"proof-have "(%x. a + x) ` (cball 0 e)<=cball a e" unfolding cball_def dist_norm by autohence "aff_dim (cball (0 :: 'n::euclidean_space) e) <= aff_dim (cball a e)"  using aff_dim_translation_eq[of a "cball 0 e"]        aff_dim_subset[of "op + a ` cball 0 e" "cball a e"] by automoreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"   using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"] centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms   by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])ultimately show ?thesis using aff_dim_subset_univ[of "cball a e"] by autoqedlemma aff_dim_open:fixes S :: "('n::euclidean_space) set"assumes "open S" "S ~= {}"shows "aff_dim S = int (DIM('n))"proof-obtain x where "x:S" using assms by autofrom this obtain e where e_def: "e>0 & cball x e <= S" using open_contains_cball[of S] assms by autofrom this have "aff_dim (cball x e) <= aff_dim S" using aff_dim_subset by autofrom this show ?thesis using aff_dim_cball[of e x] aff_dim_subset_univ[of S] e_def by autoqedlemma low_dim_interior:fixes S :: "('n::euclidean_space) set"assumes "~(aff_dim S = int (DIM('n)))"shows "interior S = {}"proof-have "aff_dim(interior S) <= aff_dim S"   using interior_subset aff_dim_subset[of "interior S" S] by autofrom this show ?thesis using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by autoqedsubsection {* Relative interior of a set *}definition "rel_interior S = {x. ? T. openin (subtopology euclidean (affine hull S)) T & x : T & T <= S}"lemma rel_interior: "rel_interior S = {x : S. ? T. open T & x : T & (T Int (affine hull S)) <= S}"  unfolding rel_interior_def[of S] openin_open[of "affine hull S"] apply autoproof-fix x T assume a: "x:S" "open T" "x : T" "(T Int (affine hull S)) <= S"hence h1: "x : T Int affine hull S" using hull_inc by autoshow "EX Tb. (EX Ta. open Ta & Tb = affine hull S Int Ta) & x : Tb & Tb <= S"apply (rule_tac x="T Int (affine hull S)" in exI)using a h1 by autoqedlemma mem_rel_interior:     "x : rel_interior S <-> (? T. open T & x : (T Int S) & (T Int (affine hull S)) <= S)"     by (auto simp add: rel_interior)lemma mem_rel_interior_ball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((ball x e) Int (affine hull S)) <= S)"  apply (simp add: rel_interior, safe)  apply (force simp add: open_contains_ball)  apply (rule_tac x="ball x e" in exI)  apply simp  donelemma rel_interior_ball:      "rel_interior S = {x : S. ? e. e>0 & ((ball x e) Int (affine hull S)) <= S}"      using mem_rel_interior_ball [of _ S] by autolemma mem_rel_interior_cball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((cball x e) Int (affine hull S)) <= S)"  apply (simp add: rel_interior, safe)  apply (force simp add: open_contains_cball)  apply (rule_tac x="ball x e" in exI)  apply (simp add: subset_trans [OF ball_subset_cball])  apply auto  donelemma rel_interior_cball: "rel_interior S = {x : S. ? e. e>0 & ((cball x e) Int (affine hull S)) <= S}"       using mem_rel_interior_cball [of _ S] by autolemma rel_interior_empty: "rel_interior {} = {}"   by (auto simp add: rel_interior_def)lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"by (metis affine_hull_eq affine_sing)lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"   unfolding rel_interior_ball affine_hull_sing apply auto   apply(rule_tac x="1 :: real" in exI) apply simp   donelemma subset_rel_interior:fixes S T :: "('n::euclidean_space) set"assumes "S<=T" "affine hull S=affine hull T"shows "rel_interior S <= rel_interior T"  using assms by (auto simp add: rel_interior_def)lemma rel_interior_subset: "rel_interior S <= S"   by (auto simp add: rel_interior_def)lemma rel_interior_subset_closure: "rel_interior S <= closure S"   using rel_interior_subset by (auto simp add: closure_def)lemma interior_subset_rel_interior: "interior S <= rel_interior S"   by (auto simp add: rel_interior interior_def)lemma interior_rel_interior:fixes S :: "('n::euclidean_space) set"assumes "aff_dim S = int(DIM('n))"shows "rel_interior S = interior S"proof -have "affine hull S = UNIV" using assms affine_hull_univ[of S] by autofrom this show ?thesis unfolding rel_interior interior_def by autoqedlemma rel_interior_open:fixes S :: "('n::euclidean_space) set"assumes "open S"shows "rel_interior S = S"by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)lemma interior_rel_interior_gen:fixes S :: "('n::euclidean_space) set"shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"by (metis interior_rel_interior low_dim_interior)lemma rel_interior_univ:fixes S :: "('n::euclidean_space) set"shows "rel_interior (affine hull S) = affine hull S"proof-have h1: "rel_interior (affine hull S) <= affine hull S" using rel_interior_subset by auto{ fix x assume x_def: "x : affine hull S"  obtain e :: real where "e=1" by auto  hence "e>0 & ball x e Int affine hull (affine hull S) <= affine hull S" using hull_hull[of _ S] by auto  hence "x : rel_interior (affine hull S)" using x_def rel_interior_ball[of "affine hull S"] by auto} from this show ?thesis using h1 by autoqedlemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"by (metis open_UNIV rel_interior_open)lemma rel_interior_convex_shrink:  fixes S :: "('a::euclidean_space) set"  assumes "convex S" "c : rel_interior S" "x : S" "0 < e" "e <= 1"  shows "x - e *⇩R (x - c) : rel_interior S"proof-(* Proof is a modified copy of the proof of similar lemma mem_interior_convex_shrink*)obtain d where "d>0" and d:"ball c d Int affine hull S <= S"  using assms(2) unfolding  mem_rel_interior_ball by auto{   fix y assume as:"dist (x - e *⇩R (x - c)) y < e * d & y : affine hull S"    have *:"y = (1 - (1 - e)) *⇩R ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) + (1 - e) *⇩R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)    have "x : affine hull S" using assms hull_subset[of S] by auto    moreover have "1 / e + - ((1 - e) / e) = 1"       using `e>0` left_diff_distrib[of "1" "(1-e)" "1/e"] by auto    ultimately have **: "(1 / e) *⇩R y - ((1 - e) / e) *⇩R x : affine hull S"        using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"] by (simp add: algebra_simps)    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[symmetric] apply(rule arg_cong[where f=norm]) using `e>0`      by(auto simp add:euclidean_eq_iff[where 'a='a] field_simps inner_simps)    also have "... = abs(1/e) * norm (x - e *⇩R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)    also have "... < d" using as[unfolded dist_norm] and `e>0`      by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)    finally have "y : S" apply(subst *)apply(rule assms(1)[unfolded convex_alt,rule_format])      apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) ** by auto} hence "ball (x - e *⇩R (x - c)) (e*d) Int affine hull S <= S" by automoreover have "0 < e*d" using `0<e` `0<d` by (rule mult_pos_pos)moreover have "c : S" using assms rel_interior_subset by automoreover hence "x - e *⇩R (x - c) : S"   using mem_convex[of S x c e] apply (simp add: algebra_simps) using assms by autoultimately show ?thesis  using mem_rel_interior_ball[of "x - e *⇩R (x - c)" S] `e>0` by autoqedlemma interior_real_semiline:fixes a :: realshows "interior {a..} = {a<..}"proof-{ fix y assume "a<y" hence "y : interior {a..}"  apply (simp add: mem_interior) apply (rule_tac x="(y-a)" in exI) apply (auto simp add: dist_norm)  done }moreover{ fix y assume "y : interior {a..}" (*hence "a<=y" using interior_subset by auto*)  from this obtain e where e_def: "e>0 & cball y e ⊆ {a..}"     using mem_interior_cball[of y "{a..}"] by auto  moreover hence "y-e : cball y e" by (auto simp add: cball_def dist_norm)  ultimately have "a<=y-e" by auto  hence "a<y" using e_def by auto} ultimately show ?thesis by autoqedlemma rel_interior_real_interval:  fixes a b :: real assumes "a < b" shows "rel_interior {a..b} = {a<..<b}"proof-  have "{a<..<b} ≠ {}" using assms unfolding set_eq_iff by (auto intro!: exI[of _ "(a + b) / 2"])  then show ?thesis    using interior_rel_interior_gen[of "{a..b}", symmetric]    by (simp split: split_if_asm add: interior_closed_interval)qedlemma rel_interior_real_semiline:  fixes a :: real shows "rel_interior {a..} = {a<..}"proof-  have *: "{a<..} ≠ {}" unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])  then show ?thesis using interior_real_semiline     interior_rel_interior_gen[of "{a..}"]     by (auto split: split_if_asm)qedsubsubsection {* Relative open sets *}definition "rel_open S <-> (rel_interior S) = S"lemma rel_open: "rel_open S <-> openin (subtopology euclidean (affine hull S)) S" unfolding rel_open_def rel_interior_def apply auto using openin_subopen[of "subtopology euclidean (affine hull S)" S] by autolemma opein_rel_interior:  "openin (subtopology euclidean (affine hull S)) (rel_interior S)"  apply (simp add: rel_interior_def)  apply (subst openin_subopen) by blastlemma affine_rel_open:  fixes S :: "('n::euclidean_space) set"  assumes "affine S" shows "rel_open S"  unfolding rel_open_def using assms rel_interior_univ[of S] affine_hull_eq[of S] by metislemma affine_closed:  fixes S :: "('n::euclidean_space) set"  assumes "affine S" shows "closed S"proof-{ assume "S ~= {}"  from this obtain L where L_def: "subspace L & affine_parallel S L"     using assms affine_parallel_subspace[of S] by auto  from this obtain "a" where a_def: "S=(op + a ` L)"     using affine_parallel_def[of L S] affine_parallel_commut by auto  have "closed L" using L_def closed_subspace by auto  hence "closed S" using closed_translation a_def by auto} from this show ?thesis by autoqedlemma closure_affine_hull:  fixes S :: "('n::euclidean_space) set"  shows "closure S <= affine hull S"  by (intro closure_minimal hull_subset affine_closed affine_affine_hull)lemma closure_same_affine_hull:  fixes S :: "('n::euclidean_space) set"  shows "affine hull (closure S) = affine hull S"proof-have "affine hull (closure S) <= affine hull S"   using hull_mono[of "closure S" "affine hull S" "affine"] closure_affine_hull[of S] hull_hull[of "affine" S] by automoreover have "affine hull (closure S) >= affine hull S"   using hull_mono[of "S" "closure S" "affine"] closure_subset by autoultimately show ?thesis by autoqedlemma closure_aff_dim:  fixes S :: "('n::euclidean_space) set"  shows "aff_dim (closure S) = aff_dim S"proof-have "aff_dim S <= aff_dim (closure S)" using aff_dim_subset closure_subset by automoreover have "aff_dim (closure S) <= aff_dim (affine hull S)"  using aff_dim_subset closure_affine_hull by automoreover have "aff_dim (affine hull S) = aff_dim S" using aff_dim_affine_hull by autoultimately show ?thesis by autoqedlemma rel_interior_closure_convex_shrink:  fixes S :: "(_::euclidean_space) set"  assumes "convex S" "c : rel_interior S" "x : closure S" "0 < e" "e <= 1"  shows "x - e *⇩R (x - c) : rel_interior S"proof-(* Proof is a modified copy of the proof of similar lemma mem_interior_closure_convex_shrink*)obtain d where "d>0" and d:"ball c d Int affine hull S <= S"  using assms(2) unfolding mem_rel_interior_ball by autohave "EX 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 zball: "z∈ball c d"    using mem_ball z_def dist_norm[of c] using y and assms(4,5) by (auto simp add:field_simps norm_minus_commute)  have "x : affine hull S" using closure_affine_hull assms by auto  moreover have "y : affine hull S" using `y : S` hull_subset[of S] by auto  moreover have "c : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto  ultimately have "z : affine hull S"    using z_def affine_affine_hull[of S]          mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]          assms by (auto simp add: field_simps)  hence "z : S" using d zball by auto  obtain d1 where "d1>0" and d1:"ball z d1 <= ball c d"    using zball open_ball[of c d] openE[of "ball c d" z] by auto  hence "(ball z d1) Int (affine hull S) <= (ball c d) Int (affine hull S)" by auto  hence "(ball z d1) Int (affine hull S) <= S" using d by auto  hence "z : rel_interior S" using mem_rel_interior_ball using `d1>0` `z : S` by auto  hence "y - e *⇩R (y - z) : rel_interior S" using rel_interior_convex_shrink[of S z y e] assms`y : S` by auto  thus ?thesis using * by autoqedsubsubsection{* Relative interior preserves under linear transformations *}lemma rel_interior_translation_aux:fixes a :: "'n::euclidean_space"shows "((%x. a + x) ` rel_interior S) <= rel_interior ((%x. a + x) ` S)"proof-{ fix x assume x_def: "x : rel_interior S"  from this obtain T where T_def: "open T & x : (T Int S) & (T Int (affine hull S)) <= S" using mem_rel_interior[of x S] by auto  from this have "open ((%x. a + x) ` T)" and    "(a + x) : (((%x. a + x) ` T) Int ((%x. a + x) ` S))" and    "(((%x. a + x) ` T) Int (affine hull ((%x. a + x) ` S))) <= ((%x. a + x) ` S)"    using affine_hull_translation[of a S] open_translation[of T a] x_def by auto  from this have "(a+x) : rel_interior ((%x. a + x) ` S)"    using mem_rel_interior[of "a+x" "((%x. a + x) ` S)"] by auto} from this show ?thesis by autoqedlemma rel_interior_translation:fixes a :: "'n::euclidean_space"shows "rel_interior ((%x. a + x) ` S) = ((%x. a + x) ` rel_interior S)"proof-have "(%x. (-a) + x) ` rel_interior ((%x. a + x) ` S) <= rel_interior S"   using rel_interior_translation_aux[of "-a" "(%x. a + x) ` S"]         translation_assoc[of "-a" "a"] by autohence "((%x. a + x) ` rel_interior S) >= rel_interior ((%x. a + x) ` S)"   using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"]   by autofrom this show ?thesis using  rel_interior_translation_aux[of a S] by autoqedlemma affine_hull_linear_image:assumes "bounded_linear f"shows "f ` (affine hull s) = affine hull f ` s"(* Proof is a modified copy of the proof of similar lemma convex_hull_linear_image*)  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 assumptionproof-  interpret f: bounded_linear f by fact  show "affine {x. f x : affine hull f ` s}"  unfolding affine_def by(auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format]) next  interpret f: bounded_linear f by fact  show "affine {x. x : f ` (affine hull s)}" using affine_affine_hull[unfolded affine_def, of s]    unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])qed autolemma rel_interior_injective_on_span_linear_image:fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"fixes S :: "('m::euclidean_space) set"assumes "bounded_linear f" and "inj_on f (span S)"shows "rel_interior (f ` S) = f ` (rel_interior S)"proof-{ fix z assume z_def: "z : rel_interior (f ` S)"  have "z : f ` S" using z_def rel_interior_subset[of "f ` S"] by auto  from this obtain x where x_def: "x : S & (f x = z)" by auto  obtain e2 where e2_def: "e2>0 & cball z e2 Int affine hull (f ` S) <= (f ` S)"    using z_def rel_interior_cball[of "f ` S"] by auto  obtain K where K_def: "K>0 & (! x. norm (f x) <= norm x * K)"   using assms RealVector.bounded_linear.pos_bounded[of f] by auto  def e1 == "1/K" hence e1_def: "e1>0 & (! x. e1 * norm (f x) <= norm x)"   using K_def pos_le_divide_eq[of e1] by auto  def e == "e1 * e2" hence "e>0" using e1_def e2_def mult_pos_pos by auto  { fix y assume y_def: "y : cball x e Int affine hull S"    from this have h1: "f y : affine hull (f ` S)"      using affine_hull_linear_image[of f S] assms by auto    from y_def have "norm (x-y)<=e1 * e2"      using cball_def[of x e] dist_norm[of x y] e_def by auto    moreover have "(f x)-(f y)=f (x-y)"       using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto    moreover have "e1 * norm (f (x-y)) <= norm (x-y)" using e1_def by auto    ultimately have "e1 * norm ((f x)-(f y)) <= e1 * e2" by auto    hence "(f y) : (cball z e2)"      using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1_def x_def by auto    hence "f y : (f ` S)" using y_def e2_def h1 by auto    hence "y : S" using assms y_def hull_subset[of S] affine_hull_subset_span         inj_on_image_mem_iff[of f "span S" S y] by auto  }  hence "z : f ` (rel_interior S)" using mem_rel_interior_cball[of x S] `e>0` x_def by auto}moreover{ fix x assume x_def: "x : rel_interior S"  from this obtain e2 where e2_def: "e2>0 & cball x e2 Int affine hull S <= S"    using rel_interior_cball[of S] by auto  have "x : S" using x_def rel_interior_subset by auto  hence *: "f x : f ` S" by auto  have "! x:span S. f x = 0 --> x = 0"    using assms subspace_span linear_conv_bounded_linear[of f]          linear_injective_on_subspace_0[of f "span S"] by auto  from this obtain e1 where e1_def: "e1>0 & (! x : span S. e1 * norm x <= norm (f x))"   using assms injective_imp_isometric[of "span S" f]         subspace_span[of S] closed_subspace[of "span S"] by auto  def e == "e1 * e2" hence "e>0" using e1_def e2_def mult_pos_pos by auto  { fix y assume y_def: "y : cball (f x) e Int affine hull (f ` S)"    from this have "y : f ` (affine hull S)" using affine_hull_linear_image[of f S] assms by auto    from this obtain xy where xy_def: "xy : affine hull S & (f xy = y)" by auto    from this y_def have "norm ((f x)-(f xy))<=e1 * e2"      using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto    moreover have "(f x)-(f xy)=f (x-xy)"       using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto    moreover have "x-xy : span S"       using subspace_sub[of "span S" x xy] subspace_span `x : S` xy_def             affine_hull_subset_span[of S] span_inc by auto    moreover hence "e1 * norm (x-xy) <= norm (f (x-xy))" using e1_def by auto    ultimately have "e1 * norm (x-xy) <= e1 * e2" by auto    hence "xy : (cball x e2)"  using cball_def[of x e2] dist_norm[of x xy] e1_def by auto    hence "y : (f ` S)" using xy_def e2_def by auto  }  hence "(f x) : rel_interior (f ` S)"     using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e>0` by auto}ultimately show ?thesis by autoqedlemma rel_interior_injective_linear_image:fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"assumes "bounded_linear f" and "inj f"shows "rel_interior (f ` S) = f ` (rel_interior S)"using assms rel_interior_injective_on_span_linear_image[of f S]      subset_inj_on[of f "UNIV" "span S"] by autosubsection{* Some Properties of subset of standard basis *}lemma affine_hull_substd_basis: assumes "d⊆Basis"  shows "affine hull (insert 0 d) =  {x::'a::euclidean_space. (∀i∈Basis. i ~: d --> x•i = 0)}" (is "affine hull (insert 0 ?A) = ?B")proof- have *:"!!A. op + (0::'a) ` A = A" "!!A. op + (- (0::'a)) ` A = A" by auto  show ?thesis unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..qedlemma affine_hull_convex_hull: "affine hull (convex hull S) = affine hull S"by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)subsection {* Openness and compactness are preserved by convex hull operation. *}lemma open_convex_hull[intro]:  fixes s :: "'a::real_normed_vector set"  assumes "open s"  shows "open(convex hull s)"  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(8)proof(rule, rule) fix a  assume "∃sa u. finite sa ∧ sa ⊆ s ∧ (∀x∈sa. 0 ≤ u x) ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = a"  then obtain t u where obt:"finite t" "t⊆s" "∀x∈t. 0 ≤ u x" "setsum u t = 1" "(∑v∈t. u v *⇩R v) = a" by auto  from assms[unfolded open_contains_cball] obtain b where b:"∀x∈s. 0 < b x ∧ cball x (b x) ⊆ s"    using bchoice[of s "λx e. e>0 ∧ cball x e ⊆ s"] by auto  have "b ` t≠{}" 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[symmetric] 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[symmetric] scaleR_right_distrib)    ultimately show "∃sa u. finite sa ∧ (∀x∈sa. x ∈ s) ∧ (∀x∈sa. 0 ≤ u x) ∧ setsum u sa = 1 ∧ (∑v∈sa. u v *⇩R v) = y"      apply(rule_tac x="(λv. v + (y - a)) ` t" in exI) apply(rule_tac x="λv. u (v - (y - a))" in exI)      using obt(1, 3) by auto  qedqedlemma compact_convex_combinations:  fixes s t :: "'a::real_normed_vector set"  assumes "compact s" "compact t"  shows "compact { (1 - u) *⇩R x + u *⇩R y | x y u. 0 ≤ u ∧ u ≤ 1 ∧ x ∈ s ∧ y ∈ t}"proof-  let ?X = "{0..1} × s × t"  let ?h = "(λz. (1 - fst z) *⇩R fst (snd z) + fst z *⇩R snd (snd z))"  have *:"{ (1 - u) *⇩R x + u *⇩R y | x y u. 0 ≤ u ∧ u ≤ 1 ∧ x ∈ s ∧ y ∈ t} = ?h ` ?X"    apply(rule set_eqI) unfolding image_iff mem_Collect_eq    apply rule apply auto    apply (rule_tac x=u in rev_bexI, 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_interval assms)    doneqedlemma finite_imp_compact_convex_hull:  fixes s :: "('a::real_normed_vector) set"  assumes "finite s" shows "compact (convex hull s)"proof (cases "s = {}")  case True thus ?thesis by simpnext  case False with assms show ?thesis  proof (induct rule: finite_ne_induct)    case (singleton x) show ?case by simp  next    case (insert x A)    let ?f = "λ(u, y::'a). u *⇩R x + (1 - u) *⇩R y"    let ?T = "{0..1::real} × (convex hull A)"    have "continuous_on ?T ?f"      unfolding split_def continuous_on by (intro ballI tendsto_intros)    moreover have "compact ?T"      by (intro compact_Times compact_interval insert)    ultimately have "compact (?f ` ?T)"      by (rule compact_continuous_image)    also have "?f ` ?T = convex hull (insert x A)"      unfolding convex_hull_insert [OF `A ≠ {}`]      apply safe      apply (rule_tac x=a in exI, simp)      apply (rule_tac x="1 - a" in exI, simp)      apply fast      apply (rule_tac x="(u, b)" in image_eqI, simp_all)      done    finally show "compact (convex hull (insert x A))" .  qedqedlemma compact_convex_hull: fixes s::"('a::euclidean_space) set"  assumes "compact s"  shows "compact(convex hull s)"proof(cases "s={}")  case True thus ?thesis using compact_empty by simpnext  case False then obtain w where "w∈s" by auto  show ?thesis unfolding caratheodory[of s]  proof(induct ("DIM('a) + 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 set_eq_iff and mem_Collect_eq proof(rule, rule)        fix x assume "∃t. finite t ∧ t ⊆ s ∧ card t ≤ Suc n ∧ x ∈ convex hull t"        then obtain t where t:"finite t" "t ⊆ s" "card t ≤ Suc n" "x ∈ convex hull t" by auto        show "x∈s" proof(cases "card t = 0")          case True 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 set_eq_iff and mem_Collect_eq proof(rule,rule)        fix x assume "∃u v c. x = (1 - c) *⇩R u + c *⇩R v ∧          0 ≤ c ∧ c ≤ 1 ∧ u ∈ s ∧ (∃t. finite t ∧ t ⊆ s ∧ card t ≤ n ∧ v ∈ convex hull t)"        then obtain u v c t where obt:"x = (1 - c) *⇩R u + c *⇩R v"          "0 ≤ c ∧ c ≤ 1" "u ∈ s" "finite t" "t ⊆ s" "card t ≤ n"  "v ∈ convex hull t" by auto        moreover have "(1 - c) *⇩R u + c *⇩R v ∈ convex hull insert u t"          apply(rule 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  qedqedsubsection {* 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)qedlemma 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 autolemma 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[symmetric])          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 autoqed (auto simp add: assms)lemma simplex_furthest_le:  fixes s :: "('a::real_inner) 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 autoqedlemma simplex_furthest_le_exists:  fixes s :: "('a::real_inner) set"  shows "finite s ==> (∀x∈(convex hull s). ∃y∈s. norm(x - a) ≤ norm(y - a))"  using simplex_furthest_le[of s] by (cases "s={}")autolemma simplex_extremal_le:  fixes s :: "('a::real_inner) 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 simp: dist_norm)  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 autoqedlemma simplex_extremal_le_exists:  fixes s :: "('a::real_inner) set"  shows "finite s ==> x ∈ convex hull s ==> y ∈ convex hull s  ==> (∃u∈s. ∃v∈s. norm(x - y) ≤ norm(u - v))"  using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")autosubsection {* 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 autolemma 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 autolemma 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 autolemma 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 autolemma 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) qedlemma 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) qedlemma 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) qedlemma 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 autolemma 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 autolemma 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 fastforcelemma 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) qedlemma 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 autolemma 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])subsubsection {* 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)[symmetric])      unfolding inner_diff_right[symmetric] 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 autoqedlemma 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 autonext  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)qedlemma separating_hyperplane_closed_0:  assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 ∉ s"  shows "∃a b. a ≠ 0 ∧ 0 < b ∧ (∀x∈s. inner a x > b)"  proof(cases "s={}")  case True  have "norm ((SOME i. i∈Basis)::'a) = 1" "(SOME i. i∈Basis) ≠ (0::'a)" defer    apply(subst norm_le_zero_iff[symmetric]) by (auto simp: SOME_Basis)  thus ?thesis apply(rule_tac x="SOME i. i∈Basis" in exI, rule_tac x=1 in exI)    using True using DIM_positive[where 'a='a] by autonext 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 qedsubsubsection {* Now set-to-set for closed/compact sets *}lemma separating_hyperplane_closed_compact:  assumes "convex (s::('a::euclidean_space) 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::"'a" 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 autonext  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  qedqedlemma separating_hyperplane_compact_closed:  fixes s :: "('a::euclidean_space) 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 qedsubsubsection {* General case without assuming closure and getting non-strict separation *}lemma separating_hyperplane_set_0:  assumes "convex s" "(0::'a::euclidean_space) ∉ s"  shows "∃a. a ≠ 0 ∧ (∀x∈s. 0 ≤ inner a x)"proof- let ?k = "λc. {x::'a. 0 ≤ inner c x}"  have "frontier (cball 0 1) ∩ (\<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[of convex, OF assms(1), symmetric, 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 del: ballE 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) qedlemma separating_hyperplane_sets:  assumes "convex s" "convex (t::('a::euclidean_space) 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    doneqedsubsection {* 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  by (auto del: tendsto_const intro!: tendsto_intros)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) qedlemma convex_hull_eq_empty[simp]: "convex hull s = {} <-> s = {}"  using hull_subset[of s convex] convex_hull_empty by autosubsection {* 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)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 autolemma 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 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[symmetric] convex_hull_scaling convex_hull_translation)subsection {* Convexity of cone hulls *}lemma convex_cone_hull:assumes "convex S"shows "convex (cone hull S)"proof-{ fix x y assume xy_def: "x : cone hull S & y : cone hull S"  hence "S ~= {}" using cone_hull_empty_iff[of S] by auto  fix u v assume uv_def: "u>=0 & v>=0 & (u :: real)+v=1"  hence *: "u *⇩R x : cone hull S & v *⇩R y : cone hull S"     using cone_cone_hull[of S] xy_def cone_def[of "cone hull S"] by auto  from * obtain cx xx where x_def: "u *⇩R x = cx *⇩R xx & (cx :: real)>=0 & xx : S"     using cone_hull_expl[of S] by auto  from * obtain cy yy where y_def: "v *⇩R y = cy *⇩R yy & (cy :: real)>=0 & yy : S"     using cone_hull_expl[of S] by auto  { assume "cx+cy<=0" hence "u *⇩R x=0 & v *⇩R y=0" using x_def y_def by auto    hence "u *⇩R x+ v *⇩R y = 0" by auto    hence "u *⇩R x+ v *⇩R y : cone hull S" using cone_hull_contains_0[of S] `S ~= {}` by auto  }  moreover  { assume "cx+cy>0"    hence "(cx/(cx+cy)) *⇩R xx + (cy/(cx+cy)) *⇩R yy : S"      using assms mem_convex_alt[of S xx yy cx cy] x_def y_def by auto    hence "cx *⇩R xx + cy *⇩R yy : cone hull S"      using mem_cone_hull[of "(cx/(cx+cy)) *⇩R xx + (cy/(cx+cy)) *⇩R yy" S "cx+cy"]      `cx+cy>0` by (auto simp add: scaleR_right_distrib)    hence "u *⇩R x+ v *⇩R y : cone hull S" using x_def y_def by auto  }  moreover have "(cx+cy<=0) | (cx+cy>0)" by auto  ultimately have "u *⇩R x+ v *⇩R y : cone hull S" by blast} from this show ?thesis unfolding convex_def by autoqedlemma cone_convex_hull:assumes "cone S"shows "cone (convex hull S)"proof-{ assume "S = {}" hence ?thesis by auto }moreover{ assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *⇩R c ` S = S)" using cone_iff[of S] assms by auto  { fix c assume "(c :: real)>0"    hence "op *⇩R c ` (convex hull S) = convex hull (op *⇩R c ` S)"       using convex_hull_scaling[of _ S] by auto    also have "...=convex hull S" using * `c>0` by auto    finally have "op *⇩R c ` (convex hull S) = convex hull S" by auto  }  hence "0 : convex hull S & (!c. c>0 --> (op *⇩R c ` (convex hull S)) = (convex hull S))"     using * hull_subset[of S convex] by auto  hence ?thesis using `S ~= {}` cone_iff[of "convex hull S"] by auto}ultimately show ?thesis by blastqedsubsection {* Convex set as intersection of halfspaces *}lemma convex_halfspace_intersection:  fixes s :: "('a::euclidean_space) set"  assumes "closed s" "convex s"  shows "s = \<Inter> {h. s ⊆ h ∧ (∃a b. h = {x. inner a x ≤ b})}"  apply(rule set_eqI, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof-  fix x  assume "∀xa. s ⊆ xa ∧ (∃a b. xa = {x. inner a x ≤ b}) --> x ∈ xa"  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 autoqed autosubsection {* 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), symmetric] by(auto simp add: Int_absorb1) qedlemma 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[symmetric] and setsum_restrict_set''[OF assms(1)] and setsum_addf[symmetric] and *    using assms(2) by assumption qedlemma radon_v_lemma:  assumes "finite s" "setsum f s = 0" "∀x. g x = (0::real) --> f x = (0::'a::euclidean_space)"  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[symmetric] and *    using assms(2) by assumption qedlemma 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, symmetric])  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[symmetric] scaleR_right.setsum [symmetric] and z_def    by(auto simp add: setsum_negf setsum_right_distrib[symmetric])  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[symmetric] scaleR_right.setsum [symmetric] and z_def using *    by(auto simp add: setsum_negf setsum_right_distrib[symmetric])  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 autoqedlemma 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 qedsubsection {* Helly's theorem *}lemma helly_induct: fixes f::"('a::euclidean_space) set set"  assumes "card f = n" "n ≥ DIM('a) + 1"  "∀s∈f. convex s" "∀t⊆f. card t = DIM('a) + 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 = DIM('a)") apply(rule Suc(5)[rule_format])  unfolding `card f = Suc n` proof-  assume ng:"n ≠ DIM('a)" 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[symmetric] 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[symmetric] gh] by auto    have *:"g ∩ h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto    have "convex hull (X ` h) ⊆ \<Inter> g" "convex hull (X ` g) ⊆ \<Inter> h"      apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  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 qedqed(auto) qed(auto)lemma helly: fixes f::"('a::euclidean_space) set set"  assumes "card f ≥ DIM('a) + 1" "∀s∈f. convex s"          "∀t⊆f. card t = DIM('a) + 1 --> \<Inter> t ≠ {}"  shows "\<Inter> f ≠{}"  apply(rule helly_induct) using assms by autosubsection {* Homeomorphism of all convex compact sets with nonempty interior *}lemma compact_frontier_line_lemma:  fixes s :: "('a::euclidean_space) 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 *:"!!x A B. x∈A ==> x∈B ==> A∩B ≠ {}" by blast  have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)    apply(rule, intro continuous_intros)    by(rule compact_interval)  moreover have "{y. ∃u≥0. u ≤ b / norm x ∧ y = u *⇩R x} ∩ s ≠ {}" apply(rule *[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[symmetric]] 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)[symmetric]    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))qedlemma starlike_compact_projective:  assumes "compact s" "cball (0::'a::euclidean_space) 1 ⊆ s "  "∀x∈s. ∀u. 0 ≤ u ∧ u < 1 --> (u *⇩R x) ∈ (s - frontier s )"  shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"proof-  have fs:"frontier s ⊆ s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp  def pi ≡ "λx::'a. 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 (intro continuous_intros)    apply simp    done  def sphere ≡ "{x::'a. 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_eqI,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, symmetric] 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[symmetric] 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[symmetric])    thus "x = y" apply(subst injpi[symmetric]) 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::'a) 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)[symmetric] 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, symmetric] 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[symmetric]] 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[symmetric]) 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_eqI, rule, erule imageE, drule hom)    prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-    fix x::"'a" 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 (intro continuous_intros)        using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto    next 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="SOME i. i∈Basis" in ballE) defer        apply(erule_tac x="SOME i. i∈Basis" in ballE)        unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]        by (auto simp: SOME_Basis)      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::"'a" 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)[symmetric] 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 qedlemma homeomorphic_convex_compact_lemma:  fixes s :: "('a::euclidean_space) set"  assumes "convex s" and "compact s" and "cball 0 1 ⊆ s"  shows "s homeomorphic (cball (0::'a) 1)"proof (rule starlike_compact_projective[OF assms(2-3)], clarify)  fix x u assume "x ∈ s" and "0 ≤ u" and "u < (1::real)"  have "open (ball (u *⇩R x) (1 - u))" by (rule open_ball)  moreover have "u *⇩R x ∈ ball (u *⇩R x) (1 - u)"    unfolding centre_in_ball using `u < 1` by simp  moreover have "ball (u *⇩R x) (1 - u) ⊆ s"  proof    fix y assume "y ∈ ball (u *⇩R x) (1 - u)"    hence "dist (u *⇩R x) y < 1 - u" unfolding mem_ball .    with `u < 1` have "inverse (1 - u) *⇩R (y - u *⇩R x) ∈ cball 0 1"      by (simp add: dist_norm inverse_eq_divide norm_minus_commute)    with assms(3) have "inverse (1 - u) *⇩R (y - u *⇩R x) ∈ s" ..    with assms(1) have "(1 - u) *⇩R ((y - u *⇩R x) /⇩R (1 - u)) + u *⇩R x ∈ s"      using `x ∈ s` `0 ≤ u` `u < 1` [THEN less_imp_le] by (rule mem_convex)    thus "y ∈ s" using `u < 1` by simp  qed  ultimately have "u *⇩R x ∈ interior s" ..  thus "u *⇩R x ∈ s - frontier s" using frontier_def and interior_subset by auto qedlemma homeomorphic_convex_compact_cball: fixes e::real and s::"('a::euclidean_space) set"  assumes "convex s" "compact s" "interior s ≠ {}" "0 < e"  shows "s homeomorphic (cball (b::'a) 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::'a"  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) qedlemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) 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 autolemma convex_epigraph_convex:  "convex s ==> convex_on s f <-> convex(epigraph s f)"by(simp add: convex_epigraph)subsubsection {* Use this to derive general bound property of convex function *}lemma convex_on:  assumes "convex s"  shows "convex_on s f <-> (∀k u x. (∀i∈{1..k::nat}. 0 ≤ u i ∧ x i ∈ s) ∧ 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 autosubsection {* Convexity of general and special intervals *}lemma convexI: (* TODO: move to Library/Convex.thy *)  assumes "!!x y u v. [|x ∈ s; y ∈ s; 0 ≤ u; 0 ≤ v; u + v = 1|] ==> u *⇩R x + v *⇩R y ∈ s"  shows "convex s"using assms unfolding convex_def by fastlemma is_interval_convex:  fixes s :: "('a::euclidean_space) set"  assumes "is_interval s" shows "convex s"proof (rule convexI)  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-) DIM_positive[where 'a='a] by (auto simp: inner_simps)qedlemma is_interval_connected:  fixes s :: "('a::euclidean_space) set"  shows "is_interval s ==> connected s"  using is_interval_convex convex_connected by autolemma convex_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"  apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto(* FIXME: rewrite these lemmas without using vec1subsection {* 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 autolemma 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) qedlemma 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 => 'a::euclidean_space"  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  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) qedlemma ivt_increasing_component_1: fixes f::"real => 'a::euclidean_space"  shows "a ≤ b ==> ∀x∈{a .. b}. continuous (at x) f   ==> f a•k ≤ y ==> y ≤ f b•k ==> ∃x∈{a..b}. (f x)•k = y"by(rule ivt_increasing_component_on_1)  (auto simp add: continuous_at_imp_continuous_on)lemma ivt_decreasing_component_on_1: fixes f::"real => 'a::euclidean_space"  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[symmetric])  using ivt_increasing_component_on_1[of a b "λx. - f x" k "- y"]  using assms using continuous_on_minus by autolemma ivt_decreasing_component_1: fixes f::"real => 'a::euclidean_space"  shows "a ≤ b ==> ∀x∈{a .. b}. continuous (at x) f    ==> f b•k ≤ y ==> y ≤ f a•k ==> ∃x∈{a..b}. (f x)•k = y"by(rule ivt_decreasing_component_on_1)  (auto simp: continuous_at_imp_continuous_on)subsection {* 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[symmetric] 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 qedlemma inner_setsum_Basis[simp]: "!!i. i ∈ Basis ==> (∑Basis) • i = 1"  by (simp add: One_def inner_setsum_left setsum_cases inner_Basis)lemma unit_interval_convex_hull:  defines "One ≡ (∑Basis)"  shows "{0::'a::ordered_euclidean_space .. One} =    convex hull {x. ∀i∈Basis. (x•i = 0) ∨ (x•i = 1)}"  (is "?int = convex hull ?points")proof -  have One[simp]: "!!i. i ∈ Basis ==> One • i = 1"    by (simp add: One_def inner_setsum_left setsum_cases inner_Basis)  have 01:"{0,One} ⊆ convex hull ?points"     apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto  { fix n x assume "x∈{0::'a::ordered_euclidean_space .. One}" "n ≤ DIM('a)" "card {i. i∈Basis ∧ x•i ≠ 0} ≤ n"  hence "x∈convex hull ?points" proof(induct n arbitrary: x)    case 0 hence "x = 0" apply(subst euclidean_eq_iff) 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. i∈Basis ∧ x•i ≠ 0} = {}")      case True hence "x = 0" apply(subst euclidean_eq_iff) by auto      thus "x∈convex hull ?points" using 01 by auto    next      case False def xi ≡ "Min ((λi. x•i) ` {i. i∈Basis ∧ x•i ≠ 0})"      have "xi ∈ (λi. x•i) ` {i. i∈Basis ∧ 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" "i∈Basis" by auto      have i:"!!j. j∈Basis ==> 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. i∈Basis ∧ x•i ≠ 0}. x•j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq        proof(erule conjE) fix j assume as:"x • j ≠ 0" "x • j ≠ 1" "j∈Basis"          hence j:"x•j ∈ {0<..<1}" using Suc(2)            by (auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])          hence "x•j ∈ op • x ` {i. i∈Basis ∧ x • i ≠ 0}" using as(3) 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: 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 = "∑j∈Basis. (if x•j = 0 then 0 else (x•j - x•i) / (1 - x•i)) *⇩R j"        case False        then have *: "x = (x•i) *⇩R (∑j∈Basis. (if x•j = 0 then 0 else 1) *⇩R j) + (1 - x•i) *⇩R ?y"          by (subst euclidean_eq_iff) (simp add: inner_simps)        { fix j :: 'a assume j:"j∈Basis"          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] using j            by(auto simp add: field_simps)          with j have "0 ≤ ?y • j ∧ ?y • j ≤ 1" by (auto simp: inner_simps) }        moreover have "i∈{j. j∈Basis ∧ x•j ≠ 0} - {j. j∈Basis ∧ ?y • j ≠ 0}"          using i01 using i'(3) by auto        hence "{j. j∈Basis ∧ x•j ≠ 0} ≠ {j. j∈Basis ∧ ?y • j ≠ 0}" using i'(3) by blast        hence **:"{j. j∈Basis ∧ ?y • j ≠ 0} ⊂ {j. j∈Basis ∧ x•j ≠ 0}"          by auto        have "card {j. j∈Basis ∧ ?y • 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="DIM('a)" 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 ballE)    by autoqedtext {* And this is a finite set of vertices. *}lemma unit_cube_convex_hull:  obtains s :: "'a::ordered_euclidean_space set" where "finite s" "{0 .. ∑Basis} = convex hull s"  apply(rule that[of "{x::'a. ∀i∈Basis. x•i=0 ∨ x•i=1}"])  apply(rule finite_subset[of _ "(λs. (∑i∈Basis. (if i∈s then 1 else 0) *⇩R i)::'a) ` Pow Basis"])  prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-  fix x::"'a" assume as:"∀i∈Basis. x • i = 0 ∨ x • i = 1"  show "x ∈ (λs. ∑i∈Basis. (if i∈s then 1 else 0) *⇩R i) ` Pow Basis"    apply(rule image_eqI[where x="{i. i∈Basis ∧ x•i = 1}"])    using as apply(subst euclidean_eq_iff) by (auto simp: inner_setsum_left_Basis)qed autotext {* Hence any cube (could do any nonempty interval). *}lemma cube_convex_hull:  assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where  "finite s" "{x - (∑i∈Basis. d*⇩Ri) .. x + (∑i∈Basis. d*⇩Ri)} = convex hull s" proof-  let ?d = "(∑i∈Basis. d*⇩Ri)::'a"  have *:"{x - ?d .. x + ?d} = (λy. x - ?d + (2 * d) *⇩R y) ` {0 .. ∑Basis}" apply(rule set_eqI, rule)    unfolding image_iff defer apply(erule bexE) proof-    fix y assume as:"y∈{x - ?d .. x + ?d}"    { fix i :: 'a assume i:"i∈Basis" have "x • i ≤ d + y • i" "y • i ≤ d + x • i"        using as[unfolded mem_interval, THEN bspec[where x=i]] i        by (auto simp: inner_simps)      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[symmetric]        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..∑Basis}" unfolding mem_interval using assms      by(auto simp add: field_simps inner_simps)    thus "∃z∈{0..∑Basis}. 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  next    fix y z assume as:"z∈{0..∑Basis}" "y = x - ?d + (2*d) *⇩R z"    have "!!i. i∈Basis ==> 0 ≤ d * (z • i) ∧ d * (z • i) ≤ d"      using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in ballE)      apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)      using assms by auto    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 ballE) using assms by (auto simp: inner_simps) qed  obtain s where "finite s" "{0::'a..∑Basis} = 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 qedsubsection {* 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) qedsubsection {* Upper bound on a ball implies upper and lower bounds *}lemma convex_bounds_lemma:  fixes x :: "'a::real_normed_vector"  assumes "convex_on (cball x e) f"  "∀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 qedsubsubsection {* Hence a convex function on an open set is continuous *}lemma real_of_nat_ge_one_iff: "1 ≤ real (n::nat) <-> 1 ≤ n"  by autolemma convex_on_continuous:  assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f"  (* FIXME: generalize to euclidean_space *)  shows "continuous_on s f"  unfolding continuous_on_eq_continuous_at[OF assms(1)] proof  note dimge1 = DIM_positive[where 'a='a]  fix x assume "x∈s"  then obtain e where e:"cball x e ⊆ s" "e>0" using assms(1) unfolding open_contains_cball by auto  def d ≡ "e / real DIM('a)"  have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto)  let ?d = "(∑i∈Basis. d *⇩R i)::'a"  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 simp: inner_setsum_left_Basis inner_simps)  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::'a. d) Basis" unfolding setsum_constant d_def using dimge1      unfolding real_eq_of_nat by auto    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=b in ballE) by (auto simp: inner_simps)  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`    unfolding mult_le_cancel_left1    apply (auto simp: real_of_nat_ge_one_iff Suc_le_eq DIM_positive)    done  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 :: 'a assume "i∈Basis" hence "x • i - d ≤ y • i"  "y • i ≤ x • i + d"        using order_trans[OF Basis_le_norm y[unfolded mem_cball dist_norm], of i] by (auto simp: inner_diff_left)  }    thus "f y ≤ k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm      by (auto simp: inner_simps)  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 autoqedsubsection {* 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) x. x ∈ closed_segment a b)"lemmas segment = open_segment_def closed_segment_defdefinition "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[symmetric] by autolemma 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 simpqedlemma 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)qedlemma midpoint_eq_endpoint:  "midpoint a b = a <-> a = b"  "midpoint a b = b <-> a = b"  unfolding midpoint_eq_iff by autolemma convex_contains_segment:  "convex s <-> (∀a∈s. ∀b∈s. closed_segment a b ⊆ s)"  unfolding convex_alt closed_segment_def by autolemma convex_imp_starlike:  "convex s ==> s ≠ {} ==> starlike s"  unfolding convex_contains_segment starlike_def by autolemma 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_eqI)    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 qedlemma 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 autolemma segment_furthest_le:  fixes a b x y :: "'a::euclidean_space"  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) qedlemma segment_bound:  fixes x a b :: "'a::euclidean_space"  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 by autolemma between:"between (a,b) (x::'a::euclidean_space) <-> dist a b = (dist a x) + (dist x b)"proof(cases "a = b")  case True thus ?thesis unfolding between_def split_conv    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 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] * 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 apply(subst euclidean_eq_iff) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4      proof(rule) fix i :: 'a assume i:"i∈Basis"          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 inner_simps)          also have "… = x•i" apply(rule divide_eq_imp[OF Fal])            unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-            apply(subst (asm) euclidean_eq_iff) using i apply(erule_tac x=i in ballE) by(auto simp add:field_simps inner_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) qedqedlemma between_midpoint: fixes a::"'a::euclidean_space" 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[!] *)    unfolding euclidean_eq_iff[where 'a='a]    by(auto simp add:field_simps inner_simps) qedlemma 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 :: "('a::euclidean_space) 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[symmetric] apply(rule arg_cong[where f=norm]) using `e>0`      by(auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)    also have "… = abs(1/e) * norm (x - e *⇩R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)    also have "… < d" using as[unfolded dist_norm] and `e>0`      by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)    finally 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) qedlemma mem_interior_closure_convex_shrink:  fixes s :: "('a::euclidean_space) 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 interior_mono[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 qedsubsection {* 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_eqI, 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 autolemma substd_simplex:  assumes d: "d ⊆ Basis"  shows "convex hull (insert 0 d) = {x. (∀i∈Basis. 0 ≤ x•i) ∧ (∑i∈d. x•i) ≤ 1 ∧ (∀i∈Basis. i ∉ d --> x•i = 0)}"  (is "convex hull (insert 0 ?p) = ?s")proof- let ?D = d  have "0 ~: ?p" using assms by (auto simp: image_def)  from d have "finite d" by (blast intro: finite_subset finite_Basis)  show ?thesis unfolding simplex[OF `finite d` `0 ~: ?p`]    apply(rule set_eqI) unfolding mem_Collect_eq apply rule    apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-    fix x::"'a::euclidean_space" and u assume as: "∀x∈?D. 0 ≤ u x"      "setsum u ?D ≤ 1" "(∑x∈?D. u x *⇩R x) = x"    have *:"∀i∈Basis. i:d --> u i = x•i" and "(∀i∈Basis. i ~: d --> x • i = 0)" using as(3)      unfolding substdbasis_expansion_unique[OF assms] by auto    hence **:"setsum u ?D = setsum (op • x) ?D"      apply-apply(rule setsum_cong2) using assms by auto    have " (∀i∈Basis. 0 ≤ x•i) ∧ setsum (op • x) ?D ≤ 1"      apply - proof(rule,rule)      fix i :: 'a assume i:"i∈Basis" have "i : d ==> 0 ≤ x•i" unfolding *[rule_format,OF i,symmetric]         apply(rule_tac as(1)[rule_format]) by auto      moreover have "i ~: d ==> 0 ≤ x•i"        using `(∀i∈Basis. i ~: d --> x • i = 0)`[rule_format, OF i] by auto      ultimately show "0 ≤ x•i" by auto    qed(insert as(2)[unfolded **], auto)    from this show " (∀i∈Basis. 0 ≤ x•i) ∧ setsum (op • x) ?D ≤ 1 & (∀i∈Basis. i ~: d --> x • i = 0)"      using `(∀i∈Basis. i ~: d --> x • i = 0)` by auto  next fix x::"'a::euclidean_space" assume as:"∀i∈Basis. 0 ≤ x • i" "setsum (op • x) ?D ≤ 1"      "(∀i∈Basis. i ~: d --> x • i = 0)"    show "∃u. (∀x∈?D. 0 ≤ u x) ∧ setsum u ?D ≤ 1 ∧ (∑x∈?D. u x *⇩R x) = x"      using as d unfolding substdbasis_expansion_unique[OF assms]      by (rule_tac x="inner x" in exI) auto  qedqedlemma std_simplex:  "convex hull (insert 0 Basis) =        {x::'a::euclidean_space . (∀i∈Basis. 0 ≤ x•i) ∧ setsum (λi. x•i) Basis ≤ 1 }"  using substd_simplex[of Basis] by autolemma interior_std_simplex:  "interior (convex hull (insert 0 Basis)) =  {x::'a::euclidean_space. (∀i∈Basis. 0 < x•i) ∧ setsum (λi. x•i) Basis < 1 }"  apply(rule set_eqI) 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::"'a" and e assume "0<e" and as:"∀xa. dist x xa < e --> (∀x∈Basis. 0 ≤ xa • x) ∧ setsum (op • xa) Basis ≤ 1"  show "(∀xa∈Basis. 0 < x • xa) ∧ setsum (op • x) Basis < 1" apply(safe) proof-    fix i :: 'a assume i:"i∈Basis" thus "0 < x • i" using as[THEN spec[where x="x - (e / 2) *⇩R i"]] and `e>0`      unfolding dist_norm      by (auto elim!:ballE[where x=i] simp: inner_simps)  next have **:"dist x (x + (e / 2) *⇩R (SOME i. i∈Basis)) < e" using  `e>0`      unfolding dist_norm by(auto intro!: mult_strict_left_mono simp: SOME_Basis)    have "!!i. i∈Basis ==> (x + (e / 2) *⇩R (SOME i. i∈Basis)) • i = x•i + (if i = (SOME i. i∈Basis) then e/2 else 0)"      by (auto simp: SOME_Basis inner_Basis inner_simps)    hence *:"setsum (op • (x + (e / 2) *⇩R (SOME i. i∈Basis))) Basis = setsum (λi. x•i + (if (SOME i. i∈Basis) = i then e/2 else 0)) Basis"      apply(rule_tac setsum_cong) by auto    have "setsum (op • x) Basis < setsum (op • (x + (e / 2) *⇩R (SOME i. i∈Basis))) Basis" unfolding * setsum_addf      using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by (auto simp: SOME_Basis)    also have "… ≤ 1" using ** apply(drule_tac as[rule_format]) by auto    finally show "setsum (op • x) Basis < 1" by auto qednext fix x::"'a" assume as:"∀i∈Basis. 0 < x • i" "setsum (op • x) Basis < 1"  guess a using UNIV_witness[where 'a='b] ..  let ?d = "(1 - setsum (op • x) Basis) / real (DIM('a))"  have "Min ((op • x) ` Basis) > 0" apply(rule Min_grI) using as(1) by auto  moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by (auto simp add: Suc_le_eq DIM_positive)  ultimately show "∃e>0. ∀y. dist x y < e --> (∀i∈Basis. 0 ≤ y • i) ∧ setsum (op • y) Basis ≤ 1"    apply(rule_tac x="min (Min ((op • x) ` Basis)) ?D" in exI) apply rule defer apply(rule,rule) proof-    fix y assume y:"dist x y < min (Min (op • x ` Basis)) ?d"    have "setsum (op • y) Basis ≤ setsum (λi. x•i + ?d) Basis" proof(rule setsum_mono)      fix i :: 'a assume i: "i∈Basis" hence "abs (y•i - x•i) < ?d" apply-apply(rule le_less_trans)        using Basis_le_norm[OF i, of "y - x"]        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute inner_diff_left)      thus "y • i ≤ x • i + ?d" by auto qed    also have "… ≤ 1" unfolding setsum_addf setsum_constant real_eq_of_nat by(auto simp add: Suc_le_eq)    finally show "(∀i∈Basis. 0 ≤ y • i) ∧ setsum (op • y) Basis ≤ 1"    proof safe fix i :: 'a assume i:"i∈Basis"      have "norm (x - y) < x•i" apply(rule less_le_trans)        apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto      thus "0 ≤ y•i" using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]        by (auto simp: inner_simps)    qed qed auto qedlemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where  "a ∈ interior(convex hull (insert 0 Basis))" proof-  let ?D = "Basis :: 'a set" let ?a = "setsum (λb::'a. inverse (2 * real DIM('a)) *⇩R b) Basis"  { fix i :: 'a assume i:"i∈Basis" have "?a • i = inverse (2 * real DIM('a))"      by (rule trans[of _ "setsum (λj. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])         (simp_all add: setsum_cases i) }  note ** = this  show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe    fix i :: 'a assume i:"i∈Basis" show "0 < ?a • i" unfolding **[OF i] by(auto simp add: Suc_le_eq DIM_positive)  next have "setsum (op • ?a) ?D = setsum (λi. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto    also have "… < 1" unfolding setsum_constant real_eq_of_nat divide_inverse[symmetric] by (auto simp add:field_simps)    finally show "setsum (op • ?a) ?D < 1" by auto qed qedlemma rel_interior_substd_simplex: assumes d: "d⊆Basis"  shows "rel_interior (convex hull (insert 0 d)) =  {x::'a::euclidean_space. (∀i∈d. 0 < x•i) ∧ (∑i∈d. x•i) < 1 ∧ (∀i∈Basis. i ~: d --> x•i = 0)}"  (is "rel_interior (convex hull (insert 0 ?p)) = ?s")(* Proof is a modified copy of the proof of similar lemma interior_std_simplex in Convex_Euclidean_Space.thy *)proof-have "finite d" apply(rule finite_subset) using assms by auto{ assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto }moreover{ assume "d~={}"have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (∀i∈Basis. i ~: d --> x•i = 0)}"   using affine_hull_convex_hull affine_hull_substd_basis assms by autohave aux: "!!x::'a. ∀i∈Basis. ((∀i∈d. 0 ≤ x•i) ∧ (∀i∈Basis. i ∉ d --> x•i = 0)) --> 0 ≤ x•i"   by auto{ fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"  from this obtain e where e0: "e>0" and       "ball x e Int {xa. (∀i∈Basis. i ~: d --> xa•i = 0)} <= convex hull (insert 0 ?p)"       using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto  hence as: "ALL xa. (dist x xa < e & (∀i∈Basis. i ~: d --> xa•i = 0)) -->    (!i : d. 0 <= xa • i) & setsum (op • xa) d <= 1"    unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto  have x0: "(∀i∈Basis. i ~: d --> x•i = 0)"    using x_def rel_interior_subset  substd_simplex[OF assms] by auto  have "(∀i∈d. 0 < x • i) & setsum (op • x) d < 1 & (∀i∈Basis. i ~: d --> x•i = 0)" apply(rule,rule)  proof-    fix i::'a assume "i∈d"    hence "∀ia∈d. 0 ≤ (x - (e / 2) *⇩R i) • ia" apply-apply(rule as[rule_format,THEN conjunct1])      unfolding dist_norm using d `e>0` x0 by (auto simp: inner_simps inner_Basis)    thus "0 < x • i" apply(erule_tac x=i in ballE) using `e>0` `i∈d` d    by (auto simp: inner_simps inner_Basis)  next obtain a where a:"a:d" using `d ~= {}` by auto    then have **:"dist x (x + (e / 2) *⇩R a) < e"      using  `e>0` norm_Basis[of a] d      unfolding dist_norm by auto    have "!!i. i∈Basis ==> (x + (e / 2) *⇩R a) • i = x•i + (if i = a then e/2 else 0)"      using a d by (auto simp: inner_simps inner_Basis)    hence *:"setsum (op • (x + (e / 2) *⇩R a)) d =      setsum (λi. x•i + (if a = i then e/2 else 0)) d" using d by (intro setsum_cong) auto    have "a ∈ Basis" using `a ∈ d` d by auto    then have h1: "(∀i∈Basis. i ~: d --> (x + (e / 2) *⇩R a) • i = 0)"      using x0 d `a∈d` by (auto simp add: inner_add_left inner_Basis)    have "setsum (op • x) d < setsum (op • (x + (e / 2) *⇩R a)) d" unfolding * setsum_addf      using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')    also have "… ≤ 1" using ** h1 as[rule_format, of "x + (e / 2) *⇩R a"] by auto    finally show "setsum (op • x) d < 1 & (∀i∈Basis. i ~: d --> x•i = 0)" using x0 by auto  qed}moreover{  fix x::"'a::euclidean_space" assume as: "x : ?s"  have "!i. ((0<x•i) | (0=x•i) --> 0<=x•i)" by auto  moreover have "!i. (i:d) | (i ~: d)" by auto  ultimately  have "!i. ( (ALL i:d. 0 < x•i) & (ALL i. i ~: d --> x•i = 0) ) --> 0 <= x•i" by metis  hence h2: "x : convex hull (insert 0 ?p)" using as assms    unfolding substd_simplex[OF assms] by fastforce  obtain a where a:"a:d" using `d ~= {}` by auto  let ?d = "(1 - setsum (op • x) d) / real (card d)"  have "0 < card d" using `d ~={}` `finite d` by (simp add: card_gt_0_iff)  have "Min ((op • x) ` d) > 0" using as `d ≠ {}` `finite d` by (simp add: Min_grI)  moreover have "?d > 0" apply(rule divide_pos_pos) using as using `0 < card d` by auto  ultimately have h3: "min (Min ((op • x) ` d)) ?d > 0" by auto  have "x : rel_interior (convex hull (insert 0 ?p))"    unfolding rel_interior_ball mem_Collect_eq h0 apply(rule,rule h2)    unfolding substd_simplex[OF assms]    apply(rule_tac x="min (Min ((op • x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball  proof-    fix y::'a assume y:"dist x y < min (Min (op • x ` d)) ?d" and y2: "∀i∈Basis. i ∉ d --> y•i = 0"    have "setsum (op • y) d ≤ setsum (λi. x•i + ?d) d"    proof(rule setsum_mono)      fix i assume "i ∈ d"      with d have i: "i ∈ Basis" by auto      have "abs (y•i - x•i) < ?d" apply(rule le_less_trans) using Basis_le_norm[OF i, of "y - x"]        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]        by (auto simp add: norm_minus_commute inner_simps)      thus "y • i ≤ x • i + ?d" by auto    qed    also have "… ≤ 1" unfolding setsum_addf setsum_constant real_eq_of_nat      using `0 < card d` by auto    finally show "setsum (op • y) d ≤ 1" .    fix i :: 'a assume i: "i ∈ Basis" thus "0 ≤ y•i"    proof(cases "i∈d") case True      have "norm (x - y) < x•i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]        using Min_gr_iff[of "op • x ` d" "norm (x - y)"] `0 < card d` `i:d`        by (simp add: card_gt_0_iff)      thus "0 ≤ y•i" using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]        by (auto simp: inner_simps)    qed(insert y2, auto)  qed} ultimately have    "!!x. (x : rel_interior (convex hull insert 0 d)) = (x ∈ {x. (ALL i:d. 0 < x • i) &    setsum (op • x) d < 1 & (∀i∈Basis. i ~: d --> x • i = 0)})" by blastfrom this have ?thesis by (rule set_eqI)} ultimately show ?thesis by blastqedlemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d⊆Basis"  obtains a::"'a::euclidean_space" where  "a : rel_interior(convex hull (insert 0 d))" proof-(* Proof is a modified copy of the proof of similar lemma interior_std_simplex_nonempty in Convex_Euclidean_Space.thy *)  let ?D = d let ?a = "setsum (λb::'a::euclidean_space. inverse (2 * real (card d)) *⇩R b) ?D"  have "finite d" apply(rule finite_subset) using assms(2) by auto  hence d1: "0 < real(card d)" using `d ~={}` by auto  { fix i assume "i:d"    have "?a • i = inverse (2 * real (card d))"      apply(rule trans[of _ "setsum (λj. if i = j then inverse (2 * real (card d)) else 0) ?D"])      unfolding inner_setsum_left      apply(rule setsum_cong2)      using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)      by (auto simp: inner_simps inner_Basis set_rev_mp[OF _ assms(2)]) }  note ** = this  show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq  proof safe fix i assume "i:d"    have "0 < inverse (2 * real (card d))" using d1 by auto    also have "...=?a • i" using **[of i] `i:d` by auto    finally show "0 < ?a • i" by auto  next have "setsum (op • ?a) ?D = setsum (λi. inverse (2 * real (card d))) ?D"      by(rule setsum_cong2, rule **)    also have "… < 1" unfolding setsum_constant real_eq_of_nat divide_real_def[symmetric]      by (auto simp add:field_simps)    finally show "setsum (op • ?a) ?D < 1" by auto  next fix i assume "i∈Basis" and "i~:d"    have "?a : (span d)"      apply (rule span_setsum[of d "(%b. b /⇩R (2 * real (card d)))" d])      using finite_subset[OF assms(2) finite_Basis]      apply blast    proof-      { fix x assume "(x :: 'a::euclidean_space): d"        hence "x : span d"          using span_superset[of _ "d"] by auto        hence "(x /⇩R (2 * real (card d))) : (span d)"          using span_mul[of x "d" "(inverse (real (card d)) / 2)"] by auto      } thus "∀x∈d. x /⇩R (2 * real (card d)) ∈ span d" by auto    qed    thus "?a • i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i∈Basis` by auto  qedqedsubsection {* Relative interior of convex set *}lemma rel_interior_convex_nonempty_aux:fixes S :: "('n::euclidean_space) set"assumes "convex S" and "0 : S"shows "rel_interior S ~= {}"proof-{ assume "S = {0}" hence ?thesis using rel_interior_sing by auto }moreover {assume "S ~= {0}"obtain B where B_def: "independent B & B<=S & (S <= span B) & card B = dim S" using basis_exists[of S] by autohence "B~={}" using B_def assms `S ~= {0}` span_empty by autohave "insert 0 B <= span B" using subspace_span[of B] subspace_0[of "span B"] span_inc by autohence "span (insert 0 B) <= span B"    using span_span[of B] span_mono[of "insert 0 B" "span B"] by blasthence "convex hull insert 0 B <= span B"    using convex_hull_subset_span[of "insert 0 B"] by autohence "span (convex hull insert 0 B) <= span B"    using span_span[of B] span_mono[of "convex hull insert 0 B" "span B"] by blasthence *: "span (convex hull insert 0 B) = span B"    using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by autohence "span (convex hull insert 0 B) = span S"    using B_def span_mono[of B S] span_mono[of S "span B"] span_span[of B] by automoreover have "0 : affine hull (convex hull insert 0 B)"    using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by autoultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"    using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]    assms  hull_subset[of S] by autoobtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = d &       f ` span B = {x. ∀i∈Basis. i ~: d --> x • i = (0::real)} &  inj_on f (span B)" and d:"d⊆Basis"    using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by autohence "bounded_linear f" using linear_conv_bounded_linear by autohave "d ~={}" using fd B_def `B ~={}` by autohave "(insert 0 d) = f ` (insert 0 B)" using fd linear_0 by autohence "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"   using convex_hull_linear_image[of f "(insert 0 d)"]   convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by automoreover have "rel_interior (f ` (convex hull insert 0 B)) =   f ` rel_interior (convex hull insert 0 B)"   apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])   using `bounded_linear f` fd * by autoultimately have "rel_interior (convex hull insert 0 B) ~= {}"   using rel_interior_substd_simplex_nonempty[OF `d~={}` d] apply auto by blastmoreover have "convex hull (insert 0 B) <= S"   using B_def assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by autoultimately have ?thesis using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto} ultimately show ?thesis by autoqedlemma rel_interior_convex_nonempty:fixes S :: "('n::euclidean_space) set"assumes "convex S"shows "rel_interior S = {} <-> S = {}"proof-{ assume "S ~= {}" from this obtain a where "a : S" by auto  hence "0 : op + (-a) ` S" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto  hence "rel_interior (op + (-a) ` S) ~= {}"    using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"]          convex_translation[of S "-a"] assms by auto  hence "rel_interior S ~= {}" using rel_interior_translation by auto} from this show ?thesis using rel_interior_empty by autoqedlemma convex_rel_interior:fixes S :: "(_::euclidean_space) set"assumes "convex S"shows "convex (rel_interior S)"proof-{ fix "x" "y" "u"  assume assm: "x:rel_interior S" "y:rel_interior S" "0<=u" "(u :: real) <= 1"  hence "x:S" using rel_interior_subset by auto  have "x - u *⇩R (x-y) : rel_interior S"  proof(cases "0=u")     case False hence "0<u" using assm by auto        thus ?thesis        using assm rel_interior_convex_shrink[of S y x u] assms `x:S` by auto     next     case True thus ?thesis using assm by auto  qed  hence "(1-u) *⇩R x + u *⇩R y : rel_interior S" by (simp add: algebra_simps)} from this show ?thesis unfolding convex_alt by autoqedlemma convex_closure_rel_interior:fixes S :: "('n::euclidean_space) set"assumes "convex S"shows "closure(rel_interior S) = closure S"proof-have h1: "closure(rel_interior S) <= closure S"   using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto{ assume "S ~= {}" from this obtain a where a_def: "a : rel_interior S"    using rel_interior_convex_nonempty assms by auto  { fix x assume x_def: "x : closure S"    { assume "x=a" hence "x : closure(rel_interior S)" using a_def unfolding closure_def by auto }    moreover    { assume "x ~= a"       { fix e :: real assume e_def: "e>0"         def e1 == "min 1 (e/norm (x - a))" hence e1_def: "e1>0 & e1<=1 & e1*norm(x-a)<=e"            using `x ~= a` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(x-a)"] by simp         hence *: "x - e1 *⇩R (x - a) : rel_interior S"            using rel_interior_closure_convex_shrink[of S a x e1] assms x_def a_def e1_def by auto         have "EX y. y:rel_interior S & y ~= x & (dist y x) <= e"            apply (rule_tac x="x - e1 *⇩R (x - a)" in exI)            using * e1_def dist_norm[of "x - e1 *⇩R (x - a)" x] `x ~= a` by simp      } hence "x islimpt rel_interior S" unfolding islimpt_approachable_le by auto      hence "x : closure(rel_interior S)" unfolding closure_def by auto    } ultimately have "x : closure(rel_interior S)" by auto  } hence ?thesis using h1 by auto}moreover{ assume "S = {}" hence "rel_interior S = {}" using rel_interior_empty by auto  hence "closure(rel_interior S) = {}" using closure_empty by auto  hence ?thesis using `S={}` by auto} ultimately show ?thesis by blastqedlemma rel_interior_same_affine_hull:  fixes S :: "('n::euclidean_space) set"  assumes "convex S"  shows "affine hull (rel_interior S) = affine hull S"by (metis assms closure_same_affine_hull convex_closure_rel_interior)lemma rel_interior_aff_dim:  fixes S :: "('n::euclidean_space) set"  assumes "convex S"  shows "aff_dim (rel_interior S) = aff_dim S"by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)lemma rel_interior_rel_interior:  fixes S :: "('n::euclidean_space) set"  assumes "convex S"  shows "rel_interior (rel_interior S) = rel_interior S"proof-have "openin (subtopology euclidean (affine hull (rel_interior S))) (rel_interior S)"  using opein_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by autofrom this show ?thesis using rel_interior_def by autoqedlemma rel_interior_rel_open:  fixes S :: "('n::euclidean_space) set"  assumes "convex S"  shows "rel_open (rel_interior S)"unfolding rel_open_def using rel_interior_rel_interior assms by autolemma convex_rel_interior_closure_aux:  fixes x y z :: "_::euclidean_space"  assumes "0 < a" "0 < b" "(a+b) *⇩R z = a *⇩R x + b *⇩R y"  obtains e where "0 < e" "e <= 1" "z = y - e *⇩R (y-x)"proof-def e == "a/(a+b)"have "z = (1 / (a + b)) *⇩R ((a + b) *⇩R z)" apply auto using assms by simpalso have "... = (1 / (a + b)) *⇩R (a *⇩R x + b *⇩R y)" using assms   scaleR_cancel_left[of "1/(a+b)" "(a + b) *⇩R z" "a *⇩R x + b *⇩R y"] by autoalso have "... = y - e *⇩R (y-x)" using e_def apply (simp add: algebra_simps)   using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"] by autofinally have "z = y - e *⇩R (y-x)" by automoreover have "0<e" using e_def assms divide_pos_pos[of a "a+b"] by automoreover have "e<=1" using e_def assms by autoultimately show ?thesis using that[of e] by autoqedlemma convex_rel_interior_closure:  fixes S :: "('n::euclidean_space) set"  assumes "convex S"  shows "rel_interior (closure S) = rel_interior S"proof-{ assume "S={}" hence ?thesis using assms rel_interior_convex_nonempty by auto }moreover{ assume "S ~= {}"  have "rel_interior (closure S) >= rel_interior S"    using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset by auto  moreover  { fix z assume z_def: "z : rel_interior (closure S)"    obtain x where x_def: "x : rel_interior S"      using `S ~= {}` assms rel_interior_convex_nonempty by auto    { assume "x=z" hence "z : rel_interior S" using x_def by auto }    moreover    { assume "x ~= z"      obtain e where e_def: "e > 0 & cball z e Int affine hull closure S <= closure S"        using z_def rel_interior_cball[of "closure S"] by auto      hence *: "0 < e/norm(z-x)" using e_def `x ~= z` divide_pos_pos[of e "norm(z-x)"] by auto      def y == "z + (e/norm(z-x)) *⇩R (z-x)"      have yball: "y : cball z e"        using mem_cball y_def dist_norm[of z y] e_def by auto      have "x : affine hull closure S"        using x_def rel_interior_subset_closure hull_inc[of x "closure S"] by auto      moreover have "z : affine hull closure S"        using z_def rel_interior_subset hull_subset[of "closure S"] by auto      ultimately have "y : affine hull closure S"        using y_def affine_affine_hull[of "closure S"]          mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto      hence "y : closure S" using e_def yball by auto      have "(1+(e/norm(z-x))) *⇩R z = (e/norm(z-x)) *⇩R x + y"        using y_def by (simp add: algebra_simps)      from this obtain e1 where "0 < e1 & e1 <= 1 & z = y - e1 *⇩R (y - x)"        using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]          by (auto simp add: algebra_simps)      hence "z : rel_interior S"        using rel_interior_closure_convex_shrink assms x_def `y : closure S` by auto    } ultimately have "z : rel_interior S" by auto  } ultimately have ?thesis by auto} ultimately show ?thesis by blastqedlemma convex_interior_closure:fixes S :: "('n::euclidean_space) set"assumes "convex S"shows "interior (closure S) = interior S"using closure_aff_dim[of S] interior_rel_interior_gen[of S] interior_rel_interior_gen[of "closure S"]      convex_rel_interior_closure[of S] assms by autolemma closure_eq_rel_interior_eq:fixes S1 S2 ::  "('n::euclidean_space) set"assumes "convex S1" "convex S2"shows "(closure S1 = closure S2) <-> (rel_interior S1 = rel_interior S2)" by (metis convex_rel_interior_closure convex_closure_rel_interior assms)lemma closure_eq_between:fixes S1 S2 ::  "('n::euclidean_space) set"assumes "convex S1" "convex S2"shows "(closure S1 = closure S2) <->      ((rel_interior S1 <= S2) & (S2 <= closure S1))" (is "?A <-> ?B")proof-have "?A --> ?B" by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)moreover have "?B --> (closure S1 <= closure S2)"     by (metis assms(1) convex_closure_rel_interior closure_mono)moreover have "?B --> (closure S1 >= closure S2)" by (metis closed_closure closure_minimal)ultimately show ?thesis by blastqedlemma open_inter_closure_rel_interior:fixes S A ::  "('n::euclidean_space) set"assumes "convex S" "open A"shows "((A Int closure S) = {}) <-> ((A Int rel_interior S) = {})"by (metis assms convex_closure_rel_interior open_inter_closure_eq_empty)definition "rel_frontier S = closure S - rel_interior S"lemma closed_affine_hull: "closed (affine hull ((S :: ('n::euclidean_space) set)))"by (metis affine_affine_hull affine_closed)lemma closed_rel_frontier: "closed(rel_frontier (S :: ('n::euclidean_space) set))"proof-have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"apply (rule closedin_diff[of "subtopology euclidean (affine hull S)""closure S" "rel_interior S"])  using closed_closedin_trans[of "affine hull S" "closure S"] closed_affine_hull[of S]  closure_affine_hull[of S] opein_rel_interior[of S] by autoshow ?thesis apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])  unfolding rel_frontier_def using * closed_affine_hull by autoqedlemma convex_rel_frontier_aff_dim:fixes S1 S2 ::  "('n::euclidean_space) set"assumes "convex S1" "convex S2" "S2 ~= {}"assumes "S1 <= rel_frontier S2"shows "aff_dim S1 < aff_dim S2"proof-have "S1 <= closure S2" using assms unfolding rel_frontier_def by autohence *: "affine hull S1 <= affine hull S2"   using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by autohence "aff_dim S1 <= aff_dim S2" using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]    aff_dim_subset[of "affine hull S1" "affine hull S2"] by automoreover{ assume eq: "aff_dim S1 = aff_dim S2"  hence "S1 ~= {}" using aff_dim_empty[of S1] aff_dim_empty[of S2] `S2 ~= {}` by auto  have **: "affine hull S1 = affine hull S2"     apply (rule affine_dim_equal) using * affine_affine_hull apply auto     using `S1 ~= {}` hull_subset[of S1] apply auto     using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] by auto  obtain a where a_def: "a : rel_interior S1"     using  `S1 ~= {}` rel_interior_convex_nonempty assms by auto  obtain T where T_def: "open T & a : T Int S1 & T Int affine hull S1 <= S1"     using mem_rel_interior[of a S1] a_def by auto  hence "a : T Int closure S2" using a_def assms unfolding rel_frontier_def by auto  from this obtain b where b_def: "b : T Int rel_interior S2"     using open_inter_closure_rel_interior[of S2 T] assms T_def by auto  hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto  hence "b : S1" using T_def b_def by auto  hence False using b_def assms unfolding rel_frontier_def by auto} ultimately show ?thesis using less_le by autoqedlemma convex_rel_interior_if:fixes S ::  "('n::euclidean_space) set"assumes "convex S"assumes "z : rel_interior S"shows "(!x:affine hull S. EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*⇩R x+ e *⇩R z : S ))"proof-obtain e1 where e1_def: "e1>0 & cball z e1 Int affine hull S <= S"    using mem_rel_interior_cball[of z S] assms by auto{ fix x assume x_def: "x:affine hull S"  { assume "x ~= z"    def m == "1+e1/norm(x-z)"    hence "m>1" using e1_def `x ~= z` divide_pos_pos[of e1 "norm (x - z)"] by auto    { fix e assume e_def: "e>1 & e<=m"      have "z : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto      hence *: "(1-e)*⇩R x+ e *⇩R z : affine hull S"         using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x_def by auto      have "norm (z + e *⇩R x - (x + e *⇩R z)) = norm ((e - 1) *⇩R (x-z))" by (simp add: algebra_simps)      also have "...= (e - 1) * norm(x-z)" using norm_scaleR e_def by auto      also have "...<=(m - 1) * norm(x-z)" using e_def mult_right_mono[of _ _ "norm(x-z)"] by auto      also have "...= (e1 / norm (x - z)) * norm (x - z)" using m_def by auto      also have "...=e1" using `x ~= z` e1_def by simp      finally have **: "norm (z + e *⇩R x - (x + e *⇩R z)) <= e1" by auto      have "(1-e)*⇩R x+ e *⇩R z : cball z e1"         using m_def ** unfolding cball_def dist_norm by (auto simp add: algebra_simps)      hence "(1-e)*⇩R x+ e *⇩R z : S" using e_def * e1_def by auto    } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*⇩R x+ e *⇩R z : S )" using `m>1` by auto  }  moreover  { assume "x=z" def m == "1+e1" hence "m>1" using e1_def by auto    { fix e assume e_def: "e>1 & e<=m"      hence "(1-e)*⇩R x+ e *⇩R z : S"        using e1_def x_def `x=z` by (auto simp add: algebra_simps)      hence "(1-e)*⇩R x+ e *⇩R z : S" using e_def by auto    } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*⇩R x+ e *⇩R z : S )" using `m>1` by auto  } ultimately have "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*⇩R x+ e *⇩R z : S )" by auto} from this show ?thesis by autoqedlemma convex_rel_interior_if2:fixes S ::  "('n::euclidean_space) set"assumes "convex S"assumes "z : rel_interior S"shows "(!x:affine hull S. EX e. e>1 & (1-e)*⇩R x+ e *⇩R z : S)"using convex_rel_interior_if[of S z] assms by autolemma convex_rel_interior_only_if:fixes S ::  "('n::euclidean_space) set"assumes "convex S" "S ~= {}"assumes "(!x:S. EX e. e>1 & (1-e)*⇩R x+ e *⇩R z : S)"shows "z : rel_interior S"proof-obtain x where x_def: "x : rel_interior S" using rel_interior_convex_nonempty assms by autohence "x:S" using rel_interior_subset by autofrom this obtain e where e_def: "e>1 & (1 - e) *⇩R x + e *⇩R z : S" using assms by autodef y == "(1 - e) *⇩R x + e *⇩R z" hence "y:S" using e_def by autodef e1 == "1/e" hence "0<e1 & e1<1" using e_def by autohence "z=y-(1-e1)*⇩R (y-x)" using e1_def y_def by (auto simp add: algebra_simps)from this show ?thesis    using rel_interior_convex_shrink[of S x y "1-e1"] `0<e1 & e1<1` `y:S` x_def assms by autoqedlemma convex_rel_interior_iff:fixes S ::  "('n::euclidean_space) set"assumes "convex S" "S ~= {}"shows "z : rel_interior S <-> (!x:S. EX e. e>1 & (1-e)*⇩R x+ e *⇩R z : S)"using assms hull_subset[of S "affine"]      convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z] by autolemma convex_rel_interior_iff2:fixes S ::  "('n::euclidean_space) set"assumes "convex S" "S ~= {}"shows "z : rel_interior S <-> (!x:affine hull S. EX e. e>1 & (1-e)*⇩R x+ e *⇩R z : S)"using assms hull_subset[of S]      convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z] by autolemma convex_interior_iff:fixes S ::  "('n::euclidean_space) set"assumes "convex S"shows "z : interior S <-> (!x. EX e. e>0 & z+ e *⇩R x : S)"proof-{ assume a: "~(aff_dim S = int DIM('n))"  { assume "z : interior S"    hence False using a interior_rel_interior_gen[of S] by auto  }  moreover  { assume r: "!x. EX e. e>0 & z+ e *⇩R x : S"    { fix x obtain e1 where e1_def: "e1>0 & z+ e1 *⇩R (x-z) : S" using r by auto      obtain e2 where e2_def: "e2>0 & z+ e2 *⇩R (z-x) : S" using r by auto      def x1 == "z+ e1 *⇩R (x-z)"         hence x1: "x1 : affine hull S" using e1_def hull_subset[of S] by auto      def x2 == "z+ e2 *⇩R (z-x)"         hence x2: "x2 : affine hull S" using e2_def hull_subset[of S] by auto      have *: "e1/(e1+e2) + e2/(e1+e2) = 1" using add_divide_distrib[of e1 e2 "e1+e2"] e1_def e2_def by simp      hence "z = (e2/(e1+e2)) *⇩R x1 + (e1/(e1+e2)) *⇩R x2"         using x1_def x2_def apply (auto simp add: algebra_simps)         using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z] by auto      hence z: "z : affine hull S"         using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]         x1 x2 affine_affine_hull[of S] * by auto      have "x1-x2 = (e1+e2) *⇩R (x-z)"         using x1_def x2_def by (auto simp add: algebra_simps)      hence "x=z+(1/(e1+e2)) *⇩R (x1-x2)" using e1_def e2_def by simp      hence "x : affine hull S" using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]          x1 x2 z affine_affine_hull[of S] by auto    } hence "affine hull S = UNIV" by auto    hence "aff_dim S = int DIM('n)" using aff_dim_affine_hull[of S] by (simp add: aff_dim_univ)    hence False using a by auto  } ultimately have ?thesis by auto}moreover{ assume a: "aff_dim S = int DIM('n)"  hence "S ~= {}" using aff_dim_empty[of S] by auto  have *: "affine hull S=UNIV" using a affine_hull_univ by auto  { assume "z : interior S"    hence "z : rel_interior S" using a interior_rel_interior_gen[of S] by auto    hence **: "(!x. EX e. e>1 & (1-e)*⇩R x+ e *⇩R z : S)"      using convex_rel_interior_iff2[of S z] assms `S~={}` * by auto    fix x obtain e1 where e1_def: "e1>1 & (1-e1)*⇩R (z-x)+ e1 *⇩R z : S"      using **[rule_format, of "z-x"] by auto    def e == "e1 - 1"    hence "(1-e1)*⇩R (z-x)+ e1 *⇩R z = z+ e *⇩R x" by (simp add: algebra_simps)    hence "e>0 & z+ e *⇩R x : S" using e1_def e_def by auto    hence "EX e. e>0 & z+ e *⇩R x : S" by auto  }  moreover  { assume r: "(!x. EX e. e>0 & z+ e *⇩R x : S)"    { fix x obtain e1 where e1_def: "e1>0 & z + e1*⇩R (z-x) : S"         using r[rule_format, of "z-x"] by auto      def e == "e1 + 1"      hence "z + e1*⇩R (z-x) = (1-e)*⇩R x+ e *⇩R z" by (simp add: algebra_simps)      hence "e > 1 & (1-e)*⇩R x+ e *⇩R z : S" using e1_def e_def by auto      hence "EX e. e>1 & (1-e)*⇩R x+ e *⇩R z : S" by auto    }    hence "z : rel_interior S" using convex_rel_interior_iff2[of S z] assms `S~={}` by auto    hence "z : interior S" using a interior_rel_interior_gen[of S] by auto  } ultimately have ?thesis by auto} ultimately show ?thesis by autoqedsubsubsection {* Relative interior and closure under common operations *}lemma rel_interior_inter_aux: "Inter {rel_interior S |S. S : I} <= Inter I"proof-{ fix y assume "y : Inter {rel_interior S |S. S : I}"  hence y_def: "!S : I. y : rel_interior S" by auto  { fix S assume "S : I" hence "y : S" using rel_interior_subset y_def by auto }  hence "y : Inter I" by auto} thus ?thesis by autoqedlemma closure_inter: "closure (Inter I) <= Inter {closure S |S. S : I}"proof-{ fix y assume "y : Inter I" hence y_def: "!S : I. y : S" by auto  { fix S assume "S : I" hence "y : closure S" using closure_subset y_def by auto }  hence "y : Inter {closure S |S. S : I}" by auto} hence "Inter I <= Inter {closure S |S. S : I}" by automoreover have "closed (Inter {closure S |S. S : I})"  unfolding closed_Inter closed_closure by autoultimately show ?thesis using closure_hull[of "Inter I"]  hull_minimal[of "Inter I" "Inter {closure S |S. S : I}" "closed"] by autoqedlemma convex_closure_rel_interior_inter:assumes "!S : I. convex (S :: ('n::euclidean_space) set)"assumes "Inter {rel_interior S |S. S : I} ~= {}"shows "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"proof-obtain x where x_def: "!S : I. x : rel_interior S" using assms by auto{ fix y assume "y : Inter {closure S |S. S : I}" hence y_def: "!S : I. y : closure S" by auto  { assume "y = x"    hence "y : closure (Inter {rel_interior S |S. S : I})"       using x_def closure_subset[of "Inter {rel_interior S |S. S : I}"] by auto  }  moreover  { assume "y ~= x"    { fix e :: real assume e_def: "0 < e"      def e1 == "min 1 (e/norm (y - x))" hence e1_def: "e1>0 & e1<=1 & e1*norm(y-x)<=e"        using `y ~= x` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(y-x)"] by simp      def z == "y - e1 *⇩R (y - x)"      { fix S assume "S : I"        hence "z : rel_interior S" using rel_interior_closure_convex_shrink[of S x y e1]           assms x_def y_def e1_def z_def by auto      } hence *: "z : Inter {rel_interior S |S. S : I}" by auto      have "EX z. z:Inter {rel_interior S |S. S : I} & z ~= y & (dist z y) <= e"           apply (rule_tac x="z" in exI) using `y ~= x` z_def * e1_def e_def dist_norm[of z y] by simp    } hence "y islimpt Inter {rel_interior S |S. S : I}" unfolding islimpt_approachable_le by blast    hence "y : closure (Inter {rel_interior S |S. S : I})" unfolding closure_def by auto  } ultimately have "y : closure (Inter {rel_interior S |S. S : I})" by auto} from this show ?thesis by autoqedlemma convex_closure_inter:assumes "!S : I. convex (S :: ('n::euclidean_space) set)"assumes "Inter {rel_interior S |S. S : I} ~= {}"shows "closure (Inter I) = Inter {closure S |S. S : I}"proof-have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"  using convex_closure_rel_interior_inter assms by automoreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)"    using rel_interior_inter_aux          closure_mono[of "Inter {rel_interior S |S. S : I}" "Inter I"] by autoultimately show ?thesis using closure_inter[of I] by autoqedlemma convex_inter_rel_interior_same_closure:assumes "!S : I. convex (S :: ('n::euclidean_space) set)"assumes "Inter {rel_interior S |S. S : I} ~= {}"shows "closure (Inter {rel_interior S |S. S : I}) = closure (Inter I)"proof-have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"  using convex_closure_rel_interior_inter assms by automoreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)"    using rel_interior_inter_aux          closure_mono[of "Inter {rel_interior S |S. S : I}" "Inter I"] by autoultimately show ?thesis using closure_inter[of I] by autoqedlemma convex_rel_interior_inter:assumes "!S : I. convex (S :: ('n::euclidean_space) set)"assumes "Inter {rel_interior S |S. S : I} ~= {}"shows "rel_interior (Inter I) <= Inter {rel_interior S |S. S : I}"proof-have "convex(Inter I)" using assms convex_Inter by automoreover have "convex(Inter {rel_interior S |S. S : I})" apply (rule convex_Inter)   using assms convex_rel_interior by autoultimately have "rel_interior (Inter {rel_interior S |S. S : I}) = rel_interior (Inter I)"   using convex_inter_rel_interior_same_closure assms   closure_eq_rel_interior_eq[of "Inter {rel_interior S |S. S : I}" "Inter I"] by blastfrom this show ?thesis using rel_interior_subset[of "Inter {rel_interior S |S. S : I}"] by autoqedlemma convex_rel_interior_finite_inter:assumes "!S : I. convex (S :: ('n::euclidean_space) set)"assumes "Inter {rel_interior S |S. S : I} ~= {}"assumes "finite I"shows "rel_interior (Inter I) = Inter {rel_interior S |S. S : I}"proof-have "Inter I ~= {}" using assms rel_interior_inter_aux[of I] by autohave "convex (Inter I)" using convex_Inter assms by auto{ assume "I={}" hence ?thesis using Inter_empty rel_interior_univ2 by auto }moreover{ assume "I ~= {}"{ fix z assume z_def: "z : Inter {rel_interior S |S. S : I}"  { fix x assume x_def: "x : Inter I"    { fix S assume S_def: "S : I" hence "z : rel_interior S" "x : S" using z_def x_def by auto      (*from this obtain e where e_def: "e>1 & (1 - e) *⇩R x + e *⇩R z : S"*)      hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*⇩R x+ e *⇩R z : S )"         using convex_rel_interior_if[of S z] S_def assms hull_subset[of S] by auto    } from this obtain mS where mS_def: "!S : I. (mS(S) > (1 :: real) &         (!e. (e>1 & e<=mS(S)) --> (1-e)*⇩R x+ e *⇩R z : S))" by metis    obtain e where e_def: "e=Min (mS ` I)" by auto    have "e : (mS ` I)" using e_def assms `I ~= {}` by simp    hence "e>(1 :: real)" using mS_def by auto    moreover have "!S : I. e<=mS(S)" using e_def assms by auto    ultimately have "EX e>1. (1 - e) *⇩R x + e *⇩R z : Inter I" using mS_def by auto  } hence "z : rel_interior (Inter I)" using convex_rel_interior_iff[of "Inter I" z]       `Inter I ~= {}` `convex (Inter I)` by auto} from this have ?thesis using convex_rel_interior_inter[of I] assms by auto} ultimately show ?thesis by blastqedlemma convex_closure_inter_two:fixes S T :: "('n::euclidean_space) set"assumes "convex S" "convex T"assumes "(rel_interior S) Int (rel_interior T) ~= {}"shows "closure (S Int T) = (closure S) Int (closure T)"using convex_closure_inter[of "{S,T}"] assms by autolemma convex_rel_interior_inter_two:fixes S T :: "('n::euclidean_space) set"assumes "convex S" "convex T"assumes "(rel_interior S) Int (rel_interior T) ~= {}"shows "rel_interior (S Int T) = (rel_interior S) Int (rel_interior T)"using convex_rel_interior_finite_inter[of "{S,T}"] assms by autolemma convex_affine_closure_inter:fixes S T :: "('n::euclidean_space) set"assumes "convex S" "affine T"assumes "(rel_interior S) Int T ~= {}"shows "closure (S Int T) = (closure S) Int T"proof-have "affine hull T = T" using assms by autohence "rel_interior T = T" using rel_interior_univ[of T] by metismoreover have "closure T = T" using assms affine_closed[of T] by autoultimately show ?thesis using convex_closure_inter_two[of S T] assms affine_imp_convex by autoqedlemma convex_affine_rel_interior_inter:fixes S T :: "('n::euclidean_space) set"assumes "convex S" "affine T"assumes "(rel_interior S) Int T ~= {}"shows "rel_interior (S Int T) = (rel_interior S) Int T"proof-have "affine hull T = T" using assms by autohence "rel_interior T = T" using rel_interior_univ[of T] by metismoreover have "closure T = T" using assms affine_closed[of T] by autoultimately show ?thesis using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by autoqedlemma subset_rel_interior_convex:fixes S T :: "('n::euclidean_space) set"assumes "convex S" "convex T"assumes "S <= closure T"assumes "~(S <= rel_frontier T)"shows "rel_interior S <= rel_interior T"proof-have *: "S Int closure T = S" using assms by autohave "~(rel_interior S <= rel_frontier T)"     using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]     closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms by autohence "(rel_interior S) Int (rel_interior (closure T)) ~= {}"     using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T] by autohence "rel_interior S Int rel_interior T = rel_interior (S Int closure T)" using assms convex_closure     convex_rel_interior_inter_two[of S "closure T"] convex_rel_interior_closure[of T] by autoalso have "...=rel_interior (S)" using * by autofinally show ?thesis by autoqedlemma rel_interior_convex_linear_image:fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"assumes "linear f"assumes "convex S"shows "f ` (rel_interior S) = rel_interior (f ` S)"proof-{ assume "S = {}" hence ?thesis using assms rel_interior_empty rel_interior_convex_nonempty by auto }moreover{ assume "S ~= {}"have *: "f ` (rel_interior S) <= f ` S" unfolding image_mono using rel_interior_subset by autohave "f ` S <= f ` (closure S)" unfolding image_mono using closure_subset by autoalso have "... = f ` (closure (rel_interior S))" using convex_closure_rel_interior assms by autoalso have "... <= closure (f ` (rel_interior S))" using closure_linear_image assms by autofinally have "closure (f ` S) = closure (f ` rel_interior S)"   using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure         closure_mono[of "f ` rel_interior S" "f ` S"] * by autohence "rel_interior (f ` S) = rel_interior (f ` rel_interior S)" using assms convex_rel_interior   linear_conv_bounded_linear[of f] convex_linear_image[of S] convex_linear_image[of "rel_interior S"]   closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"] by autohence "rel_interior (f ` S) <= f ` rel_interior S" using rel_interior_subset by automoreover{ fix z assume z_def: "z : f ` rel_interior S"  from this obtain z1 where z1_def: "z1 : rel_interior S & (f z1 = z)" by auto  { fix x assume "x : f ` S"    from this obtain x1 where x1_def: "x1 : S & (f x1 = x)" by auto    from this obtain e where e_def: "e>1 & (1 - e) *⇩R x1 + e *⇩R z1 : S"       using convex_rel_interior_iff[of S z1] `convex S` x1_def z1_def by auto    moreover have "f ((1 - e) *⇩R x1 + e *⇩R z1) = (1 - e) *⇩R x + e *⇩R z"        using x1_def z1_def `linear f` by (simp add: linear_add_cmul)    ultimately have "(1 - e) *⇩R x + e *⇩R z : f ` S"        using imageI[of "(1 - e) *⇩R x1 + e *⇩R z1" S f] by auto    hence "EX e. (e>1 & (1 - e) *⇩R x + e *⇩R z : f ` S)" using e_def by auto  } from this have "z : rel_interior (f ` S)" using convex_rel_interior_iff[of "f ` S" z] `convex S`       `linear f` `S ~= {}` convex_linear_image[of S f]  linear_conv_bounded_linear[of f] by auto} ultimately have ?thesis by auto} ultimately show ?thesis by blastqedlemma convex_linear_preimage:  assumes c:"convex S" and l:"bounded_linear f"  shows "convex(f -` S)"proof(auto simp add: convex_def)  interpret f: bounded_linear f by fact  fix x y assume xy:"f x : S" "f y : S"  fix u v ::real assume uv:"0 <= u" "0 <= v" "u + v = 1"  show "f (u *⇩R x + v *⇩R y) : S" unfolding image_iff    using bexI[of _ "u *⇩R x + v *⇩R y"] f.add f.scaleR      c[unfolded convex_def] xy uv by autoqedlemma rel_interior_convex_linear_preimage:fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"assumes "linear f"assumes "convex S"assumes "f -` (rel_interior S) ~= {}"shows "rel_interior (f -` S) = f -` (rel_interior S)"proof-have "S ~= {}" using assms rel_interior_empty by autohave nonemp: "f -` S ~= {}" by (metis assms(3) rel_interior_subset subset_empty vimage_mono)hence "S Int (range f) ~= {}" by autohave conv: "convex (f -` S)" using convex_linear_preimage assms linear_conv_bounded_linear by autohence "convex (S Int (range f))"  by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image){ fix z assume "z : f -` (rel_interior S)"  hence z_def: "f z : rel_interior S" by auto  { fix x assume "x : f -` S" from this have x_def: "f x : S" by auto    from this obtain e where e_def: "e>1 & (1-e)*⇩R (f x)+ e *⇩R (f z) : S"      using convex_rel_interior_iff[of S "f z"] z_def assms `S ~= {}` by auto    moreover have "(1-e)*⇩R (f x)+ e *⇩R (f z) = f ((1-e)*⇩R x + e *⇩R z)"      using `linear f` by (simp add: linear_def)    ultimately have "EX e. e>1 & (1-e)*⇩R x + e *⇩R z : f -` S" using e_def by auto  } hence "z : rel_interior (f -` S)"       using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto}moreover{ fix z assume z_def: "z : rel_interior (f -` S)"  { fix x assume x_def: "x: S Int (range f)"    from this obtain y where y_def: "(f y = x) & (y : f -` S)" by auto    from this obtain e where e_def: "e>1 & (1-e)*⇩R y+ e *⇩R z : f -` S"      using convex_rel_interior_iff[of "f -` S" z] z_def conv by auto    moreover have "(1-e)*⇩R x+ e *⇩R (f z) = f ((1-e)*⇩R y + e *⇩R z)"      using `linear f` y_def by (simp add: linear_def)    ultimately have "EX e. e>1 & (1-e)*⇩R x + e *⇩R (f z) : S Int (range f)"      using e_def by auto  } hence "f z : rel_interior (S Int (range f))" using `convex (S Int (range f))`    `S Int (range f) ~= {}` convex_rel_interior_iff[of "S Int (range f)" "f z"] by auto  moreover have "affine (range f)"    by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)  ultimately have "f z : rel_interior S"    using convex_affine_rel_interior_inter[of S "range f"] assms by auto  hence "z : f -` (rel_interior S)" by auto}ultimately show ?thesis by autoqedlemma convex_direct_sum:fixes S :: "('n::euclidean_space) set"fixes T :: "('m::euclidean_space) set"assumes "convex S" "convex T"shows "convex (S <*> T)"proof-{fix x assume "x : S <*> T"from this obtain xs xt where xst_def: "xs : S & xt : T & (xs,xt) = x" by autofix y assume "y : S <*> T"from this obtain ys yt where yst_def: "ys : S & yt : T & (ys,yt) = y" by autofix u v assume uv_def: "(u :: real)>=0 & (v :: real)>=0 & u+v=1"have "u *⇩R x + v *⇩R y = (u *⇩R xs + v *⇩R ys, u *⇩R xt + v *⇩R yt)" using xst_def yst_def by automoreover have "u *⇩R xs + v *⇩R ys : S"   using uv_def xst_def yst_def convex_def[of S] assms by automoreover have "u *⇩R xt + v *⇩R yt : T"   using uv_def xst_def yst_def convex_def[of T] assms by autoultimately have "u *⇩R x + v *⇩R y : S <*> T" by auto} from this show ?thesis unfolding convex_def by autoqedlemma convex_hull_direct_sum:fixes S :: "('n::euclidean_space) set"fixes T :: "('m::euclidean_space) set"shows "convex hull (S <*> T) = (convex hull S) <*> (convex hull T)"proof-{ fix x assume "x : (convex hull S) <*> (convex hull T)"  from this obtain xs xt where xst_def: "xs : convex hull S & xt : convex hull T & (xs,xt) = x" by auto  from xst_def obtain sI su where s: "finite sI & sI <= S & (ALL x:sI. 0 <= su x) & setsum su sI = 1     & (SUM v:sI. su v *⇩R v) = xs" using convex_hull_explicit[of S] by auto  from xst_def obtain tI tu where t: "finite tI & tI <= T & (ALL x:tI. 0 <= tu x) & setsum tu tI = 1     & (SUM v:tI. tu v *⇩R v) = xt" using convex_hull_explicit[of T] by auto  def I == "(sI <*> tI)"  def u == "(%i. (su (fst i))*(tu(snd i)))"  have "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *⇩R v)=     (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *⇩R vs)"     using fst_setsum[of "(%v. (su (fst v) * tu (snd v)) *⇩R v)" "sI <*> tI"]     by (simp add: split_def scaleR_prod_def setsum_cartesian_product)  also have "...=(SUM vt:tI. tu vt *⇩R (SUM vs:sI. su vs *⇩R vs))"     using setsum_commute[of "(%vt vs. (su vs * tu vt) *⇩R vs)" sI tI]     by (simp add: mult_commute scaleR_right.setsum)  also have "...=(SUM vt:tI. tu vt *⇩R xs)" using s by auto  also have "...=(SUM vt:tI. tu vt) *⇩R xs" by (simp add: scaleR_left.setsum)  also have "...=xs" using t by auto  finally have h1: "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *⇩R v)=xs" by auto  have "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *⇩R v)=     (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *⇩R vt)"     using snd_setsum[of "(%v. (su (fst v) * tu (snd v)) *⇩R v)" "sI <*> tI"]     by (simp add: split_def scaleR_prod_def setsum_cartesian_product)  also have "...=(SUM vs:sI. su vs *⇩R (SUM vt:tI. tu vt *⇩R vt))"     by (simp add: mult_commute scaleR_right.setsum)  also have "...=(SUM vs:sI. su vs *⇩R xt)" using t by auto  also have "...=(SUM vs:sI. su vs) *⇩R xt" by (simp add: scaleR_left.setsum)  also have "...=xt" using s by auto  finally have h2: "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *⇩R v)=xt" by auto  from h1 h2 have "(SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *⇩R v) = x" using xst_def by auto  moreover have "finite I & (I <= S <*> T)" using s t I_def by auto  moreover have "!i:I. 0 <= u i" using s t I_def u_def by (simp add: mult_nonneg_nonneg)  moreover have "setsum u I = 1" using u_def I_def setsum_cartesian_product[of "(% x y. (su x)*(tu y))"]     s t setsum_product[of su sI tu tI] by (auto simp add: split_def)  ultimately have "x : convex hull (S <*> T)"     apply (subst convex_hull_explicit[of "S <*> T"]) apply rule     apply (rule_tac x="I" in exI) apply (rule_tac x="u" in exI)     using I_def u_def by auto}hence "convex hull (S <*> T) >= (convex hull S) <*> (convex hull T)" by automoreover have "convex ((convex hull S) <*> (convex hull T))"   by (simp add: convex_direct_sum convex_convex_hull)ultimately show ?thesis   using hull_minimal[of "S <*> T" "(convex hull S) <*> (convex hull T)" "convex"]         hull_subset[of S convex] hull_subset[of T convex] by autoqedlemma rel_interior_direct_sum:fixes S :: "('n::euclidean_space) set"fixes T :: "('m::euclidean_space) set"assumes "convex S" "convex T"shows "rel_interior (S <*> T) = rel_interior S <*> rel_interior T"proof-{ assume "S={}" hence ?thesis apply auto using rel_interior_empty by auto }moreover{ assume "T={}" hence ?thesis apply auto using rel_interior_empty by auto }moreover {assume "S ~={}" "T ~={}"hence ri: "rel_interior S ~= {}" "rel_interior T ~= {}" using rel_interior_convex_nonempty assms by autohence "fst -` rel_interior S ~= {}" using fst_vimage_eq_Times[of "rel_interior S"] by autohence "rel_interior ((fst :: 'n * 'm => 'n) -` S) = fst -` rel_interior S"  using fst_linear `convex S` rel_interior_convex_linear_preimage[of fst S] by autohence s: "rel_interior (S <*> (UNIV :: 'm set)) = rel_interior S <*> UNIV" by (simp add: fst_vimage_eq_Times)from ri have "snd -` rel_interior T ~= {}" using snd_vimage_eq_Times[of "rel_interior T"] by autohence "rel_interior ((snd :: 'n * 'm => 'm) -` T) = snd -` rel_interior T"  using snd_linear `convex T` rel_interior_convex_linear_preimage[of snd T] by autohence t: "rel_interior ((UNIV :: 'n set) <*> T) = UNIV <*> rel_interior T" by (simp add: snd_vimage_eq_Times)from s t have *: "rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)  = rel_interior S <*> rel_interior T" by autohave "(S <*> T) = (S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T)" by autohence "rel_interior (S <*> T) = rel_interior ((S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T))" by autoalso have "...=rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)"   apply (subst convex_rel_interior_inter_two[of "S <*> (UNIV :: 'm set)" "(UNIV :: 'n set) <*> T"])   using * ri assms convex_direct_sum by autofinally have ?thesis using * by auto}ultimately show ?thesis by blastqedlemma rel_interior_scaleR:fixes S :: "('n::euclidean_space) set"assumes "c ~= 0"shows "(op *⇩R c) ` (rel_interior S) = rel_interior ((op *⇩R c) ` S)"using rel_interior_injective_linear_image[of "(op *⇩R c)" S]      linear_conv_bounded_linear[of "op *⇩R c"] linear_scaleR injective_scaleR[of c] assms by autolemma rel_interior_convex_scaleR:fixes S :: "('n::euclidean_space) set"assumes "convex S"shows "(op *⇩R c) ` (rel_interior S) = rel_interior ((op *⇩R c) ` S)"by (metis assms linear_scaleR rel_interior_convex_linear_image)lemma convex_rel_open_scaleR:fixes S :: "('n::euclidean_space) set"assumes "convex S" "rel_open S"shows "convex ((op *⇩R c) ` S) & rel_open ((op *⇩R c) ` S)"by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)lemma convex_rel_open_finite_inter:assumes "!S : I. (convex (S :: ('n::euclidean_space) set) & rel_open S)"assumes "finite I"shows "convex (Inter I) & rel_open (Inter I)"proof-{ assume "Inter {rel_interior S |S. S : I} = {}"  hence "Inter I = {}" using assms unfolding rel_open_def by auto  hence ?thesis unfolding rel_open_def using rel_interior_empty by auto}moreover{ assume "Inter {rel_interior S |S. S : I} ~= {}"  hence "rel_open (Inter I)" using assms unfolding rel_open_def    using convex_rel_interior_finite_inter[of I] by auto  hence ?thesis using convex_Inter assms by auto} ultimately show ?thesis by autoqedlemma convex_rel_open_linear_image:fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"assumes "linear f"assumes "convex S" "rel_open S"shows "convex (f ` S) & rel_open (f ` S)"by (metis assms convex_linear_image rel_interior_convex_linear_image   linear_conv_bounded_linear rel_open_def)lemma convex_rel_open_linear_preimage:fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"assumes "linear f"assumes "convex S" "rel_open S"shows "convex (f -` S) & rel_open (f -` S)"proof-{ assume "f -` (rel_interior S) = {}"  hence "f -` S = {}" using assms unfolding rel_open_def by auto  hence ?thesis unfolding rel_open_def using rel_interior_empty by auto}moreover{ assume "f -` (rel_interior S) ~= {}"  hence "rel_open (f -` S)" using assms unfolding rel_open_def    using rel_interior_convex_linear_preimage[of f S] by auto  hence ?thesis using convex_linear_preimage assms linear_conv_bounded_linear by auto} ultimately show ?thesis by autoqedlemma rel_interior_projection:fixes S :: "('m::euclidean_space*'n::euclidean_space) set"fixes f :: "'m::euclidean_space => ('n::euclidean_space) set"assumes "convex S"assumes "f = (%y. {z. (y,z) : S})"shows "(y,z) : rel_interior S <-> (y : rel_interior {y. (f y ~= {})} & z : rel_interior (f y))"proof-{ fix y assume "y : {y. (f y ~= {})}" from this obtain z where "(y,z) : S" using assms by auto  hence "EX x. x : S & y = fst x" apply (rule_tac x="(y,z)" in exI) by auto  from this obtain x where "x : S & y = fst x" by blast  hence "y : fst ` S" unfolding image_def by auto}hence "fst ` S = {y. (f y ~= {})}" unfolding fst_def using assms by autohence h1: "fst ` rel_interior S = rel_interior {y. (f y ~= {})}"   using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto{ fix y assume "y : rel_interior {y. (f y ~= {})}"  hence "y : fst ` rel_interior S" using h1 by auto  hence *: "rel_interior S Int fst -` {y} ~= {}" by auto  moreover have aff: "affine (fst -` {y})" unfolding affine_alt by (simp add: algebra_simps)  ultimately have **: "rel_interior (S Int fst -` {y}) = rel_interior S Int fst -` {y}"    using convex_affine_rel_interior_inter[of S "fst -` {y}"] assms by auto  have conv: "convex (S Int fst -` {y})" using convex_Int assms aff affine_imp_convex by auto  { fix x assume "x : f y"    hence "(y,x) : S Int (fst -` {y})" using assms by auto    moreover have "x = snd (y,x)" by auto    ultimately have "x : snd ` (S Int fst -` {y})" by blast  }  hence "snd ` (S Int fst -` {y}) = f y" using assms by auto  hence ***: "rel_interior (f y) = snd ` rel_interior (S Int fst -` {y})"    using rel_interior_convex_linear_image[of snd "S Int fst -` {y}"] snd_linear conv by auto  { fix z assume "z : rel_interior (f y)"    hence "z : snd ` rel_interior (S Int fst -` {y})" using *** by auto    moreover have "{y} = fst ` rel_interior (S Int fst -` {y})" using * ** rel_interior_subset by auto    ultimately have "(y,z) : rel_interior (S Int fst -` {y})" by force    hence "(y,z) : rel_interior S" using ** by auto  }  moreover  { fix z assume "(y,z) : rel_interior S"    hence "(y,z) : rel_interior (S Int fst -` {y})" using ** by auto    hence "z : snd ` rel_interior (S Int fst -` {y})" by (metis Range_iff snd_eq_Range)    hence "z : rel_interior (f y)" using *** by auto  }  ultimately have "!!z. (y,z) : rel_interior S <-> z : rel_interior (f y)" by auto}hence h2: "!!y z. y : rel_interior {t. f t ~= {}} ==> ((y, z) : rel_interior S) = (z : rel_interior (f y))"  by auto{ fix y z assume asm: "(y, z) : rel_interior S"  hence "y : fst ` rel_interior S" by (metis Domain_iff fst_eq_Domain)  hence "y : rel_interior {t. f t ~= {}}" using h1 by auto  hence "y : rel_interior {t. f t ~= {}} & (z : rel_interior (f y))" using h2 asm by auto} from this show ?thesis using h2 by blastqedsubsubsection {* Relative interior of convex cone *}lemma cone_rel_interior:fixes S :: "('m::euclidean_space) set"assumes "cone S"shows "cone ({0} Un (rel_interior S))"proof-{ assume "S = {}" hence ?thesis by (simp add: rel_interior_empty cone_0) }moreover{ assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *⇩R c ` S = S)" using cone_iff[of S] assms by auto  hence *: "0:({0} Un (rel_interior S)) &           (!c. c>0 --> op *⇩R c ` ({0} Un rel_interior S) = ({0} Un rel_interior S))"           by (auto simp add: rel_interior_scaleR)  hence ?thesis using cone_iff[of "{0} Un rel_interior S"] by auto}ultimately show ?thesis by blastqedlemma rel_interior_convex_cone_aux:fixes S :: "('m::euclidean_space) set"assumes "convex S"shows "(c,x) : rel_interior (cone hull ({(1 :: real)} <*> S)) <->       c>0 & x : ((op *⇩R c) ` (rel_interior S))"proof-{ assume "S={}" hence ?thesis by (simp add: rel_interior_empty cone_hull_empty) }moreover{ assume "S ~= {}" from this obtain s where "s : S" by autohave conv: "convex ({(1 :: real)} <*> S)" using convex_direct_sum[of "{(1 :: real)}" S]   assms convex_singleton[of "1 :: real"] by autodef f == "(%y. {z. (y,z) : cone hull ({(1 :: real)} <*> S)})"hence *: "(c, x) : rel_interior (cone hull ({(1 :: real)} <*> S)) =      (c : rel_interior {y. f y ~= {}} & x : rel_interior (f c))"  apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} <*> S)" f c x])  using convex_cone_hull[of "{(1 :: real)} <*> S"] conv by auto{ fix y assume "(y :: real)>=0"  hence "y *⇩R (1,s) : cone hull ({(1 :: real)} <*> S)"     using cone_hull_expl[of "{(1 :: real)} <*> S"] `s:S` by auto  hence "f y ~= {}" using f_def by auto}hence "{y. f y ~= {}} = {0..}" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by autohence **: "rel_interior {y. f y ~= {}} = {0<..}" using rel_interior_real_semiline by auto{ fix c assume "c>(0 :: real)"  hence "f c = (op *⇩R c ` S)" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto  hence "rel_interior (f c)= (op *⇩R c ` rel_interior S)"     using rel_interior_convex_scaleR[of S c] assms by auto}hence ?thesis using * ** by auto} ultimately show ?thesis by blastqedlemma rel_interior_convex_cone:fixes S :: "('m::euclidean_space) set"assumes "convex S"shows "rel_interior (cone hull ({(1 :: real)} <*> S)) =       {(c,c *⇩R x) |c x. c>0 & x : (rel_interior S)}"(is "?lhs=?rhs")proof-{ fix z assume "z:?lhs"  have *: "z=(fst z,snd z)" by auto  have "z:?rhs" using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms `z:?lhs` apply auto     apply (rule_tac x="fst z" in exI) apply (rule_tac x="x" in exI) using * by auto}moreover{ fix z assume "z:?rhs" hence "z:?lhs"  using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms by auto}ultimately show ?thesis by blastqedlemma convex_hull_finite_union:assumes "finite I"assumes "!i:I. (convex (S i) & (S i) ~= {})"shows "convex hull (Union (S ` I)) =       {setsum (%i. c i *⇩R s i) I |c s. (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)}"  (is "?lhs = ?rhs")proof-{ fix x assume "x : ?rhs"  from this obtain c s    where *: "setsum (%i. c i *⇩R s i) I=x" "(setsum c I = 1)"     "(!i:I. c i >= 0) & (!i:I. s i : S i)" by auto  hence "!i:I. s i : convex hull (Union (S ` I))" using hull_subset[of "Union (S ` I)" convex] by auto  hence "x : ?lhs" unfolding *(1)[symmetric]     apply (subst convex_setsum[of I "convex hull Union (S ` I)" c s]`