src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author huffman
Mon Nov 14 09:49:05 2011 +0100 (2011-11-14 ago)
changeset 45498 2dc373f1867a
parent 45051 c478d1876371
child 47108 2a1953f0d20d
permissions -rw-r--r--
avoid numeral-representation-specific rules in metis proof
     1 (*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
     2     Author:     Robert Himmelmann, TU Muenchen
     3     Author:     Bogdan Grechuk, University of Edinburgh
     4 *)
     5 
     6 header {* Convex sets, functions and related things. *}
     7 
     8 theory Convex_Euclidean_Space
     9 imports
    10   Topology_Euclidean_Space
    11   "~~/src/HOL/Library/Convex"
    12   "~~/src/HOL/Library/Set_Algebras"
    13 begin
    14 
    15 
    16 (* ------------------------------------------------------------------------- *)
    17 (* To be moved elsewhere                                                     *)
    18 (* ------------------------------------------------------------------------- *)
    19 
    20 lemma linear_scaleR: "linear (\<lambda>x. scaleR c x)"
    21   by (simp add: linear_def scaleR_add_right)
    22 
    23 lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>(x::'a::real_vector). scaleR c x)"
    24   by (simp add: inj_on_def)
    25 
    26 lemma linear_add_cmul:
    27 assumes "linear f"
    28 shows "f(a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
    29 using linear_add[of f] linear_cmul[of f] assms by (simp) 
    30 
    31 lemma mem_convex_2:
    32   assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v=1"
    33   shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
    34   using assms convex_def[of S] by auto
    35 
    36 lemma mem_convex_alt:
    37   assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
    38   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
    39 apply (subst mem_convex_2) 
    40 using assms apply (auto simp add: algebra_simps zero_le_divide_iff)
    41 using add_divide_distrib[of u v "u+v"] by auto
    42 
    43 lemma inj_on_image_mem_iff: "inj_on f B ==> (A <= B) ==> (f a : f`A) ==> (a : B) ==> (a : A)" 
    44 by (blast dest: inj_onD)
    45 
    46 lemma independent_injective_on_span_image:
    47   assumes iS: "independent S" 
    48      and lf: "linear f" and fi: "inj_on f (span S)"
    49   shows "independent (f ` S)"
    50 proof-
    51   {fix a assume a: "a : S" "f a : span (f ` S - {f a})"
    52     have eq: "f ` S - {f a} = f ` (S - {a})" using fi a span_inc
    53       by (auto simp add: inj_on_def)
    54     from a have "f a : f ` span (S -{a})"
    55       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
    56     moreover have "span (S -{a}) <= span S" using span_mono[of "S-{a}" S] by auto
    57     ultimately have "a : span (S -{a})" using fi a span_inc by (auto simp add: inj_on_def)
    58     with a(1) iS  have False by (simp add: dependent_def) }
    59   then show ?thesis unfolding dependent_def by blast
    60 qed
    61 
    62 lemma dim_image_eq:
    63 fixes f :: "'n::euclidean_space => 'm::euclidean_space"
    64 assumes lf: "linear f" and fi: "inj_on f (span S)" 
    65 shows "dim (f ` S) = dim (S:: ('n::euclidean_space) set)"
    66 proof-
    67 obtain B where B_def: "B<=S & independent B & S <= span B & card B = dim S" 
    68   using basis_exists[of S] by auto
    69 hence "span S = span B" using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
    70 hence "independent (f ` B)" using independent_injective_on_span_image[of B f] B_def assms by auto
    71 moreover have "card (f ` B) = card B" using assms card_image[of f B] subset_inj_on[of f "span S" B] 
    72    B_def span_inc by auto
    73 moreover have "(f ` B) <= (f ` S)" using B_def by auto
    74 ultimately have "dim (f ` S) >= dim S" 
    75   using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
    76 from this show ?thesis using dim_image_le[of f S] assms by auto
    77 qed
    78 
    79 lemma linear_injective_on_subspace_0:
    80 assumes lf: "linear f" and "subspace S"
    81   shows "inj_on f S <-> (!x : S. f x = 0 --> x = 0)"
    82 proof-
    83   have "inj_on f S <-> (!x : S. !y : S. f x = f y --> x = y)" by (simp add: inj_on_def)
    84   also have "... <-> (!x : S. !y : S. f x - f y = 0 --> x - y = 0)" by simp
    85   also have "... <-> (!x : S. !y : S. f (x - y) = 0 --> x - y = 0)"
    86     by (simp add: linear_sub[OF lf])
    87   also have "... <-> (! x : S. f x = 0 --> x = 0)" 
    88     using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
    89   finally show ?thesis .
    90 qed
    91 
    92 lemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"
    93   unfolding subspace_def by auto 
    94 
    95 lemma span_eq[simp]: "(span s = s) <-> subspace s"
    96   unfolding span_def by (rule hull_eq, rule subspace_Inter)
    97 
    98 lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
    99   by(auto simp add: inj_on_def euclidean_eq[where 'a='n])
   100   
   101 lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
   102 proof-
   103   have eq: "?S = basis ` d" by blast
   104   show ?thesis unfolding eq apply(rule finite_subset[OF _ range_basis_finite]) by auto
   105 qed
   106 
   107 lemma card_substdbasis: assumes "d \<subseteq> {..<DIM('n::euclidean_space)}"
   108   shows "card {basis i ::'n::euclidean_space | i. i : d} = card d" (is "card ?S = _")
   109 proof-
   110   have eq: "?S = basis ` d" by blast
   111   show ?thesis unfolding eq using card_image[OF basis_inj_on[of d]] assms by auto
   112 qed
   113 
   114 lemma substdbasis_expansion_unique: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
   115   shows "setsum (%i. f i *\<^sub>R basis i) d = (x::'a::euclidean_space)
   116       <-> (!i<DIM('a). (i:d --> f i = x$$i) & (i ~: d --> x $$ i = 0))"
   117 proof- have *:"\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto
   118   have **:"finite d" apply(rule finite_subset[OF assms]) by fastforce
   119   have ***:"\<And>i. (setsum (%i. f i *\<^sub>R ((basis i)::'a)) d) $$ i = (\<Sum>x\<in>d. if x = i then f x else 0)"
   120     unfolding euclidean_component_setsum euclidean_component_scaleR basis_component *
   121     apply(rule setsum_cong2) using assms by auto
   122   show ?thesis unfolding euclidean_eq[where 'a='a] *** setsum_delta[OF **] using assms by auto
   123 qed
   124 
   125 lemma independent_substdbasis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
   126   shows "independent {basis i ::'a::euclidean_space |i. i : (d :: nat set)}" (is "independent ?A")
   127 proof -
   128   have *: "{basis i |i. i < DIM('a)} = basis ` {..<DIM('a)}" by auto
   129   show ?thesis
   130     apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
   131     using independent_basis[where 'a='a] assms by (auto simp: *)
   132 qed
   133 
   134 lemma dim_cball: 
   135 assumes "0<e"
   136 shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
   137 proof-
   138 { fix x :: "'n::euclidean_space" def y == "(e/norm x) *\<^sub>R x"
   139   hence "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto
   140   moreover have "x = (norm x/e) *\<^sub>R y" using y_def assms by simp
   141   moreover hence "x = (norm x/e) *\<^sub>R y"  by auto
   142   ultimately have "x : span (cball 0 e)"
   143      using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
   144 } hence "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto 
   145 from this show ?thesis using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
   146 qed
   147 
   148 lemma indep_card_eq_dim_span:
   149 fixes B :: "('n::euclidean_space) set"
   150 assumes "independent B"
   151 shows "finite B & card B = dim (span B)" 
   152   using assms basis_card_eq_dim[of B "span B"] span_inc by auto
   153 
   154 lemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"
   155   apply(rule ccontr) by auto
   156 
   157 lemma translate_inj_on: 
   158 fixes A :: "('a::ab_group_add) set"
   159 shows "inj_on (%x. a+x) A" unfolding inj_on_def by auto
   160 
   161 lemma translation_assoc:
   162   fixes a b :: "'a::ab_group_add"
   163   shows "(\<lambda>x. b+x) ` ((\<lambda>x. a+x) ` S) = (\<lambda>x. (a+b)+x) ` S" by auto
   164 
   165 lemma translation_invert:
   166   fixes a :: "'a::ab_group_add"
   167   assumes "(\<lambda>x. a+x) ` A = (\<lambda>x. a+x) ` B"
   168   shows "A=B"
   169 proof-
   170   have "(%x. -a+x) ` ((%x. a+x) ` A) = (%x. -a+x) ` ((%x. a+x) ` B)" using assms by auto
   171   from this show ?thesis using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto 
   172 qed
   173 
   174 lemma translation_galois:
   175   fixes a :: "'a::ab_group_add"
   176   shows "T=((\<lambda>x. a+x) ` S) <-> S=((\<lambda>x. (-a)+x) ` T)"
   177   using translation_assoc[of "-a" a S] apply auto
   178   using translation_assoc[of a "-a" T] by auto
   179 
   180 lemma translation_inverse_subset:
   181   assumes "((%x. -a+x) ` V) <= (S :: 'n::ab_group_add set)" 
   182   shows "V <= ((%x. a+x) ` S)"
   183 proof-
   184 { fix x assume "x:V" hence "x-a : S" using assms by auto 
   185   hence "x : {a + v |v. v : S}" apply auto apply (rule exI[of _ "x-a"]) apply simp done 
   186   hence "x : ((%x. a+x) ` S)" by auto }
   187   from this show ?thesis by auto
   188 qed
   189 
   190 lemma basis_to_basis_subspace_isomorphism:
   191   assumes s: "subspace (S:: ('n::euclidean_space) set)"
   192   and t: "subspace (T :: ('m::euclidean_space) set)"
   193   and d: "dim S = dim T"
   194   and B: "B <= S" "independent B" "S <= span B" "card B = dim S"
   195   and C: "C <= T" "independent C" "T <= span C" "card C = dim T"
   196   shows "EX f. linear f & f ` B = C & f ` S = T & inj_on f S"
   197 proof-
   198 (* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
   199 *)
   200   from B independent_bound have fB: "finite B" by blast
   201   from C independent_bound have fC: "finite C" by blast
   202   from B(4) C(4) card_le_inj[of B C] d obtain f where
   203     f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
   204   from linear_independent_extend[OF B(2)] obtain g where
   205     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
   206   from inj_on_iff_eq_card[OF fB, of f] f(2)
   207   have "card (f ` B) = card B" by simp
   208   with B(4) C(4) have ceq: "card (f ` B) = card C" using d
   209     by simp
   210   have "g ` B = f ` B" using g(2)
   211     by (auto simp add: image_iff)
   212   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
   213   finally have gBC: "g ` B = C" .
   214   have gi: "inj_on g B" using f(2) g(2)
   215     by (auto simp add: inj_on_def)
   216   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
   217   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
   218     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
   219     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
   220     have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
   221     have "x=y" using g0[OF th1 th0] by simp }
   222   then have giS: "inj_on g S"
   223     unfolding inj_on_def by blast
   224   from span_subspace[OF B(1,3) s]
   225   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
   226   also have "\<dots> = span C" unfolding gBC ..
   227   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
   228   finally have gS: "g ` S = T" .
   229   from g(1) gS giS gBC show ?thesis by blast
   230 qed
   231 
   232 lemma closure_bounded_linear_image:
   233   assumes f: "bounded_linear f"
   234   shows "f ` (closure S) \<subseteq> closure (f ` S)"
   235   using linear_continuous_on [OF f] closed_closure closure_subset
   236   by (rule image_closure_subset)
   237 
   238 lemma closure_linear_image:
   239 fixes f :: "('m::euclidean_space) => ('n::real_normed_vector)"
   240 assumes "linear f"
   241 shows "f ` (closure S) <= closure (f ` S)"
   242   using assms unfolding linear_conv_bounded_linear
   243   by (rule closure_bounded_linear_image)
   244 
   245 lemma closure_injective_linear_image:
   246 fixes f :: "('n::euclidean_space) => ('n::euclidean_space)"
   247 assumes "linear f" "inj f"
   248 shows "f ` (closure S) = closure (f ` S)"
   249 proof-
   250 obtain f' where f'_def: "linear f' & f o f' = id & f' o f = id" 
   251    using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
   252 hence "f' ` closure (f ` S) <= closure (S)"
   253    using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto
   254 hence "f ` f' ` closure (f ` S) <= f ` closure (S)" by auto
   255 hence "closure (f ` S) <= f ` closure (S)" using image_compose[of f f' "closure (f ` S)"] f'_def by auto
   256 from this show ?thesis using closure_linear_image[of f S] assms by auto 
   257 qed
   258 
   259 lemma closure_direct_sum:
   260 shows "closure (S <*> T) = closure S <*> closure T"
   261   by (rule closure_Times)
   262 
   263 lemma closure_scaleR:
   264   fixes S :: "('a::real_normed_vector) set"
   265   shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
   266 proof
   267   show "(op *\<^sub>R c) ` (closure S) \<subseteq> closure ((op *\<^sub>R c) ` S)"
   268     using bounded_linear_scaleR_right
   269     by (rule closure_bounded_linear_image)
   270   show "closure ((op *\<^sub>R c) ` S) \<subseteq> (op *\<^sub>R c) ` (closure S)"
   271     by (intro closure_minimal image_mono closure_subset
   272       closed_scaling closed_closure)
   273 qed
   274 
   275 lemma fst_linear: "linear fst" unfolding linear_def by (simp add: algebra_simps)
   276 
   277 lemma snd_linear: "linear snd" unfolding linear_def by (simp add: algebra_simps)
   278 
   279 lemma fst_snd_linear: "linear (%(x,y). x + y)" unfolding linear_def by (simp add: algebra_simps)
   280 
   281 lemma scaleR_2:
   282   fixes x :: "'a::real_vector"
   283   shows "scaleR 2 x = x + x"
   284 unfolding one_add_one_is_two [symmetric] scaleR_left_distrib by simp
   285 
   286 lemma vector_choose_size: "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
   287   apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"]) using DIM_positive[where 'a='a] by auto
   288 
   289 lemma setsum_delta_notmem: assumes "x\<notin>s"
   290   shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
   291         "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
   292         "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
   293         "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
   294   apply(rule_tac [!] setsum_cong2) using assms by auto
   295 
   296 lemma setsum_delta'':
   297   fixes s::"'a::real_vector set" assumes "finite s"
   298   shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
   299 proof-
   300   have *:"\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" by auto
   301   show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
   302 qed
   303 
   304 lemma if_smult:"(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" by auto
   305 
   306 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
   307   (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
   308   using image_affinity_interval[of m 0 a b] by auto
   309 
   310 lemma dist_triangle_eq:
   311   fixes x y z :: "'a::real_inner"
   312   shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
   313 proof- have *:"x - y + (y - z) = x - z" by auto
   314   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
   315     by(auto simp add:norm_minus_commute) qed
   316 
   317 lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
   318 
   319 lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
   320   unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
   321 
   322 lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
   323   unfolding norm_eq_sqrt_inner by simp
   324 
   325 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
   326   unfolding norm_eq_sqrt_inner by simp
   327 
   328 
   329 subsection {* Affine set and affine hull *}
   330 
   331 definition
   332   affine :: "'a::real_vector set \<Rightarrow> bool" where
   333   "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
   334 
   335 lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
   336 unfolding affine_def by(metis eq_diff_eq')
   337 
   338 lemma affine_empty[intro]: "affine {}"
   339   unfolding affine_def by auto
   340 
   341 lemma affine_sing[intro]: "affine {x}"
   342   unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
   343 
   344 lemma affine_UNIV[intro]: "affine UNIV"
   345   unfolding affine_def by auto
   346 
   347 lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
   348   unfolding affine_def by auto 
   349 
   350 lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
   351   unfolding affine_def by auto
   352 
   353 lemma affine_affine_hull: "affine(affine hull s)"
   354   unfolding hull_def using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"]
   355   by auto
   356 
   357 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   358 by (metis affine_affine_hull hull_same)
   359 
   360 subsubsection {* Some explicit formulations (from Lars Schewe) *}
   361 
   362 lemma affine: fixes V::"'a::real_vector set"
   363   shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
   364 unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
   365 defer apply(rule, rule, rule, rule, rule) proof-
   366   fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
   367     "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
   368   thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
   369     using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]] and as(1-3) 
   370     by(auto simp add: scaleR_left_distrib[THEN sym])
   371 next
   372   fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
   373     "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
   374   def n \<equiv> "card s"
   375   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   376   thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
   377     assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
   378     then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
   379     thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
   380       by(auto simp add: setsum_clauses(2))
   381   next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
   382       case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
   383       assume IA:"\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
   384                s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
   385         as:"Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
   386            "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
   387       have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
   388         assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
   389         thus False using as(7) and `card s > 2` by (metis One_nat_def
   390           less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
   391       qed
   392       then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
   393 
   394       have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
   395       have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
   396       have **:"setsum u (s - {x}) = 1 - u x"
   397         using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
   398       have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
   399       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
   400         case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
   401           assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
   402           thus False using True by auto qed auto
   403         thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
   404         unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
   405       next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
   406         then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
   407         thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
   408           using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
   409       hence "u x + (1 - u x) = 1 \<Longrightarrow> u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa) /\<^sub>R (1 - u x)) \<in> V"
   410         apply-apply(rule as(3)[rule_format]) 
   411         unfolding  RealVector.scaleR_right.setsum using x(1) as(6) by auto
   412       thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
   413          apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
   414          using `u x \<noteq> 1` by auto 
   415     qed auto
   416   next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
   417     thus ?thesis using as(4,5) by simp
   418   qed(insert `s\<noteq>{}` `finite s`, auto)
   419 qed
   420 
   421 lemma affine_hull_explicit:
   422   "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
   423   apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq
   424   apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
   425   fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   426     apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
   427 next
   428   fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
   429   thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
   430 next
   431   show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}" unfolding affine_def
   432     apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
   433     fix u v ::real assume uv:"u + v = 1"
   434     fix x assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   435     then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
   436     fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
   437     then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
   438     have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
   439     have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
   440     show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
   441       apply(rule_tac x="sx \<union> sy" in exI)
   442       apply(rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>sy then v * uy a else 0)" in exI)
   443       unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
   444       unfolding scaleR_scaleR[THEN sym] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
   445       unfolding x y using x(1-3) y(1-3) uv by simp qed qed
   446 
   447 lemma affine_hull_finite:
   448   assumes "finite s"
   449   shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
   450   unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq apply (rule,rule)
   451   apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
   452   fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   453   thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
   454     apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
   455 next
   456   fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
   457   assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
   458   thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   459     unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
   460 
   461 subsubsection {* Stepping theorems and hence small special cases *}
   462 
   463 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   464   apply(rule hull_unique) by auto
   465 
   466 lemma affine_hull_finite_step:
   467   fixes y :: "'a::real_vector"
   468   shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
   469   "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
   470                 (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
   471 proof-
   472   show ?th1 by simp
   473   assume ?as 
   474   { assume ?lhs
   475     then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
   476     have ?rhs proof(cases "a\<in>s")
   477       case True hence *:"insert a s = s" by auto
   478       show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
   479     next
   480       case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
   481     qed  } moreover
   482   { assume ?rhs
   483     then obtain v u where vu:"setsum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
   484     have *:"\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" by auto
   485     have ?lhs proof(cases "a\<in>s")
   486       case True thus ?thesis
   487         apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
   488         unfolding setsum_clauses(2)[OF `?as`]  apply simp
   489         unfolding scaleR_left_distrib and setsum_addf 
   490         unfolding vu and * and scaleR_zero_left
   491         by (auto simp add: setsum_delta[OF `?as`])
   492     next
   493       case False 
   494       hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
   495                "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
   496       from False show ?thesis
   497         apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
   498         unfolding setsum_clauses(2)[OF `?as`] and * using vu
   499         using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
   500         using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
   501     qed }
   502   ultimately show "?lhs = ?rhs" by blast
   503 qed
   504 
   505 lemma affine_hull_2:
   506   fixes a b :: "'a::real_vector"
   507   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
   508 proof-
   509   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   510          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   511   have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
   512     using affine_hull_finite[of "{a,b}"] by auto
   513   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
   514     by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
   515   also have "\<dots> = ?rhs" unfolding * by auto
   516   finally show ?thesis by auto
   517 qed
   518 
   519 lemma affine_hull_3:
   520   fixes a b c :: "'a::real_vector"
   521   shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
   522 proof-
   523   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   524          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   525   show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
   526     unfolding * apply auto
   527     apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
   528     apply(rule_tac x=u in exI) by force
   529 qed
   530 
   531 lemma mem_affine:
   532   assumes "affine S" "x : S" "y : S" "u+v=1"
   533   shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
   534   using assms affine_def[of S] by auto
   535 
   536 lemma mem_affine_3:
   537   assumes "affine S" "x : S" "y : S" "z : S" "u+v+w=1"
   538   shows "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : S"
   539 proof-
   540 have "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : affine hull {x, y, z}"
   541   using affine_hull_3[of x y z] assms by auto
   542 moreover have " affine hull {x, y, z} <= affine hull S" 
   543   using hull_mono[of "{x, y, z}" "S"] assms by auto
   544 moreover have "affine hull S = S" 
   545   using assms affine_hull_eq[of S] by auto
   546 ultimately show ?thesis by auto 
   547 qed
   548 
   549 lemma mem_affine_3_minus:
   550   assumes "affine S" "x : S" "y : S" "z : S"
   551   shows "x + v *\<^sub>R (y-z) : S"
   552 using mem_affine_3[of S x y z 1 v "-v"] assms by (simp add: algebra_simps)
   553 
   554 
   555 subsubsection {* Some relations between affine hull and subspaces *}
   556 
   557 lemma affine_hull_insert_subset_span:
   558   shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
   559   unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
   560   apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
   561   fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
   562   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
   563   thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
   564     apply(rule_tac x="x - a" in exI)
   565     apply (rule conjI, simp)
   566     apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
   567     apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
   568     apply (rule conjI) using as(1) apply simp
   569     apply (erule conjI)
   570     using as(1)
   571     apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib setsum_subtractf scaleR_left.setsum[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
   572     unfolding as by simp qed
   573 
   574 lemma affine_hull_insert_span:
   575   assumes "a \<notin> s"
   576   shows "affine hull (insert a s) =
   577             {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
   578   apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
   579   unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
   580   fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
   581   then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y" unfolding span_explicit by auto
   582   def f \<equiv> "(\<lambda>x. x + a) ` t"
   583   have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt 
   584     by(auto simp add: setsum_reindex[unfolded inj_on_def])
   585   have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
   586   show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
   587     apply(rule_tac x="insert a f" in exI)
   588     apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
   589     using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
   590     unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
   591     by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
   592 
   593 lemma affine_hull_span:
   594   assumes "a \<in> s"
   595   shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
   596   using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
   597 
   598 subsubsection {* Parallel affine sets *}
   599 
   600 definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
   601 where "affine_parallel S T = (? a. T = ((%x. a + x) ` S))"
   602 
   603 lemma affine_parallel_expl_aux:
   604    fixes S T :: "'a::real_vector set"
   605    assumes "!x. (x : S <-> (a+x) : T)" 
   606    shows "T = ((%x. a + x) ` S)"
   607 proof-
   608 { fix x assume "x : T" hence "(-a)+x : S" using assms by auto
   609   hence " x : ((%x. a + x) ` S)" using imageI[of "-a+x" S "(%x. a+x)"] by auto}
   610 moreover have "T >= ((%x. a + x) ` S)" using assms by auto 
   611 ultimately show ?thesis by auto
   612 qed
   613 
   614 lemma affine_parallel_expl: 
   615    "affine_parallel S T = (? a. !x. (x : S <-> (a+x) : T))" 
   616    unfolding affine_parallel_def using affine_parallel_expl_aux[of S _ T] by auto
   617 
   618 lemma affine_parallel_reflex: "affine_parallel S S" unfolding affine_parallel_def apply (rule exI[of _ "0"]) by auto
   619 
   620 lemma affine_parallel_commut:
   621 assumes "affine_parallel A B" shows "affine_parallel B A" 
   622 proof-
   623 from assms obtain a where "B=((%x. a + x) ` A)" unfolding affine_parallel_def by auto 
   624 from this show ?thesis using translation_galois[of B a A] unfolding affine_parallel_def by auto
   625 qed
   626 
   627 lemma affine_parallel_assoc:
   628 assumes "affine_parallel A B" "affine_parallel B C"
   629 shows "affine_parallel A C" 
   630 proof-
   631 from assms obtain ab where "B=((%x. ab + x) ` A)" unfolding affine_parallel_def by auto 
   632 moreover 
   633 from assms obtain bc where "C=((%x. bc + x) ` B)" unfolding affine_parallel_def by auto
   634 ultimately show ?thesis using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto 
   635 qed
   636 
   637 lemma affine_translation_aux:
   638   fixes a :: "'a::real_vector"
   639   assumes "affine ((%x. a + x) ` S)" shows "affine S"
   640 proof-
   641 { fix x y u v
   642   assume xy: "x : S" "y : S" "(u :: real)+v=1"
   643   hence "(a+x):((%x. a + x) ` S)" "(a+y):((%x. a + x) ` S)" by auto
   644   hence h1: "u *\<^sub>R  (a+x) + v *\<^sub>R (a+y) : ((%x. a + x) ` S)" using xy assms unfolding affine_def by auto
   645   have "u *\<^sub>R (a+x) + v *\<^sub>R (a+y) = (u+v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)" by (simp add:algebra_simps)
   646   also have "...= a + (u *\<^sub>R x + v *\<^sub>R y)" using `u+v=1` by auto
   647   ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) : ((%x. a + x) ` S)" using h1 by auto
   648   hence "u *\<^sub>R x + v *\<^sub>R y : S" by auto
   649 } from this show ?thesis unfolding affine_def by auto
   650 qed
   651 
   652 lemma affine_translation:
   653   fixes a :: "'a::real_vector"
   654   shows "affine S <-> affine ((%x. a + x) ` S)"
   655 proof-
   656 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
   657 from this show ?thesis using affine_translation_aux by auto
   658 qed
   659 
   660 lemma parallel_is_affine:
   661 fixes S T :: "'a::real_vector set"
   662 assumes "affine S" "affine_parallel S T"
   663 shows "affine T"
   664 proof-
   665   from assms obtain a where "T=((%x. a + x) ` S)" unfolding affine_parallel_def by auto 
   666   from this show ?thesis using affine_translation assms by auto
   667 qed
   668 
   669 lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
   670   unfolding subspace_def affine_def by auto
   671 
   672 subsubsection {* Subspace parallel to an affine set *}
   673 
   674 lemma subspace_affine:
   675   shows "subspace S <-> (affine S & 0 : S)"
   676 proof-
   677 have h0: "subspace S ==> (affine S & 0 : S)" using subspace_imp_affine[of S] subspace_0 by auto
   678 { assume assm: "affine S & 0 : S"
   679   { fix c :: real 
   680     fix x assume x_def: "x : S"
   681     have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
   682     moreover have "(1-c) *\<^sub>R 0 + c *\<^sub>R x : S" using affine_alt[of S] assm x_def by auto
   683     ultimately have "c *\<^sub>R x : S" by auto
   684   } hence h1: "!c. !x : S. c *\<^sub>R x : S" by auto
   685   { fix x y assume xy_def: "x : S" "y : S"
   686     def u == "(1 :: real)/2"
   687     have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)" by auto
   688     moreover have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y" by (simp add: algebra_simps)
   689     moreover have "(1-u) *\<^sub>R x + u *\<^sub>R y : S" using affine_alt[of S] assm xy_def by auto
   690     ultimately have "(1/2) *\<^sub>R (x+y) : S" using u_def by auto
   691     moreover have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" by auto
   692     ultimately have "(x+y) : S" using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
   693   } hence "!x : S. !y : S. (x+y) : S" by auto 
   694   hence "subspace S" using h1 assm unfolding subspace_def by auto
   695 } from this show ?thesis using h0 by metis
   696 qed
   697 
   698 lemma affine_diffs_subspace:
   699   assumes "affine S" "a : S"
   700   shows "subspace ((%x. (-a)+x) ` S)"
   701 proof-
   702 have "affine ((%x. (-a)+x) ` S)" using  affine_translation assms by auto  
   703 moreover have "0 : ((%x. (-a)+x) ` S)" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
   704 ultimately show ?thesis using subspace_affine by auto 
   705 qed
   706 
   707 lemma parallel_subspace_explicit:
   708 assumes "affine S" "a : S"
   709 assumes "L == {y. ? x : S. (-a)+x=y}" 
   710 shows "subspace L & affine_parallel S L" 
   711 proof-
   712 have par: "affine_parallel S L" unfolding affine_parallel_def using assms by auto
   713 hence "affine L" using assms parallel_is_affine by auto  
   714 moreover have "0 : L" using assms apply auto using exI[of "(%x. x:S & -a+x=0)" a] by auto
   715 ultimately show ?thesis using subspace_affine par by auto 
   716 qed
   717 
   718 lemma parallel_subspace_aux:
   719 assumes "subspace A" "subspace B" "affine_parallel A B"
   720 shows "A>=B"
   721 proof-
   722 from assms obtain a where a_def: "!x. (x : A <-> (a+x) : B)" using affine_parallel_expl[of A B] by auto
   723 hence "-a : A" using assms subspace_0[of B] by auto
   724 hence "a : A" using assms subspace_neg[of A "-a"] by auto
   725 from this show ?thesis using assms a_def unfolding subspace_def by auto
   726 qed
   727 
   728 lemma parallel_subspace:
   729 assumes "subspace A" "subspace B" "affine_parallel A B"
   730 shows "A=B"
   731 proof-
   732 have "A>=B" using assms parallel_subspace_aux by auto
   733 moreover have "A<=B" using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
   734 ultimately show ?thesis by auto  
   735 qed
   736 
   737 lemma affine_parallel_subspace:
   738 assumes "affine S" "S ~= {}"
   739 shows "?!L. subspace L & affine_parallel S L" 
   740 proof-
   741 have ex: "? L. subspace L & affine_parallel S L" using assms  parallel_subspace_explicit by auto 
   742 { fix L1 L2 assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"
   743   hence "affine_parallel L1 L2" using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
   744   hence "L1=L2" using ass parallel_subspace by auto
   745 } from this show ?thesis using ex by auto
   746 qed
   747 
   748 subsection {* Cones *}
   749 
   750 definition
   751   cone :: "'a::real_vector set \<Rightarrow> bool" where
   752   "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   753 
   754 lemma cone_empty[intro, simp]: "cone {}"
   755   unfolding cone_def by auto
   756 
   757 lemma cone_univ[intro, simp]: "cone UNIV"
   758   unfolding cone_def by auto
   759 
   760 lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
   761   unfolding cone_def by auto
   762 
   763 subsubsection {* Conic hull *}
   764 
   765 lemma cone_cone_hull: "cone (cone hull s)"
   766   unfolding hull_def by auto
   767 
   768 lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
   769   apply(rule hull_eq)
   770   using cone_Inter unfolding subset_eq by auto
   771 
   772 lemma mem_cone:
   773   assumes "cone S" "x : S" "c>=0"
   774   shows "c *\<^sub>R x : S"
   775   using assms cone_def[of S] by auto
   776 
   777 lemma cone_contains_0:
   778 assumes "cone S"
   779 shows "(S ~= {}) <-> (0 : S)"
   780 proof-
   781 { assume "S ~= {}" from this obtain a where "a:S" by auto
   782   hence "0 : S" using assms mem_cone[of S a 0] by auto
   783 } from this show ?thesis by auto
   784 qed
   785 
   786 lemma cone_0: "cone {0}"
   787 unfolding cone_def by auto
   788 
   789 lemma cone_Union[intro]: "(!s:f. cone s) --> (cone (Union f))"
   790   unfolding cone_def by blast
   791 
   792 lemma cone_iff:
   793 assumes "S ~= {}"
   794 shows "cone S <-> 0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
   795 proof-
   796 { assume "cone S"
   797   { fix c assume "(c :: real)>0"
   798     { fix x assume "x : S" hence "x : (op *\<^sub>R c) ` S" unfolding image_def
   799         using `cone S` `c>0` mem_cone[of S x "1/c"]
   800         exI[of "(%t. t:S & x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"] by auto
   801     }
   802     moreover
   803     { fix x assume "x : (op *\<^sub>R c) ` S"
   804       (*from this obtain t where "t:S & x = c *\<^sub>R t" by auto*)
   805       hence "x:S" using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto
   806     }
   807     ultimately have "((op *\<^sub>R c) ` S) = S" by auto
   808   } hence "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)" using `cone S` cone_contains_0[of S] assms by auto
   809 }
   810 moreover
   811 { assume a: "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
   812   { fix x assume "x:S"
   813     fix c1 assume "(c1 :: real)>=0"
   814     hence "(c1=0) | (c1>0)" by auto
   815     hence "c1 *\<^sub>R x : S" using a `x:S` by auto
   816   }
   817  hence "cone S" unfolding cone_def by auto
   818 } ultimately show ?thesis by blast
   819 qed
   820 
   821 lemma cone_hull_empty:
   822 "cone hull {} = {}"
   823 by (metis cone_empty cone_hull_eq)
   824 
   825 lemma cone_hull_empty_iff:
   826 shows "(S = {}) <-> (cone hull S = {})"
   827 by (metis bot_least cone_hull_empty hull_subset xtrans(5))
   828 
   829 lemma cone_hull_contains_0: 
   830 shows "(S ~= {}) <-> (0 : cone hull S)"
   831 using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] by auto
   832 
   833 lemma mem_cone_hull:
   834   assumes "x : S" "c>=0"
   835   shows "c *\<^sub>R x : cone hull S"
   836 by (metis assms cone_cone_hull hull_inc mem_cone)
   837 
   838 lemma cone_hull_expl:
   839 shows "cone hull S = {c *\<^sub>R x | c x. c>=0 & x : S}" (is "?lhs = ?rhs")
   840 proof-
   841 { fix x assume "x : ?rhs"
   842   from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
   843   fix c assume c_def: "(c :: real)>=0"
   844   hence "c *\<^sub>R x = (c*cx) *\<^sub>R xx" using x_def by (simp add: algebra_simps)
   845   moreover have "(c*cx) >= 0" using c_def x_def using mult_nonneg_nonneg by auto
   846   ultimately have "c *\<^sub>R x : ?rhs" using x_def by auto
   847 } hence "cone ?rhs" unfolding cone_def by auto
   848   hence "?rhs : Collect cone" unfolding mem_Collect_eq by auto
   849 { fix x assume "x : S" hence "1 *\<^sub>R x : ?rhs" apply auto apply(rule_tac x="1" in exI) by auto
   850   hence "x : ?rhs" by auto
   851 } hence "S <= ?rhs" by auto
   852 hence "?lhs <= ?rhs" using `?rhs : Collect cone` hull_minimal[of S "?rhs" "cone"] by auto
   853 moreover
   854 { fix x assume "x : ?rhs"
   855   from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
   856   hence "xx : cone hull S" using hull_subset[of S] by auto
   857   hence "x : ?lhs" using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
   858 } ultimately show ?thesis by auto
   859 qed
   860 
   861 lemma cone_closure:
   862   fixes S :: "('a::real_normed_vector) set"
   863   assumes "cone S" shows "cone (closure S)"
   864 proof-
   865 { assume "S = {}" hence ?thesis by auto }
   866 moreover
   867 { assume "S ~= {}" hence "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
   868   hence "0:(closure S) & (!c. c>0 --> op *\<^sub>R c ` (closure S) = (closure S))"
   869      using closure_subset by (auto simp add: closure_scaleR)
   870   hence ?thesis using cone_iff[of "closure S"] by auto
   871 }
   872 ultimately show ?thesis by blast
   873 qed
   874 
   875 subsection {* Affine dependence and consequential theorems (from Lars Schewe) *}
   876 
   877 definition
   878   affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
   879   "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
   880 
   881 lemma affine_dependent_explicit:
   882   "affine_dependent p \<longleftrightarrow>
   883     (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
   884     (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   885   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
   886   apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
   887 proof-
   888   fix x s u assume as:"x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   889   have "x\<notin>s" using as(1,4) by auto
   890   show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
   891     apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
   892     unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
   893 next
   894   fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
   895   have "s \<noteq> {v}" using as(3,6) by auto
   896   thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
   897     apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
   898     unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
   899 qed
   900 
   901 lemma affine_dependent_explicit_finite:
   902   fixes s :: "'a::real_vector set" assumes "finite s"
   903   shows "affine_dependent s \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   904   (is "?lhs = ?rhs")
   905 proof
   906   have *:"\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::'a))" by auto
   907   assume ?lhs
   908   then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
   909     unfolding affine_dependent_explicit by auto
   910   thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   911     apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
   912     unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
   913 next
   914   assume ?rhs
   915   then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
   916   thus ?lhs unfolding affine_dependent_explicit using assms by auto
   917 qed
   918 
   919 subsection {* Connectedness of convex sets *}
   920 
   921 lemma connected_real_lemma:
   922   fixes f :: "real \<Rightarrow> 'a::metric_space"
   923   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
   924   and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
   925   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
   926   and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
   927   and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
   928   shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
   929 proof-
   930   let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
   931   have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
   932   have Sub: "\<exists>y. isUb UNIV ?S y"
   933     apply (rule exI[where x= b])
   934     using ab fb e12 by (auto simp add: isUb_def setle_def)
   935   from reals_complete[OF Se Sub] obtain l where
   936     l: "isLub UNIV ?S l"by blast
   937   have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
   938     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   939     by (metis linorder_linear)
   940   have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
   941     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   942     by (metis linorder_linear not_le)
   943     have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
   944     have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
   945     have "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
   946     then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
   947     {assume le2: "f l \<in> e2"
   948       from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
   949       hence lap: "l - a > 0" using alb by arith
   950       from e2[rule_format, OF le2] obtain e where
   951         e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
   952       from dst[OF alb e(1)] obtain d where
   953         d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   954       let ?d' = "min (d/2) ((l - a)/2)"
   955       have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
   956         by (simp add: min_max.less_infI2)
   957       then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
   958       then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
   959       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
   960       from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
   961       moreover
   962       have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
   963       ultimately have False using e12 alb d' by auto}
   964     moreover
   965     {assume le1: "f l \<in> e1"
   966     from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
   967       hence blp: "b - l > 0" using alb by arith
   968       from e1[rule_format, OF le1] obtain e where
   969         e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
   970       from dst[OF alb e(1)] obtain d where
   971         d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   972       have "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
   973       then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
   974       then obtain d' where d': "d' > 0" "d' < d" by metis
   975       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
   976       hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
   977       with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
   978       with l d' have False
   979         by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
   980     ultimately show ?thesis using alb by metis
   981 qed
   982 
   983 lemma convex_connected:
   984   fixes s :: "'a::real_normed_vector set"
   985   assumes "convex s" shows "connected s"
   986 proof-
   987   { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
   988     assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
   989     then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
   990     hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
   991 
   992     { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
   993       { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
   994           by (simp add: algebra_simps)
   995         assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
   996         hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
   997           unfolding * and scaleR_right_diff_distrib[THEN sym]
   998           unfolding less_divide_eq using n by auto  }
   999       hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
  1000         apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
  1001         apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
  1002 
  1003     have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
  1004       apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
  1005       using * apply(simp add: dist_norm)
  1006       using as(1,2)[unfolded open_dist] apply simp
  1007       using as(1,2)[unfolded open_dist] apply simp
  1008       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
  1009       using as(3) by auto
  1010     then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
  1011     hence False using as(4) 
  1012       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
  1013       using x1(2) x2(2) by auto  }
  1014   thus ?thesis unfolding connected_def by auto
  1015 qed
  1016 
  1017 text {* One rather trivial consequence. *}
  1018 
  1019 lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
  1020   by(simp add: convex_connected convex_UNIV)
  1021 
  1022 text {* Balls, being convex, are connected. *}
  1023 
  1024 lemma convex_box: fixes a::"'a::euclidean_space"
  1025   assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
  1026   shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
  1027   using assms unfolding convex_def by auto
  1028 
  1029 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
  1030   by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
  1031 
  1032 lemma convex_local_global_minimum:
  1033   fixes s :: "'a::real_normed_vector set"
  1034   assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
  1035   shows "\<forall>y\<in>s. f x \<le> f y"
  1036 proof(rule ccontr)
  1037   have "x\<in>s" using assms(1,3) by auto
  1038   assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
  1039   then obtain y where "y\<in>s" and y:"f x > f y" by auto
  1040   hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
  1041 
  1042   then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
  1043     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
  1044   hence "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
  1045     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
  1046   moreover
  1047   have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
  1048   have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
  1049     using u unfolding pos_less_divide_eq[OF xy] by auto
  1050   hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
  1051   ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
  1052 qed
  1053 
  1054 lemma convex_ball:
  1055   fixes x :: "'a::real_normed_vector"
  1056   shows "convex (ball x e)" 
  1057 proof(auto simp add: convex_def)
  1058   fix y z assume yz:"dist x y < e" "dist x z < e"
  1059   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
  1060   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
  1061     using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
  1062   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using convex_bound_lt[OF yz uv] by auto
  1063 qed
  1064 
  1065 lemma convex_cball:
  1066   fixes x :: "'a::real_normed_vector"
  1067   shows "convex(cball x e)"
  1068 proof(auto simp add: convex_def Ball_def)
  1069   fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
  1070   fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
  1071   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
  1072     using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
  1073   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto 
  1074 qed
  1075 
  1076 lemma connected_ball:
  1077   fixes x :: "'a::real_normed_vector"
  1078   shows "connected (ball x e)"
  1079   using convex_connected convex_ball by auto
  1080 
  1081 lemma connected_cball:
  1082   fixes x :: "'a::real_normed_vector"
  1083   shows "connected(cball x e)"
  1084   using convex_connected convex_cball by auto
  1085 
  1086 subsection {* Convex hull *}
  1087 
  1088 lemma convex_convex_hull: "convex(convex hull s)"
  1089   unfolding hull_def using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
  1090   by auto
  1091 
  1092 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
  1093 by (metis convex_convex_hull hull_same)
  1094 
  1095 lemma bounded_convex_hull:
  1096   fixes s :: "'a::real_normed_vector set"
  1097   assumes "bounded s" shows "bounded(convex hull s)"
  1098 proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
  1099   show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
  1100     unfolding subset_hull[of convex, OF convex_cball]
  1101     unfolding subset_eq mem_cball dist_norm using B by auto qed
  1102 
  1103 lemma finite_imp_bounded_convex_hull:
  1104   fixes s :: "'a::real_normed_vector set"
  1105   shows "finite s \<Longrightarrow> bounded(convex hull s)"
  1106   using bounded_convex_hull finite_imp_bounded by auto
  1107 
  1108 subsubsection {* Convex hull is "preserved" by a linear function *}
  1109 
  1110 lemma convex_hull_linear_image:
  1111   assumes "bounded_linear f"
  1112   shows "f ` (convex hull s) = convex hull (f ` s)"
  1113   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  1114   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  1115   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  1116 proof-
  1117   interpret f: bounded_linear f by fact
  1118   show "convex {x. f x \<in> convex hull f ` s}" 
  1119   unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
  1120   interpret f: bounded_linear f by fact
  1121   show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
  1122     unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  1123 qed auto
  1124 
  1125 lemma in_convex_hull_linear_image:
  1126   assumes "bounded_linear f" "x \<in> convex hull s"
  1127   shows "(f x) \<in> convex hull (f ` s)"
  1128 using convex_hull_linear_image[OF assms(1)] assms(2) by auto
  1129 
  1130 subsubsection {* Stepping theorems for convex hulls of finite sets *}
  1131 
  1132 lemma convex_hull_empty[simp]: "convex hull {} = {}"
  1133   apply(rule hull_unique) by auto
  1134 
  1135 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
  1136   apply(rule hull_unique) by auto
  1137 
  1138 lemma convex_hull_insert:
  1139   fixes s :: "'a::real_vector set"
  1140   assumes "s \<noteq> {}"
  1141   shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
  1142                                     b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
  1143  apply(rule,rule hull_minimal,rule) unfolding insert_iff prefer 3 apply rule proof-
  1144  fix x assume x:"x = a \<or> x \<in> s"
  1145  thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
  1146    apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
  1147 next
  1148   fix x assume "x\<in>?hull"
  1149   then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
  1150   have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
  1151     using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
  1152   thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
  1153     apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
  1154 next
  1155   show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
  1156     fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
  1157     from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
  1158     from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
  1159     have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
  1160     have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
  1161     proof(cases "u * v1 + v * v2 = 0")
  1162       have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
  1163       case True hence **:"u * v1 = 0" "v * v2 = 0"
  1164         using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by arith+
  1165       hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
  1166       thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
  1167     next
  1168       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)
  1169       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
  1170       also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
  1171       case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
  1172         apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
  1173         using as(1,2) obt1(1,2) obt2(1,2) by auto 
  1174       thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
  1175         apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
  1176         apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
  1177         unfolding add_divide_distrib[THEN sym] and zero_le_divide_iff
  1178         by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
  1179     qed note * = this
  1180     have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
  1181     have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
  1182     have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
  1183       apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
  1184     also have "\<dots> \<le> 1" unfolding right_distrib[THEN sym] and as(3) using u1 u2 by auto
  1185     finally 
  1186     show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
  1187       apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
  1188       using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
  1189   qed
  1190 qed
  1191 
  1192 
  1193 subsubsection {* Explicit expression for convex hull *}
  1194 
  1195 lemma convex_hull_indexed:
  1196   fixes s :: "'a::real_vector set"
  1197   shows "convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
  1198                             (setsum u {1..k} = 1) \<and>
  1199                             (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
  1200   apply(rule hull_unique) apply(rule) defer
  1201   apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
  1202 proof-
  1203   fix x assume "x\<in>s"
  1204   thus "x \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
  1205 next
  1206   fix t assume as:"s \<subseteq> t" "convex t"
  1207   show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
  1208     fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
  1209     show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
  1210       using assm(1,2) as(1) by auto qed
  1211 next
  1212   fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and xy:"x\<in>?hull" "y\<in>?hull"
  1213   from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" by auto
  1214   from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" by auto
  1215   have *:"\<And>P (x1::'a) x2 s1 s2 i.(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
  1216     "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
  1217     prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
  1218   have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
  1219   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
  1220     apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
  1221     apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
  1222     unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
  1223     unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
  1224     fix i assume i:"i \<in> {1..k1+k2}"
  1225     show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and> (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
  1226     proof(cases "i\<in>{1..k1}")
  1227       case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
  1228     next def j \<equiv> "i - k1"
  1229       case False with i have "j \<in> {1..k2}" unfolding j_def by auto
  1230       thus ?thesis unfolding j_def[symmetric] using False
  1231         using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
  1232   qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
  1233 qed
  1234 
  1235 lemma convex_hull_finite:
  1236   fixes s :: "'a::real_vector set"
  1237   assumes "finite s"
  1238   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
  1239          setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
  1240 proof(rule hull_unique, auto simp add: convex_def[of ?set])
  1241   fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x" 
  1242     apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
  1243     unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
  1244 next
  1245   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
  1246   fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
  1247   fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
  1248   { fix x assume "x\<in>s"
  1249     hence "0 \<le> 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)
  1250       by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
  1251   moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
  1252     unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
  1253   moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
  1254     unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
  1255   ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
  1256     apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
  1257 next
  1258   fix t assume t:"s \<subseteq> t" "convex t" 
  1259   fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
  1260   thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
  1261     using assms and t(1) by auto
  1262 qed
  1263 
  1264 subsubsection {* Another formulation from Lars Schewe *}
  1265 
  1266 lemma setsum_constant_scaleR:
  1267   fixes y :: "'a::real_vector"
  1268   shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
  1269 apply (cases "finite A")
  1270 apply (induct set: finite)
  1271 apply (simp_all add: algebra_simps)
  1272 done
  1273 
  1274 lemma convex_hull_explicit:
  1275   fixes p :: "'a::real_vector set"
  1276   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
  1277              (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
  1278 proof-
  1279   { fix x assume "x\<in>?lhs"
  1280     then obtain k u y where obt:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
  1281       unfolding convex_hull_indexed by auto
  1282 
  1283     have fin:"finite {1..k}" by auto
  1284     have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
  1285     { fix j assume "j\<in>{1..k}"
  1286       hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
  1287         using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
  1288         apply(rule setsum_nonneg) using obt(1) by auto } 
  1289     moreover
  1290     have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
  1291       unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
  1292     moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
  1293       using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
  1294       unfolding scaleR_left.setsum using obt(3) by auto
  1295     ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
  1296       apply(rule_tac x="y ` {1..k}" in exI)
  1297       apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
  1298     hence "x\<in>?rhs" by auto  }
  1299   moreover
  1300   { fix y assume "y\<in>?rhs"
  1301     then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
  1302 
  1303     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
  1304     
  1305     { fix i::nat assume "i\<in>{1..card s}"
  1306       hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
  1307       hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
  1308     moreover have *:"finite {1..card s}" by auto
  1309     { fix y assume "y\<in>s"
  1310       then obtain i where "i\<in>{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"] by auto
  1311       hence "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}" apply auto using f(1)[unfolded inj_on_def] apply(erule_tac x=x in ballE) by auto
  1312       hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
  1313       hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
  1314             "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
  1315         by (auto simp add: setsum_constant_scaleR)   }
  1316 
  1317     hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
  1318       unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
  1319       unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
  1320       using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
  1321     
  1322     ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
  1323       apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastforce
  1324     hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
  1325   ultimately show ?thesis unfolding set_eq_iff by blast
  1326 qed
  1327 
  1328 subsubsection {* A stepping theorem for that expansion *}
  1329 
  1330 lemma convex_hull_finite_step:
  1331   fixes s :: "'a::real_vector set" assumes "finite s"
  1332   shows "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
  1333      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?lhs = ?rhs")
  1334 proof(rule, case_tac[!] "a\<in>s")
  1335   assume "a\<in>s" hence *:"insert a s = s" by auto
  1336   assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
  1337 next
  1338   assume ?lhs then obtain u where u:"\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
  1339   assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
  1340     apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
  1341 next
  1342   assume "a\<in>s" hence *:"insert a s = s" by auto
  1343   have fin:"finite (insert a s)" using assms by auto
  1344   assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
  1345   show ?lhs apply(rule_tac x="\<lambda>x. (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]
  1346     unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
  1347 next
  1348   assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
  1349   moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
  1350     apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
  1351   ultimately show ?lhs apply(rule_tac x="\<lambda>x. if a = x then v else u x" in exI)  unfolding setsum_clauses(2)[OF assms] by auto
  1352 qed
  1353 
  1354 subsubsection {* Hence some special cases *}
  1355 
  1356 lemma convex_hull_2:
  1357   "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
  1358 proof- have *:"\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" by auto have **:"finite {b}" by auto
  1359 show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
  1360   apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
  1361   apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
  1362 
  1363 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
  1364   unfolding convex_hull_2
  1365 proof(rule Collect_cong) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
  1366   fix x show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
  1367     unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
  1368 
  1369 lemma convex_hull_3:
  1370   "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
  1371 proof-
  1372   have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
  1373   have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
  1374     by (auto simp add: field_simps)
  1375   show ?thesis unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
  1376     unfolding convex_hull_finite_step[OF fin(3)] apply(rule Collect_cong) apply simp apply auto
  1377     apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
  1378     apply(rule_tac x="1 - v - w" in exI) apply simp apply(rule_tac x=v in exI) apply simp apply(rule_tac x="\<lambda>x. w" in exI) by simp qed
  1379 
  1380 lemma convex_hull_3_alt:
  1381   "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
  1382 proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
  1383   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)
  1384     apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
  1385 
  1386 subsection {* Relations among closure notions and corresponding hulls *}
  1387 
  1388 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
  1389   unfolding affine_def convex_def by auto
  1390 
  1391 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
  1392   using subspace_imp_affine affine_imp_convex by auto
  1393 
  1394 lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
  1395 by (metis hull_minimal span_inc subspace_imp_affine subspace_span)
  1396 
  1397 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
  1398 by (metis hull_minimal span_inc subspace_imp_convex subspace_span)
  1399 
  1400 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
  1401 by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
  1402 
  1403 
  1404 lemma affine_dependent_imp_dependent:
  1405   shows "affine_dependent s \<Longrightarrow> dependent s"
  1406   unfolding affine_dependent_def dependent_def 
  1407   using affine_hull_subset_span by auto
  1408 
  1409 lemma dependent_imp_affine_dependent:
  1410   assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
  1411   shows "affine_dependent (insert a s)"
  1412 proof-
  1413   from assms(1)[unfolded dependent_explicit] obtain S u v 
  1414     where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
  1415   def t \<equiv> "(\<lambda>x. x + a) ` S"
  1416 
  1417   have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
  1418   have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
  1419   have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
  1420 
  1421   hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
  1422   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
  1423     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1424   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
  1425     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
  1426   moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
  1427     apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
  1428   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
  1429     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1430   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
  1431     unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
  1432     using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
  1433   hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
  1434     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *)
  1435   ultimately show ?thesis unfolding affine_dependent_explicit
  1436     apply(rule_tac x="insert a t" in exI) by auto 
  1437 qed
  1438 
  1439 lemma convex_cone:
  1440   "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)" (is "?lhs = ?rhs")
  1441 proof-
  1442   { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
  1443     hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
  1444     hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
  1445       apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
  1446       apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
  1447   thus ?thesis unfolding convex_def cone_def by blast
  1448 qed
  1449 
  1450 lemma affine_dependent_biggerset: fixes s::"('a::euclidean_space) set"
  1451   assumes "finite s" "card s \<ge> DIM('a) + 2"
  1452   shows "affine_dependent s"
  1453 proof-
  1454   have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
  1455   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1456   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1457     apply(rule card_image) unfolding inj_on_def by auto
  1458   also have "\<dots> > DIM('a)" using assms(2)
  1459     unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1460   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1461     apply(rule dependent_imp_affine_dependent)
  1462     apply(rule dependent_biggerset) by auto qed
  1463 
  1464 lemma affine_dependent_biggerset_general:
  1465   assumes "finite (s::('a::euclidean_space) set)" "card s \<ge> dim s + 2"
  1466   shows "affine_dependent s"
  1467 proof-
  1468   from assms(2) have "s \<noteq> {}" by auto
  1469   then obtain a where "a\<in>s" by auto
  1470   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1471   have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1472     apply(rule card_image) unfolding inj_on_def by auto
  1473   have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
  1474     apply(rule subset_le_dim) unfolding subset_eq
  1475     using `a\<in>s` by (auto simp add:span_superset span_sub)
  1476   also have "\<dots> < dim s + 1" by auto
  1477   also have "\<dots> \<le> card (s - {a})" using assms
  1478     using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1479   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1480     apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
  1481 
  1482 subsection {* Caratheodory's theorem. *}
  1483 
  1484 lemma convex_hull_caratheodory: fixes p::"('a::euclidean_space) set"
  1485   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
  1486   (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  1487   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
  1488 proof(rule,rule)
  1489   fix y let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
  1490   assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
  1491   then obtain N where "?P N" by auto
  1492   hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
  1493   then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
  1494   then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
  1495 
  1496   have "card s \<le> DIM('a) + 1" proof(rule ccontr, simp only: not_le)
  1497     assume "DIM('a) + 1 < card s"
  1498     hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
  1499     then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
  1500       using affine_dependent_explicit_finite[OF obt(1)] by auto
  1501     def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
  1502     have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
  1503       assume as:"\<forall>x\<in>s. 0 \<le> w x"
  1504       hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
  1505       hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
  1506         using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
  1507       thus False using wv(1) by auto
  1508     qed hence "i\<noteq>{}" unfolding i_def by auto
  1509 
  1510     hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
  1511       using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
  1512     have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
  1513       fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
  1514       show"0 \<le> u v + t * w v" proof(cases "w v < 0")
  1515         case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
  1516           using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
  1517         case True hence "t \<le> u v / (- w v)" using `v\<in>s`
  1518           unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
  1519         thus ?thesis unfolding real_0_le_add_iff
  1520           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
  1521       qed qed
  1522 
  1523     obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
  1524       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
  1525     hence a:"a\<in>s" "u a + t * w a = 0" by auto
  1526     have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
  1527       unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
  1528     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
  1529       unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
  1530     moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y" 
  1531       unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
  1532       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
  1533     ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
  1534       apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a
  1535       by (auto simp add: * scaleR_left_distrib)
  1536     thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
  1537   thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1
  1538     \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" using obt by auto
  1539 qed auto
  1540 
  1541 lemma caratheodory:
  1542  "convex hull p = {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
  1543       card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
  1544   unfolding set_eq_iff apply(rule, rule) unfolding mem_Collect_eq proof-
  1545   fix x assume "x \<in> convex hull p"
  1546   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
  1547      "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"unfolding convex_hull_caratheodory by auto
  1548   thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  1549     apply(rule_tac x=s in exI) using hull_subset[of s convex]
  1550   using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
  1551 next
  1552   fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  1553   then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s" by auto
  1554   thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
  1555 qed
  1556 
  1557 
  1558 subsection {* Some Properties of Affine Dependent Sets *}
  1559 
  1560 lemma affine_independent_empty: "~(affine_dependent {})"
  1561   by (simp add: affine_dependent_def)
  1562 
  1563 lemma affine_independent_sing:
  1564 shows "~(affine_dependent {a})"
  1565  by (simp add: affine_dependent_def)
  1566 
  1567 lemma affine_hull_translation:
  1568 "affine hull ((%x. a + x) `  S) = (%x. a + x) ` (affine hull S)"
  1569 proof-
  1570 have "affine ((%x. a + x) ` (affine hull S))" using affine_translation affine_affine_hull by auto
  1571 moreover have "(%x. a + x) `  S <= (%x. a + x) ` (affine hull S)" using hull_subset[of S] by auto
  1572 ultimately have h1: "affine hull ((%x. a + x) `  S) <= (%x. a + x) ` (affine hull S)" by (metis hull_minimal)
  1573 have "affine((%x. -a + x) ` (affine hull ((%x. a + x) `  S)))"  using affine_translation affine_affine_hull by auto
  1574 moreover 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 auto 
  1575 moreover have "S=(%x. -a + x) ` (%x. a + x) `  S" using  translation_assoc[of "-a" a] by auto
  1576 ultimately have "(%x. -a + x) ` (affine hull ((%x. a + x) `  S)) >= (affine hull S)" by (metis hull_minimal)
  1577 hence "affine hull ((%x. a + x) `  S) >= (%x. a + x) ` (affine hull S)" by auto
  1578 from this show ?thesis using h1 by auto
  1579 qed
  1580 
  1581 lemma affine_dependent_translation:
  1582   assumes "affine_dependent S"
  1583   shows "affine_dependent ((%x. a + x) ` S)"
  1584 proof-
  1585 obtain x where x_def: "x : S & x : affine hull (S - {x})" using assms affine_dependent_def by auto
  1586 have "op + a ` (S - {x}) = op + a ` S - {a + x}" by auto
  1587 hence "a+x : affine hull ((%x. a + x) ` S - {a+x})" using  affine_hull_translation[of a "S-{x}"] x_def by auto
  1588 moreover have "a+x : (%x. a + x) ` S" using x_def by auto  
  1589 ultimately show ?thesis unfolding affine_dependent_def by auto 
  1590 qed
  1591 
  1592 lemma affine_dependent_translation_eq:
  1593   "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"
  1594 proof-
  1595 { assume "affine_dependent ((%x. a + x) ` S)" 
  1596   hence "affine_dependent S" using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] by auto  
  1597 } from this show ?thesis using affine_dependent_translation by auto
  1598 qed
  1599 
  1600 lemma affine_hull_0_dependent:
  1601   assumes "0 : affine hull S"
  1602   shows "dependent S"
  1603 proof-
  1604 obtain s u where s_u_def: "finite s & s ~= {} & s <= S & setsum u s = 1 & (SUM v:s. u v *\<^sub>R v) = 0" using assms affine_hull_explicit[of S] by auto
  1605 hence "EX v:s. u v ~= 0" using setsum_not_0[of "u" "s"] by auto 
  1606 hence "finite s & s <= S & (EX v:s. u v ~= 0 & (SUM v:s. u v *\<^sub>R v) = 0)" using s_u_def by auto
  1607 from this show ?thesis unfolding dependent_explicit[of S] by auto
  1608 qed
  1609 
  1610 lemma affine_dependent_imp_dependent2:
  1611   assumes "affine_dependent (insert 0 S)"
  1612   shows "dependent S"
  1613 proof-
  1614 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 blast
  1615 hence "x : span (insert 0 S - {x})" using affine_hull_subset_span by auto
  1616 moreover have "span (insert 0 S - {x}) = span (S - {x})" using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
  1617 ultimately have "x : span (S - {x})" by auto
  1618 hence "(x~=0) ==> dependent S" using x_def dependent_def by auto
  1619 moreover
  1620 { assume "x=0" hence "0 : affine hull S" using x_def hull_mono[of "S - {0}" S] by auto
  1621                hence "dependent S" using affine_hull_0_dependent by auto  
  1622 } ultimately show ?thesis by auto
  1623 qed
  1624 
  1625 lemma affine_dependent_iff_dependent:
  1626   assumes "a ~: S"
  1627   shows "affine_dependent (insert a S) <-> dependent ((%x. -a + x) ` S)" 
  1628 proof-
  1629 have "(op + (- a) ` S)={x - a| x . x : S}" by auto
  1630 from this show ?thesis using affine_dependent_translation_eq[of "(insert a S)" "-a"] 
  1631       affine_dependent_imp_dependent2 assms 
  1632       dependent_imp_affine_dependent[of a S] by auto
  1633 qed
  1634 
  1635 lemma affine_dependent_iff_dependent2:
  1636   assumes "a : S"
  1637   shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"
  1638 proof-
  1639 have "insert a (S - {a})=S" using assms by auto
  1640 from this show ?thesis using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto 
  1641 qed
  1642 
  1643 lemma affine_hull_insert_span_gen:
  1644   shows "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)" 
  1645 proof-
  1646 have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)" by auto
  1647 { assume "a ~: s" hence ?thesis using affine_hull_insert_span[of a s] h1 by auto}  
  1648 moreover
  1649 { assume a1: "a : s"
  1650   have "EX x. x:s & -a+x=0" apply (rule exI[of _ a]) using a1 by auto
  1651   hence "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s" by auto
  1652   hence "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)" 
  1653     using span_insert_0[of "op + (- a) ` (s - {a})"] by auto
  1654   moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))" by auto 
  1655   moreover have "insert a (s - {a})=(insert a s)" using assms by auto
  1656   ultimately have ?thesis using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
  1657 } 
  1658 ultimately show ?thesis by auto  
  1659 qed
  1660 
  1661 lemma affine_hull_span2:
  1662   assumes "a : s"
  1663   shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` (s-{a}))"
  1664   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
  1665 
  1666 lemma affine_hull_span_gen:
  1667   assumes "a : affine hull s"
  1668   shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` s)"
  1669 proof-
  1670 have "affine hull (insert a s) = affine hull s" using hull_redundant[of a affine s] assms by auto
  1671 from this show ?thesis using affine_hull_insert_span_gen[of a "s"] by auto
  1672 qed
  1673 
  1674 lemma affine_hull_span_0:
  1675   assumes "0 : affine hull S"
  1676   shows "affine hull S = span S"
  1677 using affine_hull_span_gen[of "0" S] assms by auto
  1678 
  1679 
  1680 lemma extend_to_affine_basis:
  1681 fixes S V :: "('n::euclidean_space) set"
  1682 assumes "~(affine_dependent S)" "S <= V" "S~={}"
  1683 shows "? T. ~(affine_dependent T) & S<=T & T<=V & affine hull T = affine hull V"
  1684 proof-
  1685 obtain a where a_def: "a : S" using assms by auto
  1686 hence h0: "independent  ((%x. -a + x) ` (S-{a}))" using affine_dependent_iff_dependent2 assms by auto
  1687 from this obtain B 
  1688    where B_def: "(%x. -a+x) ` (S - {a}) <= B & B <= (%x. -a+x) ` V & independent B & (%x. -a+x) ` V <= span B" 
  1689    using maximal_independent_subset_extend[of "(%x. -a+x) ` (S-{a})" "(%x. -a + x) ` V"] assms by blast
  1690 def T == "(%x. a+x) ` (insert 0 B)" hence "T=insert a ((%x. a+x) ` B)" by auto
  1691 hence "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 auto
  1692 hence "V <= affine hull T" using B_def assms translation_inverse_subset[of a V "span B"] by auto
  1693 moreover have "T<=V" using T_def B_def a_def assms by auto
  1694 ultimately have "affine hull T = affine hull V" 
  1695     by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
  1696 moreover have "S<=T" using T_def B_def translation_inverse_subset[of a "S-{a}" B] by auto
  1697 moreover have "~(affine_dependent T)" using T_def affine_dependent_translation_eq[of "insert 0 B"] affine_dependent_imp_dependent2 B_def by auto
  1698 ultimately show ?thesis using `T<=V` by auto
  1699 qed
  1700 
  1701 lemma affine_basis_exists: 
  1702 fixes V :: "('n::euclidean_space) set"
  1703 shows "? B. B <= V & ~(affine_dependent B) & affine hull V = affine hull B"
  1704 proof-
  1705 { assume empt: "V={}" have "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)" using empt affine_independent_empty by auto
  1706 }
  1707 moreover
  1708 { assume nonempt: "V~={}" obtain x where "x:V" using nonempt by auto
  1709   hence "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)"
  1710   using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}:: ('n::euclidean_space) set" V] by auto
  1711 }
  1712 ultimately show ?thesis by auto
  1713 qed
  1714 
  1715 subsection {* Affine Dimension of a Set *}
  1716 
  1717 definition "aff_dim V = (SOME d :: int. ? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1))"
  1718 
  1719 lemma aff_dim_basis_exists:
  1720   fixes V :: "('n::euclidean_space) set" 
  1721   shows "? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
  1722 proof-
  1723 obtain B where B_def: "~(affine_dependent B) & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
  1724 from 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 auto
  1725 qed
  1726 
  1727 lemma affine_hull_nonempty: "(S ~= {}) <-> affine hull S ~= {}"
  1728 proof-
  1729 have "(S = {}) ==> affine hull S = {}"using affine_hull_empty by auto 
  1730 moreover have "affine hull S = {} ==> S = {}" unfolding hull_def by auto
  1731 ultimately show "(S ~= {}) <-> affine hull S ~= {}" by blast
  1732 qed
  1733 
  1734 lemma aff_dim_parallel_subspace_aux:
  1735 fixes B :: "('n::euclidean_space) set"
  1736 assumes "~(affine_dependent B)" "a:B"
  1737 shows "finite B & ((card B) - 1 = dim (span ((%x. -a+x) ` (B-{a}))))" 
  1738 proof-
  1739 have "independent ((%x. -a + x) ` (B-{a}))" using affine_dependent_iff_dependent2 assms by auto
  1740 hence 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
  1741 { assume emp: "(%x. -a + x) ` (B - {a}) = {}" 
  1742   have "B=insert a ((%x. a + x) ` (%x. -a + x) ` (B - {a}))" using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
  1743   hence "B={a}" using emp by auto
  1744   hence ?thesis using assms fin by auto  
  1745 }
  1746 moreover
  1747 { assume "(%x. -a + x) ` (B - {a}) ~= {}"
  1748   hence "card ((%x. -a + x) ` (B - {a}))>0" using fin by auto
  1749   moreover have h1: "card ((%x. -a + x) ` (B-{a})) = card (B-{a})"  
  1750      apply (rule card_image) using translate_inj_on by auto
  1751   ultimately have "card (B-{a})>0" by auto
  1752   hence "finite(B-{a})" using card_gt_0_iff[of "(B - {a})"] by auto
  1753   moreover hence "(card (B-{a})= (card B) - 1)" using card_Diff_singleton assms by auto
  1754   ultimately have ?thesis using fin h1 by auto
  1755 } ultimately show ?thesis by auto
  1756 qed
  1757 
  1758 lemma aff_dim_parallel_subspace:
  1759 fixes V L :: "('n::euclidean_space) set"
  1760 assumes "V ~= {}"
  1761 assumes "subspace L" "affine_parallel (affine hull V) L"
  1762 shows "aff_dim V=int(dim L)"
  1763 proof-
  1764 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 auto
  1765 hence "B~={}" using assms B_def  affine_hull_nonempty[of V] affine_hull_nonempty[of B] by auto 
  1766 from this obtain a where a_def: "a : B" by auto
  1767 def Lb == "span ((%x. -a+x) ` (B-{a}))"
  1768   moreover have "affine_parallel (affine hull B) Lb"
  1769      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
  1770   moreover have "subspace Lb" using Lb_def subspace_span by auto
  1771   moreover have "affine hull B ~= {}" using assms B_def affine_hull_nonempty[of V] by auto
  1772   ultimately have "L=Lb" using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B_def by auto 
  1773   hence "dim L=dim Lb" by auto 
  1774   moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def B_def by auto
  1775 (*  hence "card B=dim Lb+1" using `B~={}` card_gt_0_iff[of B] by auto *)
  1776   ultimately show ?thesis using B_def `B~={}` card_gt_0_iff[of B] by auto
  1777 qed
  1778 
  1779 lemma aff_independent_finite:
  1780 fixes B :: "('n::euclidean_space) set"
  1781 assumes "~(affine_dependent B)"
  1782 shows "finite B"
  1783 proof-
  1784 { assume "B~={}" from this obtain a where "a:B" by auto 
  1785   hence ?thesis using aff_dim_parallel_subspace_aux assms by auto 
  1786 } from this show ?thesis by auto
  1787 qed
  1788 
  1789 lemma independent_finite:
  1790 fixes B :: "('n::euclidean_space) set"
  1791 assumes "independent B" 
  1792 shows "finite B"
  1793 using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms by auto
  1794 
  1795 lemma subspace_dim_equal:
  1796 assumes "subspace (S :: ('n::euclidean_space) set)" "subspace T" "S <= T" "dim S >= dim T"
  1797 shows "S=T"
  1798 proof- 
  1799 obtain B where B_def: "B <= S & independent B & S <= span B & (card B = dim S)" using basis_exists[of S] by auto
  1800 hence "span B <= S" using span_mono[of B S] span_eq[of S] assms by metis 
  1801 hence "span B = S" using B_def by auto
  1802 have "dim S = dim T" using assms dim_subset[of S T] by auto
  1803 hence "T <= span B" using card_eq_dim[of B T] B_def independent_finite assms by auto
  1804 from this show ?thesis using assms `span B=S` by auto
  1805 qed
  1806 
  1807 lemma span_substd_basis:  assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  1808   shows "(span {basis i | i. i : d}) = {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
  1809   (is "span ?A = ?B")
  1810 proof-
  1811 have "?A <= ?B" by auto
  1812 moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .
  1813 ultimately have "span ?A <= ?B" using span_mono[of "?A" "?B"] span_eq[of "?B"] by blast
  1814 moreover have "card d <= dim (span ?A)" using independent_card_le_dim[of "?A" "span ?A"] 
  1815    independent_substdbasis[OF assms] card_substdbasis[OF assms] span_inc[of "?A"] by auto
  1816 moreover hence "dim ?B <= dim (span ?A)" using dim_substandard[OF assms] by auto
  1817 ultimately show ?thesis using s subspace_dim_equal[of "span ?A" "?B"] 
  1818   subspace_span[of "?A"] by auto
  1819 qed
  1820 
  1821 lemma basis_to_substdbasis_subspace_isomorphism:
  1822 fixes B :: "('a::euclidean_space) set" 
  1823 assumes "independent B"
  1824 shows "EX f d. card d = card B & linear f & f ` B = {basis i::'a |i. i : (d :: nat set)} & 
  1825        f ` span B = {x. ALL i<DIM('a). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B) \<and> d\<subseteq>{..<DIM('a)}" 
  1826 proof-
  1827   have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto
  1828   def d \<equiv> "{..<dim B}" have t:"card d = dim B" unfolding d_def by auto
  1829   have "dim B <= DIM('a)" using dim_subset_UNIV[of B] by auto
  1830   hence d:"d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
  1831   let ?t = "{x::'a::euclidean_space. !i<DIM('a). i ~: d --> x$$i = 0}"
  1832   have "EX f. linear f & f ` B = {basis i |i. i : d} &
  1833     f ` span B = ?t & inj_on f (span B)"
  1834     apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "{basis i |i. i : d}"])
  1835     apply(rule subspace_span) apply(rule subspace_substandard) defer
  1836     apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)
  1837     unfolding span_substd_basis[OF d,THEN sym] card_substdbasis[OF d] apply(rule span_inc)
  1838     apply(rule independent_substdbasis[OF d]) apply(rule,assumption)
  1839     unfolding t[THEN sym] span_substd_basis[OF d] dim_substandard[OF d] by auto
  1840   from this t `card B=dim B` show ?thesis using d by auto 
  1841 qed
  1842 
  1843 lemma aff_dim_empty:
  1844 fixes S :: "('n::euclidean_space) set" 
  1845 shows "S = {} <-> aff_dim S = -1"
  1846 proof-
  1847 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 auto
  1848 moreover hence "S={} <-> B={}" using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
  1849 ultimately show ?thesis using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
  1850 qed
  1851 
  1852 lemma aff_dim_affine_hull:
  1853 shows "aff_dim (affine hull S)=aff_dim S" 
  1854 unfolding aff_dim_def using hull_hull[of _ S] by auto 
  1855 
  1856 lemma aff_dim_affine_hull2:
  1857 assumes "affine hull S=affine hull T"
  1858 shows "aff_dim S=aff_dim T" unfolding aff_dim_def using assms by auto
  1859 
  1860 lemma aff_dim_unique: 
  1861 fixes B V :: "('n::euclidean_space) set" 
  1862 assumes "(affine hull B=affine hull V) & ~(affine_dependent B)"
  1863 shows "of_nat(card B) = aff_dim V+1"
  1864 proof-
  1865 { assume "B={}" hence "V={}" using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms by auto
  1866   hence "aff_dim V = (-1::int)"  using aff_dim_empty by auto  
  1867   hence ?thesis using `B={}` by auto
  1868 }
  1869 moreover
  1870 { assume "B~={}" from this obtain a where a_def: "a:B" by auto 
  1871   def Lb == "span ((%x. -a+x) ` (B-{a}))"
  1872   have "affine_parallel (affine hull B) Lb"
  1873      using Lb_def affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"] 
  1874      unfolding affine_parallel_def by auto
  1875   moreover have "subspace Lb" using Lb_def subspace_span by auto
  1876   ultimately have "aff_dim B=int(dim Lb)" using aff_dim_parallel_subspace[of B Lb] `B~={}` by auto 
  1877   moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def assms by auto
  1878   ultimately have "(of_nat(card B) = aff_dim B+1)" using  `B~={}` card_gt_0_iff[of B] by auto
  1879   hence ?thesis using aff_dim_affine_hull2 assms by auto
  1880 } ultimately show ?thesis by blast
  1881 qed
  1882 
  1883 lemma aff_dim_affine_independent: 
  1884 fixes B :: "('n::euclidean_space) set" 
  1885 assumes "~(affine_dependent B)"
  1886 shows "of_nat(card B) = aff_dim B+1"
  1887   using aff_dim_unique[of B B] assms by auto
  1888 
  1889 lemma aff_dim_sing: 
  1890 fixes a :: "'n::euclidean_space" 
  1891 shows "aff_dim {a}=0"
  1892   using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
  1893 
  1894 lemma aff_dim_inner_basis_exists:
  1895   fixes V :: "('n::euclidean_space) set" 
  1896   shows "? B. B<=V & (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
  1897 proof-
  1898 obtain B where B_def: "~(affine_dependent B) & B<=V & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
  1899 moreover hence "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
  1900 ultimately show ?thesis by auto
  1901 qed
  1902 
  1903 lemma aff_dim_le_card:
  1904 fixes V :: "('n::euclidean_space) set" 
  1905 assumes "finite V"
  1906 shows "aff_dim V <= of_nat(card V) - 1"
  1907  proof-
  1908  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 
  1909  moreover hence "card B <= card V" using assms card_mono by auto
  1910  ultimately show ?thesis by auto
  1911 qed
  1912 
  1913 lemma aff_dim_parallel_eq:
  1914 fixes S T :: "('n::euclidean_space) set"
  1915 assumes "affine_parallel (affine hull S) (affine hull T)"
  1916 shows "aff_dim S=aff_dim T"
  1917 proof-
  1918 { assume "T~={}" "S~={}" 
  1919   from this obtain L where L_def: "subspace L & affine_parallel (affine hull T) L" 
  1920        using affine_parallel_subspace[of "affine hull T"] affine_affine_hull[of T] affine_hull_nonempty by auto
  1921   hence "aff_dim T = int(dim L)" using aff_dim_parallel_subspace `T~={}` by auto
  1922   moreover have "subspace L & affine_parallel (affine hull S) L" 
  1923      using L_def affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
  1924   moreover hence "aff_dim S = int(dim L)" using aff_dim_parallel_subspace `S~={}` by auto 
  1925   ultimately have ?thesis by auto
  1926 }
  1927 moreover
  1928 { assume "S={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
  1929   hence ?thesis using aff_dim_empty by auto
  1930 }
  1931 moreover
  1932 { assume "T={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
  1933   hence ?thesis using aff_dim_empty by auto
  1934 }
  1935 ultimately show ?thesis by blast
  1936 qed
  1937 
  1938 lemma aff_dim_translation_eq:
  1939 fixes a :: "'n::euclidean_space"
  1940 shows "aff_dim ((%x. a + x) ` S)=aff_dim S" 
  1941 proof-
  1942 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 auto
  1943 from this show ?thesis using  aff_dim_parallel_eq[of S "(%x. a + x) ` S"] by auto 
  1944 qed
  1945 
  1946 lemma aff_dim_affine:
  1947 fixes S L :: "('n::euclidean_space) set"
  1948 assumes "S ~= {}" "affine S"
  1949 assumes "subspace L" "affine_parallel S L"
  1950 shows "aff_dim S=int(dim L)" 
  1951 proof-
  1952 have 1: "(affine hull S) = S" using assms affine_hull_eq[of S] by auto 
  1953 hence "affine_parallel (affine hull S) L" using assms by (simp add:1)
  1954 from this show ?thesis using assms aff_dim_parallel_subspace[of S L] by blast 
  1955 qed
  1956 
  1957 lemma dim_affine_hull:
  1958 fixes S :: "('n::euclidean_space) set"
  1959 shows "dim (affine hull S)=dim S"
  1960 proof-
  1961 have "dim (affine hull S)>=dim S" using dim_subset by auto
  1962 moreover have "dim(span S) >= dim (affine hull S)" using dim_subset affine_hull_subset_span by auto
  1963 moreover have "dim(span S)=dim S" using dim_span by auto
  1964 ultimately show ?thesis by auto
  1965 qed
  1966 
  1967 lemma aff_dim_subspace:
  1968 fixes S :: "('n::euclidean_space) set"
  1969 assumes "S ~= {}" "subspace S"
  1970 shows "aff_dim S=int(dim S)" using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] by auto 
  1971 
  1972 lemma aff_dim_zero:
  1973 fixes S :: "('n::euclidean_space) set"
  1974 assumes "0 : affine hull S"
  1975 shows "aff_dim S=int(dim S)"
  1976 proof-
  1977 have "subspace(affine hull S)" using subspace_affine[of "affine hull S"] affine_affine_hull assms by auto
  1978 hence "aff_dim (affine hull S) =int(dim (affine hull S))" using assms aff_dim_subspace[of "affine hull S"] by auto  
  1979 from this show ?thesis using aff_dim_affine_hull[of S] dim_affine_hull[of S] by auto
  1980 qed
  1981 
  1982 lemma aff_dim_univ: "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))"
  1983   using aff_dim_subspace[of "(UNIV :: ('n::euclidean_space) set)"]
  1984     dim_UNIV[where 'a="'n::euclidean_space"] by auto
  1985 
  1986 lemma aff_dim_geq:
  1987   fixes V :: "('n::euclidean_space) set"
  1988   shows "aff_dim V >= -1"
  1989 proof-
  1990 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 auto
  1991 from this show ?thesis by auto
  1992 qed
  1993 
  1994 lemma independent_card_le_aff_dim: 
  1995   assumes "(B::('n::euclidean_space) set) <= V"
  1996   assumes "~(affine_dependent B)" 
  1997   shows "int(card B) <= aff_dim V+1"
  1998 proof-
  1999 { assume "B~={}" 
  2000   from this obtain T where T_def: "~(affine_dependent T) & B<=T & T<=V & affine hull T = affine hull V" 
  2001   using assms extend_to_affine_basis[of B V] by auto
  2002   hence "of_nat(card T) = aff_dim V+1" using aff_dim_unique by auto
  2003   hence ?thesis using T_def card_mono[of T B] aff_independent_finite[of T] by auto
  2004 }
  2005 moreover
  2006 { assume "B={}"
  2007   moreover have "-1<= aff_dim V" using aff_dim_geq by auto
  2008   ultimately have ?thesis by auto
  2009 }  ultimately show ?thesis by blast
  2010 qed
  2011 
  2012 lemma aff_dim_subset:
  2013   fixes S T :: "('n::euclidean_space) set"
  2014   assumes "S <= T"
  2015   shows "aff_dim S <= aff_dim T"
  2016 proof-
  2017 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 auto
  2018 moreover hence "int (card B) <= aff_dim T + 1" using assms independent_card_le_aff_dim[of B T] by auto
  2019 ultimately show ?thesis by auto
  2020 qed
  2021 
  2022 lemma aff_dim_subset_univ:
  2023 fixes S :: "('n::euclidean_space) set"
  2024 shows "aff_dim S <= int(DIM('n))"
  2025 proof - 
  2026   have "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))" using aff_dim_univ by auto
  2027   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 auto
  2028 qed
  2029 
  2030 lemma affine_dim_equal:
  2031 assumes "affine (S :: ('n::euclidean_space) set)" "affine T" "S ~= {}" "S <= T" "aff_dim S = aff_dim T"
  2032 shows "S=T"
  2033 proof-
  2034 obtain a where "a : S" using assms by auto 
  2035 hence "a : T" using assms by auto
  2036 def LS == "{y. ? x : S. (-a)+x=y}"
  2037 hence ls: "subspace LS & affine_parallel S LS" using assms parallel_subspace_explicit[of S a LS] `a : S` by auto 
  2038 hence h1: "int(dim LS) = aff_dim S" using assms aff_dim_affine[of S LS] by auto
  2039 have "T ~= {}" using assms by auto
  2040 def LT == "{y. ? x : T. (-a)+x=y}" 
  2041 hence lt: "subspace LT & affine_parallel T LT" using assms parallel_subspace_explicit[of T a LT] `a : T` by auto
  2042 hence "int(dim LT) = aff_dim T" using assms aff_dim_affine[of T LT] `T ~= {}` by auto 
  2043 hence "dim LS = dim LT" using h1 assms by auto
  2044 moreover have "LS <= LT" using LS_def LT_def assms by auto
  2045 ultimately have "LS=LT" using subspace_dim_equal[of LS LT] ls lt by auto
  2046 moreover have "S = {x. ? y : LS. a+y=x}" using LS_def by auto 
  2047 moreover have "T = {x. ? y : LT. a+y=x}" using LT_def by auto
  2048 ultimately show ?thesis by auto 
  2049 qed
  2050 
  2051 lemma affine_hull_univ:
  2052 fixes S :: "('n::euclidean_space) set"
  2053 assumes "aff_dim S = int(DIM('n))"
  2054 shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
  2055 proof-
  2056 have "S ~= {}" using assms aff_dim_empty[of S] by auto
  2057 have h0: "S <= affine hull S" using hull_subset[of S _] by auto
  2058 have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S" using aff_dim_univ assms by auto
  2059 hence h2: "aff_dim (affine hull S) <= aff_dim (UNIV :: ('n::euclidean_space) set)" using aff_dim_subset_univ[of "affine hull S"] assms h0 by auto  
  2060 have h3: "aff_dim S <= aff_dim (affine hull S)" using h0 aff_dim_subset[of S "affine hull S"] assms by auto
  2061 hence h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" using h0 h1 h2 by auto
  2062 from 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 auto
  2063 qed
  2064 
  2065 lemma aff_dim_convex_hull:
  2066 fixes S :: "('n::euclidean_space) set"
  2067 shows "aff_dim (convex hull S)=aff_dim S"
  2068   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S] 
  2069   hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"] 
  2070   aff_dim_subset[of "convex hull S" "affine hull S"] by auto
  2071 
  2072 lemma aff_dim_cball:
  2073 fixes a :: "'n::euclidean_space" 
  2074 assumes "0<e"
  2075 shows "aff_dim (cball a e) = int (DIM('n))"
  2076 proof-
  2077 have "(%x. a + x) ` (cball 0 e)<=cball a e" unfolding cball_def dist_norm by auto
  2078 hence "aff_dim (cball (0 :: 'n::euclidean_space) e) <= aff_dim (cball a e)"
  2079   using aff_dim_translation_eq[of a "cball 0 e"] 
  2080         aff_dim_subset[of "op + a ` cball 0 e" "cball a e"] by auto
  2081 moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))" 
  2082    using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"] centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms 
  2083    by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
  2084 ultimately show ?thesis using aff_dim_subset_univ[of "cball a e"] by auto 
  2085 qed
  2086 
  2087 lemma aff_dim_open:
  2088 fixes S :: "('n::euclidean_space) set"
  2089 assumes "open S" "S ~= {}"
  2090 shows "aff_dim S = int (DIM('n))"
  2091 proof-
  2092 obtain x where "x:S" using assms by auto
  2093 from this obtain e where e_def: "e>0 & cball x e <= S" using open_contains_cball[of S] assms by auto
  2094 from this have "aff_dim (cball x e) <= aff_dim S" using aff_dim_subset by auto
  2095 from this show ?thesis using aff_dim_cball[of e x] aff_dim_subset_univ[of S] e_def by auto     
  2096 qed
  2097 
  2098 lemma low_dim_interior:
  2099 fixes S :: "('n::euclidean_space) set"
  2100 assumes "~(aff_dim S = int (DIM('n)))"
  2101 shows "interior S = {}"
  2102 proof-
  2103 have "aff_dim(interior S) <= aff_dim S" 
  2104    using interior_subset aff_dim_subset[of "interior S" S] by auto 
  2105 from this show ?thesis using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto   
  2106 qed
  2107 
  2108 subsection {* Relative interior of a set *}
  2109 
  2110 definition "rel_interior S = {x. ? T. openin (subtopology euclidean (affine hull S)) T & x : T & T <= S}"
  2111 
  2112 lemma rel_interior: "rel_interior S = {x : S. ? T. open T & x : T & (T Int (affine hull S)) <= S}"
  2113   unfolding rel_interior_def[of S] openin_open[of "affine hull S"] apply auto
  2114 proof-
  2115 fix x T assume a: "x:S" "open T" "x : T" "(T Int (affine hull S)) <= S"
  2116 hence h1: "x : T Int affine hull S" using hull_inc by auto
  2117 show "EX Tb. (EX Ta. open Ta & Tb = affine hull S Int Ta) & x : Tb & Tb <= S"
  2118 apply (rule_tac x="T Int (affine hull S)" in exI)
  2119 using a h1 by auto
  2120 qed
  2121 
  2122 lemma mem_rel_interior: 
  2123      "x : rel_interior S <-> (? T. open T & x : (T Int S) & (T Int (affine hull S)) <= S)" 
  2124      by (auto simp add: rel_interior)
  2125 
  2126 lemma mem_rel_interior_ball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((ball x e) Int (affine hull S)) <= S)"
  2127   apply (simp add: rel_interior, safe)
  2128   apply (force simp add: open_contains_ball)
  2129   apply (rule_tac x="ball x e" in exI)
  2130   apply simp
  2131   done
  2132 
  2133 lemma rel_interior_ball: 
  2134       "rel_interior S = {x : S. ? e. e>0 & ((ball x e) Int (affine hull S)) <= S}" 
  2135       using mem_rel_interior_ball [of _ S] by auto 
  2136 
  2137 lemma mem_rel_interior_cball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((cball x e) Int (affine hull S)) <= S)"
  2138   apply (simp add: rel_interior, safe) 
  2139   apply (force simp add: open_contains_cball)
  2140   apply (rule_tac x="ball x e" in exI)
  2141   apply (simp add: subset_trans [OF ball_subset_cball])
  2142   apply auto
  2143   done
  2144 
  2145 lemma 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 auto
  2146 
  2147 lemma rel_interior_empty: "rel_interior {} = {}" 
  2148    by (auto simp add: rel_interior_def) 
  2149 
  2150 lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"
  2151 by (metis affine_hull_eq affine_sing)
  2152 
  2153 lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"
  2154    unfolding rel_interior_ball affine_hull_sing apply auto
  2155    apply(rule_tac x="1 :: real" in exI) apply simp
  2156    done
  2157 
  2158 lemma subset_rel_interior:
  2159 fixes S T :: "('n::euclidean_space) set"
  2160 assumes "S<=T" "affine hull S=affine hull T"
  2161 shows "rel_interior S <= rel_interior T"
  2162   using assms by (auto simp add: rel_interior_def)  
  2163 
  2164 lemma rel_interior_subset: "rel_interior S <= S" 
  2165    by (auto simp add: rel_interior_def)
  2166 
  2167 lemma rel_interior_subset_closure: "rel_interior S <= closure S" 
  2168    using rel_interior_subset by (auto simp add: closure_def) 
  2169 
  2170 lemma interior_subset_rel_interior: "interior S <= rel_interior S" 
  2171    by (auto simp add: rel_interior interior_def)
  2172 
  2173 lemma interior_rel_interior:
  2174 fixes S :: "('n::euclidean_space) set"
  2175 assumes "aff_dim S = int(DIM('n))"
  2176 shows "rel_interior S = interior S"
  2177 proof -
  2178 have "affine hull S = UNIV" using assms affine_hull_univ[of S] by auto 
  2179 from this show ?thesis unfolding rel_interior interior_def by auto
  2180 qed
  2181 
  2182 lemma rel_interior_open:
  2183 fixes S :: "('n::euclidean_space) set"
  2184 assumes "open S"
  2185 shows "rel_interior S = S"
  2186 by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
  2187 
  2188 lemma interior_rel_interior_gen:
  2189 fixes S :: "('n::euclidean_space) set"
  2190 shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
  2191 by (metis interior_rel_interior low_dim_interior)
  2192 
  2193 lemma rel_interior_univ: 
  2194 fixes S :: "('n::euclidean_space) set"
  2195 shows "rel_interior (affine hull S) = affine hull S"
  2196 proof-
  2197 have h1: "rel_interior (affine hull S) <= affine hull S" using rel_interior_subset by auto 
  2198 { fix x assume x_def: "x : affine hull S"
  2199   obtain e :: real where "e=1" by auto
  2200   hence "e>0 & ball x e Int affine hull (affine hull S) <= affine hull S" using hull_hull[of _ S] by auto
  2201   hence "x : rel_interior (affine hull S)" using x_def rel_interior_ball[of "affine hull S"] by auto
  2202 } from this show ?thesis using h1 by auto 
  2203 qed
  2204 
  2205 lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
  2206 by (metis open_UNIV rel_interior_open)
  2207 
  2208 lemma rel_interior_convex_shrink:
  2209   fixes S :: "('a::euclidean_space) set"
  2210   assumes "convex S" "c : rel_interior S" "x : S" "0 < e" "e <= 1"
  2211   shows "x - e *\<^sub>R (x - c) : rel_interior S"
  2212 proof- 
  2213 (* Proof is a modified copy of the proof of similar lemma mem_interior_convex_shrink 
  2214 *)
  2215 obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
  2216   using assms(2) unfolding  mem_rel_interior_ball by auto
  2217 {   fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d & y : affine hull S"
  2218     have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
  2219     have "x : affine hull S" using assms hull_subset[of S] by auto
  2220     moreover have "1 / e + - ((1 - e) / e) = 1" 
  2221        using `e>0` left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
  2222     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x : affine hull S"
  2223         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)     
  2224     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
  2225       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
  2226       by(auto simp add:euclidean_eq[where 'a='a] field_simps) 
  2227     also have "... = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
  2228     also have "... < d" using as[unfolded dist_norm] and `e>0`
  2229       by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
  2230     finally have "y : S" apply(subst *) 
  2231 apply(rule assms(1)[unfolded convex_alt,rule_format])
  2232       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) ** by auto
  2233 } hence "ball (x - e *\<^sub>R (x - c)) (e*d) Int affine hull S <= S" by auto
  2234 moreover have "0 < e*d" using `0<e` `0<d` by (rule mult_pos_pos)
  2235 moreover have "c : S" using assms rel_interior_subset by auto
  2236 moreover hence "x - e *\<^sub>R (x - c) : S"
  2237    using mem_convex[of S x c e] apply (simp add: algebra_simps) using assms by auto
  2238 ultimately show ?thesis 
  2239   using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e>0` by auto
  2240 qed
  2241 
  2242 lemma interior_real_semiline:
  2243 fixes a :: real
  2244 shows "interior {a..} = {a<..}"
  2245 proof-
  2246 { fix y assume "a<y" hence "y : interior {a..}"
  2247   apply (simp add: mem_interior) apply (rule_tac x="(y-a)" in exI) apply (auto simp add: dist_norm) 
  2248   done }
  2249 moreover
  2250 { fix y assume "y : interior {a..}" (*hence "a<=y" using interior_subset by auto*)
  2251   from this obtain e where e_def: "e>0 & cball y e \<subseteq> {a..}" 
  2252      using mem_interior_cball[of y "{a..}"] by auto
  2253   moreover hence "y-e : cball y e" by (auto simp add: cball_def dist_norm) 
  2254   ultimately have "a<=y-e" by auto
  2255   hence "a<y" using e_def by auto
  2256 } ultimately show ?thesis by auto
  2257 qed
  2258 
  2259 lemma rel_interior_real_interval:
  2260   fixes a b :: real assumes "a < b" shows "rel_interior {a..b} = {a<..<b}"
  2261 proof-
  2262   have "{a<..<b} \<noteq> {}" using assms unfolding set_eq_iff by (auto intro!: exI[of _ "(a + b) / 2"])
  2263   then show ?thesis
  2264     using interior_rel_interior_gen[of "{a..b}", symmetric]
  2265     by (simp split: split_if_asm add: interior_closed_interval)
  2266 qed
  2267 
  2268 lemma rel_interior_real_semiline:
  2269   fixes a :: real shows "rel_interior {a..} = {a<..}"
  2270 proof-
  2271   have *: "{a<..} \<noteq> {}" unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
  2272   then show ?thesis using interior_real_semiline
  2273      interior_rel_interior_gen[of "{a..}"]
  2274      by (auto split: split_if_asm)
  2275 qed
  2276 
  2277 subsubsection {* Relative open sets *}
  2278 
  2279 definition "rel_open S <-> (rel_interior S) = S"
  2280 
  2281 lemma rel_open: "rel_open S <-> openin (subtopology euclidean (affine hull S)) S"
  2282  unfolding rel_open_def rel_interior_def apply auto
  2283  using openin_subopen[of "subtopology euclidean (affine hull S)" S] by auto
  2284 
  2285 lemma opein_rel_interior: 
  2286   "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
  2287   apply (simp add: rel_interior_def)
  2288   apply (subst openin_subopen) by blast
  2289 
  2290 lemma affine_rel_open: 
  2291   fixes S :: "('n::euclidean_space) set"
  2292   assumes "affine S" shows "rel_open S" 
  2293   unfolding rel_open_def using assms rel_interior_univ[of S] affine_hull_eq[of S] by metis
  2294 
  2295 lemma affine_closed: 
  2296   fixes S :: "('n::euclidean_space) set"
  2297   assumes "affine S" shows "closed S"
  2298 proof-
  2299 { assume "S ~= {}"
  2300   from this obtain L where L_def: "subspace L & affine_parallel S L"
  2301      using assms affine_parallel_subspace[of S] by auto
  2302   from this obtain "a" where a_def: "S=(op + a ` L)" 
  2303      using affine_parallel_def[of L S] affine_parallel_commut by auto 
  2304   have "closed L" using L_def closed_subspace by auto
  2305   hence "closed S" using closed_translation a_def by auto
  2306 } from this show ?thesis by auto
  2307 qed
  2308 
  2309 lemma closure_affine_hull:
  2310   fixes S :: "('n::euclidean_space) set"
  2311   shows "closure S <= affine hull S"
  2312   by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
  2313 
  2314 lemma closure_same_affine_hull:
  2315   fixes S :: "('n::euclidean_space) set"
  2316   shows "affine hull (closure S) = affine hull S"
  2317 proof-
  2318 have "affine hull (closure S) <= affine hull S"
  2319    using hull_mono[of "closure S" "affine hull S" "affine"] closure_affine_hull[of S] hull_hull[of "affine" S] by auto
  2320 moreover have "affine hull (closure S) >= affine hull S"  
  2321    using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
  2322 ultimately show ?thesis by auto  
  2323 qed
  2324 
  2325 lemma closure_aff_dim: 
  2326   fixes S :: "('n::euclidean_space) set"
  2327   shows "aff_dim (closure S) = aff_dim S"
  2328 proof-
  2329 have "aff_dim S <= aff_dim (closure S)" using aff_dim_subset closure_subset by auto
  2330 moreover have "aff_dim (closure S) <= aff_dim (affine hull S)" 
  2331   using aff_dim_subset closure_affine_hull by auto
  2332 moreover have "aff_dim (affine hull S) = aff_dim S" using aff_dim_affine_hull by auto
  2333 ultimately show ?thesis by auto
  2334 qed
  2335 
  2336 lemma rel_interior_closure_convex_shrink:
  2337   fixes S :: "(_::euclidean_space) set"
  2338   assumes "convex S" "c : rel_interior S" "x : closure S" "0 < e" "e <= 1"
  2339   shows "x - e *\<^sub>R (x - c) : rel_interior S"
  2340 proof- 
  2341 (* Proof is a modified copy of the proof of similar lemma mem_interior_closure_convex_shrink
  2342 *)
  2343 obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
  2344   using assms(2) unfolding mem_rel_interior_ball by auto
  2345 have "EX y : S. norm (y - x) * (1 - e) < e * d" proof(cases "x : S")
  2346     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  2347     case False hence x:"x islimpt S" using assms(3)[unfolded closure_def] by auto
  2348     show ?thesis proof(cases "e=1")
  2349       case True obtain y where "y : S" "y ~= x" "dist y x < 1"
  2350         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  2351       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  2352       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  2353         using `e<=1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  2354       then obtain y where "y : S" "y ~= x" "dist y x < e * d / (1 - e)"
  2355         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  2356       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  2357   then obtain y where "y : S" and y:"norm (y - x) * (1 - e) < e * d" by auto
  2358   def z == "c + ((1 - e) / e) *\<^sub>R (x - y)"
  2359   have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
  2360   have zball: "z\<in>ball c d"
  2361     using mem_ball z_def dist_norm[of c] using y and assms(4,5) by (auto simp add:field_simps norm_minus_commute)
  2362   have "x : affine hull S" using closure_affine_hull assms by auto
  2363   moreover have "y : affine hull S" using `y : S` hull_subset[of S] by auto
  2364   moreover have "c : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
  2365   ultimately have "z : affine hull S" 
  2366     using z_def affine_affine_hull[of S] 
  2367           mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"] 
  2368           assms by (auto simp add: field_simps)
  2369   hence "z : S" using d zball by auto
  2370   obtain d1 where "d1>0" and d1:"ball z d1 <= ball c d"
  2371     using zball open_ball[of c d] openE[of "ball c d" z] by auto
  2372   hence "(ball z d1) Int (affine hull S) <= (ball c d) Int (affine hull S)" by auto
  2373   hence "(ball z d1) Int (affine hull S) <= S" using d by auto 
  2374   hence "z : rel_interior S" using mem_rel_interior_ball using `d1>0` `z : S` by auto
  2375   hence "y - e *\<^sub>R (y - z) : rel_interior S" using rel_interior_convex_shrink[of S z y e] assms`y : S` by auto
  2376   thus ?thesis using * by auto 
  2377 qed
  2378 
  2379 subsubsection{* Relative interior preserves under linear transformations *}
  2380 
  2381 lemma rel_interior_translation_aux:
  2382 fixes a :: "'n::euclidean_space"
  2383 shows "((%x. a + x) ` rel_interior S) <= rel_interior ((%x. a + x) ` S)"
  2384 proof-
  2385 { fix x assume x_def: "x : rel_interior S"
  2386   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 
  2387   from this have "open ((%x. a + x) ` T)" and 
  2388     "(a + x) : (((%x. a + x) ` T) Int ((%x. a + x) ` S))" and 
  2389     "(((%x. a + x) ` T) Int (affine hull ((%x. a + x) ` S))) <= ((%x. a + x) ` S)" 
  2390     using affine_hull_translation[of a S] open_translation[of T a] x_def by auto 
  2391   from this have "(a+x) : rel_interior ((%x. a + x) ` S)" 
  2392     using mem_rel_interior[of "a+x" "((%x. a + x) ` S)"] by auto 
  2393 } from this show ?thesis by auto 
  2394 qed
  2395 
  2396 lemma rel_interior_translation:
  2397 fixes a :: "'n::euclidean_space"
  2398 shows "rel_interior ((%x. a + x) ` S) = ((%x. a + x) ` rel_interior S)"
  2399 proof-
  2400 have "(%x. (-a) + x) ` rel_interior ((%x. a + x) ` S) <= rel_interior S" 
  2401    using rel_interior_translation_aux[of "-a" "(%x. a + x) ` S"] 
  2402          translation_assoc[of "-a" "a"] by auto
  2403 hence "((%x. a + x) ` rel_interior S) >= rel_interior ((%x. a + x) ` S)" 
  2404    using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"] 
  2405    by auto
  2406 from this show ?thesis using  rel_interior_translation_aux[of a S] by auto 
  2407 qed
  2408 
  2409 
  2410 lemma affine_hull_linear_image:
  2411 assumes "bounded_linear f"
  2412 shows "f ` (affine hull s) = affine hull f ` s"
  2413 (* Proof is a modified copy of the proof of similar lemma convex_hull_linear_image
  2414 *)
  2415   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  2416   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  2417   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  2418 proof-
  2419   interpret f: bounded_linear f by fact
  2420   show "affine {x. f x : affine hull f ` s}" 
  2421   unfolding affine_def by(auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format]) next
  2422   interpret f: bounded_linear f by fact
  2423   show "affine {x. x : f ` (affine hull s)}" using affine_affine_hull[unfolded affine_def, of s] 
  2424     unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  2425 qed auto
  2426 
  2427 
  2428 lemma rel_interior_injective_on_span_linear_image:
  2429 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  2430 fixes S :: "('m::euclidean_space) set"
  2431 assumes "bounded_linear f" and "inj_on f (span S)"
  2432 shows "rel_interior (f ` S) = f ` (rel_interior S)"
  2433 proof-
  2434 { fix z assume z_def: "z : rel_interior (f ` S)"
  2435   have "z : f ` S" using z_def rel_interior_subset[of "f ` S"] by auto
  2436   from this obtain x where x_def: "x : S & (f x = z)" by auto
  2437   obtain e2 where e2_def: "e2>0 & cball z e2 Int affine hull (f ` S) <= (f ` S)" 
  2438     using z_def rel_interior_cball[of "f ` S"] by auto
  2439   obtain K where K_def: "K>0 & (! x. norm (f x) <= norm x * K)" 
  2440    using assms RealVector.bounded_linear.pos_bounded[of f] by auto
  2441   def e1 == "1/K" hence e1_def: "e1>0 & (! x. e1 * norm (f x) <= norm x)" 
  2442    using K_def pos_le_divide_eq[of e1] by auto
  2443   def e == "e1 * e2" hence "e>0" using e1_def e2_def mult_pos_pos by auto 
  2444   { fix y assume y_def: "y : cball x e Int affine hull S"
  2445     from this have h1: "f y : affine hull (f ` S)" 
  2446       using affine_hull_linear_image[of f S] assms by auto 
  2447     from y_def have "norm (x-y)<=e1 * e2" 
  2448       using cball_def[of x e] dist_norm[of x y] e_def by auto
  2449     moreover have "(f x)-(f y)=f (x-y)"
  2450        using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
  2451     moreover have "e1 * norm (f (x-y)) <= norm (x-y)" using e1_def by auto
  2452     ultimately have "e1 * norm ((f x)-(f y)) <= e1 * e2" by auto
  2453     hence "(f y) : (cball z e2)" 
  2454       using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1_def x_def by auto
  2455     hence "f y : (f ` S)" using y_def e2_def h1 by auto
  2456     hence "y : S" using assms y_def hull_subset[of S] affine_hull_subset_span 
  2457          inj_on_image_mem_iff[of f "span S" S y] by auto
  2458   } 
  2459   hence "z : f ` (rel_interior S)" using mem_rel_interior_cball[of x S] `e>0` x_def by auto
  2460 } 
  2461 moreover
  2462 { fix x assume x_def: "x : rel_interior S"
  2463   from this obtain e2 where e2_def: "e2>0 & cball x e2 Int affine hull S <= S" 
  2464     using rel_interior_cball[of S] by auto
  2465   have "x : S" using x_def rel_interior_subset by auto
  2466   hence *: "f x : f ` S" by auto
  2467   have "! x:span S. f x = 0 --> x = 0" 
  2468     using assms subspace_span linear_conv_bounded_linear[of f] 
  2469           linear_injective_on_subspace_0[of f "span S"] by auto
  2470   from this obtain e1 where e1_def: "e1>0 & (! x : span S. e1 * norm x <= norm (f x))" 
  2471    using assms injective_imp_isometric[of "span S" f] 
  2472          subspace_span[of S] closed_subspace[of "span S"] by auto
  2473   def e == "e1 * e2" hence "e>0" using e1_def e2_def mult_pos_pos by auto 
  2474   { fix y assume y_def: "y : cball (f x) e Int affine hull (f ` S)"
  2475     from this have "y : f ` (affine hull S)" using affine_hull_linear_image[of f S] assms by auto 
  2476     from this obtain xy where xy_def: "xy : affine hull S & (f xy = y)" by auto
  2477     from this y_def have "norm ((f x)-(f xy))<=e1 * e2" 
  2478       using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
  2479     moreover have "(f x)-(f xy)=f (x-xy)"
  2480        using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
  2481     moreover have "x-xy : span S" 
  2482        using subspace_sub[of "span S" x xy] subspace_span `x : S` xy_def 
  2483              affine_hull_subset_span[of S] span_inc by auto
  2484     moreover hence "e1 * norm (x-xy) <= norm (f (x-xy))" using e1_def by auto
  2485     ultimately have "e1 * norm (x-xy) <= e1 * e2" by auto
  2486     hence "xy : (cball x e2)"  using cball_def[of x e2] dist_norm[of x xy] e1_def by auto
  2487     hence "y : (f ` S)" using xy_def e2_def by auto
  2488   } 
  2489   hence "(f x) : rel_interior (f ` S)" 
  2490      using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e>0` by auto
  2491 } 
  2492 ultimately show ?thesis by auto
  2493 qed
  2494 
  2495 lemma rel_interior_injective_linear_image:
  2496 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  2497 assumes "bounded_linear f" and "inj f"
  2498 shows "rel_interior (f ` S) = f ` (rel_interior S)"
  2499 using assms rel_interior_injective_on_span_linear_image[of f S] 
  2500       subset_inj_on[of f "UNIV" "span S"] by auto
  2501 
  2502 subsection{* Some Properties of subset of standard basis *}
  2503 
  2504 lemma affine_hull_substd_basis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  2505   shows "affine hull (insert 0 {basis i | i. i : d}) =
  2506   {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
  2507  (is "affine hull (insert 0 ?A) = ?B")
  2508 proof- have *:"\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A" by auto
  2509   show ?thesis unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,THEN sym] * ..
  2510 qed
  2511 
  2512 lemma affine_hull_convex_hull: "affine hull (convex hull S) = affine hull S"
  2513 by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
  2514 
  2515 subsection {* Openness and compactness are preserved by convex hull operation. *}
  2516 
  2517 lemma open_convex_hull[intro]:
  2518   fixes s :: "'a::real_normed_vector set"
  2519   assumes "open s"
  2520   shows "open(convex hull s)"
  2521   unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(8)
  2522 proof(rule, rule) fix a
  2523   assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
  2524   then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
  2525 
  2526   from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
  2527     using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
  2528   have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
  2529 
  2530   show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
  2531     apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
  2532   proof-
  2533     show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
  2534       using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
  2535   next  fix y assume "y \<in> cball a (Min i)"
  2536     hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
  2537     { fix x assume "x\<in>t"
  2538       hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
  2539       hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
  2540       moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
  2541       ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
  2542     moreover
  2543     have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
  2544     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
  2545       unfolding setsum_reindex[OF *] o_def using obt(4) by auto
  2546     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
  2547       unfolding setsum_reindex[OF *] o_def using obt(4,5)
  2548       by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
  2549     ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
  2550       apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
  2551       using obt(1, 3) by auto
  2552   qed
  2553 qed
  2554 
  2555 lemma compact_convex_combinations:
  2556   fixes s t :: "'a::real_normed_vector set"
  2557   assumes "compact s" "compact t"
  2558   shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
  2559 proof-
  2560   let ?X = "{0..1} \<times> s \<times> t"
  2561   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  2562   have *:"{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
  2563     apply(rule set_eqI) unfolding image_iff mem_Collect_eq
  2564     apply rule apply auto
  2565     apply (rule_tac x=u in rev_bexI, simp)
  2566     apply (erule rev_bexI, erule rev_bexI, simp)
  2567     by auto
  2568   have "continuous_on ({0..1} \<times> s \<times> t)
  2569      (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  2570     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  2571   thus ?thesis unfolding *
  2572     apply (rule compact_continuous_image)
  2573     apply (intro compact_Times compact_interval assms)
  2574     done
  2575 qed
  2576 
  2577 lemma finite_imp_compact_convex_hull:
  2578   fixes s :: "('a::real_normed_vector) set"
  2579   assumes "finite s" shows "compact (convex hull s)"
  2580 proof (cases "s = {}")
  2581   case True thus ?thesis by simp
  2582 next
  2583   case False with assms show ?thesis
  2584   proof (induct rule: finite_ne_induct)
  2585     case (singleton x) show ?case by simp
  2586   next
  2587     case (insert x A)
  2588     let ?f = "\<lambda>(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
  2589     let ?T = "{0..1::real} \<times> (convex hull A)"
  2590     have "continuous_on ?T ?f"
  2591       unfolding split_def continuous_on by (intro ballI tendsto_intros)
  2592     moreover have "compact ?T"
  2593       by (intro compact_Times compact_interval insert)
  2594     ultimately have "compact (?f ` ?T)"
  2595       by (rule compact_continuous_image)
  2596     also have "?f ` ?T = convex hull (insert x A)"
  2597       unfolding convex_hull_insert [OF `A \<noteq> {}`]
  2598       apply safe
  2599       apply (rule_tac x=a in exI, simp)
  2600       apply (rule_tac x="1 - a" in exI, simp)
  2601       apply fast
  2602       apply (rule_tac x="(u, b)" in image_eqI, simp_all)
  2603       done
  2604     finally show "compact (convex hull (insert x A))" .
  2605   qed
  2606 qed
  2607 
  2608 lemma compact_convex_hull: fixes s::"('a::euclidean_space) set"
  2609   assumes "compact s"  shows "compact(convex hull s)"
  2610 proof(cases "s={}")
  2611   case True thus ?thesis using compact_empty by simp
  2612 next
  2613   case False then obtain w where "w\<in>s" by auto
  2614   show ?thesis unfolding caratheodory[of s]
  2615   proof(induct ("DIM('a) + 1"))
  2616     have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
  2617       using compact_empty by auto
  2618     case 0 thus ?case unfolding * by simp
  2619   next
  2620     case (Suc n)
  2621     show ?case proof(cases "n=0")
  2622       case True have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
  2623         unfolding set_eq_iff and mem_Collect_eq proof(rule, rule)
  2624         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2625         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  2626         show "x\<in>s" proof(cases "card t = 0")
  2627           case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
  2628         next
  2629           case False hence "card t = Suc 0" using t(3) `n=0` by auto
  2630           then obtain a where "t = {a}" unfolding card_Suc_eq by auto
  2631           thus ?thesis using t(2,4) by simp
  2632         qed
  2633       next
  2634         fix x assume "x\<in>s"
  2635         thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2636           apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
  2637       qed thus ?thesis using assms by simp
  2638     next
  2639       case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
  2640         { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
  2641         0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
  2642         unfolding set_eq_iff and mem_Collect_eq proof(rule,rule)
  2643         fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  2644           0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
  2645         then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
  2646           "0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n"  "v \<in> convex hull t" by auto
  2647         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
  2648           apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
  2649           using obt(7) and hull_mono[of t "insert u t"] by auto
  2650         ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2651           apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
  2652       next
  2653         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2654         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  2655         let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  2656           0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
  2657         show ?P proof(cases "card t = Suc n")
  2658           case False hence "card t \<le> n" using t(3) by auto
  2659           thus ?P apply(rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) using `w\<in>s` and t
  2660             by(auto intro!: exI[where x=t])
  2661         next
  2662           case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
  2663           show ?P proof(cases "u={}")
  2664             case True hence "x=a" using t(4)[unfolded au] by auto
  2665             show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
  2666               using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
  2667           next
  2668             case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
  2669               using t(4)[unfolded au convex_hull_insert[OF False]] by auto
  2670             have *:"1 - vx = ux" using obt(3) by auto
  2671             show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
  2672               using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
  2673               by(auto intro!: exI[where x=u])
  2674           qed
  2675         qed
  2676       qed
  2677       thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
  2678     qed
  2679   qed
  2680 qed
  2681 
  2682 subsection {* Extremal points of a simplex are some vertices. *}
  2683 
  2684 lemma dist_increases_online:
  2685   fixes a b d :: "'a::real_inner"
  2686   assumes "d \<noteq> 0"
  2687   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
  2688 proof(cases "inner a d - inner b d > 0")
  2689   case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
  2690     apply(rule_tac add_pos_pos) using assms by auto
  2691   thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  2692     by (simp add: algebra_simps inner_commute)
  2693 next
  2694   case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
  2695     apply(rule_tac add_pos_nonneg) using assms by auto
  2696   thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  2697     by (simp add: algebra_simps inner_commute)
  2698 qed
  2699 
  2700 lemma norm_increases_online:
  2701   fixes d :: "'a::real_inner"
  2702   shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
  2703   using dist_increases_online[of d a 0] unfolding dist_norm by auto
  2704 
  2705 lemma simplex_furthest_lt:
  2706   fixes s::"'a::real_inner set" assumes "finite s"
  2707   shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
  2708 proof(induct_tac rule: finite_induct[of s])
  2709   fix x s assume as:"finite s" "x\<notin>s" "\<forall>x\<in>convex hull s. x \<notin> s \<longrightarrow> (\<exists>y\<in>convex hull s. norm (x - a) < norm (y - a))"
  2710   show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow> (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
  2711   proof(rule,rule,cases "s = {}")
  2712     case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
  2713     obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *\<^sub>R x + v *\<^sub>R b"
  2714       using y(1)[unfolded convex_hull_insert[OF False]] by auto
  2715     show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
  2716     proof(cases "y\<in>convex hull s")
  2717       case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
  2718         using as(3)[THEN bspec[where x=y]] and y(2) by auto
  2719       thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
  2720     next
  2721       case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
  2722         assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
  2723         thus ?thesis using False and obt(4) by auto
  2724       next
  2725         assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
  2726         thus ?thesis using y(2) by auto
  2727       next
  2728         assume "u\<noteq>0" "v\<noteq>0"
  2729         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
  2730         have "x\<noteq>b" proof(rule ccontr) 
  2731           assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
  2732             using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
  2733           thus False using obt(4) and False by simp qed
  2734         hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
  2735         show ?thesis using dist_increases_online[OF *, of a y]
  2736         proof(erule_tac disjE)
  2737           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
  2738           hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
  2739             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  2740           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
  2741             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  2742             apply(rule_tac x="u + w" in exI) apply rule defer 
  2743             apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  2744           ultimately show ?thesis by auto
  2745         next
  2746           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
  2747           hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
  2748             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  2749           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
  2750             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  2751             apply(rule_tac x="u - w" in exI) apply rule defer 
  2752             apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  2753           ultimately show ?thesis by auto
  2754         qed
  2755       qed auto
  2756     qed
  2757   qed auto
  2758 qed (auto simp add: assms)
  2759 
  2760 lemma simplex_furthest_le:
  2761   fixes s :: "('a::real_inner) set"
  2762   assumes "finite s" "s \<noteq> {}"
  2763   shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
  2764 proof-
  2765   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  2766   then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
  2767     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
  2768     unfolding dist_commute[of a] unfolding dist_norm by auto
  2769   thus ?thesis proof(cases "x\<in>s")
  2770     case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
  2771       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
  2772     thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
  2773   qed auto
  2774 qed
  2775 
  2776 lemma simplex_furthest_le_exists:
  2777   fixes s :: "('a::real_inner) set"
  2778   shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
  2779   using simplex_furthest_le[of s] by (cases "s={}")auto
  2780 
  2781 lemma simplex_extremal_le:
  2782   fixes s :: "('a::real_inner) set"
  2783   assumes "finite s" "s \<noteq> {}"
  2784   shows "\<exists>u\<in>s. \<exists>v\<in>s. \<forall>x\<in>convex hull s. \<forall>y \<in> convex hull s. norm(x - y) \<le> norm(u - v)"
  2785 proof-
  2786   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  2787   then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
  2788     "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
  2789     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
  2790   thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
  2791     assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
  2792       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
  2793     thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
  2794   next
  2795     assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
  2796       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
  2797     thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
  2798       by (auto simp add: norm_minus_commute)
  2799   qed auto
  2800 qed 
  2801 
  2802 lemma simplex_extremal_le_exists:
  2803   fixes s :: "('a::real_inner) set"
  2804   shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
  2805   \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
  2806   using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
  2807 
  2808 subsection {* Closest point of a convex set is unique, with a continuous projection. *}
  2809 
  2810 definition
  2811   closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a" where
  2812  "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
  2813 
  2814 lemma closest_point_exists:
  2815   assumes "closed s" "s \<noteq> {}"
  2816   shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
  2817   unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
  2818   using distance_attains_inf[OF assms(1,2), of a] by auto
  2819 
  2820 lemma closest_point_in_set:
  2821   "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
  2822   by(meson closest_point_exists)
  2823 
  2824 lemma closest_point_le:
  2825   "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
  2826   using closest_point_exists[of s] by auto
  2827 
  2828 lemma closest_point_self:
  2829   assumes "x \<in> s"  shows "closest_point s x = x"
  2830   unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
  2831   using assms by auto
  2832 
  2833 lemma closest_point_refl:
  2834  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
  2835   using closest_point_in_set[of s x] closest_point_self[of x s] by auto
  2836 
  2837 lemma closer_points_lemma:
  2838   assumes "inner y z > 0"
  2839   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
  2840 proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
  2841   thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
  2842     fix v assume "0<v" "v \<le> inner y z / inner z z"
  2843     thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
  2844       by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
  2845   qed(rule divide_pos_pos, auto) qed
  2846 
  2847 lemma closer_point_lemma:
  2848   assumes "inner (y - x) (z - x) > 0"
  2849   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
  2850 proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
  2851     using closer_points_lemma[OF assms] by auto
  2852   show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
  2853     unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
  2854 
  2855 lemma any_closest_point_dot:
  2856   assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  2857   shows "inner (a - x) (y - x) \<le> 0"
  2858 proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
  2859   then obtain u where u:"u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
  2860   let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
  2861   thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
  2862 
  2863 lemma any_closest_point_unique:
  2864   fixes x :: "'a::real_inner"
  2865   assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
  2866   "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
  2867   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)]
  2868   unfolding norm_pths(1) and norm_le_square
  2869   by (auto simp add: algebra_simps)
  2870 
  2871 lemma closest_point_unique:
  2872   assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  2873   shows "x = closest_point s a"
  2874   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
  2875   using closest_point_exists[OF assms(2)] and assms(3) by auto
  2876 
  2877 lemma closest_point_dot:
  2878   assumes "convex s" "closed s" "x \<in> s"
  2879   shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
  2880   apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
  2881   using closest_point_exists[OF assms(2)] and assms(3) by auto
  2882 
  2883 lemma closest_point_lt:
  2884   assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
  2885   shows "dist a (closest_point s a) < dist a x"
  2886   apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
  2887   apply(rule closest_point_unique[OF assms(1-3), of a])
  2888   using closest_point_le[OF assms(2), of _ a] by fastforce
  2889 
  2890 lemma closest_point_lipschitz:
  2891   assumes "convex s" "closed s" "s \<noteq> {}"
  2892   shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
  2893 proof-
  2894   have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
  2895        "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
  2896     apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
  2897     using closest_point_exists[OF assms(2-3)] by auto
  2898   thus ?thesis unfolding dist_norm and norm_le
  2899     using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
  2900     by (simp add: inner_add inner_diff inner_commute) qed
  2901 
  2902 lemma continuous_at_closest_point:
  2903   assumes "convex s" "closed s" "s \<noteq> {}"
  2904   shows "continuous (at x) (closest_point s)"
  2905   unfolding continuous_at_eps_delta 
  2906   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
  2907 
  2908 lemma continuous_on_closest_point:
  2909   assumes "convex s" "closed s" "s \<noteq> {}"
  2910   shows "continuous_on t (closest_point s)"
  2911 by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
  2912 
  2913 subsubsection {* Various point-to-set separating/supporting hyperplane theorems. *}
  2914 
  2915 lemma supporting_hyperplane_closed_point:
  2916   fixes z :: "'a::{real_inner,heine_borel}"
  2917   assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
  2918   shows "\<exists>a b. \<exists>y\<in>s. inner a z < b \<and> (inner a y = b) \<and> (\<forall>x\<in>s. inner a x \<ge> b)"
  2919 proof-
  2920   from distance_attains_inf[OF assms(2-3)] obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x" by auto
  2921   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)
  2922     apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
  2923     show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
  2924       unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
  2925   next
  2926     fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
  2927       using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
  2928     assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
  2929       "v>0" "v\<le>1" "dist (y + v *\<^sub>R (x - y)) z < dist y z" using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
  2930     thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
  2931   qed auto
  2932 qed
  2933 
  2934 lemma separating_hyperplane_closed_point:
  2935   fixes z :: "'a::{real_inner,heine_borel}"
  2936   assumes "convex s" "closed s" "z \<notin> s"
  2937   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
  2938 proof(cases "s={}")
  2939   case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
  2940     using less_le_trans[OF _ inner_ge_zero[of z]] by auto
  2941 next
  2942   case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
  2943     using distance_attains_inf[OF assms(2) False] by auto
  2944   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
  2945     apply rule defer apply rule proof-
  2946     fix x assume "x\<in>s"
  2947     have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
  2948       assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
  2949       then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
  2950       thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
  2951         using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
  2952         using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
  2953     moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
  2954     hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
  2955     ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
  2956       unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
  2957   qed(insert `y\<in>s` `z\<notin>s`, auto)
  2958 qed
  2959 
  2960 lemma separating_hyperplane_closed_0:
  2961   assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 \<notin> s"
  2962   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
  2963   proof(cases "s={}")
  2964   case True have "norm ((basis 0)::'a) = 1" by auto
  2965   hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
  2966     apply(subst norm_le_zero_iff[THEN sym]) by auto
  2967   thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
  2968     using True using DIM_positive[where 'a='a] by auto
  2969 next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
  2970     apply - apply(erule exE)+ unfolding inner_zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
  2971 
  2972 subsubsection {* Now set-to-set for closed/compact sets *}
  2973 
  2974 lemma separating_hyperplane_closed_compact:
  2975   assumes "convex (s::('a::euclidean_space) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
  2976   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  2977 proof(cases "s={}")
  2978   case True
  2979   obtain b where b:"b>0" "\<forall>x\<in>t. norm x \<le> b" using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
  2980   obtain z::"'a" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
  2981   hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
  2982   then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
  2983     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
  2984   thus ?thesis using True by auto
  2985 next
  2986   case False then obtain y where "y\<in>s" by auto
  2987   obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
  2988     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
  2989     using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
  2990   hence ab:"\<forall>x\<in>s. \<forall>y\<in>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)
  2991   def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
  2992   show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
  2993     apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
  2994     from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
  2995       apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
  2996     hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
  2997     fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
  2998   next
  2999     fix x assume "x\<in>s" 
  3000     hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
  3001       using ab[THEN bspec[where x=x]] by auto
  3002     thus "k + b / 2 < inner a x" using `0 < b` by auto
  3003   qed
  3004 qed
  3005 
  3006 lemma separating_hyperplane_compact_closed:
  3007   fixes s :: "('a::euclidean_space) set"
  3008   assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
  3009   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  3010 proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
  3011     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
  3012   thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
  3013 
  3014 subsubsection {* General case without assuming closure and getting non-strict separation *}
  3015 
  3016 lemma separating_hyperplane_set_0:
  3017   assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
  3018   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
  3019 proof- let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
  3020   have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
  3021     apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
  3022     defer apply(rule,rule,erule conjE) proof-
  3023     fix f assume as:"f \<subseteq> ?k ` s" "finite f"
  3024     obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
  3025     then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
  3026       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
  3027       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
  3028       using subset_hull[of convex, OF assms(1), THEN sym, of c] by auto
  3029     hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)" apply(rule_tac x="inverse(norm a) *\<^sub>R a" in exI)
  3030        using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
  3031        apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
  3032        by(auto simp add: inner_commute del: ballE elim!: ballE)
  3033     thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
  3034   qed(insert closed_halfspace_ge, auto)
  3035   then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
  3036   thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
  3037 
  3038 lemma separating_hyperplane_sets:
  3039   assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
  3040   shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
  3041 proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
  3042   obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x" 
  3043     using assms(3-5) by auto 
  3044   hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
  3045     by (force simp add: inner_diff)
  3046   thus ?thesis
  3047     apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
  3048     apply auto
  3049     apply (rule Sup[THEN isLubD2]) 
  3050     prefer 4
  3051     apply (rule Sup_least) 
  3052      using assms(3-5) apply (auto simp add: setle_def)
  3053     apply metis
  3054     done
  3055 qed
  3056 
  3057 subsection {* More convexity generalities *}
  3058 
  3059 lemma convex_closure:
  3060   fixes s :: "'a::real_normed_vector set"
  3061   assumes "convex s" shows "convex(closure s)"
  3062   unfolding convex_def Ball_def closure_sequential
  3063   apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
  3064   apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
  3065   apply(rule assms[unfolded convex_def, rule_format]) prefer 6
  3066   by (auto del: tendsto_const intro!: tendsto_intros)
  3067 
  3068 lemma convex_interior:
  3069   fixes s :: "'a::real_normed_vector set"
  3070   assumes "convex s" shows "convex(interior s)"
  3071   unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
  3072   fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
  3073   fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
  3074   show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
  3075     apply rule unfolding subset_eq defer apply rule proof-
  3076     fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
  3077     hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
  3078       apply(rule_tac assms[unfolded convex_alt, rule_format])
  3079       using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
  3080     thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
  3081 
  3082 lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
  3083   using hull_subset[of s convex] convex_hull_empty by auto
  3084 
  3085 subsection {* Moving and scaling convex hulls. *}
  3086 
  3087 lemma convex_hull_translation_lemma:
  3088   "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
  3089 by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono)
  3090 
  3091 lemma convex_hull_bilemma: fixes neg
  3092   assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
  3093   shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
  3094   \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
  3095   using assms by(metis subset_antisym) 
  3096 
  3097 lemma convex_hull_translation:
  3098   "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
  3099   apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
  3100 
  3101 lemma convex_hull_scaling_lemma:
  3102  "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  3103 by (metis convex_convex_hull convex_scaling hull_subset subset_hull subset_image_iff)
  3104 
  3105 lemma convex_hull_scaling:
  3106   "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  3107   apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
  3108   unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
  3109 
  3110 lemma convex_hull_affinity:
  3111   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
  3112 by(simp only: image_image[THEN sym] convex_hull_scaling convex_hull_translation)
  3113 
  3114 subsection {* Convexity of cone hulls *}
  3115 
  3116 lemma convex_cone_hull:
  3117 assumes "convex S"
  3118 shows "convex (cone hull S)"
  3119 proof-
  3120 { fix x y assume xy_def: "x : cone hull S & y : cone hull S"
  3121   hence "S ~= {}" using cone_hull_empty_iff[of S] by auto
  3122   fix u v assume uv_def: "u>=0 & v>=0 & (u :: real)+v=1"
  3123   hence *: "u *\<^sub>R x : cone hull S & v *\<^sub>R y : cone hull S"
  3124      using cone_cone_hull[of S] xy_def cone_def[of "cone hull S"] by auto
  3125   from * obtain cx xx where x_def: "u *\<^sub>R x = cx *\<^sub>R xx & (cx :: real)>=0 & xx : S"
  3126      using cone_hull_expl[of S] by auto
  3127   from * obtain cy yy where y_def: "v *\<^sub>R y = cy *\<^sub>R yy & (cy :: real)>=0 & yy : S"
  3128      using cone_hull_expl[of S] by auto
  3129   { assume "cx+cy<=0" hence "u *\<^sub>R x=0 & v *\<^sub>R y=0" using x_def y_def by auto
  3130     hence "u *\<^sub>R x+ v *\<^sub>R y = 0" by auto
  3131     hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using cone_hull_contains_0[of S] `S ~= {}` by auto
  3132   }
  3133   moreover
  3134   { assume "cx+cy>0"
  3135     hence "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy : S"
  3136       using assms mem_convex_alt[of S xx yy cx cy] x_def y_def by auto
  3137     hence "cx *\<^sub>R xx + cy *\<^sub>R yy : cone hull S"
  3138       using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"]
  3139       `cx+cy>0` by (auto simp add: scaleR_right_distrib)
  3140     hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using x_def y_def by auto
  3141   }
  3142   moreover have "(cx+cy<=0) | (cx+cy>0)" by auto
  3143   ultimately have "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" by blast
  3144 } from this show ?thesis unfolding convex_def by auto
  3145 qed
  3146 
  3147 lemma cone_convex_hull:
  3148 assumes "cone S"
  3149 shows "cone (convex hull S)"
  3150 proof-
  3151 { assume "S = {}" hence ?thesis by auto }
  3152 moreover
  3153 { assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  3154   { fix c assume "(c :: real)>0"
  3155     hence "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
  3156        using convex_hull_scaling[of _ S] by auto
  3157     also have "...=convex hull S" using * `c>0` by auto
  3158     finally have "op *\<^sub>R c ` (convex hull S) = convex hull S" by auto
  3159   }
  3160   hence "0 : convex hull S & (!c. c>0 --> (op *\<^sub>R c ` (convex hull S)) = (convex hull S))"
  3161      using * hull_subset[of S convex] by auto
  3162   hence ?thesis using `S ~= {}` cone_iff[of "convex hull S"] by auto
  3163 }
  3164 ultimately show ?thesis by blast
  3165 qed
  3166 
  3167 subsection {* Convex set as intersection of halfspaces *}
  3168 
  3169 lemma convex_halfspace_intersection:
  3170   fixes s :: "('a::euclidean_space) set"
  3171   assumes "closed s" "convex s"
  3172   shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
  3173   apply(rule set_eqI, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
  3174   fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
  3175   hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
  3176   thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
  3177     apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
  3178 qed auto
  3179 
  3180 subsection {* Radon's theorem (from Lars Schewe) *}
  3181 
  3182 lemma radon_ex_lemma:
  3183   assumes "finite c" "affine_dependent c"
  3184   shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
  3185 proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
  3186   thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
  3187     and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
  3188 
  3189 lemma radon_s_lemma:
  3190   assumes "finite s" "setsum f s = (0::real)"
  3191   shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
  3192 proof- have *:"\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" by auto
  3193   show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  3194     using assms(2) by assumption qed
  3195 
  3196 lemma radon_v_lemma:
  3197   assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
  3198   shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
  3199 proof-
  3200   have *:"\<And>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 
  3201   show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  3202     using assms(2) by assumption qed
  3203 
  3204 lemma radon_partition:
  3205   assumes "finite c" "affine_dependent c"
  3206   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
  3207   obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
  3208   have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
  3209   def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
  3210   have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
  3211     case False hence "u v < 0" by auto
  3212     thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
  3213       case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
  3214     next
  3215       case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
  3216       thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
  3217   qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
  3218 
  3219   hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
  3220   moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
  3221     "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
  3222     using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
  3223   hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
  3224    "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)" 
  3225     unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
  3226   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
  3227     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
  3228 
  3229   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
  3230     apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
  3231     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
  3232     by(auto simp add: setsum_negf setsum_right_distrib[THEN sym])
  3233   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
  3234     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
  3235   hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
  3236     apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
  3237     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
  3238     by(auto simp add: setsum_negf setsum_right_distrib[THEN sym])
  3239   ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
  3240 qed
  3241 
  3242 lemma radon: assumes "affine_dependent c"
  3243   obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
  3244 proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
  3245   hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
  3246   from radon_partition[OF *] guess m .. then guess p ..
  3247   thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
  3248 
  3249 subsection {* Helly's theorem *}
  3250 
  3251 lemma helly_induct: fixes f::"('a::euclidean_space) set set"
  3252   assumes "card f = n" "n \<ge> DIM('a) + 1"
  3253   "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  3254   shows "\<Inter> f \<noteq> {}"
  3255 using assms proof(induct n arbitrary: f)
  3256 case (Suc n)
  3257 have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
  3258 show "\<Inter> f \<noteq> {}" apply(cases "n = DIM('a)") apply(rule Suc(5)[rule_format])
  3259   unfolding `card f = Suc n` proof-
  3260   assume ng:"n \<noteq> DIM('a)" hence "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" apply(rule_tac bchoice) unfolding ex_in_conv
  3261     apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
  3262     defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
  3263   then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
  3264   show ?thesis proof(cases "inj_on X f")
  3265     case False then obtain s t where st:"s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t" unfolding inj_on_def by auto
  3266     hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
  3267     show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
  3268       apply(rule, rule X[rule_format]) using X st by auto
  3269   next case True then obtain m p where mp:"m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
  3270       using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
  3271       unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
  3272     have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
  3273     then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
  3274     hence "f \<union> (g \<union> h) = f" by auto
  3275     hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
  3276       unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
  3277     have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
  3278     have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
  3279       apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  unfolding subset_eq
  3280       apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
  3281       fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
  3282       thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
  3283       fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
  3284       thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
  3285     qed(auto)
  3286     thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
  3287 qed(auto) qed(auto)
  3288 
  3289 lemma helly: fixes f::"('a::euclidean_space) set set"
  3290   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
  3291           "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  3292   shows "\<Inter> f \<noteq>{}"
  3293   apply(rule helly_induct) using assms by auto
  3294 
  3295 subsection {* Homeomorphism of all convex compact sets with nonempty interior *}
  3296 
  3297 lemma compact_frontier_line_lemma:
  3298   fixes s :: "('a::euclidean_space) set"
  3299   assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
  3300   obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
  3301 proof-
  3302   obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
  3303   let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
  3304   have A:"?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
  3305     by auto
  3306   have *:"\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
  3307   have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
  3308     apply(rule, intro continuous_intros)
  3309     by(rule compact_interval)
  3310   moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}" apply(rule *[OF _ assms(2)])
  3311     unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
  3312   ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
  3313     "y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
  3314 
  3315   have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
  3316   { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
  3317     hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
  3318       using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
  3319     hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
  3320       apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
  3321       using as(1) `u\<ge>0` by(auto simp add:field_simps) 
  3322     hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
  3323   } note u_max = this
  3324 
  3325   have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[THEN sym]
  3326     prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
  3327     fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
  3328     hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
  3329     thus False using u_max[OF _ as] by auto
  3330   qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
  3331   thus ?thesis by(metis that[of u] u_max obt(1))
  3332 qed
  3333 
  3334 lemma starlike_compact_projective:
  3335   assumes "compact s" "cball (0::'a::euclidean_space) 1 \<subseteq> s "
  3336   "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
  3337   shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
  3338 proof-
  3339   have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
  3340   def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
  3341   have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
  3342     using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
  3343   have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
  3344 
  3345   have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
  3346     apply rule unfolding pi_def
  3347     apply (intro continuous_intros)
  3348     apply simp
  3349     done
  3350   def sphere \<equiv> "{x::'a. norm x = 1}"
  3351   have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x" unfolding pi_def sphere_def by auto
  3352 
  3353   have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
  3354   have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
  3355     fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
  3356     hence "x\<noteq>0" using `0\<notin>frontier s` by auto
  3357     obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
  3358       using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
  3359     have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
  3360       assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
  3361       assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
  3362         using v and x and fs unfolding inverse_less_1_iff by auto qed
  3363     show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
  3364       assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
  3365         using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
  3366 
  3367   have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
  3368     apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
  3369     apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_eqI,rule) 
  3370     unfolding inj_on_def prefer 3 apply(rule,rule,rule)
  3371   proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
  3372     thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
  3373   next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
  3374     then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
  3375       using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
  3376     thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
  3377   next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
  3378     hence xys:"x\<in>s" "y\<in>s" using fs by auto
  3379     from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
  3380     from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
  3381     from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto 
  3382     have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
  3383       unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
  3384     hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
  3385       using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
  3386       using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
  3387       using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
  3388     thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
  3389   qed(insert `0 \<notin> frontier s`, auto)
  3390   then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
  3391     "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
  3392   
  3393   have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
  3394     apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
  3395 
  3396   { fix x assume as:"x \<in> cball (0::'a) 1"
  3397     have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
  3398       case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
  3399       thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
  3400         apply(rule_tac fs[unfolded subset_eq, rule_format])
  3401         unfolding surf(5)[THEN sym] by auto
  3402     next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
  3403         unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
  3404 
  3405   { fix x assume "x\<in>s"
  3406     hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
  3407       case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
  3408     next let ?a = "inverse (norm (surf (pi x)))"
  3409       case False hence invn:"inverse (norm x) \<noteq> 0" by auto
  3410       from False have pix:"pi x\<in>sphere" using pi(1) by auto
  3411       hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
  3412       hence **:"norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
  3413       hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
  3414         apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
  3415       have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
  3416       hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
  3417         unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
  3418       moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))" 
  3419         unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
  3420       moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
  3421       hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
  3422         using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
  3423         using False `x\<in>s` by(auto simp add:field_simps)
  3424       ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
  3425         apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
  3426         unfolding pi(2)[OF `?a > 0`] by auto
  3427     qed } note hom2 = this
  3428 
  3429   show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
  3430     apply(rule compact_cball) defer apply(rule set_eqI, rule, erule imageE, drule hom)
  3431     prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
  3432     fix x::"'a" assume as:"x \<in> cball 0 1"
  3433     thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
  3434       case False thus ?thesis apply (intro continuous_intros)
  3435         using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
  3436     next obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
  3437       hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer 
  3438         apply(erule_tac x="basis 0" in ballE)
  3439         unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
  3440         by auto
  3441       case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
  3442         apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
  3443         unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
  3444         fix e and x::"'a" assume as:"norm x < e / B" "0 < norm x" "0<e"
  3445         hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
  3446         hence "norm (surf (pi x)) \<le> B" using B fs by auto
  3447         hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
  3448         also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
  3449         also have "\<dots> = e" using `B>0` by auto
  3450         finally show "norm x * norm (surf (pi x)) < e" by assumption
  3451       qed(insert `B>0`, auto) qed
  3452   next { fix x assume as:"surf (pi x) = 0"
  3453       have "x = 0" proof(rule ccontr)
  3454         assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
  3455         hence "surf (pi x) \<in> frontier s" using surf(5) by auto
  3456         thus False using `0\<notin>frontier s` unfolding as by simp qed
  3457     } note surf_0 = this
  3458     show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
  3459       fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
  3460       thus "x=y" proof(cases "x=0 \<or> y=0") 
  3461         case True thus ?thesis using as by(auto elim: surf_0) next
  3462         case False
  3463         hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
  3464           using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
  3465         moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
  3466         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 
  3467         moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
  3468         ultimately show ?thesis using injpi by auto qed qed
  3469   qed auto qed
  3470 
  3471 lemma homeomorphic_convex_compact_lemma:
  3472   fixes s :: "('a::euclidean_space) set"
  3473   assumes "convex s" and "compact s" and "cball 0 1 \<subseteq> s"
  3474   shows "s homeomorphic (cball (0::'a) 1)"
  3475 proof (rule starlike_compact_projective[OF assms(2-3)], clarify)
  3476   fix x u assume "x \<in> s" and "0 \<le> u" and "u < (1::real)"
  3477   have "open (ball (u *\<^sub>R x) (1 - u))" by (rule open_ball)
  3478   moreover have "u *\<^sub>R x \<in> ball (u *\<^sub>R x) (1 - u)"
  3479     unfolding centre_in_ball using `u < 1` by simp
  3480   moreover have "ball (u *\<^sub>R x) (1 - u) \<subseteq> s"
  3481   proof
  3482     fix y assume "y \<in> ball (u *\<^sub>R x) (1 - u)"
  3483     hence "dist (u *\<^sub>R x) y < 1 - u" unfolding mem_ball .
  3484     with `u < 1` have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> cball 0 1"
  3485       by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
  3486     with assms(3) have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s" ..
  3487     with assms(1) have "(1 - u) *\<^sub>R ((y - u *\<^sub>R x) /\<^sub>R (1 - u)) + u *\<^sub>R x \<in> s"
  3488       using `x \<in> s` `0 \<le> u` `u < 1` [THEN less_imp_le] by (rule mem_convex)
  3489     thus "y \<in> s" using `u < 1` by simp
  3490   qed
  3491   ultimately have "u *\<^sub>R x \<in> interior s" ..
  3492   thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
  3493 
  3494 lemma homeomorphic_convex_compact_cball: fixes e::real and s::"('a::euclidean_space) set"
  3495   assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
  3496   shows "s homeomorphic (cball (b::'a) e)"
  3497 proof- obtain a where "a\<in>interior s" using assms(3) by auto
  3498   then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
  3499   let ?d = "inverse d" and ?n = "0::'a"
  3500   have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
  3501     apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
  3502     apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
  3503     by(auto simp add: mult_right_le_one_le)
  3504   hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
  3505     using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
  3506     using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
  3507   thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
  3508     apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
  3509     using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
  3510 
  3511 lemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) set"
  3512   assumes "convex s" "compact s" "interior s \<noteq> {}"
  3513           "convex t" "compact t" "interior t \<noteq> {}"
  3514   shows "s homeomorphic t"
  3515   using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
  3516 
  3517 subsection {* Epigraphs of convex functions *}
  3518 
  3519 definition "epigraph s (f::_ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
  3520 
  3521 lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
  3522 
  3523 (** This might break sooner or later. In fact it did already once. **)
  3524 lemma convex_epigraph: 
  3525   "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
  3526   unfolding convex_def convex_on_def
  3527   unfolding Ball_def split_paired_All epigraph_def
  3528   unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
  3529   apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
  3530   apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
  3531   apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)
  3532 
  3533 lemma convex_epigraphI:
  3534   "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex(epigraph s f)"
  3535 unfolding convex_epigraph by auto
  3536 
  3537 lemma convex_epigraph_convex:
  3538   "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
  3539 by(simp add: convex_epigraph)
  3540 
  3541 subsubsection {* Use this to derive general bound property of convex function *}
  3542 
  3543 lemma convex_on:
  3544   assumes "convex s"
  3545   shows "convex_on s f \<longleftrightarrow> (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
  3546    f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
  3547   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
  3548   unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
  3549   apply safe
  3550   apply (drule_tac x=k in spec)
  3551   apply (drule_tac x=u in spec)
  3552   apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
  3553   apply simp
  3554   using assms[unfolded convex] apply simp
  3555   apply(rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
  3556   defer apply(rule setsum_mono) apply(erule_tac x=i in allE) unfolding real_scaleR_def
  3557   apply(rule mult_left_mono)using assms[unfolded convex] by auto
  3558 
  3559 
  3560 subsection {* Convexity of general and special intervals *}
  3561 
  3562 lemma convexI: (* TODO: move to Library/Convex.thy *)
  3563   assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
  3564   shows "convex s"
  3565 using assms unfolding convex_def by fast
  3566 
  3567 lemma is_interval_convex:
  3568   fixes s :: "('a::euclidean_space) set"
  3569   assumes "is_interval s" shows "convex s"
  3570 proof (rule convexI)
  3571   fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
  3572   hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
  3573   { fix a b assume "\<not> b \<le> u * a + v * b"
  3574     hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
  3575     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)
  3576     hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
  3577   } moreover
  3578   { fix a b assume "\<not> u * a + v * b \<le> a"
  3579     hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
  3580     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
  3581     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
  3582   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
  3583     using as(3-) DIM_positive[where 'a='a] by auto qed
  3584 
  3585 lemma is_interval_connected:
  3586   fixes s :: "('a::euclidean_space) set"
  3587   shows "is_interval s \<Longrightarrow> connected s"
  3588   using is_interval_convex convex_connected by auto
  3589 
  3590 lemma convex_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"
  3591   apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
  3592 
  3593 (* FIXME: rewrite these lemmas without using vec1
  3594 subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
  3595 
  3596 lemma is_interval_1:
  3597   "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b \<longrightarrow> x \<in> s)"
  3598   unfolding is_interval_def forall_1 by auto
  3599 
  3600 lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
  3601   apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
  3602   apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
  3603   fix a b x assume as:"connected s" "a \<in> s" "b \<in> s" "dest_vec1 a \<le> dest_vec1 x" "dest_vec1 x \<le> dest_vec1 b" "x\<notin>s"
  3604   hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
  3605   let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
  3606   { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
  3607     using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
  3608   moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
  3609   hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
  3610   ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
  3611     apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
  3612     apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
  3613     by(auto simp add: field_simps) qed
  3614 
  3615 lemma is_interval_convex_1:
  3616   "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
  3617 by(metis is_interval_convex convex_connected is_interval_connected_1)
  3618 
  3619 lemma convex_connected_1:
  3620   "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
  3621 by(metis is_interval_convex convex_connected is_interval_connected_1)
  3622 *)
  3623 subsection {* Another intermediate value theorem formulation *}
  3624 
  3625 lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3626   assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
  3627   shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
  3628 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
  3629     using assms(1) by auto
  3630   thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
  3631     using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
  3632     using assms by(auto intro!: imageI) qed
  3633 
  3634 lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3635   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
  3636    \<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
  3637 by(rule ivt_increasing_component_on_1)
  3638   (auto simp add: continuous_at_imp_continuous_on)
  3639 
  3640 lemma ivt_decreasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3641   assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
  3642   shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
  3643   apply(subst neg_equal_iff_equal[THEN sym])
  3644   using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"]
  3645   using assms using continuous_on_minus by auto
  3646 
  3647 lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3648   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
  3649     \<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
  3650 by(rule ivt_decreasing_component_on_1)
  3651   (auto simp: continuous_at_imp_continuous_on)
  3652 
  3653 subsection {* A bound within a convex hull, and so an interval *}
  3654 
  3655 lemma convex_on_convex_hull_bound:
  3656   assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
  3657   shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
  3658   fix x assume "x\<in>convex hull s"
  3659   then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
  3660     unfolding convex_hull_indexed mem_Collect_eq by auto
  3661   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
  3662     unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
  3663     using assms(2) obt(1) by auto
  3664   thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
  3665     unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
  3666 
  3667 lemma unit_interval_convex_hull:
  3668   "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
  3669   (is "?int = convex hull ?points")
  3670 proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
  3671   { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n" 
  3672   hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
  3673     case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
  3674     thus "x\<in>convex hull ?points" using 01 by auto
  3675   next
  3676     case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
  3677       case True hence "x = 0" apply(subst euclidean_eq) by auto
  3678       thus "x\<in>convex hull ?points" using 01 by auto
  3679     next
  3680       case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
  3681       have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
  3682       then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
  3683       have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
  3684         unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
  3685         defer apply(rule_tac x=j in bexI) using i' by auto
  3686       have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
  3687         using i'(2-) `x$$i \<noteq> 0` by auto
  3688       show ?thesis proof(cases "x$$i=1")
  3689         case True have "\<forall>j\<in>{i. i<DIM('a) \<and> x$$i \<noteq> 0}. x$$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
  3690         proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
  3691           hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
  3692           hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto 
  3693           hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
  3694           thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
  3695         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
  3696           by auto
  3697       next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
  3698         case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
  3699           apply(subst euclidean_eq) by(auto simp add: field_simps)
  3700         { fix j assume j:"j<DIM('a)"
  3701           have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
  3702             apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
  3703             using Suc(2)[unfolded mem_interval, rule_format, of j] using j
  3704             by(auto simp add:field_simps)
  3705           hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
  3706         moreover have "i\<in>{j. j<DIM('a) \<and> x$$j \<noteq> 0} - {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}"
  3707           using i01 using i'(3) by auto
  3708         hence "{j. j<DIM('a) \<and> x$$j \<noteq> 0} \<noteq> {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}" using i'(3) by blast
  3709         hence **:"{j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<subset> {j. j<DIM('a) \<and> x$$j \<noteq> 0}" apply - apply rule 
  3710           by auto
  3711         have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
  3712           using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
  3713         ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
  3714           apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
  3715           unfolding mem_interval using i01 Suc(3) by auto
  3716       qed qed qed } note * = this
  3717   have **:"DIM('a) = card {..<DIM('a)}" by auto
  3718   show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
  3719     apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **) 
  3720     apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
  3721     unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
  3722     by auto qed
  3723 
  3724 text {* And this is a finite set of vertices. *}
  3725 
  3726 lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
  3727   apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
  3728   apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
  3729   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
  3730   fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
  3731   show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
  3732     apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
  3733     using as apply(subst euclidean_eq) by auto qed auto
  3734 
  3735 text {* Hence any cube (could do any nonempty interval). *}
  3736 
  3737 lemma cube_convex_hull:
  3738   assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
  3739   "finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
  3740   let ?d = "(\<chi>\<chi> i. d)::'a"
  3741   have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_eqI, rule)
  3742     unfolding image_iff defer apply(erule bexE) proof-
  3743     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
  3744     { fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
  3745         using as[unfolded mem_interval, THEN spec[where x=i]] i
  3746         by auto
  3747       hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
  3748         apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
  3749         using assms by(auto simp add: field_simps)
  3750       hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
  3751             "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
  3752     hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
  3753       by(auto simp add: field_simps)
  3754     thus "\<exists>z\<in>{0..\<chi>\<chi> i.1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI) 
  3755       using assms by auto
  3756   next
  3757     fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z" 
  3758     have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
  3759       using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
  3760       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
  3761       using assms by auto
  3762     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
  3763       apply(erule_tac x=i in allE) using assms by auto qed
  3764   obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
  3765   thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
  3766 
  3767 subsection {* Bounded convex function on open set is continuous *}
  3768 
  3769 lemma convex_on_bounded_continuous:
  3770   fixes s :: "('a::real_normed_vector) set"
  3771   assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
  3772   shows "continuous_on s f"
  3773   apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
  3774   fix x e assume "x\<in>s" "(0::real) < e"
  3775   def B \<equiv> "abs b + 1"
  3776   have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
  3777     unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
  3778   obtain k where "k>0"and k:"cball x k \<subseteq> s" using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]] using `x\<in>s` by auto
  3779   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
  3780     apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
  3781     fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)" 
  3782     show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
  3783       case False def t \<equiv> "k / norm (y - x)"
  3784       have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
  3785       have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
  3786         apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute) 
  3787       { def w \<equiv> "x + t *\<^sub>R (y - x)"
  3788         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  3789           unfolding t_def using `k>0` by auto
  3790         have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x" by (auto simp add: algebra_simps)
  3791         also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps)
  3792         finally have w:"(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
  3793         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) 
  3794         hence "(f w - f x) / t < e"
  3795           using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps) 
  3796         hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
  3797           using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
  3798           using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
  3799       moreover 
  3800       { def w \<equiv> "x - t *\<^sub>R (y - x)"
  3801         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  3802           unfolding t_def using `k>0` by auto
  3803         have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x" by (auto simp add: algebra_simps)
  3804         also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
  3805         finally have w:"(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
  3806         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) 
  3807         hence *:"(f w - f y) / t < e" using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`] using `t>0` by(auto simp add:field_simps) 
  3808         have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y" 
  3809           using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
  3810           using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
  3811         also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding divide_inverse by (auto simp add:field_simps)
  3812         also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
  3813         finally have "f x - f y < e" by auto }
  3814       ultimately show ?thesis by auto 
  3815     qed(insert `0<e`, auto) 
  3816   qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
  3817 
  3818 subsection {* Upper bound on a ball implies upper and lower bounds *}
  3819 
  3820 lemma convex_bounds_lemma:
  3821   fixes x :: "'a::real_normed_vector"
  3822   assumes "convex_on (cball x e) f"  "\<forall>y \<in> cball x e. f y \<le> b"
  3823   shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
  3824   apply(rule) proof(cases "0 \<le> e") case True
  3825   fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *\<^sub>R x - y"
  3826   have *:"x - (2 *\<^sub>R x - y) = y - x" by (simp add: scaleR_2)
  3827   have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
  3828   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
  3829   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
  3830     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
  3831 next case False fix y assume "y\<in>cball x e" 
  3832   hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
  3833   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
  3834 
  3835 subsubsection {* Hence a convex function on an open set is continuous *}
  3836 
  3837 lemma convex_on_continuous:
  3838   assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f" 
  3839   (* FIXME: generalize to euclidean_space *)
  3840   shows "continuous_on s f"
  3841   unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
  3842   note dimge1 = DIM_positive[where 'a='a]
  3843   fix x assume "x\<in>s"
  3844   then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
  3845   def d \<equiv> "e / real DIM('a)"
  3846   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
  3847   let ?d = "(\<chi>\<chi> i. d)::'a"
  3848   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
  3849   have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by auto
  3850   hence "c\<noteq>{}" using c by auto
  3851   def k \<equiv> "Max (f ` c)"
  3852   have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
  3853     apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
  3854     fix z assume z:"z\<in>{x - ?d..x + ?d}"
  3855     have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
  3856       unfolding real_eq_of_nat by auto
  3857     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
  3858       using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
  3859   hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
  3860     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
  3861   have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 by auto
  3862   hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
  3863   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
  3864   hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
  3865     fix y assume y:"y\<in>cball x d"
  3866     { fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i"  "y $$ i \<le> x $$ i + d" 
  3867         using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by auto  }
  3868     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
  3869       by auto qed
  3870   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
  3871     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
  3872     apply force
  3873     done
  3874   thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
  3875     using `d>0` by auto 
  3876 qed
  3877 
  3878 subsection {* Line segments, Starlike Sets, etc. *}
  3879 
  3880 (* Use the same overloading tricks as for intervals, so that 
  3881    segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
  3882 
  3883 definition
  3884   midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a" where
  3885   "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
  3886 
  3887 definition
  3888   open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
  3889   "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real.  0 < u \<and> u < 1}"
  3890 
  3891 definition
  3892   closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
  3893   "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
  3894 
  3895 definition "between = (\<lambda> (a,b) x. x \<in> closed_segment a b)"
  3896 
  3897 lemmas segment = open_segment_def closed_segment_def
  3898 
  3899 definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
  3900 
  3901 lemma midpoint_refl: "midpoint x x = x"
  3902   unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto
  3903 
  3904 lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
  3905 
  3906 lemma midpoint_eq_iff: "midpoint a b = c \<longleftrightarrow> a + b = c + c"
  3907 proof -
  3908   have "midpoint a b = c \<longleftrightarrow> scaleR 2 (midpoint a b) = scaleR 2 c"
  3909     by simp
  3910   thus ?thesis
  3911     unfolding midpoint_def scaleR_2 [symmetric] by simp
  3912 qed
  3913 
  3914 lemma dist_midpoint:
  3915   fixes a b :: "'a::real_normed_vector" shows
  3916   "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
  3917   "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
  3918   "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
  3919   "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
  3920 proof-
  3921   have *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
  3922   have **:"\<And>x y::'a. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2" by auto
  3923   note scaleR_right_distrib [simp]
  3924   show ?t1 unfolding midpoint_def dist_norm apply (rule **)
  3925     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3926   show ?t2 unfolding midpoint_def dist_norm apply (rule *)
  3927     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3928   show ?t3 unfolding midpoint_def dist_norm apply (rule *)
  3929     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3930   show ?t4 unfolding midpoint_def dist_norm apply (rule **)
  3931     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3932 qed
  3933 
  3934 lemma midpoint_eq_endpoint:
  3935   "midpoint a b = a \<longleftrightarrow> a = b"
  3936   "midpoint a b = b \<longleftrightarrow> a = b"
  3937   unfolding midpoint_eq_iff by auto
  3938 
  3939 lemma convex_contains_segment:
  3940   "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. closed_segment a b \<subseteq> s)"
  3941   unfolding convex_alt closed_segment_def by auto
  3942 
  3943 lemma convex_imp_starlike:
  3944   "convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
  3945   unfolding convex_contains_segment starlike_def by auto
  3946 
  3947 lemma segment_convex_hull:
  3948  "closed_segment a b = convex hull {a,b}" proof-
  3949   have *:"\<And>x. {x} \<noteq> {}" by auto
  3950   have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
  3951   show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_eqI)
  3952     unfolding mem_Collect_eq apply(rule,erule exE) 
  3953     apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
  3954     apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
  3955 
  3956 lemma convex_segment: "convex (closed_segment a b)"
  3957   unfolding segment_convex_hull by(rule convex_convex_hull)
  3958 
  3959 lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
  3960   unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto
  3961 
  3962 lemma segment_furthest_le:
  3963   fixes a b x y :: "'a::euclidean_space"
  3964   assumes "x \<in> closed_segment a b" shows "norm(y - x) \<le> norm(y - a) \<or>  norm(y - x) \<le> norm(y - b)" proof-
  3965   obtain z where "z\<in>{a, b}" "norm (x - y) \<le> norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
  3966     using assms[unfolded segment_convex_hull] by auto
  3967   thus ?thesis by(auto simp add:norm_minus_commute) qed
  3968 
  3969 lemma segment_bound:
  3970   fixes x a b :: "'a::euclidean_space"
  3971   assumes "x \<in> closed_segment a b"
  3972   shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
  3973   using segment_furthest_le[OF assms, of a]
  3974   using segment_furthest_le[OF assms, of b]
  3975   by (auto simp add:norm_minus_commute) 
  3976 
  3977 lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
  3978 
  3979 lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
  3980   unfolding between_def by auto
  3981 
  3982 lemma between:"between (a,b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
  3983 proof(cases "a = b")
  3984   case True thus ?thesis unfolding between_def split_conv
  3985     by(auto simp add:segment_refl dist_commute) next
  3986   case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto 
  3987   have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
  3988   show ?thesis unfolding between_def split_conv closed_segment_def mem_Collect_eq
  3989     apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
  3990       fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" 
  3991       hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
  3992         unfolding as(1) by(auto simp add:algebra_simps)
  3993       show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
  3994         unfolding norm_minus_commute[of x a] * using as(2,3)
  3995         by(auto simp add: field_simps)
  3996     next assume as:"dist a b = dist a x + dist x b"
  3997       have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2]
  3998         unfolding as[unfolded dist_norm] norm_ge_zero by auto 
  3999       thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
  4000         unfolding dist_norm apply(subst euclidean_eq) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
  4001       proof(rule,rule) fix i assume i:"i<DIM('a)"
  4002           have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
  4003             ((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
  4004             using Fal by(auto simp add: field_simps)
  4005           also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
  4006             unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
  4007             apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps)
  4008           finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i" 
  4009             by auto
  4010         qed(insert Fal2, auto) qed qed
  4011 
  4012 lemma between_midpoint: fixes a::"'a::euclidean_space" shows
  4013   "between (a,b) (midpoint a b)" (is ?t1) 
  4014   "between (b,a) (midpoint a b)" (is ?t2)
  4015 proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
  4016   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
  4017     unfolding euclidean_eq[where 'a='a]
  4018     by(auto simp add:field_simps) qed
  4019 
  4020 lemma between_mem_convex_hull:
  4021   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
  4022   unfolding between_mem_segment segment_convex_hull ..
  4023 
  4024 subsection {* Shrinking towards the interior of a convex set *}
  4025 
  4026 lemma mem_interior_convex_shrink:
  4027   fixes s :: "('a::euclidean_space) set"
  4028   assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
  4029   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  4030 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  4031   show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
  4032     apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
  4033     fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
  4034     have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
  4035     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
  4036       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
  4037       by(auto simp add: euclidean_eq[where 'a='a] field_simps) 
  4038     also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
  4039     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
  4040       by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
  4041     finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
  4042       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
  4043   qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
  4044 
  4045 lemma mem_interior_closure_convex_shrink:
  4046   fixes s :: "('a::euclidean_space) set"
  4047   assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
  4048   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  4049 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  4050   have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
  4051     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  4052     case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
  4053     show ?thesis proof(cases "e=1")
  4054       case True obtain y where "y\<in>s" "y \<noteq> x" "dist y x < 1"
  4055         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  4056       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  4057       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  4058         using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  4059       then obtain y where "y\<in>s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
  4060         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  4061       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  4062   then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
  4063   def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
  4064   have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
  4065   have "z\<in>interior s" apply(rule interior_mono[OF d,unfolded subset_eq,rule_format])
  4066     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
  4067     by(auto simp add:field_simps norm_minus_commute)
  4068   thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink) 
  4069     using assms(1,4-5) `y\<in>s` by auto qed
  4070 
  4071 subsection {* Some obvious but surprisingly hard simplex lemmas *}
  4072 
  4073 lemma simplex:
  4074   assumes "finite s" "0 \<notin> s"
  4075   shows "convex hull (insert 0 s) =  { y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y)}"
  4076   unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_eqI, rule) unfolding mem_Collect_eq
  4077   apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
  4078   apply(rule_tac x=u in exI) defer apply(rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI) using assms(2)
  4079   unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
  4080 
  4081 lemma substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4082   shows "convex hull (insert 0 { basis i | i. i : d}) =
  4083         {x::'a::euclidean_space . (!i<DIM('a). 0 <= x$$i) & setsum (%i. x$$i) d <= 1 &
  4084   (!i<DIM('a). i ~: d --> x$$i = 0)}" 
  4085   (is "convex hull (insert 0 ?p) = ?s")
  4086 (* Proof is a modified copy of the proof of similar lemma std_simplex in Convex_Euclidean_Space.thy *)
  4087 proof- let ?D = d (*"{..<DIM('a::euclidean_space)}"*)
  4088   have "0 ~: ?p" using assms by (auto simp: image_def)
  4089   have "{(basis i)::'n::euclidean_space |i. i \<in> ?D} = basis ` ?D" by auto
  4090   note sumbas = this setsum_reindex[OF basis_inj_on[of d], unfolded o_def, OF assms]
  4091   show ?thesis unfolding simplex[OF finite_substdbasis `0 ~: ?p`] 
  4092     apply(rule set_eqI) unfolding mem_Collect_eq apply rule
  4093     apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
  4094     fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x"
  4095       "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
  4096     have *:"\<forall>i<DIM('a). i:d --> u (basis i) = x$$i" and "(!i<DIM('a). i ~: d --> x $$ i = 0)" using as(3) 
  4097       unfolding sumbas unfolding substdbasis_expansion_unique[OF assms] by auto
  4098     hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $$ x) ?D" unfolding sumbas 
  4099       apply-apply(rule setsum_cong2) using assms by auto
  4100     have " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1" 
  4101       apply - proof(rule,rule,rule)
  4102       fix i assume i:"i<DIM('a)" have "i : d ==> 0 \<le> x$$i" unfolding *[rule_format,OF i,THEN sym] 
  4103          apply(rule_tac as(1)[rule_format]) by auto
  4104       moreover have "i ~: d ==> 0 \<le> x$$i" 
  4105         using `(!i<DIM('a). i ~: d --> x $$ i = 0)`[rule_format, OF i] by auto
  4106       ultimately show "0 \<le> x$$i" by auto
  4107     qed(insert as(2)[unfolded **], auto)
  4108     from this show " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1 & (!i<DIM('a). i ~: d --> x $$ i = 0)" 
  4109       using `(!i<DIM('a). i ~: d --> x $$ i = 0)` by auto
  4110   next fix x::"'a::euclidean_space" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
  4111       "(!i<DIM('a). i ~: d --> x $$ i = 0)"
  4112     show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and>
  4113       setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
  4114       apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE)
  4115       using as(1) apply(erule_tac x=i in allE) unfolding sumbas apply safe unfolding not_less basis_zero
  4116       unfolding substdbasis_expansion_unique[OF assms] euclidean_component_def[THEN sym]
  4117       using as(2,3) by(auto simp add:dot_basis not_less)
  4118   qed qed
  4119 
  4120 lemma std_simplex:
  4121   "convex hull (insert 0 { basis i | i. i<DIM('a)}) =
  4122         {x::'a::euclidean_space . (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} \<le> 1 }"
  4123   using substd_simplex[of "{..<DIM('a)}"] by auto
  4124 
  4125 lemma interior_std_simplex:
  4126   "interior (convex hull (insert 0 { basis i| i. i<DIM('a)})) =
  4127   {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 < x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} < 1 }"
  4128   apply(rule set_eqI) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
  4129   unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
  4130   fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x<DIM('a). 0 \<le> xa $$ x) \<and> setsum (op $$ xa) {..<DIM('a)} \<le> 1"
  4131   show "(\<forall>xa<DIM('a). 0 < x $$ xa) \<and> setsum (op $$ x) {..<DIM('a)} < 1" apply(safe) proof-
  4132     fix i assume i:"i<DIM('a)" thus "0 < x $$ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
  4133       unfolding dist_norm by (auto elim!:allE[where x=i])
  4134   next have **:"dist x (x + (e / 2) *\<^sub>R basis 0) < e" using  `e>0`
  4135       unfolding dist_norm by(auto intro!: mult_strict_left_mono)
  4136     have "\<And>i. i<DIM('a) \<Longrightarrow> (x + (e / 2) *\<^sub>R basis 0) $$ i = x$$i + (if i = 0 then e/2 else 0)"
  4137       by auto
  4138     hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)} = setsum (\<lambda>i. x$$i + (if 0 = i then e/2 else 0)) {..<DIM('a)}"
  4139       apply(rule_tac setsum_cong) by auto
  4140     have "setsum (op $$ x) {..<DIM('a)} < setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)}" unfolding * setsum_addf
  4141       using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by auto
  4142     also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
  4143     finally show "setsum (op $$ x) {..<DIM('a)} < 1" by auto qed
  4144 next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 < x $$ i" "setsum (op $$ x) {..<DIM('a)} < 1"
  4145   guess a using UNIV_witness[where 'a='b] ..
  4146   let ?d = "(1 - setsum (op $$ x) {..<DIM('a)}) / real (DIM('a))"
  4147   have "Min ((op $$ x) ` {..<DIM('a)}) > 0" apply(rule Min_grI) using as(1) by auto
  4148   moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by(auto simp add: Suc_le_eq)
  4149   ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
  4150     apply(rule_tac x="min (Min ((op $$ x) ` {..<DIM('a)})) ?D" in exI) apply rule defer apply(rule,rule) proof-
  4151     fix y assume y:"dist x y < min (Min (op $$ x ` {..<DIM('a)})) ?d"
  4152     have "setsum (op $$ y) {..<DIM('a)} \<le> setsum (\<lambda>i. x$$i + ?d) {..<DIM('a)}" proof(rule setsum_mono)
  4153       fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
  4154         using component_le_norm[of "y - x" i]
  4155         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
  4156       thus "y $$ i \<le> x $$ i + ?d" by auto qed
  4157     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant real_eq_of_nat by(auto simp add: Suc_le_eq)
  4158     finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1" 
  4159     proof safe fix i assume i:"i<DIM('a)"
  4160       have "norm (x - y) < x$$i" apply(rule less_le_trans) 
  4161         apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
  4162       thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
  4163     qed qed auto qed
  4164 
  4165 lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
  4166   "a \<in> interior(convex hull (insert 0 {basis i | i . i<DIM('a)}))" proof-
  4167   let ?D = "{..<DIM('a)}" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) {(basis i) | i. i<DIM('a)}"
  4168   have *:"{basis i :: 'a | i. i<DIM('a)} = basis ` ?D" by auto
  4169   { fix i assume i:"i<DIM('a)" have "?a $$ i = inverse (2 * real DIM('a))"
  4170       unfolding euclidean_component_setsum * and setsum_reindex[OF basis_inj] and o_def
  4171       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"]) apply(rule setsum_cong2)
  4172       defer apply(subst setsum_delta') unfolding euclidean_component_def using i by(auto simp add:dot_basis) }
  4173   note ** = this
  4174   show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe
  4175     fix i assume i:"i<DIM('a)" show "0 < ?a $$ i" unfolding **[OF i] by(auto simp add: Suc_le_eq)
  4176   next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
  4177     also have "\<dots> < 1" unfolding setsum_constant real_eq_of_nat divide_inverse[THEN sym] by (auto simp add:field_simps)
  4178     finally show "setsum (op $$ ?a) ?D < 1" by auto qed qed
  4179 
  4180 lemma rel_interior_substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4181   shows "rel_interior (convex hull (insert 0 { basis i| i. i : d})) =
  4182   {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x$$i) & setsum (%i. x$$i) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)}"
  4183   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
  4184 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex in Convex_Euclidean_Space.thy *)
  4185 proof-
  4186 have "finite d" apply(rule finite_subset) using assms by auto
  4187 { assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq[of _ 0] by auto }
  4188 moreover
  4189 { assume "d~={}"
  4190 have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}" 
  4191    using affine_hull_convex_hull affine_hull_substd_basis assms by auto 
  4192 have aux: "!x::'n::euclidean_space. !i. ((! i:d. 0 <= x$$i) & (!i. i ~: d --> x$$i = 0))--> 0 <= x$$i" by auto
  4193 { fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"
  4194   from this obtain e where e0: "e>0" and 
  4195        "ball x e Int {xa. (!i<DIM('a). i ~: d --> xa$$i = 0)} <= convex hull (insert 0 ?p)" 
  4196        using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto   
  4197   hence as: "ALL xa. (dist x xa < e & (!i<DIM('a). i ~: d --> xa$$i = 0)) -->
  4198     (!i : d. 0 <= xa $$ i) & setsum (op $$ xa) d <= 1"
  4199     unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
  4200   have x0: "(!i<DIM('a). i ~: d --> x$$i = 0)" 
  4201     using x_def rel_interior_subset  substd_simplex[OF assms] by auto
  4202   have "(!i : d. 0 < x $$ i) & setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" apply(rule,rule) 
  4203   proof-
  4204     fix i::nat assume "i:d" 
  4205     hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R basis i) $$ ia" apply-apply(rule as[rule_format,THEN conjunct1])
  4206       unfolding dist_norm using assms `e>0` x0 by auto
  4207     thus "0 < x $$ i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` assms by auto
  4208   next obtain a where a:"a:d" using `d ~= {}` by auto
  4209     have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e"
  4210       using  `e>0` and Euclidean_Space.norm_basis[of a]
  4211       unfolding dist_norm by auto
  4212     have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $$ i = x$$i + (if i = a then e/2 else 0)"
  4213       unfolding euclidean_simps using a assms by auto
  4214     hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d =
  4215       setsum (\<lambda>i. x$$i + (if a = i then e/2 else 0)) d" by(rule_tac setsum_cong, auto)
  4216     have h1: "(ALL i<DIM('a). i ~: d --> (x + (e / 2) *\<^sub>R basis a) $$ i = 0)"
  4217       using as[THEN spec[where x="x + (e / 2) *\<^sub>R basis a"]] `a:d` using x0
  4218       by(auto elim:allE[where x=a])
  4219     have "setsum (op $$ x) d < setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d" unfolding * setsum_addf
  4220       using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')
  4221     also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R basis a"] by auto
  4222     finally show "setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" using x0 by auto 
  4223   qed
  4224 }
  4225 moreover
  4226 {
  4227   fix x::"'a::euclidean_space" assume as: "x : ?s"
  4228   have "!i. ((0<x$$i) | (0=x$$i) --> 0<=x$$i)" by auto
  4229   moreover have "!i. (i:d) | (i ~: d)" by auto
  4230   ultimately 
  4231   have "!i. ( (ALL i:d. 0 < x$$i) & (ALL i. i ~: d --> x$$i = 0) ) --> 0 <= x$$i" by metis
  4232   hence h2: "x : convex hull (insert 0 ?p)" using as assms 
  4233     unfolding substd_simplex[OF assms] by fastforce 
  4234   obtain a where a:"a:d" using `d ~= {}` by auto
  4235   let ?d = "(1 - setsum (op $$ x) d) / real (card d)"
  4236   have "0 < card d" using `d ~={}` `finite d` by (simp add: card_gt_0_iff)
  4237   have "Min ((op $$ x) ` d) > 0" using as `d \<noteq> {}` `finite d` by (simp add: Min_grI)
  4238   moreover have "?d > 0" apply(rule divide_pos_pos) using as using `0 < card d` by auto
  4239   ultimately have h3: "min (Min ((op $$ x) ` d)) ?d > 0" by auto
  4240 
  4241   have "x : rel_interior (convex hull (insert 0 ?p))"
  4242     unfolding rel_interior_ball mem_Collect_eq h0 apply(rule,rule h2)
  4243     unfolding substd_simplex[OF assms]
  4244     apply(rule_tac x="min (Min ((op $$ x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball
  4245   proof- fix y::'a assume y:"dist x y < min (Min (op $$ x ` d)) ?d" and y2:"(!i<DIM('a). i ~: d --> y$$i = 0)"
  4246     have "setsum (op $$ y) d \<le> setsum (\<lambda>i. x$$i + ?d) d" proof(rule setsum_mono)
  4247       fix i assume i:"i\<in>d"
  4248       have "abs (y$$i - x$$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
  4249         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
  4250         by(auto simp add: norm_minus_commute)
  4251       thus "y $$ i \<le> x $$ i + ?d" by auto qed
  4252     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant real_eq_of_nat
  4253       using `0 < card d` by auto
  4254     finally show "setsum (op $$ y) d \<le> 1" .
  4255      
  4256     fix i assume "i<DIM('a)" thus "0 \<le> y$$i" 
  4257     proof(cases "i\<in>d") case True
  4258       have "norm (x - y) < x$$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
  4259         using Min_gr_iff[of "op $$ x ` d" "norm (x - y)"] `0 < card d` `i:d`
  4260         by (simp add: card_gt_0_iff)
  4261       thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format] by auto
  4262     qed(insert y2, auto)
  4263   qed
  4264 } ultimately have
  4265     "!!x :: 'a::euclidean_space. (x : rel_interior (convex hull insert 0 {basis i |i. i : d})) =
  4266     (x : {x. (ALL i:d. 0 < x $$ i) &
  4267     setsum (op $$ x) d < 1 & (ALL i<DIM('a). i ~: d --> x $$ i = 0)})" by blast
  4268 from this have ?thesis by (rule set_eqI)
  4269 } ultimately show ?thesis by blast
  4270 qed
  4271 
  4272 lemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4273   obtains a::"'a::euclidean_space" where
  4274   "a : rel_interior(convex hull (insert 0 {basis i | i . i : d}))" proof-
  4275 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex_nonempty in Convex_Euclidean_Space.thy *)
  4276   let ?D = d let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
  4277   have *:"{basis i :: 'a | i. i \<in> ?D} = basis ` ?D" by auto
  4278   have "finite d" apply(rule finite_subset) using assms(2) by auto
  4279   hence d1: "0 < real(card d)" using `d ~={}` by auto
  4280   { fix i assume "i:d" have "?a $$ i = inverse (2 * real (card d))"
  4281       unfolding * setsum_reindex[OF basis_inj_on, OF assms(2)] o_def
  4282       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"]) 
  4283       unfolding euclidean_component_setsum
  4284       apply(rule setsum_cong2)
  4285       using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)
  4286       by (auto simp add: Euclidean_Space.basis_component[of i])}
  4287   note ** = this
  4288   show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
  4289   proof safe fix i assume "i:d" 
  4290     have "0 < inverse (2 * real (card d))" using d1 by auto
  4291     also have "...=?a $$ i" using **[of i] `i:d` by auto
  4292     finally show "0 < ?a $$ i" by auto
  4293   next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D" 
  4294       by(rule setsum_cong2, rule **) 
  4295     also have "\<dots> < 1" unfolding setsum_constant real_eq_of_nat divide_real_def[THEN sym]
  4296       by (auto simp add:field_simps)
  4297     finally show "setsum (op $$ ?a) ?D < 1" by auto
  4298   next fix i assume "i<DIM('a)" and "i~:d"
  4299     have "?a : (span {basis i | i. i : d})" 
  4300       apply (rule span_setsum[of "{basis i |i. i : d}" "(%b. b /\<^sub>R (2 * real (card d)))" "{basis i |i. i : d}"]) 
  4301       using finite_substdbasis[of d] apply blast 
  4302     proof-
  4303       { fix x assume "(x :: 'a::euclidean_space): {basis i |i. i : d}"
  4304         hence "x : span {basis i |i. i : d}" 
  4305           using span_superset[of _ "{basis i |i. i : d}"] by auto
  4306         hence "(x /\<^sub>R (2 * real (card d))) : (span {basis i |i. i : d})"
  4307           using span_mul[of x "{basis i |i. i : d}" "(inverse (real (card d)) / 2)"] by auto
  4308       } thus "\<forall>x\<in>{basis i |i. i \<in> d}. x /\<^sub>R (2 * real (card d)) \<in> span {basis i ::'a |i. i \<in> d}" by auto
  4309     qed
  4310     thus "?a $$ i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i<DIM('a)` by auto
  4311   qed
  4312 qed
  4313 
  4314 subsection {* Relative interior of convex set *}
  4315 
  4316 lemma rel_interior_convex_nonempty_aux: 
  4317 fixes S :: "('n::euclidean_space) set" 
  4318 assumes "convex S" and "0 : S"
  4319 shows "rel_interior S ~= {}"
  4320 proof-
  4321 { assume "S = {0}" hence ?thesis using rel_interior_sing by auto }
  4322 moreover { 
  4323 assume "S ~= {0}"
  4324 obtain B where B_def: "independent B & B<=S & (S <= span B) & card B = dim S" using basis_exists[of S] by auto
  4325 hence "B~={}" using B_def assms `S ~= {0}` span_empty by auto
  4326 have "insert 0 B <= span B" using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
  4327 hence "span (insert 0 B) <= span B" 
  4328     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
  4329 hence "convex hull insert 0 B <= span B" 
  4330     using convex_hull_subset_span[of "insert 0 B"] by auto
  4331 hence "span (convex hull insert 0 B) <= span B"
  4332     using span_span[of B] span_mono[of "convex hull insert 0 B" "span B"] by blast
  4333 hence *: "span (convex hull insert 0 B) = span B" 
  4334     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  4335 hence "span (convex hull insert 0 B) = span S"
  4336     using B_def span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
  4337 moreover have "0 : affine hull (convex hull insert 0 B)"
  4338     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  4339 ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
  4340     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"] 
  4341     assms  hull_subset[of S] by auto
  4342 obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = {basis i |i. i : (d :: nat set)} & 
  4343        f ` span B = {x. ALL i<DIM('n). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B)" and d:"d\<subseteq>{..<DIM('n)}"
  4344     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by auto
  4345 hence "bounded_linear f" using linear_conv_bounded_linear by auto
  4346 have "d ~={}" using fd B_def `B ~={}` by auto
  4347 have "(insert 0 {basis i |i. i : d}) = f ` (insert 0 B)" using fd linear_0 by auto
  4348 hence "(convex hull (insert 0 {basis i |i. i : d})) = f ` (convex hull (insert 0 B))"
  4349    using convex_hull_linear_image[of f "(insert 0 {basis i |i. i : d})"] 
  4350    convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by auto
  4351 moreover have "rel_interior (f ` (convex hull insert 0 B)) = 
  4352    f ` rel_interior (convex hull insert 0 B)"
  4353    apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
  4354    using `bounded_linear f` fd * by auto
  4355 ultimately have "rel_interior (convex hull insert 0 B) ~= {}"
  4356    using rel_interior_substd_simplex_nonempty[OF `d~={}` d] apply auto by blast 
  4357 moreover have "convex hull (insert 0 B) <= S"
  4358    using B_def assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
  4359 ultimately have ?thesis using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
  4360 } ultimately show ?thesis by auto
  4361 qed
  4362 
  4363 lemma rel_interior_convex_nonempty:
  4364 fixes S :: "('n::euclidean_space) set"
  4365 assumes "convex S"
  4366 shows "rel_interior S = {} <-> S = {}"
  4367 proof-
  4368 { assume "S ~= {}" from this obtain a where "a : S" by auto
  4369   hence "0 : op + (-a) ` S" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
  4370   hence "rel_interior (op + (-a) ` S) ~= {}"  
  4371     using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"] 
  4372           convex_translation[of S "-a"] assms by auto 
  4373   hence "rel_interior S ~= {}" using rel_interior_translation by auto
  4374 } from this show ?thesis using rel_interior_empty by auto
  4375 qed
  4376 
  4377 lemma convex_rel_interior:
  4378 fixes S :: "(_::euclidean_space) set"
  4379 assumes "convex S"
  4380 shows "convex (rel_interior S)"
  4381 proof-
  4382 { fix "x" "y" "u"
  4383   assume assm: "x:rel_interior S" "y:rel_interior S" "0<=u" "(u :: real) <= 1"
  4384   hence "x:S" using rel_interior_subset by auto
  4385   have "x - u *\<^sub>R (x-y) : rel_interior S"
  4386   proof(cases "0=u")
  4387      case False hence "0<u" using assm by auto 
  4388         thus ?thesis
  4389         using assm rel_interior_convex_shrink[of S y x u] assms `x:S` by auto
  4390      next
  4391      case True thus ?thesis using assm by auto
  4392   qed
  4393   hence "(1-u) *\<^sub>R x + u *\<^sub>R y : rel_interior S" by (simp add: algebra_simps)
  4394 } from this show ?thesis unfolding convex_alt by auto
  4395 qed
  4396 
  4397 lemma convex_closure_rel_interior: 
  4398 fixes S :: "('n::euclidean_space) set" 
  4399 assumes "convex S"
  4400 shows "closure(rel_interior S) = closure S"
  4401 proof-
  4402 have h1: "closure(rel_interior S) <= closure S" 
  4403    using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
  4404 { assume "S ~= {}" from this obtain a where a_def: "a : rel_interior S" 
  4405     using rel_interior_convex_nonempty assms by auto
  4406   { fix x assume x_def: "x : closure S"
  4407     { assume "x=a" hence "x : closure(rel_interior S)" using a_def unfolding closure_def by auto }
  4408     moreover
  4409     { assume "x ~= a"
  4410        { fix e :: real assume e_def: "e>0" 
  4411          def e1 == "min 1 (e/norm (x - a))" hence e1_def: "e1>0 & e1<=1 & e1*norm(x-a)<=e"
  4412             using `x ~= a` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(x-a)"] by simp 
  4413          hence *: "x - e1 *\<^sub>R (x - a) : rel_interior S"
  4414             using rel_interior_closure_convex_shrink[of S a x e1] assms x_def a_def e1_def by auto
  4415          have "EX y. y:rel_interior S & y ~= x & (dist y x) <= e"
  4416             apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
  4417             using * e1_def dist_norm[of "x - e1 *\<^sub>R (x - a)" x] `x ~= a` by simp
  4418       } hence "x islimpt rel_interior S" unfolding islimpt_approachable_le by auto 
  4419       hence "x : closure(rel_interior S)" unfolding closure_def by auto 
  4420     } ultimately have "x : closure(rel_interior S)" by auto
  4421   } hence ?thesis using h1 by auto
  4422 }
  4423 moreover
  4424 { assume "S = {}" hence "rel_interior S = {}" using rel_interior_empty by auto
  4425   hence "closure(rel_interior S) = {}" using closure_empty by auto 
  4426   hence ?thesis using `S={}` by auto 
  4427 } ultimately show ?thesis by blast
  4428 qed
  4429 
  4430 lemma rel_interior_same_affine_hull:
  4431   fixes S :: "('n::euclidean_space) set"
  4432   assumes "convex S"
  4433   shows "affine hull (rel_interior S) = affine hull S"
  4434 by (metis assms closure_same_affine_hull convex_closure_rel_interior)
  4435 
  4436 lemma rel_interior_aff_dim: 
  4437   fixes S :: "('n::euclidean_space) set"
  4438   assumes "convex S"
  4439   shows "aff_dim (rel_interior S) = aff_dim S"
  4440 by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
  4441 
  4442 lemma rel_interior_rel_interior:
  4443   fixes S :: "('n::euclidean_space) set"
  4444   assumes "convex S"
  4445   shows "rel_interior (rel_interior S) = rel_interior S"
  4446 proof-
  4447 have "openin (subtopology euclidean (affine hull (rel_interior S))) (rel_interior S)"
  4448   using opein_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
  4449 from this show ?thesis using rel_interior_def by auto
  4450 qed
  4451 
  4452 lemma rel_interior_rel_open:
  4453   fixes S :: "('n::euclidean_space) set"
  4454   assumes "convex S"
  4455   shows "rel_open (rel_interior S)"
  4456 unfolding rel_open_def using rel_interior_rel_interior assms by auto
  4457 
  4458 lemma convex_rel_interior_closure_aux:
  4459   fixes x y z :: "_::euclidean_space"
  4460   assumes "0 < a" "0 < b" "(a+b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
  4461   obtains e where "0 < e" "e <= 1" "z = y - e *\<^sub>R (y-x)"
  4462 proof-
  4463 def e == "a/(a+b)"
  4464 have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)" apply auto using assms by simp
  4465 also have "... = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)" using assms
  4466    scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"] by auto
  4467 also have "... = y - e *\<^sub>R (y-x)" using e_def apply (simp add: algebra_simps)
  4468    using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"] by auto
  4469 finally have "z = y - e *\<^sub>R (y-x)" by auto
  4470 moreover have "0<e" using e_def assms divide_pos_pos[of a "a+b"] by auto
  4471 moreover have "e<=1" using e_def assms by auto
  4472 ultimately show ?thesis using that[of e] by auto
  4473 qed
  4474 
  4475 lemma convex_rel_interior_closure: 
  4476   fixes S :: "('n::euclidean_space) set" 
  4477   assumes "convex S"
  4478   shows "rel_interior (closure S) = rel_interior S"
  4479 proof-
  4480 { assume "S={}" hence ?thesis using assms rel_interior_convex_nonempty by auto }
  4481 moreover
  4482 { assume "S ~= {}"
  4483   have "rel_interior (closure S) >= rel_interior S" 
  4484     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset by auto
  4485   moreover
  4486   { fix z assume z_def: "z : rel_interior (closure S)"
  4487     obtain x where x_def: "x : rel_interior S" 
  4488       using `S ~= {}` assms rel_interior_convex_nonempty by auto  
  4489     { assume "x=z" hence "z : rel_interior S" using x_def by auto }
  4490     moreover
  4491     { assume "x ~= z"
  4492       obtain e where e_def: "e > 0 & cball z e Int affine hull closure S <= closure S" 
  4493         using z_def rel_interior_cball[of "closure S"] by auto
  4494       hence *: "0 < e/norm(z-x)" using e_def `x ~= z` divide_pos_pos[of e "norm(z-x)"] by auto 
  4495       def y == "z + (e/norm(z-x)) *\<^sub>R (z-x)"
  4496       have yball: "y : cball z e"
  4497         using mem_cball y_def dist_norm[of z y] e_def by auto 
  4498       have "x : affine hull closure S" 
  4499         using x_def rel_interior_subset_closure hull_inc[of x "closure S"] by auto
  4500       moreover have "z : affine hull closure S" 
  4501         using z_def rel_interior_subset hull_subset[of "closure S"] by auto
  4502       ultimately have "y : affine hull closure S" 
  4503         using y_def affine_affine_hull[of "closure S"] 
  4504           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
  4505       hence "y : closure S" using e_def yball by auto
  4506       have "(1+(e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
  4507         using y_def by (simp add: algebra_simps) 
  4508       from this obtain e1 where "0 < e1 & e1 <= 1 & z = y - e1 *\<^sub>R (y - x)"
  4509         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y] 
  4510           by (auto simp add: algebra_simps)
  4511       hence "z : rel_interior S" 
  4512         using rel_interior_closure_convex_shrink assms x_def `y : closure S` by auto
  4513     } ultimately have "z : rel_interior S" by auto
  4514   } ultimately have ?thesis by auto
  4515 } ultimately show ?thesis by blast
  4516 qed
  4517 
  4518 lemma convex_interior_closure: 
  4519 fixes S :: "('n::euclidean_space) set" 
  4520 assumes "convex S"
  4521 shows "interior (closure S) = interior S"
  4522 using closure_aff_dim[of S] interior_rel_interior_gen[of S] interior_rel_interior_gen[of "closure S"] 
  4523       convex_rel_interior_closure[of S] assms by auto
  4524 
  4525 lemma closure_eq_rel_interior_eq:
  4526 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4527 assumes "convex S1" "convex S2"
  4528 shows "(closure S1 = closure S2) <-> (rel_interior S1 = rel_interior S2)"
  4529  by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
  4530 
  4531 
  4532 lemma closure_eq_between:
  4533 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4534 assumes "convex S1" "convex S2"
  4535 shows "(closure S1 = closure S2) <-> 
  4536       ((rel_interior S1 <= S2) & (S2 <= closure S1))" (is "?A <-> ?B")
  4537 proof-
  4538 have "?A --> ?B" by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
  4539 moreover have "?B --> (closure S1 <= closure S2)" 
  4540      by (metis assms(1) convex_closure_rel_interior closure_mono)
  4541 moreover have "?B --> (closure S1 >= closure S2)" by (metis closed_closure closure_minimal)
  4542 ultimately show ?thesis by blast
  4543 qed
  4544 
  4545 lemma open_inter_closure_rel_interior:
  4546 fixes S A ::  "('n::euclidean_space) set" 
  4547 assumes "convex S" "open A"
  4548 shows "((A Int closure S) = {}) <-> ((A Int rel_interior S) = {})"
  4549 by (metis assms convex_closure_rel_interior open_inter_closure_eq_empty) 
  4550 
  4551 definition "rel_frontier S = closure S - rel_interior S"
  4552 
  4553 lemma closed_affine_hull: "closed (affine hull ((S :: ('n::euclidean_space) set)))"
  4554 by (metis affine_affine_hull affine_closed)
  4555 
  4556 lemma closed_rel_frontier: "closed(rel_frontier (S :: ('n::euclidean_space) set))"
  4557 proof-
  4558 have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)" 
  4559 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] 
  4560   closure_affine_hull[of S] opein_rel_interior[of S] by auto 
  4561 show ?thesis apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"]) 
  4562   unfolding rel_frontier_def using * closed_affine_hull by auto 
  4563 qed
  4564  
  4565 
  4566 lemma convex_rel_frontier_aff_dim:
  4567 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4568 assumes "convex S1" "convex S2" "S2 ~= {}"
  4569 assumes "S1 <= rel_frontier S2"
  4570 shows "aff_dim S1 < aff_dim S2" 
  4571 proof-
  4572 have "S1 <= closure S2" using assms unfolding rel_frontier_def by auto
  4573 hence *: "affine hull S1 <= affine hull S2" 
  4574    using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by auto
  4575 hence "aff_dim S1 <= aff_dim S2" using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] 
  4576     aff_dim_subset[of "affine hull S1" "affine hull S2"] by auto
  4577 moreover
  4578 { assume eq: "aff_dim S1 = aff_dim S2"
  4579   hence "S1 ~= {}" using aff_dim_empty[of S1] aff_dim_empty[of S2] `S2 ~= {}` by auto
  4580   have **: "affine hull S1 = affine hull S2" 
  4581      apply (rule affine_dim_equal) using * affine_affine_hull apply auto
  4582      using `S1 ~= {}` hull_subset[of S1] apply auto
  4583      using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] by auto
  4584   obtain a where a_def: "a : rel_interior S1"
  4585      using  `S1 ~= {}` rel_interior_convex_nonempty assms by auto
  4586   obtain T where T_def: "open T & a : T Int S1 & T Int affine hull S1 <= S1"
  4587      using mem_rel_interior[of a S1] a_def by auto
  4588   hence "a : T Int closure S2" using a_def assms unfolding rel_frontier_def by auto
  4589   from this obtain b where b_def: "b : T Int rel_interior S2" 
  4590      using open_inter_closure_rel_interior[of S2 T] assms T_def by auto
  4591   hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto
  4592   hence "b : S1" using T_def b_def by auto
  4593   hence False using b_def assms unfolding rel_frontier_def by auto
  4594 } ultimately show ?thesis using less_le by auto
  4595 qed
  4596 
  4597 
  4598 lemma convex_rel_interior_if:
  4599 fixes S ::  "('n::euclidean_space) set" 
  4600 assumes "convex S"
  4601 assumes "z : rel_interior S"
  4602 shows "(!x:affine hull S. EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S ))"
  4603 proof-
  4604 obtain e1 where e1_def: "e1>0 & cball z e1 Int affine hull S <= S" 
  4605     using mem_rel_interior_cball[of z S] assms by auto
  4606 { fix x assume x_def: "x:affine hull S"
  4607   { assume "x ~= z"
  4608     def m == "1+e1/norm(x-z)" 
  4609     hence "m>1" using e1_def `x ~= z` divide_pos_pos[of e1 "norm (x - z)"] by auto 
  4610     { fix e assume e_def: "e>1 & e<=m"
  4611       have "z : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
  4612       hence *: "(1-e)*\<^sub>R x+ e *\<^sub>R z : affine hull S"
  4613          using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x_def by auto
  4614       have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x-z))" by (simp add: algebra_simps)
  4615       also have "...= (e - 1) * norm(x-z)" using norm_scaleR e_def by auto  
  4616       also have "...<=(m - 1) * norm(x-z)" using e_def mult_right_mono[of _ _ "norm(x-z)"] by auto
  4617       also have "...= (e1 / norm (x - z)) * norm (x - z)" using m_def by auto
  4618       also have "...=e1" using `x ~= z` e1_def by simp
  4619       finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) <= e1" by auto
  4620       have "(1-e)*\<^sub>R x+ e *\<^sub>R z : cball z e1"
  4621          using m_def ** unfolding cball_def dist_norm by (auto simp add: algebra_simps)
  4622       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def * e1_def by auto
  4623     } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" using `m>1` by auto
  4624   }
  4625   moreover
  4626   { assume "x=z" def m == "1+e1" hence "m>1" using e1_def by auto
  4627     { fix e assume e_def: "e>1 & e<=m"
  4628       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S"
  4629         using e1_def x_def `x=z` by (auto simp add: algebra_simps)
  4630       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def by auto
  4631     } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" using `m>1` by auto
  4632   } ultimately have "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" by auto
  4633 } from this show ?thesis by auto 
  4634 qed
  4635 
  4636 lemma convex_rel_interior_if2:
  4637 fixes S ::  "('n::euclidean_space) set" 
  4638 assumes "convex S"
  4639 assumes "z : rel_interior S"
  4640 shows "(!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4641 using convex_rel_interior_if[of S z] assms by auto
  4642 
  4643 lemma convex_rel_interior_only_if:
  4644 fixes S ::  "('n::euclidean_space) set" 
  4645 assumes "convex S" "S ~= {}"
  4646 assumes "(!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4647 shows "z : rel_interior S" 
  4648 proof-
  4649 obtain x where x_def: "x : rel_interior S" using rel_interior_convex_nonempty assms by auto
  4650 hence "x:S" using rel_interior_subset by auto
  4651 from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S" using assms by auto
  4652 def y == "(1 - e) *\<^sub>R x + e *\<^sub>R z" hence "y:S" using e_def by auto
  4653 def e1 == "1/e" hence "0<e1 & e1<1" using e_def by auto
  4654 hence "z=y-(1-e1)*\<^sub>R (y-x)" using e1_def y_def by (auto simp add: algebra_simps)
  4655 from this show ?thesis 
  4656     using rel_interior_convex_shrink[of S x y "1-e1"] `0<e1 & e1<1` `y:S` x_def assms by auto
  4657 qed
  4658 
  4659 lemma convex_rel_interior_iff:
  4660 fixes S ::  "('n::euclidean_space) set" 
  4661 assumes "convex S" "S ~= {}"
  4662 shows "z : rel_interior S <-> (!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4663 using assms hull_subset[of S "affine"] 
  4664       convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z] by auto
  4665 
  4666 lemma convex_rel_interior_iff2:
  4667 fixes S ::  "('n::euclidean_space) set" 
  4668 assumes "convex S" "S ~= {}"
  4669 shows "z : rel_interior S <-> (!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4670 using assms hull_subset[of S] 
  4671       convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z] by auto
  4672 
  4673 
  4674 lemma convex_interior_iff:
  4675 fixes S ::  "('n::euclidean_space) set" 
  4676 assumes "convex S"
  4677 shows "z : interior S <-> (!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
  4678 proof-
  4679 { assume a: "~(aff_dim S = int DIM('n))"
  4680   { assume "z : interior S"
  4681     hence False using a interior_rel_interior_gen[of S] by auto
  4682   }
  4683   moreover
  4684   { assume r: "!x. EX e. e>0 & z+ e *\<^sub>R x : S"
  4685     { fix x obtain e1 where e1_def: "e1>0 & z+ e1 *\<^sub>R (x-z) : S" using r by auto
  4686       obtain e2 where e2_def: "e2>0 & z+ e2 *\<^sub>R (z-x) : S" using r by auto
  4687       def x1 == "z+ e1 *\<^sub>R (x-z)"
  4688          hence x1: "x1 : affine hull S" using e1_def hull_subset[of S] by auto
  4689       def x2 == "z+ e2 *\<^sub>R (z-x)"
  4690          hence x2: "x2 : affine hull S" using e2_def hull_subset[of S] by auto
  4691       have *: "e1/(e1+e2) + e2/(e1+e2) = 1" using add_divide_distrib[of e1 e2 "e1+e2"] e1_def e2_def by simp
  4692       hence "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
  4693          using x1_def x2_def apply (auto simp add: algebra_simps)
  4694          using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z] by auto
  4695       hence z: "z : affine hull S" 
  4696          using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]  
  4697          x1 x2 affine_affine_hull[of S] * by auto
  4698       have "x1-x2 = (e1+e2) *\<^sub>R (x-z)"
  4699          using x1_def x2_def by (auto simp add: algebra_simps)
  4700       hence "x=z+(1/(e1+e2)) *\<^sub>R (x1-x2)" using e1_def e2_def by simp
  4701       hence "x : affine hull S" using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"] 
  4702           x1 x2 z affine_affine_hull[of S] by auto
  4703     } hence "affine hull S = UNIV" by auto
  4704     hence "aff_dim S = int DIM('n)" using aff_dim_affine_hull[of S] by (simp add: aff_dim_univ)
  4705     hence False using a by auto
  4706   } ultimately have ?thesis by auto
  4707 }
  4708 moreover
  4709 { assume a: "aff_dim S = int DIM('n)"
  4710   hence "S ~= {}" using aff_dim_empty[of S] by auto
  4711   have *: "affine hull S=UNIV" using a affine_hull_univ by auto
  4712   { assume "z : interior S"
  4713     hence "z : rel_interior S" using a interior_rel_interior_gen[of S] by auto
  4714     hence **: "(!x. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4715       using convex_rel_interior_iff2[of S z] assms `S~={}` * by auto
  4716     fix x obtain e1 where e1_def: "e1>1 & (1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z : S"
  4717       using **[rule_format, of "z-x"] by auto
  4718     def e == "e1 - 1"
  4719     hence "(1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z = z+ e *\<^sub>R x" by (simp add: algebra_simps)
  4720     hence "e>0 & z+ e *\<^sub>R x : S" using e1_def e_def by auto
  4721     hence "EX e. e>0 & z+ e *\<^sub>R x : S" by auto
  4722   }
  4723   moreover
  4724   { assume r: "(!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
  4725     { fix x obtain e1 where e1_def: "e1>0 & z + e1*\<^sub>R (z-x) : S"
  4726          using r[rule_format, of "z-x"] by auto
  4727       def e == "e1 + 1"
  4728       hence "z + e1*\<^sub>R (z-x) = (1-e)*\<^sub>R x+ e *\<^sub>R z" by (simp add: algebra_simps)
  4729       hence "e > 1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e1_def e_def by auto
  4730       hence "EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" by auto
  4731     }
  4732     hence "z : rel_interior S" using convex_rel_interior_iff2[of S z] assms `S~={}` by auto
  4733     hence "z : interior S" using a interior_rel_interior_gen[of S] by auto
  4734   } ultimately have ?thesis by auto
  4735 } ultimately show ?thesis by auto
  4736 qed
  4737 
  4738 subsubsection {* Relative interior and closure under common operations *}
  4739 
  4740 lemma rel_interior_inter_aux: "Inter {rel_interior S |S. S : I} <= Inter I"
  4741 proof- 
  4742 { fix y assume "y : Inter {rel_interior S |S. S : I}"
  4743   hence y_def: "!S : I. y : rel_interior S" by auto
  4744   { fix S assume "S : I" hence "y : S" using rel_interior_subset y_def by auto }
  4745   hence "y : Inter I" by auto
  4746 } thus ?thesis by auto
  4747 qed
  4748 
  4749 lemma closure_inter: "closure (Inter I) <= Inter {closure S |S. S : I}"
  4750 proof- 
  4751 { fix y assume "y : Inter I" hence y_def: "!S : I. y : S" by auto
  4752   { fix S assume "S : I" hence "y : closure S" using closure_subset y_def by auto }
  4753   hence "y : Inter {closure S |S. S : I}" by auto
  4754 } hence "Inter I <= Inter {closure S |S. S : I}" by auto
  4755 moreover have "closed (Inter {closure S |S. S : I})"
  4756   unfolding closed_Inter closed_closure by auto
  4757 ultimately show ?thesis using closure_hull[of "Inter I"]
  4758   hull_minimal[of "Inter I" "Inter {closure S |S. S : I}" "closed"] by auto
  4759 qed
  4760 
  4761 lemma convex_closure_rel_interior_inter: 
  4762 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4763 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4764 shows "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
  4765 proof-
  4766 obtain x where x_def: "!S : I. x : rel_interior S" using assms by auto
  4767 { fix y assume "y : Inter {closure S |S. S : I}" hence y_def: "!S : I. y : closure S" by auto
  4768   { assume "y = x" 
  4769     hence "y : closure (Inter {rel_interior S |S. S : I})"
  4770        using x_def closure_subset[of "Inter {rel_interior S |S. S : I}"] by auto
  4771   }
  4772   moreover
  4773   { assume "y ~= x"
  4774     { fix e :: real assume e_def: "0 < e"
  4775       def e1 == "min 1 (e/norm (y - x))" hence e1_def: "e1>0 & e1<=1 & e1*norm(y-x)<=e"
  4776         using `y ~= x` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(y-x)"] by simp 
  4777       def z == "y - e1 *\<^sub>R (y - x)"
  4778       { fix S assume "S : I" 
  4779         hence "z : rel_interior S" using rel_interior_closure_convex_shrink[of S x y e1] 
  4780            assms x_def y_def e1_def z_def by auto
  4781       } hence *: "z : Inter {rel_interior S |S. S : I}" by auto
  4782       have "EX z. z:Inter {rel_interior S |S. S : I} & z ~= y & (dist z y) <= e"
  4783            apply (rule_tac x="z" in exI) using `y ~= x` z_def * e1_def e_def dist_norm[of z y] by simp
  4784     } hence "y islimpt Inter {rel_interior S |S. S : I}" unfolding islimpt_approachable_le by blast 
  4785     hence "y : closure (Inter {rel_interior S |S. S : I})" unfolding closure_def by auto
  4786   } ultimately have "y : closure (Inter {rel_interior S |S. S : I})" by auto
  4787 } from this show ?thesis by auto
  4788 qed
  4789 
  4790 
  4791 lemma convex_closure_inter: 
  4792 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4793 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4794 shows "closure (Inter I) = Inter {closure S |S. S : I}"
  4795 proof-
  4796 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
  4797   using convex_closure_rel_interior_inter assms by auto
  4798 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
  4799     using rel_interior_inter_aux 
  4800           closure_mono[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
  4801 ultimately show ?thesis using closure_inter[of I] by auto
  4802 qed
  4803 
  4804 lemma convex_inter_rel_interior_same_closure: 
  4805 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4806 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4807 shows "closure (Inter {rel_interior S |S. S : I}) = closure (Inter I)"
  4808 proof-
  4809 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
  4810   using convex_closure_rel_interior_inter assms by auto
  4811 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
  4812     using rel_interior_inter_aux 
  4813           closure_mono[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
  4814 ultimately show ?thesis using closure_inter[of I] by auto
  4815 qed
  4816 
  4817 lemma convex_rel_interior_inter: 
  4818 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4819 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4820 shows "rel_interior (Inter I) <= Inter {rel_interior S |S. S : I}"
  4821 proof-
  4822 have "convex(Inter I)" using assms convex_Inter by auto
  4823 moreover have "convex(Inter {rel_interior S |S. S : I})" apply (rule convex_Inter)
  4824    using assms convex_rel_interior by auto 
  4825 ultimately have "rel_interior (Inter {rel_interior S |S. S : I}) = rel_interior (Inter I)"
  4826    using convex_inter_rel_interior_same_closure assms 
  4827    closure_eq_rel_interior_eq[of "Inter {rel_interior S |S. S : I}" "Inter I"] by blast
  4828 from this show ?thesis using rel_interior_subset[of "Inter {rel_interior S |S. S : I}"] by auto
  4829 qed
  4830 
  4831 lemma convex_rel_interior_finite_inter: 
  4832 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4833 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4834 assumes "finite I"
  4835 shows "rel_interior (Inter I) = Inter {rel_interior S |S. S : I}"
  4836 proof-
  4837 have "Inter I ~= {}" using assms rel_interior_inter_aux[of I] by auto
  4838 have "convex (Inter I)" using convex_Inter assms by auto
  4839 { assume "I={}" hence ?thesis using Inter_empty rel_interior_univ2 by auto }
  4840 moreover
  4841 { assume "I ~= {}"
  4842 { fix z assume z_def: "z : Inter {rel_interior S |S. S : I}"
  4843   { fix x assume x_def: "x : Inter I"
  4844     { fix S assume S_def: "S : I" hence "z : rel_interior S" "x : S" using z_def x_def by auto 
  4845       (*from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S"*)
  4846       hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )"
  4847          using convex_rel_interior_if[of S z] S_def assms hull_subset[of S] by auto
  4848     } from this obtain mS where mS_def: "!S : I. (mS(S) > (1 :: real) & 
  4849          (!e. (e>1 & e<=mS(S)) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S))" by metis
  4850     obtain e where e_def: "e=Min (mS ` I)" by auto 
  4851     have "e : (mS ` I)" using e_def assms `I ~= {}` by simp
  4852     hence "e>(1 :: real)" using mS_def by auto
  4853     moreover have "!S : I. e<=mS(S)" using e_def assms by auto
  4854     ultimately have "EX e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z : Inter I" using mS_def by auto
  4855   } hence "z : rel_interior (Inter I)" using convex_rel_interior_iff[of "Inter I" z]
  4856        `Inter I ~= {}` `convex (Inter I)` by auto
  4857 } from this have ?thesis using convex_rel_interior_inter[of I] assms by auto
  4858 } ultimately show ?thesis by blast
  4859 qed
  4860 
  4861 lemma convex_closure_inter_two: 
  4862 fixes S T :: "('n::euclidean_space) set"
  4863 assumes "convex S" "convex T"
  4864 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
  4865 shows "closure (S Int T) = (closure S) Int (closure T)" 
  4866 using convex_closure_inter[of "{S,T}"] assms by auto
  4867 
  4868 lemma convex_rel_interior_inter_two: 
  4869 fixes S T :: "('n::euclidean_space) set"
  4870 assumes "convex S" "convex T"
  4871 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
  4872 shows "rel_interior (S Int T) = (rel_interior S) Int (rel_interior T)" 
  4873 using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
  4874 
  4875 
  4876 lemma convex_affine_closure_inter: 
  4877 fixes S T :: "('n::euclidean_space) set"
  4878 assumes "convex S" "affine T"
  4879 assumes "(rel_interior S) Int T ~= {}"
  4880 shows "closure (S Int T) = (closure S) Int T"
  4881 proof- 
  4882 have "affine hull T = T" using assms by auto
  4883 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
  4884 moreover have "closure T = T" using assms affine_closed[of T] by auto
  4885 ultimately show ?thesis using convex_closure_inter_two[of S T] assms affine_imp_convex by auto 
  4886 qed
  4887 
  4888 lemma convex_affine_rel_interior_inter: 
  4889 fixes S T :: "('n::euclidean_space) set"
  4890 assumes "convex S" "affine T"
  4891 assumes "(rel_interior S) Int T ~= {}"
  4892 shows "rel_interior (S Int T) = (rel_interior S) Int T"
  4893 proof- 
  4894 have "affine hull T = T" using assms by auto
  4895 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
  4896 moreover have "closure T = T" using assms affine_closed[of T] by auto
  4897 ultimately show ?thesis using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto 
  4898 qed
  4899 
  4900 lemma subset_rel_interior_convex:
  4901 fixes S T :: "('n::euclidean_space) set"
  4902 assumes "convex S" "convex T"
  4903 assumes "S <= closure T"
  4904 assumes "~(S <= rel_frontier T)"
  4905 shows "rel_interior S <= rel_interior T"
  4906 proof-
  4907 have *: "S Int closure T = S" using assms by auto
  4908 have "~(rel_interior S <= rel_frontier T)"
  4909      using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T] 
  4910      closure_closed convex_closure_rel_interior[of S] closure_subset[of S] assms by auto
  4911 hence "(rel_interior S) Int (rel_interior (closure T)) ~= {}" 
  4912      using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T] by auto
  4913 hence "rel_interior S Int rel_interior T = rel_interior (S Int closure T)" using assms convex_closure  
  4914      convex_rel_interior_inter_two[of S "closure T"] convex_rel_interior_closure[of T] by auto
  4915 also have "...=rel_interior (S)" using * by auto
  4916 finally show ?thesis by auto
  4917 qed
  4918 
  4919 
  4920 lemma rel_interior_convex_linear_image:
  4921 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  4922 assumes "linear f"
  4923 assumes "convex S"
  4924 shows "f ` (rel_interior S) = rel_interior (f ` S)"
  4925 proof-
  4926 { assume "S = {}" hence ?thesis using assms rel_interior_empty rel_interior_convex_nonempty by auto }
  4927 moreover
  4928 { assume "S ~= {}"
  4929 have *: "f ` (rel_interior S) <= f ` S" unfolding image_mono using rel_interior_subset by auto
  4930 have "f ` S <= f ` (closure S)" unfolding image_mono using closure_subset by auto
  4931 also have "... = f ` (closure (rel_interior S))" using convex_closure_rel_interior assms by auto  
  4932 also have "... <= closure (f ` (rel_interior S))" using closure_linear_image assms by auto 
  4933 finally have "closure (f ` S) = closure (f ` rel_interior S)"
  4934    using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure 
  4935          closure_mono[of "f ` rel_interior S" "f ` S"] * by auto
  4936 hence "rel_interior (f ` S) = rel_interior (f ` rel_interior S)" using assms convex_rel_interior
  4937    linear_conv_bounded_linear[of f] convex_linear_image[of S] convex_linear_image[of "rel_interior S"] 
  4938    closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"] by auto
  4939 hence "rel_interior (f ` S) <= f ` rel_interior S" using rel_interior_subset by auto
  4940 moreover
  4941 { fix z assume z_def: "z : f ` rel_interior S"
  4942   from this obtain z1 where z1_def: "z1 : rel_interior S & (f z1 = z)" by auto
  4943   { fix x assume "x : f ` S"
  4944     from this obtain x1 where x1_def: "x1 : S & (f x1 = x)" by auto
  4945     from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x1 + e *\<^sub>R z1 : S"
  4946        using convex_rel_interior_iff[of S z1] `convex S` x1_def z1_def by auto
  4947     moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
  4948         using x1_def z1_def `linear f` by (simp add: linear_add_cmul)
  4949     ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
  4950         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
  4951     hence "EX e. (e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S)" using e_def by auto
  4952   } from this have "z : rel_interior (f ` S)" using convex_rel_interior_iff[of "f ` S" z] `convex S` 
  4953        `linear f` `S ~= {}` convex_linear_image[of S f]  linear_conv_bounded_linear[of f] by auto
  4954 } ultimately have ?thesis by auto
  4955 } ultimately show ?thesis by blast
  4956 qed
  4957 
  4958 
  4959 lemma convex_linear_preimage:
  4960   assumes c:"convex S" and l:"bounded_linear f"
  4961   shows "convex(f -` S)"
  4962 proof(auto simp add: convex_def)
  4963   interpret f: bounded_linear f by fact
  4964   fix x y assume xy:"f x : S" "f y : S"
  4965   fix u v ::real assume uv:"0 <= u" "0 <= v" "u + v = 1"
  4966   show "f (u *\<^sub>R x + v *\<^sub>R y) : S" unfolding image_iff
  4967     using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR
  4968       c[unfolded convex_def] xy uv by auto
  4969 qed
  4970 
  4971 
  4972 lemma rel_interior_convex_linear_preimage:
  4973 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  4974 assumes "linear f"
  4975 assumes "convex S"
  4976 assumes "f -` (rel_interior S) ~= {}"
  4977 shows "rel_interior (f -` S) = f -` (rel_interior S)"
  4978 proof-
  4979 have "S ~= {}" using assms rel_interior_empty by auto
  4980 have nonemp: "f -` S ~= {}" by (metis assms(3) rel_interior_subset subset_empty vimage_mono) 
  4981 hence "S Int (range f) ~= {}" by auto
  4982 have conv: "convex (f -` S)" using convex_linear_preimage assms linear_conv_bounded_linear by auto
  4983 hence "convex (S Int (range f))"
  4984   by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
  4985 { fix z assume "z : f -` (rel_interior S)"
  4986   hence z_def: "f z : rel_interior S" by auto
  4987   { fix x assume "x : f -` S" from this have x_def: "f x : S" by auto
  4988     from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) : S"
  4989       using convex_rel_interior_iff[of S "f z"] z_def assms `S ~= {}` by auto
  4990     moreover have "(1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R x + e *\<^sub>R z)"
  4991       using `linear f` by (simp add: linear_def)
  4992     ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R z : f -` S" using e_def by auto
  4993   } hence "z : rel_interior (f -` S)" 
  4994        using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
  4995 } 
  4996 moreover
  4997 { fix z assume z_def: "z : rel_interior (f -` S)" 
  4998   { fix x assume x_def: "x: S Int (range f)"
  4999     from this obtain y where y_def: "(f y = x) & (y : f -` S)" by auto
  5000     from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R y+ e *\<^sub>R z : f -` S"
  5001       using convex_rel_interior_iff[of "f -` S" z] z_def conv by auto
  5002     moreover have "(1-e)*\<^sub>R x+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R y + e *\<^sub>R z)"
  5003       using `linear f` y_def by (simp add: linear_def)
  5004     ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R (f z) : S Int (range f)"
  5005       using e_def by auto
  5006   } hence "f z : rel_interior (S Int (range f))" using `convex (S Int (range f))`
  5007     `S Int (range f) ~= {}` convex_rel_interior_iff[of "S Int (range f)" "f z"] by auto
  5008   moreover have "affine (range f)"
  5009     by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
  5010   ultimately have "f z : rel_interior S" 
  5011     using convex_affine_rel_interior_inter[of S "range f"] assms by auto
  5012   hence "z : f -` (rel_interior S)" by auto
  5013 }
  5014 ultimately show ?thesis by auto
  5015 qed
  5016     
  5017 
  5018 lemma convex_direct_sum:
  5019 fixes S :: "('n::euclidean_space) set"
  5020 fixes T :: "('m::euclidean_space) set"
  5021 assumes "convex S" "convex T"
  5022 shows "convex (S <*> T)"
  5023 proof-
  5024 {
  5025 fix x assume "x : S <*> T"
  5026 from this obtain xs xt where xst_def: "xs : S & xt : T & (xs,xt) = x" by auto
  5027 fix y assume "y : S <*> T"
  5028 from this obtain ys yt where yst_def: "ys : S & yt : T & (ys,yt) = y" by auto
  5029 fix u v assume uv_def: "(u :: real)>=0 & (v :: real)>=0 & u+v=1"
  5030 have "u *\<^sub>R x + v *\<^sub>R y = (u *\<^sub>R xs + v *\<^sub>R ys, u *\<^sub>R xt + v *\<^sub>R yt)" using xst_def yst_def by auto
  5031 moreover have "u *\<^sub>R xs + v *\<^sub>R ys : S"
  5032    using uv_def xst_def yst_def convex_def[of S] assms by auto
  5033 moreover have "u *\<^sub>R xt + v *\<^sub>R yt : T"
  5034    using uv_def xst_def yst_def convex_def[of T] assms by auto
  5035 ultimately have "u *\<^sub>R x + v *\<^sub>R y : S <*> T" by auto
  5036 } from this show ?thesis unfolding convex_def by auto
  5037 qed
  5038 
  5039 
  5040 lemma convex_hull_direct_sum:
  5041 fixes S :: "('n::euclidean_space) set"
  5042 fixes T :: "('m::euclidean_space) set"
  5043 shows "convex hull (S <*> T) = (convex hull S) <*> (convex hull T)"
  5044 proof-
  5045 { fix x assume "x : (convex hull S) <*> (convex hull T)"
  5046   from this obtain xs xt where xst_def: "xs : convex hull S & xt : convex hull T & (xs,xt) = x" by auto
  5047   from xst_def obtain sI su where s: "finite sI & sI <= S & (ALL x:sI. 0 <= su x) & setsum su sI = 1 
  5048      & (SUM v:sI. su v *\<^sub>R v) = xs" using convex_hull_explicit[of S] by auto
  5049   from xst_def obtain tI tu where t: "finite tI & tI <= T & (ALL x:tI. 0 <= tu x) & setsum tu tI = 1 
  5050      & (SUM v:tI. tu v *\<^sub>R v) = xt" using convex_hull_explicit[of T] by auto
  5051   def I == "(sI <*> tI)"
  5052   def u == "(%i. (su (fst i))*(tu(snd i)))"
  5053   have "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
  5054      (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vs)"
  5055      using fst_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
  5056      by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
  5057   also have "...=(SUM vt:tI. tu vt *\<^sub>R (SUM vs:sI. su vs *\<^sub>R vs))"
  5058      using setsum_commute[of "(%vt vs. (su vs * tu vt) *\<^sub>R vs)" sI tI]
  5059      by (simp add: mult_commute scaleR_right.setsum)
  5060   also have "...=(SUM vt:tI. tu vt *\<^sub>R xs)" using s by auto
  5061   also have "...=(SUM vt:tI. tu vt) *\<^sub>R xs" by (simp add: scaleR_left.setsum)
  5062   also have "...=xs" using t by auto
  5063   finally have h1: "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xs" by auto
  5064   have "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
  5065      (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vt)"
  5066      using snd_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
  5067      by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
  5068   also have "...=(SUM vs:sI. su vs *\<^sub>R (SUM vt:tI. tu vt *\<^sub>R vt))"
  5069      by (simp add: mult_commute scaleR_right.setsum)
  5070   also have "...=(SUM vs:sI. su vs *\<^sub>R xt)" using t by auto
  5071   also have "...=(SUM vs:sI. su vs) *\<^sub>R xt" by (simp add: scaleR_left.setsum)
  5072   also have "...=xt" using s by auto
  5073   finally have h2: "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xt" by auto
  5074   from h1 h2 have "(SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v) = x" using xst_def by auto
  5075 
  5076   moreover have "finite I & (I <= S <*> T)" using s t I_def by auto 
  5077   moreover have "!i:I. 0 <= u i" using s t I_def u_def by (simp add: mult_nonneg_nonneg)
  5078   moreover have "setsum u I = 1" using u_def I_def setsum_cartesian_product[of "(% x y. (su x)*(tu y))"] 
  5079      s t setsum_product[of su sI tu tI] by (auto simp add: split_def)
  5080   ultimately have "x : convex hull (S <*> T)" 
  5081      apply (subst convex_hull_explicit[of "S <*> T"]) apply rule
  5082      apply (rule_tac x="I" in exI) apply (rule_tac x="u" in exI)
  5083      using I_def u_def by auto
  5084 }
  5085 hence "convex hull (S <*> T) >= (convex hull S) <*> (convex hull T)" by auto
  5086 moreover have "convex ((convex hull S) <*> (convex hull T))" 
  5087    by (simp add: convex_direct_sum convex_convex_hull)
  5088 ultimately show ?thesis 
  5089    using hull_minimal[of "S <*> T" "(convex hull S) <*> (convex hull T)" "convex"] 
  5090          hull_subset[of S convex] hull_subset[of T convex] by auto
  5091 qed
  5092 
  5093 lemma rel_interior_direct_sum:
  5094 fixes S :: "('n::euclidean_space) set"
  5095 fixes T :: "('m::euclidean_space) set"
  5096 assumes "convex S" "convex T"
  5097 shows "rel_interior (S <*> T) = rel_interior S <*> rel_interior T"
  5098 proof-
  5099 { assume "S={}" hence ?thesis apply auto using rel_interior_empty by auto }
  5100 moreover
  5101 { assume "T={}" hence ?thesis apply auto using rel_interior_empty by auto }
  5102 moreover {
  5103 assume "S ~={}" "T ~={}"
  5104 hence ri: "rel_interior S ~= {}" "rel_interior T ~= {}" using rel_interior_convex_nonempty assms by auto
  5105 hence "fst -` rel_interior S ~= {}" using fst_vimage_eq_Times[of "rel_interior S"] by auto
  5106 hence "rel_interior ((fst :: 'n * 'm => 'n) -` S) = fst -` rel_interior S"
  5107   using fst_linear `convex S` rel_interior_convex_linear_preimage[of fst S] by auto 
  5108 hence s: "rel_interior (S <*> (UNIV :: 'm set)) = rel_interior S <*> UNIV" by (simp add: fst_vimage_eq_Times)
  5109 from ri have "snd -` rel_interior T ~= {}" using snd_vimage_eq_Times[of "rel_interior T"] by auto
  5110 hence "rel_interior ((snd :: 'n * 'm => 'm) -` T) = snd -` rel_interior T"
  5111   using snd_linear `convex T` rel_interior_convex_linear_preimage[of snd T] by auto 
  5112 hence t: "rel_interior ((UNIV :: 'n set) <*> T) = UNIV <*> rel_interior T" by (simp add: snd_vimage_eq_Times)
  5113 from s t have *: "rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T) 
  5114   = rel_interior S <*> rel_interior T" by auto
  5115 have "(S <*> T) = (S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T)" by auto
  5116 hence "rel_interior (S <*> T) = rel_interior ((S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T))" by auto
  5117 also have "...=rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)" 
  5118    apply (subst convex_rel_interior_inter_two[of "S <*> (UNIV :: 'm set)" "(UNIV :: 'n set) <*> T"]) 
  5119    using * ri assms convex_direct_sum by auto
  5120 finally have ?thesis using * by auto
  5121 }
  5122 ultimately show ?thesis by blast
  5123 qed
  5124 
  5125 lemma rel_interior_scaleR: 
  5126 fixes S :: "('n::euclidean_space) set"
  5127 assumes "c ~= 0"
  5128 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  5129 using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
  5130       linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms by auto
  5131 
  5132 lemma rel_interior_convex_scaleR: 
  5133 fixes S :: "('n::euclidean_space) set"
  5134 assumes "convex S"
  5135 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  5136 by (metis assms linear_scaleR rel_interior_convex_linear_image)
  5137 
  5138 lemma convex_rel_open_scaleR: 
  5139 fixes S :: "('n::euclidean_space) set"
  5140 assumes "convex S" "rel_open S"
  5141 shows "convex ((op *\<^sub>R c) ` S) & rel_open ((op *\<^sub>R c) ` S)"
  5142 by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
  5143 
  5144 
  5145 lemma convex_rel_open_finite_inter: 
  5146 assumes "!S : I. (convex (S :: ('n::euclidean_space) set) & rel_open S)"
  5147 assumes "finite I"
  5148 shows "convex (Inter I) & rel_open (Inter I)"
  5149 proof-
  5150 { assume "Inter {rel_interior S |S. S : I} = {}"
  5151   hence "Inter I = {}" using assms unfolding rel_open_def by auto
  5152   hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
  5153 }
  5154 moreover
  5155 { assume "Inter {rel_interior S |S. S : I} ~= {}"
  5156   hence "rel_open (Inter I)" using assms unfolding rel_open_def
  5157     using convex_rel_interior_finite_inter[of I] by auto
  5158   hence ?thesis using convex_Inter assms by auto
  5159 } ultimately show ?thesis by auto
  5160 qed
  5161 
  5162 lemma convex_rel_open_linear_image:
  5163 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  5164 assumes "linear f"
  5165 assumes "convex S" "rel_open S"
  5166 shows "convex (f ` S) & rel_open (f ` S)"
  5167 by (metis assms convex_linear_image rel_interior_convex_linear_image 
  5168    linear_conv_bounded_linear rel_open_def)
  5169 
  5170 lemma convex_rel_open_linear_preimage:
  5171 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  5172 assumes "linear f"
  5173 assumes "convex S" "rel_open S"
  5174 shows "convex (f -` S) & rel_open (f -` S)" 
  5175 proof-
  5176 { assume "f -` (rel_interior S) = {}"
  5177   hence "f -` S = {}" using assms unfolding rel_open_def by auto
  5178   hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
  5179 }
  5180 moreover
  5181 { assume "f -` (rel_interior S) ~= {}"
  5182   hence "rel_open (f -` S)" using assms unfolding rel_open_def
  5183     using rel_interior_convex_linear_preimage[of f S] by auto
  5184   hence ?thesis using convex_linear_preimage assms linear_conv_bounded_linear by auto
  5185 } ultimately show ?thesis by auto
  5186 qed
  5187 
  5188 lemma rel_interior_projection:
  5189 fixes S :: "('m::euclidean_space*'n::euclidean_space) set"
  5190 fixes f :: "'m::euclidean_space => ('n::euclidean_space) set"
  5191 assumes "convex S"
  5192 assumes "f = (%y. {z. (y,z) : S})"
  5193 shows "(y,z) : rel_interior S <-> (y : rel_interior {y. (f y ~= {})} & z : rel_interior (f y))"
  5194 proof-
  5195 { fix y assume "y : {y. (f y ~= {})}" from this obtain z where "(y,z) : S" using assms by auto
  5196   hence "EX x. x : S & y = fst x" apply (rule_tac x="(y,z)" in exI) by auto
  5197   from this obtain x where "x : S & y = fst x" by blast
  5198   hence "y : fst ` S" unfolding image_def by auto
  5199 }
  5200 hence "fst ` S = {y. (f y ~= {})}" unfolding fst_def using assms by auto
  5201 hence h1: "fst ` rel_interior S = rel_interior {y. (f y ~= {})}"
  5202    using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
  5203 { fix y assume "y : rel_interior {y. (f y ~= {})}"
  5204   hence "y : fst ` rel_interior S" using h1 by auto
  5205   hence *: "rel_interior S Int fst -` {y} ~= {}" by auto
  5206   moreover have aff: "affine (fst -` {y})" unfolding affine_alt by (simp add: algebra_simps)
  5207   ultimately have **: "rel_interior (S Int fst -` {y}) = rel_interior S Int fst -` {y}"
  5208     using convex_affine_rel_interior_inter[of S "fst -` {y}"] assms by auto
  5209   have conv: "convex (S Int fst -` {y})" using convex_Int assms aff affine_imp_convex by auto
  5210   { fix x assume "x : f y"
  5211     hence "(y,x) : S Int (fst -` {y})" using assms by auto
  5212     moreover have "x = snd (y,x)" by auto
  5213     ultimately have "x : snd ` (S Int fst -` {y})" by blast
  5214   }
  5215   hence "snd ` (S Int fst -` {y}) = f y" using assms by auto
  5216   hence ***: "rel_interior (f y) = snd ` rel_interior (S Int fst -` {y})"
  5217     using rel_interior_convex_linear_image[of snd "S Int fst -` {y}"] snd_linear conv by auto 
  5218   { fix z assume "z : rel_interior (f y)"
  5219     hence "z : snd ` rel_interior (S Int fst -` {y})" using *** by auto
  5220     moreover have "{y} = fst ` rel_interior (S Int fst -` {y})" using * ** rel_interior_subset by auto   
  5221     ultimately have "(y,z) : rel_interior (S Int fst -` {y})" by force
  5222     hence "(y,z) : rel_interior S" using ** by auto
  5223   }
  5224   moreover
  5225   { fix z assume "(y,z) : rel_interior S"
  5226     hence "(y,z) : rel_interior (S Int fst -` {y})" using ** by auto
  5227     hence "z : snd ` rel_interior (S Int fst -` {y})" by (metis Range_iff snd_eq_Range) 
  5228     hence "z : rel_interior (f y)" using *** by auto
  5229   }
  5230   ultimately have "!!z. (y,z) : rel_interior S <-> z : rel_interior (f y)" by auto
  5231 } 
  5232 hence h2: "!!y z. y : rel_interior {t. f t ~= {}} ==> ((y, z) : rel_interior S) = (z : rel_interior (f y))"
  5233   by auto
  5234 { fix y z assume asm: "(y, z) : rel_interior S"
  5235   hence "y : fst ` rel_interior S" by (metis Domain_iff fst_eq_Domain)
  5236   hence "y : rel_interior {t. f t ~= {}}" using h1 by auto
  5237   hence "y : rel_interior {t. f t ~= {}} & (z : rel_interior (f y))" using h2 asm by auto
  5238 } from this show ?thesis using h2 by blast
  5239 qed
  5240 
  5241 subsubsection {* Relative interior of convex cone *}
  5242 
  5243 lemma cone_rel_interior:
  5244 fixes S :: "('m::euclidean_space) set"
  5245 assumes "cone S"
  5246 shows "cone ({0} Un (rel_interior S))"
  5247 proof-
  5248 { assume "S = {}" hence ?thesis by (simp add: rel_interior_empty cone_0) }
  5249 moreover
  5250 { assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  5251   hence *: "0:({0} Un (rel_interior S)) &
  5252            (!c. c>0 --> op *\<^sub>R c ` ({0} Un rel_interior S) = ({0} Un rel_interior S))"
  5253            by (auto simp add: rel_interior_scaleR)
  5254   hence ?thesis using cone_iff[of "{0} Un rel_interior S"] by auto
  5255 }
  5256 ultimately show ?thesis by blast
  5257 qed
  5258 
  5259 lemma rel_interior_convex_cone_aux:
  5260 fixes S :: "('m::euclidean_space) set"
  5261 assumes "convex S"
  5262 shows "(c,x) : rel_interior (cone hull ({(1 :: real)} <*> S)) <-> 
  5263        c>0 & x : ((op *\<^sub>R c) ` (rel_interior S))"
  5264 proof-
  5265 { assume "S={}" hence ?thesis by (simp add: rel_interior_empty cone_hull_empty) } 
  5266 moreover
  5267 { assume "S ~= {}" from this obtain s where "s : S" by auto
  5268 have conv: "convex ({(1 :: real)} <*> S)" using convex_direct_sum[of "{(1 :: real)}" S] 
  5269    assms convex_singleton[of "1 :: real"] by auto
  5270 def f == "(%y. {z. (y,z) : cone hull ({(1 :: real)} <*> S)})"
  5271 hence *: "(c, x) : rel_interior (cone hull ({(1 :: real)} <*> S)) =
  5272       (c : rel_interior {y. f y ~= {}} & x : rel_interior (f c))"
  5273   apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} <*> S)" f c x])
  5274   using convex_cone_hull[of "{(1 :: real)} <*> S"] conv by auto
  5275 { fix y assume "(y :: real)>=0"
  5276   hence "y *\<^sub>R (1,s) : cone hull ({(1 :: real)} <*> S)"
  5277      using cone_hull_expl[of "{(1 :: real)} <*> S"] `s:S` by auto
  5278   hence "f y ~= {}" using f_def by auto
  5279 }
  5280 hence "{y. f y ~= {}} = {0..}" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
  5281 hence **: "rel_interior {y. f y ~= {}} = {0<..}" using rel_interior_real_semiline by auto
  5282 { fix c assume "c>(0 :: real)"
  5283   hence "f c = (op *\<^sub>R c ` S)" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
  5284   hence "rel_interior (f c)= (op *\<^sub>R c ` rel_interior S)"
  5285      using rel_interior_convex_scaleR[of S c] assms by auto
  5286 }
  5287 hence ?thesis using * ** by auto
  5288 } ultimately show ?thesis by blast
  5289 qed
  5290 
  5291 
  5292 lemma rel_interior_convex_cone:
  5293 fixes S :: "('m::euclidean_space) set"
  5294 assumes "convex S"
  5295 shows "rel_interior (cone hull ({(1 :: real)} <*> S)) = 
  5296        {(c,c *\<^sub>R x) |c x. c>0 & x : (rel_interior S)}"
  5297 (is "?lhs=?rhs")
  5298 proof-
  5299 { fix z assume "z:?lhs" 
  5300   have *: "z=(fst z,snd z)" by auto 
  5301   have "z:?rhs" using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms `z:?lhs` apply auto
  5302      apply (rule_tac x="fst z" in exI) apply (rule_tac x="x" in exI) using * by auto
  5303 }
  5304 moreover
  5305 { fix z assume "z:?rhs" hence "z:?lhs" 
  5306   using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms by auto
  5307 }
  5308 ultimately show ?thesis by blast
  5309 qed
  5310 
  5311 lemma convex_hull_finite_union:
  5312 assumes "finite I"
  5313 assumes "!i:I. (convex (S i) & (S i) ~= {})"
  5314 shows "convex hull (Union (S ` I)) = 
  5315        {setsum (%i. c i *\<^sub>R s i) I |c s. (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)}"
  5316   (is "?lhs = ?rhs")
  5317 proof-
  5318 { fix x assume "x : ?rhs" 
  5319   from this obtain c s 
  5320     where *: "setsum (%i. c i *\<^sub>R s i) I=x" "(setsum c I = 1)"
  5321      "(!i:I. c i >= 0) & (!i:I. s i : S i)" by auto
  5322   hence "!i:I. s i : convex hull (Union (S ` I))" using hull_subset[of "Union (S ` I)" convex] by auto
  5323   hence "x : ?lhs" unfolding *(1)[THEN sym]
  5324      apply (subst convex_setsum[of I "convex hull Union (S ` I)" c s])
  5325      using * assms convex_convex_hull by auto
  5326 } hence "?lhs >= ?rhs" by auto
  5327 
  5328 { fix i assume "i:I"
  5329     from this assms have "EX p. p : S i" by auto
  5330 } 
  5331 from this obtain p where p_def: "!i:I. p i : S i" by metis
  5332 
  5333 { fix i assume "i:I"
  5334   { fix x assume "x : S i"
  5335     def c == "(%j. if (j=i) then (1::real) else 0)"
  5336     hence *: "setsum c I = 1" using `finite I` `i:I` setsum_delta[of I i "(%(j::'a). (1::real))"] by auto
  5337     def s == "(%j. if (j=i) then x else p j)"
  5338     hence "!j. c j *\<^sub>R s j = (if (j=i) then x else 0)" using c_def by (auto simp add: algebra_simps)
  5339     hence "x = setsum (%i. c i *\<^sub>R s i) I"
  5340        using s_def c_def `finite I` `i:I` setsum_delta[of I i "(%(j::'a). x)"] by auto 
  5341     hence "x : ?rhs" apply auto
  5342       apply (rule_tac x="c" in exI) 
  5343       apply (rule_tac x="s" in exI) using * c_def s_def p_def `x : S i` by auto 
  5344   } hence "?rhs >= S i" by auto
  5345 } hence *: "?rhs >= Union (S ` I)" by auto
  5346 
  5347 { fix u v assume uv: "(u :: real)>=0 & v>=0 & u+v=1"
  5348   fix x y assume xy: "(x : ?rhs) & (y : ?rhs)"
  5349   from xy obtain c s where xc: "x=setsum (%i. c i *\<^sub>R s i) I &
  5350      (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)" by auto
  5351   from xy obtain d t where yc: "y=setsum (%i. d i *\<^sub>R t i) I &
  5352      (!i:I. d i >= 0) & (setsum d I = 1) & (!i:I. t i : S i)" by auto
  5353   def e == "(%i. u * (c i)+v * (d i))"
  5354   have ge0: "!i:I. e i >= 0"  using e_def xc yc uv by (simp add: mult_nonneg_nonneg)
  5355   have "setsum (%i. u * c i) I = u * setsum c I" by (simp add: setsum_right_distrib)
  5356   moreover have "setsum (%i. v * d i) I = v * setsum d I" by (simp add: setsum_right_distrib)
  5357   ultimately have sum1: "setsum e I = 1" using e_def xc yc uv by (simp add: setsum_addf)
  5358   def q == "(%i. if (e i = 0) then (p i) 
  5359                  else (u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))"
  5360   { fix i assume "i:I"
  5361     { assume "e i = 0" hence "q i : S i" using `i:I` p_def q_def by auto }
  5362     moreover
  5363     { assume "e i ~= 0" 
  5364       hence "q i : S i" using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"] 
  5365          mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
  5366          assms q_def e_def `i:I` `e i ~= 0` xc yc uv by auto
  5367     } ultimately have "q i : S i" by auto
  5368   } hence qs: "!i:I. q i : S i" by auto
  5369   { fix i assume "i:I"
  5370     { assume "e i = 0" 
  5371       have ge: "u * (c i) >= 0 & v * (d i) >= 0" using xc yc uv `i:I` by (simp add: mult_nonneg_nonneg)
  5372       moreover hence "u * (c i) <= 0 & v * (d i) <= 0" using `e i = 0` e_def `i:I` by simp 
  5373       ultimately have "u * (c i) = 0 & v * (d i) = 0" by auto
  5374       hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
  5375          using `e i = 0` by auto
  5376     }
  5377     moreover
  5378     { assume "e i ~= 0"
  5379       hence "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
  5380          using q_def by auto
  5381       hence "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
  5382              = (e i) *\<^sub>R (q i)" by auto
  5383       hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
  5384          using `e i ~= 0` by (simp add: algebra_simps)
  5385     } ultimately have 
  5386       "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by blast
  5387   } hence *: "!i:I.
  5388     (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by auto
  5389   have "u *\<^sub>R x + v *\<^sub>R y =
  5390        setsum (%i. (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i)) I"
  5391           using xc yc by (simp add: algebra_simps scaleR_right.setsum setsum_addf)
  5392   also have "...=setsum (%i. (e i) *\<^sub>R (q i)) I" using * by auto
  5393   finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (%i. (e i) *\<^sub>R (q i)) I" by auto
  5394   hence "u *\<^sub>R x + v *\<^sub>R y : ?rhs" using ge0 sum1 qs by auto
  5395 } hence "convex ?rhs" unfolding convex_def by auto
  5396 from this show ?thesis using `?lhs >= ?rhs` * 
  5397    hull_minimal[of "Union (S ` I)" "?rhs" "convex"] by blast
  5398 qed
  5399 
  5400 lemma convex_hull_union_two:
  5401 fixes S T :: "('m::euclidean_space) set"
  5402 assumes "convex S" "S ~= {}" "convex T" "T ~= {}"
  5403 shows "convex hull (S Un T) = {u *\<^sub>R s + v *\<^sub>R t |u v s t. u>=0 & v>=0 & u+v=1 & s:S & t:T}"
  5404   (is "?lhs = ?rhs")
  5405 proof-
  5406 def I == "{(1::nat),2}"
  5407 def s == "(%i. (if i=(1::nat) then S else T))"
  5408 have "Union (s ` I) = S Un T" using s_def I_def by auto
  5409 hence "convex hull (Union (s ` I)) = convex hull (S Un T)" by auto
  5410 moreover have "convex hull Union (s ` I) =
  5411     {SUM i:I. c i *\<^sub>R sa i |c sa. (ALL i:I. 0 <= c i) & setsum c I = 1 & (ALL i:I. sa i : s i)}"
  5412     apply (subst convex_hull_finite_union[of I s]) using assms s_def I_def by auto
  5413 moreover have 
  5414   "{SUM i:I. c i *\<^sub>R sa i |c sa. (ALL i:I. 0 <= c i) & setsum c I = 1 & (ALL i:I. sa i : s i)} <=
  5415   ?rhs"
  5416   using s_def I_def by auto
  5417 ultimately have "?lhs<=?rhs" by auto 
  5418 { fix x assume "x : ?rhs" 
  5419   from this obtain u v s t 
  5420     where *: "x=u *\<^sub>R s + v *\<^sub>R t & u>=0 & v>=0 & u+v=1 & s:S & t:T" by auto
  5421   hence "x : convex hull {s,t}" using convex_hull_2[of s t] by auto
  5422   hence "x : convex hull (S Un T)" using * hull_mono[of "{s, t}" "S Un T"] by auto
  5423 } hence "?lhs >= ?rhs" by blast
  5424 from this show ?thesis using `?lhs<=?rhs` by auto
  5425 qed
  5426 
  5427 subsection {* Convexity on direct sums *}
  5428 
  5429 lemma closure_sum:
  5430   fixes S T :: "('n::euclidean_space) set"
  5431   shows "closure S \<oplus> closure T \<subseteq> closure (S \<oplus> T)"
  5432 proof-
  5433   have "(closure S) \<oplus> (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
  5434     by (simp add: set_plus_image)
  5435   also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
  5436     using closure_direct_sum by auto
  5437   also have "... \<subseteq> closure (S \<oplus> T)"
  5438     using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
  5439     by (auto simp: set_plus_image)
  5440   finally show ?thesis
  5441     by auto
  5442 qed
  5443 
  5444 lemma convex_oplus:
  5445 fixes S T :: "('n::euclidean_space) set"
  5446 assumes "convex S" "convex T"
  5447 shows "convex (S \<oplus> T)"
  5448 proof-
  5449 have "{x + y |x y. x