src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author wenzelm
Wed Dec 29 17:34:41 2010 +0100 (2010-12-29)
changeset 41413 64cd30d6b0b8
parent 40897 1eb1b2f9d062
child 41959 b460124855b8
permissions -rw-r--r--
explicit file specifications -- avoid secondary load path;
     1 (*  Title:      HOL/Library/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 Topology_Euclidean_Space Convex "~~/src/HOL/Library/Set_Algebras"
    10 begin
    11 
    12 
    13 (* ------------------------------------------------------------------------- *)
    14 (* To be moved elsewhere                                                     *)
    15 (* ------------------------------------------------------------------------- *)
    16 
    17 lemma linear_scaleR: "linear (%(x :: 'n::euclidean_space). scaleR c x)"
    18   by (metis linear_conv_bounded_linear scaleR.bounded_linear_right)
    19 
    20 lemma injective_scaleR: 
    21 assumes "(c :: real) ~= 0"
    22 shows "inj (%(x :: 'n::euclidean_space). scaleR c x)"
    23   by (metis assms injI real_vector.scale_cancel_left)
    24 
    25 lemma linear_add_cmul:
    26 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)" 
    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 card_ge1: assumes "d ~= {}" "finite d" shows "card d >= 1"
    44 by (metis Suc_eq_plus1 assms(1) assms(2) card_eq_0_iff fact_ge_one_nat fact_num_eq_if_nat one_le_mult_iff plus_nat.add_0)
    45 
    46 lemma inj_on_image_mem_iff: "inj_on f B ==> (A <= B) ==> (f a : f`A) ==> (a : B) ==> (a : A)" 
    47 by (blast dest: inj_onD)
    48 
    49 lemma independent_injective_on_span_image:
    50   assumes iS: "independent (S::(_::euclidean_space) set)" 
    51      and lf: "linear f" and fi: "inj_on f (span S)"
    52   shows "independent (f ` S)"
    53 proof-
    54   {fix a assume a: "a : S" "f a : span (f ` S - {f a})"
    55     have eq: "f ` S - {f a} = f ` (S - {a})" using fi a span_inc
    56       by (auto simp add: inj_on_def)
    57     from a have "f a : f ` span (S -{a})"
    58       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
    59     moreover have "span (S -{a}) <= span S" using span_mono[of "S-{a}" S] by auto
    60     ultimately have "a : span (S -{a})" using fi a span_inc by (auto simp add: inj_on_def)
    61     with a(1) iS  have False by (simp add: dependent_def) }
    62   then show ?thesis unfolding dependent_def by blast
    63 qed
    64 
    65 lemma dim_image_eq:
    66 fixes f :: "'n::euclidean_space => 'm::euclidean_space"
    67 assumes lf: "linear f" and fi: "inj_on f (span S)" 
    68 shows "dim (f ` S) = dim (S:: ('n::euclidean_space) set)"
    69 proof-
    70 obtain B where B_def: "B<=S & independent B & S <= span B & card B = dim S" 
    71   using basis_exists[of S] by auto
    72 hence "span S = span B" using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
    73 hence "independent (f ` B)" using independent_injective_on_span_image[of B f] B_def assms by auto
    74 moreover have "card (f ` B) = card B" using assms card_image[of f B] subset_inj_on[of f "span S" B] 
    75    B_def span_inc by auto
    76 moreover have "(f ` B) <= (f ` S)" using B_def by auto
    77 ultimately have "dim (f ` S) >= dim S" 
    78   using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
    79 from this show ?thesis using dim_image_le[of f S] assms by auto
    80 qed
    81 
    82 lemma linear_injective_on_subspace_0:
    83 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
    84 assumes lf: "linear f" and "subspace S"
    85   shows "inj_on f S <-> (!x : S. f x = 0 --> x = 0)"
    86 proof-
    87   have "inj_on f S <-> (!x : S. !y : S. f x = f y --> x = y)" by (simp add: inj_on_def)
    88   also have "... <-> (!x : S. !y : S. f x - f y = 0 --> x - y = 0)" by simp
    89   also have "... <-> (!x : S. !y : S. f (x - y) = 0 --> x - y = 0)"
    90     by (simp add: linear_sub[OF lf])
    91   also have "... <-> (! x : S. f x = 0 --> x = 0)" 
    92     using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
    93   finally show ?thesis .
    94 qed
    95 
    96 lemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"
    97   unfolding subspace_def by auto 
    98 
    99 lemma span_eq[simp]: "(span s = s) <-> subspace s"
   100 proof-
   101   { fix f assume "f <= subspace"
   102     hence "subspace (Inter f)" using subspace_Inter[of f] unfolding subset_eq mem_def by auto  }
   103   thus ?thesis using hull_eq[unfolded mem_def, of subspace s] span_def by auto
   104 qed
   105 
   106 lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
   107   by(auto simp add: inj_on_def euclidean_eq[where 'a='n])
   108   
   109 lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
   110 proof-
   111   have eq: "?S = basis ` d" by blast
   112   show ?thesis unfolding eq apply(rule finite_subset[OF _ range_basis_finite]) by auto
   113 qed
   114 
   115 lemma card_substdbasis: assumes "d \<subseteq> {..<DIM('n::euclidean_space)}"
   116   shows "card {basis i ::'n::euclidean_space | i. i : d} = card d" (is "card ?S = _")
   117 proof-
   118   have eq: "?S = basis ` d" by blast
   119   show ?thesis unfolding eq using card_image[OF basis_inj_on[of d]] assms by auto
   120 qed
   121 
   122 lemma substdbasis_expansion_unique: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
   123   shows "setsum (%i. f i *\<^sub>R basis i) d = (x::'a::euclidean_space)
   124       <-> (!i<DIM('a). (i:d --> f i = x$$i) & (i ~: d --> x $$ i = 0))"
   125 proof- have *:"\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto
   126   have **:"finite d" apply(rule finite_subset[OF assms]) by fastsimp
   127   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)"
   128     unfolding euclidean_component.setsum euclidean_scaleR basis_component *
   129     apply(rule setsum_cong2) using assms by auto
   130   show ?thesis unfolding euclidean_eq[where 'a='a] *** setsum_delta[OF **] using assms by auto
   131 qed
   132 
   133 lemma independent_substdbasis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
   134   shows "independent {basis i ::'a::euclidean_space |i. i : (d :: nat set)}" (is "independent ?A")
   135 proof -
   136   have *: "{basis i |i. i < DIM('a)} = basis ` {..<DIM('a)}" by auto
   137   show ?thesis
   138     apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
   139     using independent_basis[where 'a='a] assms by (auto simp: *)
   140 qed
   141 
   142 lemma dim_cball: 
   143 assumes "0<e"
   144 shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
   145 proof-
   146 { fix x :: "'n::euclidean_space" def y == "(e/norm x) *\<^sub>R x"
   147   hence "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto
   148   moreover have "x = (norm x/e) *\<^sub>R y" using y_def assms by simp
   149   moreover hence "x = (norm x/e) *\<^sub>R y"  by auto
   150   ultimately have "x : span (cball 0 e)"
   151      using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
   152 } hence "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto 
   153 from this show ?thesis using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
   154 qed
   155 
   156 lemma indep_card_eq_dim_span:
   157 fixes B :: "('n::euclidean_space) set"
   158 assumes "independent B"
   159 shows "finite B & card B = dim (span B)" 
   160   using assms basis_card_eq_dim[of B "span B"] span_inc by auto
   161 
   162 lemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"
   163   apply(rule ccontr) by auto
   164 
   165 lemma translate_inj_on: 
   166 fixes A :: "('n::euclidean_space) set"
   167 shows "inj_on (%x. a+x) A" unfolding inj_on_def by auto
   168 
   169 lemma translation_assoc:
   170   fixes a b :: "'a::ab_group_add"
   171   shows "(\<lambda>x. b+x) ` ((\<lambda>x. a+x) ` S) = (\<lambda>x. (a+b)+x) ` S" by auto
   172 
   173 lemma translation_invert:
   174   fixes a :: "'a::ab_group_add"
   175   assumes "(\<lambda>x. a+x) ` A = (\<lambda>x. a+x) ` B"
   176   shows "A=B"
   177 proof-
   178   have "(%x. -a+x) ` ((%x. a+x) ` A) = (%x. -a+x) ` ((%x. a+x) ` B)" using assms by auto
   179   from this show ?thesis using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto 
   180 qed
   181 
   182 lemma translation_galois:
   183   fixes a :: "'a::ab_group_add"
   184   shows "T=((\<lambda>x. a+x) ` S) <-> S=((\<lambda>x. (-a)+x) ` T)"
   185   using translation_assoc[of "-a" a S] apply auto
   186   using translation_assoc[of a "-a" T] by auto
   187 
   188 lemma translation_inverse_subset:
   189   assumes "((%x. -a+x) ` V) <= (S :: 'n::ab_group_add set)" 
   190   shows "V <= ((%x. a+x) ` S)"
   191 proof-
   192 { fix x assume "x:V" hence "x-a : S" using assms by auto 
   193   hence "x : {a + v |v. v : S}" apply auto apply (rule exI[of _ "x-a"]) apply simp done 
   194   hence "x : ((%x. a+x) ` S)" by auto }
   195   from this show ?thesis by auto
   196 qed
   197 
   198 lemma basis_0[simp]:"(basis i::'a::euclidean_space) = 0 \<longleftrightarrow> i\<ge>DIM('a)"
   199   using norm_basis[of i, where 'a='a] unfolding norm_eq_zero[where 'a='a,THEN sym] by auto
   200 
   201 lemma basis_to_basis_subspace_isomorphism:
   202   assumes s: "subspace (S:: ('n::euclidean_space) set)"
   203   and t: "subspace (T :: ('m::euclidean_space) set)"
   204   and d: "dim S = dim T"
   205   and B: "B <= S" "independent B" "S <= span B" "card B = dim S"
   206   and C: "C <= T" "independent C" "T <= span C" "card C = dim T"
   207   shows "EX f. linear f & f ` B = C & f ` S = T & inj_on f S"
   208 proof-
   209 (* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
   210 *)
   211   from B independent_bound have fB: "finite B" by blast
   212   from C independent_bound have fC: "finite C" by blast
   213   from B(4) C(4) card_le_inj[of B C] d obtain f where
   214     f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
   215   from linear_independent_extend[OF B(2)] obtain g where
   216     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
   217   from inj_on_iff_eq_card[OF fB, of f] f(2)
   218   have "card (f ` B) = card B" by simp
   219   with B(4) C(4) have ceq: "card (f ` B) = card C" using d
   220     by simp
   221   have "g ` B = f ` B" using g(2)
   222     by (auto simp add: image_iff)
   223   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
   224   finally have gBC: "g ` B = C" .
   225   have gi: "inj_on g B" using f(2) g(2)
   226     by (auto simp add: inj_on_def)
   227   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
   228   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
   229     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
   230     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
   231     have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
   232     have "x=y" using g0[OF th1 th0] by simp }
   233   then have giS: "inj_on g S"
   234     unfolding inj_on_def by blast
   235   from span_subspace[OF B(1,3) s]
   236   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
   237   also have "\<dots> = span C" unfolding gBC ..
   238   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
   239   finally have gS: "g ` S = T" .
   240   from g(1) gS giS gBC show ?thesis by blast
   241 qed
   242 
   243 lemma closure_linear_image:
   244 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
   245 assumes "linear f"
   246 shows "f ` (closure S) <= closure (f ` S)"
   247 using image_closure_subset[of S f "closure (f ` S)"] assms linear_conv_bounded_linear[of f] 
   248 linear_continuous_on[of f "closure S"] closed_closure[of "f ` S"] closure_subset[of "f ` S"] by auto
   249 
   250 lemma closure_injective_linear_image:
   251 fixes f :: "('n::euclidean_space) => ('n::euclidean_space)"
   252 assumes "linear f" "inj f"
   253 shows "f ` (closure S) = closure (f ` S)"
   254 proof-
   255 obtain f' where f'_def: "linear f' & f o f' = id & f' o f = id" 
   256    using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
   257 hence "f' ` closure (f ` S) <= closure (S)"
   258    using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto
   259 hence "f ` f' ` closure (f ` S) <= f ` closure (S)" by auto
   260 hence "closure (f ` S) <= f ` closure (S)" using image_compose[of f f' "closure (f ` S)"] f'_def by auto
   261 from this show ?thesis using closure_linear_image[of f S] assms by auto 
   262 qed
   263 
   264 lemma closure_direct_sum:
   265 fixes S :: "('n::euclidean_space) set"
   266 fixes T :: "('m::euclidean_space) set"
   267 shows "closure (S <*> T) = closure S <*> closure T"
   268 proof-
   269 { fix x assume "x : closure S <*> closure T"
   270   from this obtain xs xt where xst_def: "xs : closure S & xt : closure T & (xs,xt) = x" by auto
   271   { fix ee assume ee_def: "(ee :: real) > 0"
   272     def e == "ee/2" hence e_def: "(e :: real)>0 & 2*e=ee" using ee_def by auto
   273     from this obtain e where e_def: "(e :: real)>0 & 2*e=ee" by auto
   274     obtain ys where ys_def: "ys : S & (dist ys xs < e)"
   275       using e_def xst_def closure_approachable[of xs S] by auto
   276     obtain yt where yt_def: "yt : T & (dist yt xt < e)"
   277       using e_def xst_def closure_approachable[of xt T] by auto
   278     from ys_def yt_def have "dist (ys,yt) (xs,xt) < sqrt (2*e^2)" 
   279       unfolding dist_norm apply (auto simp add: norm_Pair) 
   280       using mult_strict_mono'[of "norm (ys - xs)" e "norm (ys - xs)" e] e_def
   281       mult_strict_mono'[of "norm (yt - xt)" e "norm (yt - xt)" e] by (simp add: power2_eq_square)
   282     hence "((ys,yt) : S <*> T) & (dist (ys,yt) x < 2*e)"
   283       using e_def sqrt_add_le_add_sqrt[of "e^2" "e^2"] xst_def ys_def yt_def by auto
   284     hence "EX y: S <*> T. dist y x < ee" using e_def by auto
   285   } hence "x : closure (S <*> T)" using closure_approachable[of x "S <*> T"] by auto
   286 }
   287 hence "closure (S <*> T) >= closure S <*> closure T" by auto
   288 moreover have "closed (closure S <*> closure T)" using closed_Times by auto
   289 ultimately show ?thesis using closure_minimal[of "S <*> T" "closure S <*> closure T"]
   290   closure_subset[of S] closure_subset[of T] by auto
   291 qed
   292 
   293 lemma closure_scaleR: 
   294 fixes S :: "('n::euclidean_space) set"
   295 shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
   296 proof-
   297 { assume "c ~= 0" hence ?thesis using closure_injective_linear_image[of "(op *\<^sub>R c)" S]
   298       linear_scaleR injective_scaleR by auto 
   299 }
   300 moreover
   301 { assume zero: "c=0 & S ~= {}"
   302   hence "closure S ~= {}" using closure_subset by auto
   303   hence "op *\<^sub>R c ` (closure S) = {0}" using zero by auto
   304   moreover have "op *\<^sub>R 0 ` S = {0}" using zero by auto
   305   ultimately have ?thesis using zero by auto
   306 }
   307 moreover
   308 { assume "S={}" hence ?thesis by auto }
   309 ultimately show ?thesis by blast
   310 qed
   311 
   312 lemma fst_linear: "linear fst" unfolding linear_def by (simp add: algebra_simps)
   313 
   314 lemma snd_linear: "linear snd" unfolding linear_def by (simp add: algebra_simps)
   315 
   316 lemma fst_snd_linear: "linear (%(x,y). x + y)" unfolding linear_def by (simp add: algebra_simps)
   317 
   318 lemma scaleR_2:
   319   fixes x :: "'a::real_vector"
   320   shows "scaleR 2 x = x + x"
   321 unfolding one_add_one_is_two [symmetric] scaleR_left_distrib by simp
   322 
   323 declare euclidean_simps[simp]
   324 
   325 lemma vector_choose_size: "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
   326   apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"]) using DIM_positive[where 'a='a] by auto
   327 
   328 lemma setsum_delta_notmem: assumes "x\<notin>s"
   329   shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
   330         "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
   331         "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
   332         "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
   333   apply(rule_tac [!] setsum_cong2) using assms by auto
   334 
   335 lemma setsum_delta'':
   336   fixes s::"'a::real_vector set" assumes "finite s"
   337   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)"
   338 proof-
   339   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
   340   show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
   341 qed
   342 
   343 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
   344 
   345 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
   346   (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})"
   347   using image_affinity_interval[of m 0 a b] by auto
   348 
   349 lemma dist_triangle_eq:
   350   fixes x y z :: "'a::euclidean_space"
   351   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)"
   352 proof- have *:"x - y + (y - z) = x - z" by auto
   353   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
   354     by(auto simp add:norm_minus_commute) qed
   355 
   356 lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
   357 
   358 lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
   359   unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
   360 
   361 lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
   362   unfolding norm_eq_sqrt_inner by simp
   363 
   364 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
   365   unfolding norm_eq_sqrt_inner by simp
   366 
   367 
   368 
   369 subsection {* Affine set and affine hull.*}
   370 
   371 definition
   372   affine :: "'a::real_vector set \<Rightarrow> bool" where
   373   "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)"
   374 
   375 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)"
   376 unfolding affine_def by(metis eq_diff_eq')
   377 
   378 lemma affine_empty[intro]: "affine {}"
   379   unfolding affine_def by auto
   380 
   381 lemma affine_sing[intro]: "affine {x}"
   382   unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
   383 
   384 lemma affine_UNIV[intro]: "affine UNIV"
   385   unfolding affine_def by auto
   386 
   387 lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
   388   unfolding affine_def by auto 
   389 
   390 lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
   391   unfolding affine_def by auto
   392 
   393 lemma affine_affine_hull: "affine(affine hull s)"
   394   unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
   395   unfolding mem_def by auto
   396 
   397 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   398 by (metis affine_affine_hull hull_same mem_def)
   399 
   400 lemma setsum_restrict_set'': assumes "finite A"
   401   shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
   402   unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
   403 
   404 subsection {* Some explicit formulations (from Lars Schewe). *}
   405 
   406 lemma affine: fixes V::"'a::real_vector set"
   407   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)"
   408 unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
   409 defer apply(rule, rule, rule, rule, rule) proof-
   410   fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
   411     "\<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"
   412   thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
   413     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) 
   414     by(auto simp add: scaleR_left_distrib[THEN sym])
   415 next
   416   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"
   417     "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
   418   def n \<equiv> "card s"
   419   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   420   thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
   421     assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
   422     then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
   423     thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
   424       by(auto simp add: setsum_clauses(2))
   425   next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
   426       case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
   427       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;
   428                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
   429         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"
   430            "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
   431       have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
   432         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
   433         thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
   434           less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
   435       then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
   436 
   437       have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
   438       have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
   439       have **:"setsum u (s - {x}) = 1 - u x"
   440         using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
   441       have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
   442       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
   443         case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
   444           assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
   445           thus False using True by auto qed auto
   446         thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
   447         unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
   448       next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
   449         then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
   450         thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
   451           using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
   452       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"
   453         apply-apply(rule as(3)[rule_format]) 
   454         unfolding  RealVector.scaleR_right.setsum using x(1) as(6) by auto
   455       thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
   456          apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
   457          using `u x \<noteq> 1` by auto 
   458     qed auto
   459   next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
   460     thus ?thesis using as(4,5) by simp
   461   qed(insert `s\<noteq>{}` `finite s`, auto)
   462 qed
   463 
   464 lemma affine_hull_explicit:
   465   "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}"
   466   apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
   467   apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
   468   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"
   469     apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
   470 next
   471   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" 
   472   thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
   473 next
   474   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
   475     apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
   476     fix u v ::real assume uv:"u + v = 1"
   477     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"
   478     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
   479     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"
   480     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
   481     have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
   482     have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
   483     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"
   484       apply(rule_tac x="sx \<union> sy" in exI)
   485       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)
   486       unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
   487       unfolding scaleR_scaleR[THEN sym] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
   488       unfolding x y using x(1-3) y(1-3) uv by simp qed qed
   489 
   490 lemma affine_hull_finite:
   491   assumes "finite s"
   492   shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
   493   unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq apply (rule,rule)
   494   apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
   495   fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   496   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"
   497     apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
   498 next
   499   fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
   500   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"
   501   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)
   502     unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
   503 
   504 subsection {* Stepping theorems and hence small special cases. *}
   505 
   506 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   507   apply(rule hull_unique) unfolding mem_def by auto
   508 
   509 lemma affine_hull_finite_step:
   510   fixes y :: "'a::real_vector"
   511   shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
   512   "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>
   513                 (\<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)")
   514 proof-
   515   show ?th1 by simp
   516   assume ?as 
   517   { assume ?lhs
   518     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
   519     have ?rhs proof(cases "a\<in>s")
   520       case True hence *:"insert a s = s" by auto
   521       show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
   522     next
   523       case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
   524     qed  } moreover
   525   { assume ?rhs
   526     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
   527     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
   528     have ?lhs proof(cases "a\<in>s")
   529       case True thus ?thesis
   530         apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
   531         unfolding setsum_clauses(2)[OF `?as`]  apply simp
   532         unfolding scaleR_left_distrib and setsum_addf 
   533         unfolding vu and * and scaleR_zero_left
   534         by (auto simp add: setsum_delta[OF `?as`])
   535     next
   536       case False 
   537       hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
   538                "\<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
   539       from False show ?thesis
   540         apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
   541         unfolding setsum_clauses(2)[OF `?as`] and * using vu
   542         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)]
   543         using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
   544     qed }
   545   ultimately show "?lhs = ?rhs" by blast
   546 qed
   547 
   548 lemma affine_hull_2:
   549   fixes a b :: "'a::real_vector"
   550   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
   551 proof-
   552   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   553          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   554   have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
   555     using affine_hull_finite[of "{a,b}"] by auto
   556   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
   557     by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
   558   also have "\<dots> = ?rhs" unfolding * by auto
   559   finally show ?thesis by auto
   560 qed
   561 
   562 lemma affine_hull_3:
   563   fixes a b c :: "'a::real_vector"
   564   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")
   565 proof-
   566   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   567          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   568   show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
   569     unfolding * apply auto
   570     apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
   571     apply(rule_tac x=u in exI) by(auto intro!: exI)
   572 qed
   573 
   574 lemma mem_affine:
   575   assumes "affine S" "x : S" "y : S" "u+v=1"
   576   shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
   577   using assms affine_def[of S] by auto
   578 
   579 lemma mem_affine_3:
   580   assumes "affine S" "x : S" "y : S" "z : S" "u+v+w=1"
   581   shows "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : S"
   582 proof-
   583 have "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : affine hull {x, y, z}"
   584   using affine_hull_3[of x y z] assms by auto
   585 moreover have " affine hull {x, y, z} <= affine hull S" 
   586   using hull_mono[of "{x, y, z}" "S"] assms by auto
   587 moreover have "affine hull S = S" 
   588   using assms affine_hull_eq[of S] by auto
   589 ultimately show ?thesis by auto 
   590 qed
   591 
   592 lemma mem_affine_3_minus:
   593   assumes "affine S" "x : S" "y : S" "z : S"
   594   shows "x + v *\<^sub>R (y-z) : S"
   595 using mem_affine_3[of S x y z 1 v "-v"] assms by (simp add: algebra_simps)
   596 
   597 
   598 subsection {* Some relations between affine hull and subspaces. *}
   599 
   600 lemma affine_hull_insert_subset_span:
   601   fixes a :: "'a::euclidean_space"
   602   shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
   603   unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
   604   apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
   605   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"
   606   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
   607   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)"
   608     apply(rule_tac x="x - a" in exI)
   609     apply (rule conjI, simp)
   610     apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
   611     apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
   612     apply (rule conjI) using as(1) apply simp
   613     apply (erule conjI)
   614     using as(1)
   615     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)
   616     unfolding as by simp qed
   617 
   618 lemma affine_hull_insert_span:
   619   fixes a :: "'a::euclidean_space"
   620   assumes "a \<notin> s"
   621   shows "affine hull (insert a s) =
   622             {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
   623   apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
   624   unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
   625   fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
   626   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
   627   def f \<equiv> "(\<lambda>x. x + a) ` t"
   628   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 
   629     by(auto simp add: setsum_reindex[unfolded inj_on_def])
   630   have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
   631   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"
   632     apply(rule_tac x="insert a f" in exI)
   633     apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
   634     using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
   635     unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
   636     by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
   637 
   638 lemma affine_hull_span:
   639   fixes a :: "'a::euclidean_space"
   640   assumes "a \<in> s"
   641   shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
   642   using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
   643 
   644 subsection{* Parallel Affine Sets *}
   645 
   646 definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
   647 where "affine_parallel S T = (? a. T = ((%x. a + x) ` S))"
   648 
   649 lemma affine_parallel_expl_aux:
   650    fixes S T :: "'a::real_vector set"
   651    assumes "!x. (x : S <-> (a+x) : T)" 
   652    shows "T = ((%x. a + x) ` S)"
   653 proof-
   654 { fix x assume "x : T" hence "(-a)+x : S" using assms by auto
   655   hence " x : ((%x. a + x) ` S)" using imageI[of "-a+x" S "(%x. a+x)"] by auto}
   656 moreover have "T >= ((%x. a + x) ` S)" using assms by auto 
   657 ultimately show ?thesis by auto
   658 qed
   659 
   660 lemma affine_parallel_expl: 
   661    "affine_parallel S T = (? a. !x. (x : S <-> (a+x) : T))" 
   662    unfolding affine_parallel_def using affine_parallel_expl_aux[of S _ T] by auto
   663 
   664 lemma affine_parallel_reflex: "affine_parallel S S" unfolding affine_parallel_def apply (rule exI[of _ "0"]) by auto
   665 
   666 lemma affine_parallel_commut:
   667 assumes "affine_parallel A B" shows "affine_parallel B A" 
   668 proof-
   669 from assms obtain a where "B=((%x. a + x) ` A)" unfolding affine_parallel_def by auto 
   670 from this show ?thesis using translation_galois[of B a A] unfolding affine_parallel_def by auto
   671 qed
   672 
   673 lemma affine_parallel_assoc:
   674 assumes "affine_parallel A B" "affine_parallel B C"
   675 shows "affine_parallel A C" 
   676 proof-
   677 from assms obtain ab where "B=((%x. ab + x) ` A)" unfolding affine_parallel_def by auto 
   678 moreover 
   679 from assms obtain bc where "C=((%x. bc + x) ` B)" unfolding affine_parallel_def by auto
   680 ultimately show ?thesis using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto 
   681 qed
   682 
   683 lemma affine_translation_aux:
   684   fixes a :: "'a::real_vector"
   685   assumes "affine ((%x. a + x) ` S)" shows "affine S"
   686 proof-
   687 { fix x y u v
   688   assume xy: "x : S" "y : S" "(u :: real)+v=1"
   689   hence "(a+x):((%x. a + x) ` S)" "(a+y):((%x. a + x) ` S)" by auto
   690   hence h1: "u *\<^sub>R  (a+x) + v *\<^sub>R (a+y) : ((%x. a + x) ` S)" using xy assms unfolding affine_def by auto
   691   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)
   692   also have "...= a + (u *\<^sub>R x + v *\<^sub>R y)" using `u+v=1` by auto
   693   ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) : ((%x. a + x) ` S)" using h1 by auto
   694   hence "u *\<^sub>R x + v *\<^sub>R y : S" by auto
   695 } from this show ?thesis unfolding affine_def by auto
   696 qed
   697 
   698 lemma affine_translation:
   699   fixes a :: "'a::real_vector"
   700   shows "affine S <-> affine ((%x. a + x) ` S)"
   701 proof-
   702 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
   703 from this show ?thesis using affine_translation_aux by auto
   704 qed
   705 
   706 lemma parallel_is_affine:
   707 fixes S T :: "'a::real_vector set"
   708 assumes "affine S" "affine_parallel S T"
   709 shows "affine T"
   710 proof-
   711   from assms obtain a where "T=((%x. a + x) ` S)" unfolding affine_parallel_def by auto 
   712   from this show ?thesis using affine_translation assms by auto
   713 qed
   714 
   715 lemma subspace_imp_affine:
   716   fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> affine s"
   717   unfolding subspace_def affine_def by auto
   718 
   719 subsection{* Subspace Parallel to an Affine Set *}
   720 
   721 lemma subspace_affine:
   722   fixes S :: "('n::euclidean_space) set"
   723   shows "subspace S <-> (affine S & 0 : S)"
   724 proof-
   725 have h0: "subspace S ==> (affine S & 0 : S)" using subspace_imp_affine[of S] subspace_0 by auto
   726 { assume assm: "affine S & 0 : S"
   727   { fix c :: real 
   728     fix x assume x_def: "x : S"
   729     have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
   730     moreover have "(1-c) *\<^sub>R 0 + c *\<^sub>R x : S" using affine_alt[of S] assm x_def by auto
   731     ultimately have "c *\<^sub>R x : S" by auto
   732   } hence h1: "!c. !x : S. c *\<^sub>R x : S" by auto
   733   { fix x y assume xy_def: "x : S" "y : S"
   734     def u == "(1 :: real)/2"
   735     have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)" by auto
   736     moreover have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y" by (simp add: algebra_simps)
   737     moreover have "(1-u) *\<^sub>R x + u *\<^sub>R y : S" using affine_alt[of S] assm xy_def by auto
   738     ultimately have "(1/2) *\<^sub>R (x+y) : S" using u_def by auto
   739     moreover have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" by auto
   740     ultimately have "(x+y) : S" using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
   741   } hence "!x : S. !y : S. (x+y) : S" by auto 
   742   hence "subspace S" using h1 assm unfolding subspace_def by auto
   743 } from this show ?thesis using h0 by metis
   744 qed
   745 
   746 lemma affine_diffs_subspace:
   747   fixes S :: "('n::euclidean_space) set"
   748   assumes "affine S" "a : S"
   749   shows "subspace ((%x. (-a)+x) ` S)"
   750 proof-
   751 have "affine ((%x. (-a)+x) ` S)" using  affine_translation assms by auto  
   752 moreover have "0 : ((%x. (-a)+x) ` S)" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
   753 ultimately show ?thesis using subspace_affine by auto 
   754 qed
   755 
   756 lemma parallel_subspace_explicit:
   757 fixes a :: "'n::euclidean_space"
   758 assumes "affine S" "a : S"
   759 assumes "L == {y. ? x : S. (-a)+x=y}" 
   760 shows "subspace L & affine_parallel S L" 
   761 proof-
   762 have par: "affine_parallel S L" unfolding affine_parallel_def using assms by auto
   763 hence "affine L" using assms parallel_is_affine by auto  
   764 moreover have "0 : L" using assms apply auto using exI[of "(%x. x:S & -a+x=0)" a] by auto
   765 ultimately show ?thesis using subspace_affine par by auto 
   766 qed
   767 
   768 lemma parallel_subspace_aux:
   769 fixes A B :: "('n::euclidean_space) set"
   770 assumes "subspace A" "subspace B" "affine_parallel A B"
   771 shows "A>=B"
   772 proof-
   773 from assms obtain a where a_def: "!x. (x : A <-> (a+x) : B)" using affine_parallel_expl[of A B] by auto
   774 hence "-a : A" using assms subspace_0[of B] by auto
   775 hence "a : A" using assms subspace_neg[of A "-a"] by auto
   776 from this show ?thesis using assms a_def unfolding subspace_def by auto
   777 qed
   778 
   779 lemma parallel_subspace:
   780 fixes A B :: "('n::euclidean_space) set"
   781 assumes "subspace A" "subspace B" "affine_parallel A B"
   782 shows "A=B"
   783 proof-
   784 have "A>=B" using assms parallel_subspace_aux by auto
   785 moreover have "A<=B" using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
   786 ultimately show ?thesis by auto  
   787 qed
   788 
   789 lemma affine_parallel_subspace:
   790 fixes S :: "('n::euclidean_space) set"
   791 assumes "affine S" "S ~= {}"
   792 shows "?!L. subspace L & affine_parallel S L" 
   793 proof-
   794 have ex: "? L. subspace L & affine_parallel S L" using assms  parallel_subspace_explicit by auto 
   795 { fix L1 L2 assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"
   796   hence "affine_parallel L1 L2" using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
   797   hence "L1=L2" using ass parallel_subspace by auto
   798 } from this show ?thesis using ex by auto
   799 qed
   800 
   801 subsection {* Cones. *}
   802 
   803 definition
   804   cone :: "'a::real_vector set \<Rightarrow> bool" where
   805   "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   806 
   807 lemma cone_empty[intro, simp]: "cone {}"
   808   unfolding cone_def by auto
   809 
   810 lemma cone_univ[intro, simp]: "cone UNIV"
   811   unfolding cone_def by auto
   812 
   813 lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
   814   unfolding cone_def by auto
   815 
   816 subsection {* Conic hull. *}
   817 
   818 lemma cone_cone_hull: "cone (cone hull s)"
   819   unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
   820   by (auto simp add: mem_def)
   821 
   822 lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
   823   apply(rule hull_eq[unfolded mem_def])
   824   using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
   825 
   826 lemma mem_cone:
   827   assumes "cone S" "x : S" "c>=0"
   828   shows "c *\<^sub>R x : S"
   829   using assms cone_def[of S] by auto
   830 
   831 lemma cone_contains_0:
   832 fixes S :: "('m::euclidean_space) set"
   833 assumes "cone S"
   834 shows "(S ~= {}) <-> (0 : S)"
   835 proof-
   836 { assume "S ~= {}" from this obtain a where "a:S" by auto
   837   hence "0 : S" using assms mem_cone[of S a 0] by auto
   838 } from this show ?thesis by auto
   839 qed
   840 
   841 lemma cone_0:
   842 shows "cone {(0 :: 'm::euclidean_space)}"
   843 unfolding cone_def by auto
   844 
   845 lemma cone_Union[intro]: "(!s:f. cone s) --> (cone (Union f))"
   846   unfolding cone_def by blast
   847 
   848 lemma cone_iff:
   849 fixes S :: "('m::euclidean_space) set"
   850 assumes "S ~= {}"
   851 shows "cone S <-> 0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
   852 proof-
   853 { assume "cone S"
   854   { fix c assume "(c :: real)>0"
   855     { fix x assume "x : S" hence "x : (op *\<^sub>R c) ` S" unfolding image_def
   856         using `cone S` `c>0` mem_cone[of S x "1/c"]
   857         exI[of "(%t. t:S & x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"] by auto
   858     }
   859     moreover
   860     { fix x assume "x : (op *\<^sub>R c) ` S"
   861       (*from this obtain t where "t:S & x = c *\<^sub>R t" by auto*)
   862       hence "x:S" using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto
   863     }
   864     ultimately have "((op *\<^sub>R c) ` S) = S" by auto
   865   } hence "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)" using `cone S` cone_contains_0[of S] assms by auto
   866 }
   867 moreover
   868 { assume a: "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
   869   { fix x assume "x:S"
   870     fix c1 assume "(c1 :: real)>=0"
   871     hence "(c1=0) | (c1>0)" by auto
   872     hence "c1 *\<^sub>R x : S" using a `x:S` by auto
   873   }
   874  hence "cone S" unfolding cone_def by auto
   875 } ultimately show ?thesis by blast
   876 qed
   877 
   878 lemma cone_hull_empty:
   879 "cone hull {} = {}"
   880 by (metis cone_empty cone_hull_eq)
   881 
   882 lemma cone_hull_empty_iff:
   883 fixes S :: "('m::euclidean_space) set"
   884 shows "(S = {}) <-> (cone hull S = {})"
   885 by (metis bot_least cone_hull_empty hull_subset xtrans(5))
   886 
   887 lemma cone_hull_contains_0: 
   888 fixes S :: "('m::euclidean_space) set"
   889 shows "(S ~= {}) <-> (0 : cone hull S)"
   890 using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] by auto
   891 
   892 lemma mem_cone_hull:
   893   assumes "x : S" "c>=0"
   894   shows "c *\<^sub>R x : cone hull S"
   895 by (metis assms cone_cone_hull hull_inc mem_cone mem_def)
   896 
   897 lemma cone_hull_expl:
   898 fixes S :: "('m::euclidean_space) set"
   899 shows "cone hull S = {c *\<^sub>R x | c x. c>=0 & x : S}" (is "?lhs = ?rhs")
   900 proof-
   901 { fix x assume "x : ?rhs"
   902   from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
   903   fix c assume c_def: "(c :: real)>=0"
   904   hence "c *\<^sub>R x = (c*cx) *\<^sub>R xx" using x_def by (simp add: algebra_simps)
   905   moreover have "(c*cx) >= 0" using c_def x_def using mult_nonneg_nonneg by auto
   906   ultimately have "c *\<^sub>R x : ?rhs" using x_def by auto
   907 } hence "cone ?rhs" unfolding cone_def by auto
   908   hence "?rhs : cone" unfolding mem_def by auto
   909 { fix x assume "x : S" hence "1 *\<^sub>R x : ?rhs" apply auto apply(rule_tac x="1" in exI) by auto
   910   hence "x : ?rhs" by auto
   911 } hence "S <= ?rhs" by auto
   912 hence "?lhs <= ?rhs" using `?rhs : cone` hull_minimal[of S "?rhs" "cone"] by auto
   913 moreover
   914 { fix x assume "x : ?rhs"
   915   from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
   916   hence "xx : cone hull S" using hull_subset[of S] by auto
   917   hence "x : ?lhs" using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
   918 } ultimately show ?thesis by auto
   919 qed
   920 
   921 lemma cone_closure:
   922 fixes S :: "('m::euclidean_space) set"
   923 assumes "cone S"
   924 shows "cone (closure S)"
   925 proof-
   926 { assume "S = {}" hence ?thesis by auto }
   927 moreover
   928 { assume "S ~= {}" hence "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
   929   hence "0:(closure S) & (!c. c>0 --> op *\<^sub>R c ` (closure S) = (closure S))"
   930      using closure_subset by (auto simp add: closure_scaleR)
   931   hence ?thesis using cone_iff[of "closure S"] by auto
   932 }
   933 ultimately show ?thesis by blast
   934 qed
   935 
   936 subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
   937 
   938 definition
   939   affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
   940   "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
   941 
   942 lemma affine_dependent_explicit:
   943   "affine_dependent p \<longleftrightarrow>
   944     (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
   945     (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   946   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
   947   apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
   948 proof-
   949   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"
   950   have "x\<notin>s" using as(1,4) by auto
   951   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"
   952     apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
   953     unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
   954 next
   955   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"
   956   have "s \<noteq> {v}" using as(3,6) by auto
   957   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" 
   958     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)
   959     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
   960 qed
   961 
   962 lemma affine_dependent_explicit_finite:
   963   fixes s :: "'a::real_vector set" assumes "finite s"
   964   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)"
   965   (is "?lhs = ?rhs")
   966 proof
   967   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
   968   assume ?lhs
   969   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"
   970     unfolding affine_dependent_explicit by auto
   971   thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   972     apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
   973     unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
   974 next
   975   assume ?rhs
   976   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
   977   thus ?lhs unfolding affine_dependent_explicit using assms by auto
   978 qed
   979 
   980 subsection {* A general lemma. *}
   981 
   982 lemma convex_connected:
   983   fixes s :: "'a::real_normed_vector set"
   984   assumes "convex s" shows "connected s"
   985 proof-
   986   { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
   987     assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
   988     then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
   989     hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
   990 
   991     { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
   992       { 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"
   993           by (simp add: algebra_simps)
   994         assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
   995         hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
   996           unfolding * and scaleR_right_diff_distrib[THEN sym]
   997           unfolding less_divide_eq using n by auto  }
   998       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"
   999         apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
  1000         apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
  1001 
  1002     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"
  1003       apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
  1004       using * apply(simp add: dist_norm)
  1005       using as(1,2)[unfolded open_dist] apply simp
  1006       using as(1,2)[unfolded open_dist] apply simp
  1007       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
  1008       using as(3) by auto
  1009     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
  1010     hence False using as(4) 
  1011       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
  1012       using x1(2) x2(2) by auto  }
  1013   thus ?thesis unfolding connected_def by auto
  1014 qed
  1015 
  1016 subsection {* One rather trivial consequence. *}
  1017 
  1018 lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
  1019   by(simp add: convex_connected convex_UNIV)
  1020 
  1021 subsection {* Balls, being convex, are connected. *}
  1022 
  1023 lemma convex_box: fixes a::"'a::euclidean_space"
  1024   assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
  1025   shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
  1026   using assms unfolding convex_def by(auto simp add:euclidean_simps)
  1027 
  1028 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
  1029   by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
  1030 
  1031 lemma convex_local_global_minimum:
  1032   fixes s :: "'a::real_normed_vector set"
  1033   assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
  1034   shows "\<forall>y\<in>s. f x \<le> f y"
  1035 proof(rule ccontr)
  1036   have "x\<in>s" using assms(1,3) by auto
  1037   assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
  1038   then obtain y where "y\<in>s" and y:"f x > f y" by auto
  1039   hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
  1040 
  1041   then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
  1042     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
  1043   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`
  1044     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
  1045   moreover
  1046   have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
  1047   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]
  1048     using u unfolding pos_less_divide_eq[OF xy] by auto
  1049   hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
  1050   ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
  1051 qed
  1052 
  1053 lemma convex_ball:
  1054   fixes x :: "'a::real_normed_vector"
  1055   shows "convex (ball x e)" 
  1056 proof(auto simp add: convex_def)
  1057   fix y z assume yz:"dist x y < e" "dist x z < e"
  1058   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
  1059   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
  1060     using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
  1061   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using convex_bound_lt[OF yz uv] by auto
  1062 qed
  1063 
  1064 lemma convex_cball:
  1065   fixes x :: "'a::real_normed_vector"
  1066   shows "convex(cball x e)"
  1067 proof(auto simp add: convex_def Ball_def)
  1068   fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
  1069   fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
  1070   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
  1071     using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
  1072   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto 
  1073 qed
  1074 
  1075 lemma connected_ball:
  1076   fixes x :: "'a::real_normed_vector"
  1077   shows "connected (ball x e)"
  1078   using convex_connected convex_ball by auto
  1079 
  1080 lemma connected_cball:
  1081   fixes x :: "'a::real_normed_vector"
  1082   shows "connected(cball x e)"
  1083   using convex_connected convex_cball by auto
  1084 
  1085 subsection {* Convex hull. *}
  1086 
  1087 lemma convex_convex_hull: "convex(convex hull s)"
  1088   unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
  1089   unfolding mem_def by auto
  1090 
  1091 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
  1092 by (metis convex_convex_hull hull_same mem_def)
  1093 
  1094 lemma bounded_convex_hull:
  1095   fixes s :: "'a::real_normed_vector set"
  1096   assumes "bounded s" shows "bounded(convex hull s)"
  1097 proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
  1098   show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
  1099     unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
  1100     unfolding subset_eq mem_cball dist_norm using B by auto qed
  1101 
  1102 lemma finite_imp_bounded_convex_hull:
  1103   fixes s :: "'a::real_normed_vector set"
  1104   shows "finite s \<Longrightarrow> bounded(convex hull s)"
  1105   using bounded_convex_hull finite_imp_bounded by auto
  1106 
  1107 subsection {* Convex hull is "preserved" by a linear function. *}
  1108 
  1109 lemma convex_hull_linear_image:
  1110   assumes "bounded_linear f"
  1111   shows "f ` (convex hull s) = convex hull (f ` s)"
  1112   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  1113   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  1114   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  1115 proof-
  1116   interpret f: bounded_linear f by fact
  1117   show "convex {x. f x \<in> convex hull f ` s}" 
  1118   unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
  1119   interpret f: bounded_linear f by fact
  1120   show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
  1121     unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  1122 qed auto
  1123 
  1124 lemma in_convex_hull_linear_image:
  1125   assumes "bounded_linear f" "x \<in> convex hull s"
  1126   shows "(f x) \<in> convex hull (f ` s)"
  1127 using convex_hull_linear_image[OF assms(1)] assms(2) by auto
  1128 
  1129 subsection {* Stepping theorems for convex hulls of finite sets. *}
  1130 
  1131 lemma convex_hull_empty[simp]: "convex hull {} = {}"
  1132   apply(rule hull_unique) unfolding mem_def by auto
  1133 
  1134 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
  1135   apply(rule hull_unique) unfolding mem_def by auto
  1136 
  1137 lemma convex_hull_insert:
  1138   fixes s :: "'a::real_vector set"
  1139   assumes "s \<noteq> {}"
  1140   shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
  1141                                     b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
  1142  apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
  1143  fix x assume x:"x = a \<or> x \<in> s"
  1144  thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
  1145    apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
  1146 next
  1147   fix x assume "x\<in>?hull"
  1148   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
  1149   have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
  1150     using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
  1151   thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
  1152     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
  1153 next
  1154   show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
  1155     fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
  1156     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
  1157     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
  1158     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)
  1159     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)"
  1160     proof(cases "u * v1 + v * v2 = 0")
  1161       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)
  1162       case True hence **:"u * v1 = 0" "v * v2 = 0"
  1163         using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by arith+
  1164       hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
  1165       thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
  1166     next
  1167       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)
  1168       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
  1169       also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
  1170       case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
  1171         apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
  1172         using as(1,2) obt1(1,2) obt2(1,2) by auto 
  1173       thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
  1174         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
  1175         apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
  1176         unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
  1177         by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
  1178     qed note * = this
  1179     have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
  1180     have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
  1181     have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
  1182       apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
  1183     also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
  1184     finally 
  1185     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)
  1186       apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
  1187       using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
  1188   qed
  1189 qed
  1190 
  1191 
  1192 subsection {* Explicit expression for convex hull. *}
  1193 
  1194 lemma convex_hull_indexed:
  1195   fixes s :: "'a::real_vector set"
  1196   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>
  1197                             (setsum u {1..k} = 1) \<and>
  1198                             (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
  1199   apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
  1200   apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
  1201 proof-
  1202   fix x assume "x\<in>s"
  1203   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
  1204 next
  1205   fix t assume as:"s \<subseteq> t" "convex t"
  1206   show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
  1207     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"
  1208     show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
  1209       using assm(1,2) as(1) by auto qed
  1210 next
  1211   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"
  1212   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
  1213   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
  1214   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)"
  1215     "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
  1216     prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
  1217   have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
  1218   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
  1219     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)
  1220     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)
  1221     unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
  1222     unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
  1223     fix i assume i:"i \<in> {1..k1+k2}"
  1224     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"
  1225     proof(cases "i\<in>{1..k1}")
  1226       case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
  1227     next def j \<equiv> "i - k1"
  1228       case False with i have "j \<in> {1..k2}" unfolding j_def by auto
  1229       thus ?thesis unfolding j_def[symmetric] using False
  1230         using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
  1231   qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
  1232 qed
  1233 
  1234 lemma convex_hull_finite:
  1235   fixes s :: "'a::real_vector set"
  1236   assumes "finite s"
  1237   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
  1238          setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
  1239 proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
  1240   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" 
  1241     apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
  1242     unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
  1243 next
  1244   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
  1245   fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
  1246   fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
  1247   { fix x assume "x\<in>s"
  1248     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)
  1249       by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
  1250   moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
  1251     unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
  1252   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)"
  1253     unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
  1254   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)"
  1255     apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
  1256 next
  1257   fix t assume t:"s \<subseteq> t" "convex t" 
  1258   fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
  1259   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]]
  1260     using assms and t(1) by auto
  1261 qed
  1262 
  1263 subsection {* Another formulation from Lars Schewe. *}
  1264 
  1265 lemma setsum_constant_scaleR:
  1266   fixes y :: "'a::real_vector"
  1267   shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
  1268 apply (cases "finite A")
  1269 apply (induct set: finite)
  1270 apply (simp_all add: algebra_simps)
  1271 done
  1272 
  1273 lemma convex_hull_explicit:
  1274   fixes p :: "'a::real_vector set"
  1275   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
  1276              (\<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")
  1277 proof-
  1278   { fix x assume "x\<in>?lhs"
  1279     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"
  1280       unfolding convex_hull_indexed by auto
  1281 
  1282     have fin:"finite {1..k}" by auto
  1283     have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
  1284     { fix j assume "j\<in>{1..k}"
  1285       hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
  1286         using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
  1287         apply(rule setsum_nonneg) using obt(1) by auto } 
  1288     moreover
  1289     have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
  1290       unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
  1291     moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
  1292       using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
  1293       unfolding scaleR_left.setsum using obt(3) by auto
  1294     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"
  1295       apply(rule_tac x="y ` {1..k}" in exI)
  1296       apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
  1297     hence "x\<in>?rhs" by auto  }
  1298   moreover
  1299   { fix y assume "y\<in>?rhs"
  1300     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
  1301 
  1302     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
  1303     
  1304     { fix i::nat assume "i\<in>{1..card s}"
  1305       hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
  1306       hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
  1307     moreover have *:"finite {1..card s}" by auto
  1308     { fix y assume "y\<in>s"
  1309       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
  1310       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
  1311       hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
  1312       hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
  1313             "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
  1314         by (auto simp add: setsum_constant_scaleR)   }
  1315 
  1316     hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
  1317       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] 
  1318       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"]
  1319       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
  1320     
  1321     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"
  1322       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 fastsimp
  1323     hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
  1324   ultimately show ?thesis unfolding set_eq_iff by blast
  1325 qed
  1326 
  1327 subsection {* A stepping theorem for that expansion. *}
  1328 
  1329 lemma convex_hull_finite_step:
  1330   fixes s :: "'a::real_vector set" assumes "finite s"
  1331   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)
  1332      \<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")
  1333 proof(rule, case_tac[!] "a\<in>s")
  1334   assume "a\<in>s" hence *:"insert a s = s" by auto
  1335   assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
  1336 next
  1337   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
  1338   assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
  1339     apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
  1340 next
  1341   assume "a\<in>s" hence *:"insert a s = s" by auto
  1342   have fin:"finite (insert a s)" using assms by auto
  1343   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
  1344   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]
  1345     unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
  1346 next
  1347   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
  1348   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)"
  1349     apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
  1350   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
  1351 qed
  1352 
  1353 subsection {* Hence some special cases. *}
  1354 
  1355 lemma convex_hull_2:
  1356   "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}"
  1357 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
  1358 show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
  1359   apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
  1360   apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
  1361 
  1362 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
  1363   unfolding convex_hull_2 unfolding Collect_def 
  1364 proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
  1365   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)"
  1366     unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
  1367 
  1368 lemma convex_hull_3:
  1369   "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}"
  1370 proof-
  1371   have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
  1372   have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
  1373          "\<And>x y z ::_::euclidean_space. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: field_simps)
  1374   show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
  1375     unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
  1376     apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
  1377     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
  1378 
  1379 lemma convex_hull_3_alt:
  1380   "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}"
  1381 proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
  1382   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)
  1383     apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
  1384 
  1385 subsection {* Relations among closure notions and corresponding hulls. *}
  1386 
  1387 text {* TODO: Generalize linear algebra concepts defined in @{text
  1388 Euclidean_Space.thy} so that we can generalize these lemmas. *}
  1389 
  1390 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
  1391   unfolding affine_def convex_def by auto
  1392 
  1393 lemma subspace_imp_convex:
  1394   fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> convex s"
  1395   using subspace_imp_affine affine_imp_convex by auto
  1396 
  1397 lemma affine_hull_subset_span:
  1398   fixes s :: "(_::euclidean_space) set" shows "(affine hull s) \<subseteq> (span s)"
  1399 by (metis hull_minimal mem_def span_inc subspace_imp_affine subspace_span)
  1400 
  1401 lemma convex_hull_subset_span:
  1402   fixes s :: "(_::euclidean_space) set" shows "(convex hull s) \<subseteq> (span s)"
  1403 by (metis hull_minimal mem_def span_inc subspace_imp_convex subspace_span)
  1404 
  1405 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
  1406 by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset mem_def)
  1407 
  1408 
  1409 lemma affine_dependent_imp_dependent:
  1410   fixes s :: "(_::euclidean_space) set" shows "affine_dependent s \<Longrightarrow> dependent s"
  1411   unfolding affine_dependent_def dependent_def 
  1412   using affine_hull_subset_span by auto
  1413 
  1414 lemma dependent_imp_affine_dependent:
  1415   fixes s :: "(_::euclidean_space) set"
  1416   assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
  1417   shows "affine_dependent (insert a s)"
  1418 proof-
  1419   from assms(1)[unfolded dependent_explicit] obtain S u v 
  1420     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
  1421   def t \<equiv> "(\<lambda>x. x + a) ` S"
  1422 
  1423   have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
  1424   have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
  1425   have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
  1426 
  1427   hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
  1428   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
  1429     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1430   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
  1431     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
  1432   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"
  1433     apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
  1434   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)"
  1435     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1436   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
  1437     unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
  1438     using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
  1439   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"
  1440     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *)
  1441   ultimately show ?thesis unfolding affine_dependent_explicit
  1442     apply(rule_tac x="insert a t" in exI) by auto 
  1443 qed
  1444 
  1445 lemma convex_cone:
  1446   "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")
  1447 proof-
  1448   { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
  1449     hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
  1450     hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
  1451       apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
  1452       apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
  1453   thus ?thesis unfolding convex_def cone_def by blast
  1454 qed
  1455 
  1456 lemma affine_dependent_biggerset: fixes s::"('a::euclidean_space) set"
  1457   assumes "finite s" "card s \<ge> DIM('a) + 2"
  1458   shows "affine_dependent s"
  1459 proof-
  1460   have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
  1461   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1462   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1463     apply(rule card_image) unfolding inj_on_def by auto
  1464   also have "\<dots> > DIM('a)" using assms(2)
  1465     unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1466   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1467     apply(rule dependent_imp_affine_dependent)
  1468     apply(rule dependent_biggerset) by auto qed
  1469 
  1470 lemma affine_dependent_biggerset_general:
  1471   assumes "finite (s::('a::euclidean_space) set)" "card s \<ge> dim s + 2"
  1472   shows "affine_dependent s"
  1473 proof-
  1474   from assms(2) have "s \<noteq> {}" by auto
  1475   then obtain a where "a\<in>s" by auto
  1476   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1477   have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1478     apply(rule card_image) unfolding inj_on_def by auto
  1479   have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
  1480     apply(rule subset_le_dim) unfolding subset_eq
  1481     using `a\<in>s` by (auto simp add:span_superset span_sub)
  1482   also have "\<dots> < dim s + 1" by auto
  1483   also have "\<dots> \<le> card (s - {a})" using assms
  1484     using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1485   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1486     apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
  1487 
  1488 subsection {* Caratheodory's theorem. *}
  1489 
  1490 lemma convex_hull_caratheodory: fixes p::"('a::euclidean_space) set"
  1491   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
  1492   (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  1493   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
  1494 proof(rule,rule)
  1495   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"
  1496   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"
  1497   then obtain N where "?P N" by auto
  1498   hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
  1499   then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
  1500   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
  1501 
  1502   have "card s \<le> DIM('a) + 1" proof(rule ccontr, simp only: not_le)
  1503     assume "DIM('a) + 1 < card s"
  1504     hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
  1505     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"
  1506       using affine_dependent_explicit_finite[OF obt(1)] by auto
  1507     def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
  1508     have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
  1509       assume as:"\<forall>x\<in>s. 0 \<le> w x"
  1510       hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
  1511       hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
  1512         using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
  1513       thus False using wv(1) by auto
  1514     qed hence "i\<noteq>{}" unfolding i_def by auto
  1515 
  1516     hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
  1517       using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
  1518     have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
  1519       fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
  1520       show"0 \<le> u v + t * w v" proof(cases "w v < 0")
  1521         case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
  1522           using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
  1523         case True hence "t \<le> u v / (- w v)" using `v\<in>s`
  1524           unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
  1525         thus ?thesis unfolding real_0_le_add_iff
  1526           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
  1527       qed qed
  1528 
  1529     obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
  1530       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
  1531     hence a:"a\<in>s" "u a + t * w a = 0" by auto
  1532     have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
  1533       unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
  1534     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
  1535       unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
  1536     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" 
  1537       unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
  1538       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
  1539     ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
  1540       apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a
  1541       by (auto simp add: * scaleR_left_distrib)
  1542     thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
  1543   thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1
  1544     \<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
  1545 qed auto
  1546 
  1547 lemma caratheodory:
  1548  "convex hull p = {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
  1549       card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
  1550   unfolding set_eq_iff apply(rule, rule) unfolding mem_Collect_eq proof-
  1551   fix x assume "x \<in> convex hull p"
  1552   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
  1553      "\<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
  1554   thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  1555     apply(rule_tac x=s in exI) using hull_subset[of s convex]
  1556   using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
  1557 next
  1558   fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  1559   then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s" by auto
  1560   thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
  1561 qed
  1562 
  1563 
  1564 subsection {* Some Properties of Affine Dependent Sets *}
  1565 
  1566 lemma affine_independent_empty: "~(affine_dependent {})"
  1567   by (simp add: affine_dependent_def)
  1568 
  1569 lemma affine_independent_sing:
  1570 fixes a :: "'n::euclidean_space" 
  1571 shows "~(affine_dependent {a})"
  1572  by (simp add: affine_dependent_def)
  1573 
  1574 lemma affine_hull_translation:
  1575 "affine hull ((%x. a + x) `  S) = (%x. a + x) ` (affine hull S)"
  1576 proof-
  1577 have "affine ((%x. a + x) ` (affine hull S))" using affine_translation affine_affine_hull by auto
  1578 moreover have "(%x. a + x) `  S <= (%x. a + x) ` (affine hull S)" using hull_subset[of S] by auto
  1579 ultimately have h1: "affine hull ((%x. a + x) `  S) <= (%x. a + x) ` (affine hull S)" by (metis hull_minimal mem_def)
  1580 have "affine((%x. -a + x) ` (affine hull ((%x. a + x) `  S)))"  using affine_translation affine_affine_hull by auto
  1581 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 
  1582 moreover have "S=(%x. -a + x) ` (%x. a + x) `  S" using  translation_assoc[of "-a" a] by auto
  1583 ultimately have "(%x. -a + x) ` (affine hull ((%x. a + x) `  S)) >= (affine hull S)" by (metis hull_minimal mem_def)
  1584 hence "affine hull ((%x. a + x) `  S) >= (%x. a + x) ` (affine hull S)" by auto
  1585 from this show ?thesis using h1 by auto
  1586 qed
  1587 
  1588 lemma affine_dependent_translation:
  1589   assumes "affine_dependent S"
  1590   shows "affine_dependent ((%x. a + x) ` S)"
  1591 proof-
  1592 obtain x where x_def: "x : S & x : affine hull (S - {x})" using assms affine_dependent_def by auto
  1593 have "op + a ` (S - {x}) = op + a ` S - {a + x}" by auto
  1594 hence "a+x : affine hull ((%x. a + x) ` S - {a+x})" using  affine_hull_translation[of a "S-{x}"] x_def by auto
  1595 moreover have "a+x : (%x. a + x) ` S" using x_def by auto  
  1596 ultimately show ?thesis unfolding affine_dependent_def by auto 
  1597 qed
  1598 
  1599 lemma affine_dependent_translation_eq:
  1600   "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"
  1601 proof-
  1602 { assume "affine_dependent ((%x. a + x) ` S)" 
  1603   hence "affine_dependent S" using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] by auto  
  1604 } from this show ?thesis using affine_dependent_translation by auto
  1605 qed
  1606 
  1607 lemma affine_hull_0_dependent:
  1608   fixes S ::  "('n::euclidean_space) set"
  1609   assumes "0 : affine hull S"
  1610   shows "dependent S"
  1611 proof-
  1612 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
  1613 hence "EX v:s. u v ~= 0" using setsum_not_0[of "u" "s"] by auto 
  1614 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
  1615 from this show ?thesis unfolding dependent_explicit[of S] by auto
  1616 qed
  1617 
  1618 lemma affine_dependent_imp_dependent2:
  1619   fixes S :: "('n::euclidean_space) set" 
  1620   assumes "affine_dependent (insert 0 S)"
  1621   shows "dependent S"
  1622 proof-
  1623 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
  1624 hence "x : span (insert 0 S - {x})" using affine_hull_subset_span by auto
  1625 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
  1626 ultimately have "x : span (S - {x})" by auto
  1627 hence "(x~=0) ==> dependent S" using x_def dependent_def by auto
  1628 moreover
  1629 { assume "x=0" hence "0 : affine hull S" using x_def hull_mono[of "S - {0}" S] by auto
  1630                hence "dependent S" using affine_hull_0_dependent by auto  
  1631 } ultimately show ?thesis by auto
  1632 qed
  1633 
  1634 lemma affine_dependent_iff_dependent:
  1635   fixes S :: "('n::euclidean_space) set" 
  1636   assumes "a ~: S"
  1637   shows "affine_dependent (insert a S) <-> dependent ((%x. -a + x) ` S)" 
  1638 proof-
  1639 have "(op + (- a) ` S)={x - a| x . x : S}" by auto
  1640 from this show ?thesis using affine_dependent_translation_eq[of "(insert a S)" "-a"] 
  1641       affine_dependent_imp_dependent2 assms 
  1642       dependent_imp_affine_dependent[of a S] by auto
  1643 qed
  1644 
  1645 lemma affine_dependent_iff_dependent2:
  1646   fixes S :: "('n::euclidean_space) set" 
  1647   assumes "a : S"
  1648   shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"
  1649 proof-
  1650 have "insert a (S - {a})=S" using assms by auto
  1651 from this show ?thesis using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto 
  1652 qed
  1653 
  1654 lemma affine_hull_insert_span_gen:
  1655   fixes a :: "_::euclidean_space"
  1656   shows "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)" 
  1657 proof-
  1658 have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)" by auto
  1659 { assume "a ~: s" hence ?thesis using affine_hull_insert_span[of a s] h1 by auto}  
  1660 moreover
  1661 { assume a1: "a : s"
  1662   have "EX x. x:s & -a+x=0" apply (rule exI[of _ a]) using a1 by auto
  1663   hence "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s" by auto
  1664   hence "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)" 
  1665     using span_insert_0[of "op + (- a) ` (s - {a})"] by auto
  1666   moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))" by auto 
  1667   moreover have "insert a (s - {a})=(insert a s)" using assms by auto
  1668   ultimately have ?thesis using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
  1669 } 
  1670 ultimately show ?thesis by auto  
  1671 qed
  1672 
  1673 lemma affine_hull_span2:
  1674   fixes a :: "_::euclidean_space"
  1675   assumes "a : s"
  1676   shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` (s-{a}))"
  1677   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
  1678 
  1679 lemma affine_hull_span_gen:
  1680   fixes a :: "_::euclidean_space"
  1681   assumes "a : affine hull s"
  1682   shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` s)"
  1683 proof-
  1684 have "affine hull (insert a s) = affine hull s" using hull_redundant[of a affine s] assms by auto
  1685 from this show ?thesis using affine_hull_insert_span_gen[of a "s"] by auto
  1686 qed
  1687 
  1688 lemma affine_hull_span_0:
  1689   assumes "(0 :: _::euclidean_space) : affine hull S"
  1690   shows "affine hull S = span S"
  1691 using affine_hull_span_gen[of "0" S] assms by auto
  1692 
  1693 
  1694 lemma extend_to_affine_basis:
  1695 fixes S V :: "('n::euclidean_space) set"
  1696 assumes "~(affine_dependent S)" "S <= V" "S~={}"
  1697 shows "? T. ~(affine_dependent T) & S<=T & T<=V & affine hull T = affine hull V"
  1698 proof-
  1699 obtain a where a_def: "a : S" using assms by auto
  1700 hence h0: "independent  ((%x. -a + x) ` (S-{a}))" using affine_dependent_iff_dependent2 assms by auto
  1701 from this obtain B 
  1702    where B_def: "(%x. -a+x) ` (S - {a}) <= B & B <= (%x. -a+x) ` V & independent B & (%x. -a+x) ` V <= span B" 
  1703    using maximal_independent_subset_extend[of "(%x. -a+x) ` (S-{a})" "(%x. -a + x) ` V"] assms by blast
  1704 def T == "(%x. a+x) ` (insert 0 B)" hence "T=insert a ((%x. a+x) ` B)" by auto
  1705 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
  1706 hence "V <= affine hull T" using B_def assms translation_inverse_subset[of a V "span B"] by auto
  1707 moreover have "T<=V" using T_def B_def a_def assms by auto
  1708 ultimately have "affine hull T = affine hull V" 
  1709     by (metis Int_absorb1 Int_absorb2 Int_commute Int_lower2 assms hull_hull hull_mono) 
  1710 moreover have "S<=T" using T_def B_def translation_inverse_subset[of a "S-{a}" B] by auto
  1711 moreover have "~(affine_dependent T)" using T_def affine_dependent_translation_eq[of "insert 0 B"] affine_dependent_imp_dependent2 B_def by auto
  1712 ultimately show ?thesis using `T<=V` by auto
  1713 qed
  1714 
  1715 lemma affine_basis_exists: 
  1716 fixes V :: "('n::euclidean_space) set"
  1717 shows "? B. B <= V & ~(affine_dependent B) & affine hull V = affine hull B"
  1718 proof-
  1719 { assume empt: "V={}" have "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)" using empt affine_independent_empty by auto
  1720 }
  1721 moreover
  1722 { assume nonempt: "V~={}" obtain x where "x:V" using nonempt by auto
  1723   hence "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)"
  1724   using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}:: ('n::euclidean_space) set" V] by auto
  1725 }
  1726 ultimately show ?thesis by auto
  1727 qed
  1728 
  1729 subsection {* Affine Dimension of a Set *}
  1730 
  1731 definition "aff_dim V = (SOME d :: int. ? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1))"
  1732 
  1733 lemma aff_dim_basis_exists:
  1734   fixes V :: "('n::euclidean_space) set" 
  1735   shows "? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
  1736 proof-
  1737 obtain B where B_def: "~(affine_dependent B) & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
  1738 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
  1739 qed
  1740 
  1741 lemma affine_hull_nonempty: "(S ~= {}) <-> affine hull S ~= {}"
  1742 proof-
  1743 fix S have "(S = {}) ==> affine hull S = {}"using affine_hull_empty by auto 
  1744 moreover have "affine hull S = {} ==> S = {}" unfolding hull_def by auto
  1745 ultimately show "(S ~= {}) <-> affine hull S ~= {}" by blast
  1746 qed
  1747 
  1748 lemma aff_dim_parallel_subspace_aux:
  1749 fixes B :: "('n::euclidean_space) set"
  1750 assumes "~(affine_dependent B)" "a:B"
  1751 shows "finite B & ((card B) - 1 = dim (span ((%x. -a+x) ` (B-{a}))))" 
  1752 proof-
  1753 have "independent ((%x. -a + x) ` (B-{a}))" using affine_dependent_iff_dependent2 assms by auto
  1754 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
  1755 { assume emp: "(%x. -a + x) ` (B - {a}) = {}" 
  1756   have "B=insert a ((%x. a + x) ` (%x. -a + x) ` (B - {a}))" using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
  1757   hence "B={a}" using emp by auto
  1758   hence ?thesis using assms fin by auto  
  1759 }
  1760 moreover
  1761 { assume "(%x. -a + x) ` (B - {a}) ~= {}"
  1762   hence "card ((%x. -a + x) ` (B - {a}))>0" using fin by auto
  1763   moreover have h1: "card ((%x. -a + x) ` (B-{a})) = card (B-{a})"  
  1764      apply (rule card_image) using translate_inj_on by auto
  1765   ultimately have "card (B-{a})>0" by auto
  1766   hence "finite(B-{a})" using card_gt_0_iff[of "(B - {a})"] by auto
  1767   moreover hence "(card (B-{a})= (card B) - 1)" using card_Diff_singleton assms by auto
  1768   ultimately have ?thesis using fin h1 by auto
  1769 } ultimately show ?thesis by auto
  1770 qed
  1771 
  1772 lemma aff_dim_parallel_subspace:
  1773 fixes V L :: "('n::euclidean_space) set"
  1774 assumes "V ~= {}"
  1775 assumes "subspace L" "affine_parallel (affine hull V) L"
  1776 shows "aff_dim V=int(dim L)"
  1777 proof-
  1778 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
  1779 hence "B~={}" using assms B_def  affine_hull_nonempty[of V] affine_hull_nonempty[of B] by auto 
  1780 from this obtain a where a_def: "a : B" by auto
  1781 def Lb == "span ((%x. -a+x) ` (B-{a}))"
  1782   moreover have "affine_parallel (affine hull B) Lb"
  1783      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
  1784   moreover have "subspace Lb" using Lb_def subspace_span by auto
  1785   moreover have "affine hull B ~= {}" using assms B_def affine_hull_nonempty[of V] by auto
  1786   ultimately have "L=Lb" using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B_def by auto 
  1787   hence "dim L=dim Lb" by auto 
  1788   moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def B_def by auto
  1789 (*  hence "card B=dim Lb+1" using `B~={}` card_gt_0_iff[of B] by auto *)
  1790   ultimately show ?thesis using B_def `B~={}` card_gt_0_iff[of B] by auto
  1791 qed
  1792 
  1793 lemma aff_independent_finite:
  1794 fixes B :: "('n::euclidean_space) set"
  1795 assumes "~(affine_dependent B)"
  1796 shows "finite B"
  1797 proof-
  1798 { assume "B~={}" from this obtain a where "a:B" by auto 
  1799   hence ?thesis using aff_dim_parallel_subspace_aux assms by auto 
  1800 } from this show ?thesis by auto
  1801 qed
  1802 
  1803 lemma independent_finite:
  1804 fixes B :: "('n::euclidean_space) set"
  1805 assumes "independent B" 
  1806 shows "finite B"
  1807 using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms by auto
  1808 
  1809 lemma subspace_dim_equal:
  1810 assumes "subspace (S :: ('n::euclidean_space) set)" "subspace T" "S <= T" "dim S >= dim T"
  1811 shows "S=T"
  1812 proof- 
  1813 obtain B where B_def: "B <= S & independent B & S <= span B & (card B = dim S)" using basis_exists[of S] by auto
  1814 hence "span B <= S" using span_mono[of B S] span_eq[of S] assms by metis 
  1815 hence "span B = S" using B_def by auto
  1816 have "dim S = dim T" using assms dim_subset[of S T] by auto
  1817 hence "T <= span B" using card_eq_dim[of B T] B_def independent_finite assms by auto
  1818 from this show ?thesis using assms `span B=S` by auto
  1819 qed
  1820 
  1821 lemma span_substd_basis:  assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  1822   shows "(span {basis i | i. i : d}) = {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
  1823   (is "span ?A = ?B")
  1824 proof-
  1825 have "?A <= ?B" by auto
  1826 moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .
  1827 ultimately have "span ?A <= ?B" using span_mono[of "?A" "?B"] span_eq[of "?B"] by blast
  1828 moreover have "card d <= dim (span ?A)" using independent_card_le_dim[of "?A" "span ?A"] 
  1829    independent_substdbasis[OF assms] card_substdbasis[OF assms] span_inc[of "?A"] by auto
  1830 moreover hence "dim ?B <= dim (span ?A)" using dim_substandard[OF assms] by auto
  1831 ultimately show ?thesis using s subspace_dim_equal[of "span ?A" "?B"] 
  1832   subspace_span[of "?A"] by auto
  1833 qed
  1834 
  1835 lemma basis_to_substdbasis_subspace_isomorphism:
  1836 fixes B :: "('a::euclidean_space) set" 
  1837 assumes "independent B"
  1838 shows "EX f d. card d = card B & linear f & f ` B = {basis i::'a |i. i : (d :: nat set)} & 
  1839        f ` span B = {x. ALL i<DIM('a). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B) \<and> d\<subseteq>{..<DIM('a)}" 
  1840 proof-
  1841   have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto
  1842   def d \<equiv> "{..<dim B}" have t:"card d = dim B" unfolding d_def by auto
  1843   have "dim B <= DIM('a)" using dim_subset_UNIV[of B] by auto
  1844   hence d:"d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
  1845   let ?t = "{x::'a::euclidean_space. !i<DIM('a). i ~: d --> x$$i = 0}"
  1846   have "EX f. linear f & f ` B = {basis i |i. i : d} &
  1847     f ` span B = ?t & inj_on f (span B)"
  1848     apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "{basis i |i. i : d}"])
  1849     apply(rule subspace_span) apply(rule subspace_substandard) defer
  1850     apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)
  1851     unfolding span_substd_basis[OF d,THEN sym] card_substdbasis[OF d] apply(rule span_inc)
  1852     apply(rule independent_substdbasis[OF d]) apply(rule,assumption)
  1853     unfolding t[THEN sym] span_substd_basis[OF d] dim_substandard[OF d] by auto
  1854   from this t `card B=dim B` show ?thesis using d by auto 
  1855 qed
  1856 
  1857 lemma aff_dim_empty:
  1858 fixes S :: "('n::euclidean_space) set" 
  1859 shows "S = {} <-> aff_dim S = -1"
  1860 proof-
  1861 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
  1862 moreover hence "S={} <-> B={}" using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
  1863 ultimately show ?thesis using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
  1864 qed
  1865 
  1866 lemma aff_dim_affine_hull:
  1867 fixes S :: "('n::euclidean_space) set"
  1868 shows "aff_dim (affine hull S)=aff_dim S" 
  1869 unfolding aff_dim_def using hull_hull[of _ S] by auto 
  1870 
  1871 lemma aff_dim_affine_hull2:
  1872 fixes S T :: "('n::euclidean_space) set"
  1873 assumes "affine hull S=affine hull T"
  1874 shows "aff_dim S=aff_dim T" unfolding aff_dim_def using assms by auto
  1875 
  1876 lemma aff_dim_unique: 
  1877 fixes B V :: "('n::euclidean_space) set" 
  1878 assumes "(affine hull B=affine hull V) & ~(affine_dependent B)"
  1879 shows "of_nat(card B) = aff_dim V+1"
  1880 proof-
  1881 { assume "B={}" hence "V={}" using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms by auto
  1882   hence "aff_dim V = (-1::int)"  using aff_dim_empty by auto  
  1883   hence ?thesis using `B={}` by auto
  1884 }
  1885 moreover
  1886 { assume "B~={}" from this obtain a where a_def: "a:B" by auto 
  1887   def Lb == "span ((%x. -a+x) ` (B-{a}))"
  1888   have "affine_parallel (affine hull B) Lb"
  1889      using Lb_def affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"] 
  1890      unfolding affine_parallel_def by auto
  1891   moreover have "subspace Lb" using Lb_def subspace_span by auto
  1892   ultimately have "aff_dim B=int(dim Lb)" using aff_dim_parallel_subspace[of B Lb] `B~={}` by auto 
  1893   moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def assms by auto
  1894   ultimately have "(of_nat(card B) = aff_dim B+1)" using  `B~={}` card_gt_0_iff[of B] by auto
  1895   hence ?thesis using aff_dim_affine_hull2 assms by auto
  1896 } ultimately show ?thesis by blast
  1897 qed
  1898 
  1899 lemma aff_dim_affine_independent: 
  1900 fixes B :: "('n::euclidean_space) set" 
  1901 assumes "~(affine_dependent B)"
  1902 shows "of_nat(card B) = aff_dim B+1"
  1903   using aff_dim_unique[of B B] assms by auto
  1904 
  1905 lemma aff_dim_sing: 
  1906 fixes a :: "'n::euclidean_space" 
  1907 shows "aff_dim {a}=0"
  1908   using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
  1909 
  1910 lemma aff_dim_inner_basis_exists:
  1911   fixes V :: "('n::euclidean_space) set" 
  1912   shows "? B. B<=V & (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
  1913 proof-
  1914 obtain B where B_def: "~(affine_dependent B) & B<=V & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
  1915 moreover hence "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
  1916 ultimately show ?thesis by auto
  1917 qed
  1918 
  1919 lemma aff_dim_le_card:
  1920 fixes V :: "('n::euclidean_space) set" 
  1921 assumes "finite V"
  1922 shows "aff_dim V <= of_nat(card V) - 1"
  1923  proof-
  1924  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 
  1925  moreover hence "card B <= card V" using assms card_mono by auto
  1926  ultimately show ?thesis by auto
  1927 qed
  1928 
  1929 lemma aff_dim_parallel_eq:
  1930 fixes S T :: "('n::euclidean_space) set"
  1931 assumes "affine_parallel (affine hull S) (affine hull T)"
  1932 shows "aff_dim S=aff_dim T"
  1933 proof-
  1934 { assume "T~={}" "S~={}" 
  1935   from this obtain L where L_def: "subspace L & affine_parallel (affine hull T) L" 
  1936        using affine_parallel_subspace[of "affine hull T"] affine_affine_hull[of T] affine_hull_nonempty by auto
  1937   hence "aff_dim T = int(dim L)" using aff_dim_parallel_subspace `T~={}` by auto
  1938   moreover have "subspace L & affine_parallel (affine hull S) L" 
  1939      using L_def affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
  1940   moreover hence "aff_dim S = int(dim L)" using aff_dim_parallel_subspace `S~={}` by auto 
  1941   ultimately have ?thesis by auto
  1942 }
  1943 moreover
  1944 { assume "S={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
  1945   hence ?thesis using aff_dim_empty by auto
  1946 }
  1947 moreover
  1948 { assume "T={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
  1949   hence ?thesis using aff_dim_empty by auto
  1950 }
  1951 ultimately show ?thesis by blast
  1952 qed
  1953 
  1954 lemma aff_dim_translation_eq:
  1955 fixes a :: "'n::euclidean_space"
  1956 shows "aff_dim ((%x. a + x) ` S)=aff_dim S" 
  1957 proof-
  1958 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
  1959 from this show ?thesis using  aff_dim_parallel_eq[of S "(%x. a + x) ` S"] by auto 
  1960 qed
  1961 
  1962 lemma aff_dim_affine:
  1963 fixes S L :: "('n::euclidean_space) set"
  1964 assumes "S ~= {}" "affine S"
  1965 assumes "subspace L" "affine_parallel S L"
  1966 shows "aff_dim S=int(dim L)" 
  1967 proof-
  1968 have 1: "(affine hull S) = S" using assms affine_hull_eq[of S] by auto 
  1969 hence "affine_parallel (affine hull S) L" using assms by (simp add:1)
  1970 from this show ?thesis using assms aff_dim_parallel_subspace[of S L] by blast 
  1971 qed
  1972 
  1973 lemma dim_affine_hull:
  1974 fixes S :: "('n::euclidean_space) set"
  1975 shows "dim (affine hull S)=dim S"
  1976 proof-
  1977 have "dim (affine hull S)>=dim S" using dim_subset by auto
  1978 moreover have "dim(span S) >= dim (affine hull S)" using dim_subset affine_hull_subset_span by auto
  1979 moreover have "dim(span S)=dim S" using dim_span by auto
  1980 ultimately show ?thesis by auto
  1981 qed
  1982 
  1983 lemma aff_dim_subspace:
  1984 fixes S :: "('n::euclidean_space) set"
  1985 assumes "S ~= {}" "subspace S"
  1986 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 
  1987 
  1988 lemma aff_dim_zero:
  1989 fixes S :: "('n::euclidean_space) set"
  1990 assumes "0 : affine hull S"
  1991 shows "aff_dim S=int(dim S)"
  1992 proof-
  1993 have "subspace(affine hull S)" using subspace_affine[of "affine hull S"] affine_affine_hull assms by auto
  1994 hence "aff_dim (affine hull S) =int(dim (affine hull S))" using assms aff_dim_subspace[of "affine hull S"] by auto  
  1995 from this show ?thesis using aff_dim_affine_hull[of S] dim_affine_hull[of S] by auto
  1996 qed
  1997 
  1998 lemma aff_dim_univ: "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))"
  1999   using aff_dim_subspace[of "(UNIV :: ('n::euclidean_space) set)"]
  2000     dim_UNIV[where 'a="'n::euclidean_space"] by auto
  2001 
  2002 lemma aff_dim_geq:
  2003   fixes V :: "('n::euclidean_space) set"
  2004   shows "aff_dim V >= -1"
  2005 proof-
  2006 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
  2007 from this show ?thesis by auto
  2008 qed
  2009 
  2010 lemma independent_card_le_aff_dim: 
  2011   assumes "(B::('n::euclidean_space) set) <= V"
  2012   assumes "~(affine_dependent B)" 
  2013   shows "int(card B) <= aff_dim V+1"
  2014 proof-
  2015 { assume "B~={}" 
  2016   from this obtain T where T_def: "~(affine_dependent T) & B<=T & T<=V & affine hull T = affine hull V" 
  2017   using assms extend_to_affine_basis[of B V] by auto
  2018   hence "of_nat(card T) = aff_dim V+1" using aff_dim_unique by auto
  2019   hence ?thesis using T_def card_mono[of T B] aff_independent_finite[of T] by auto
  2020 }
  2021 moreover
  2022 { assume "B={}"
  2023   moreover have "-1<= aff_dim V" using aff_dim_geq by auto
  2024   ultimately have ?thesis by auto
  2025 }  ultimately show ?thesis by blast
  2026 qed
  2027 
  2028 lemma aff_dim_subset:
  2029   fixes S T :: "('n::euclidean_space) set"
  2030   assumes "S <= T"
  2031   shows "aff_dim S <= aff_dim T"
  2032 proof-
  2033 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
  2034 moreover hence "int (card B) <= aff_dim T + 1" using assms independent_card_le_aff_dim[of B T] by auto
  2035 ultimately show ?thesis by auto
  2036 qed
  2037 
  2038 lemma aff_dim_subset_univ:
  2039 fixes S :: "('n::euclidean_space) set"
  2040 shows "aff_dim S <= int(DIM('n))"
  2041 proof - 
  2042   have "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))" using aff_dim_univ by auto
  2043   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
  2044 qed
  2045 
  2046 lemma affine_dim_equal:
  2047 assumes "affine (S :: ('n::euclidean_space) set)" "affine T" "S ~= {}" "S <= T" "aff_dim S = aff_dim T"
  2048 shows "S=T"
  2049 proof-
  2050 obtain a where "a : S" using assms by auto 
  2051 hence "a : T" using assms by auto
  2052 def LS == "{y. ? x : S. (-a)+x=y}"
  2053 hence ls: "subspace LS & affine_parallel S LS" using assms parallel_subspace_explicit[of S a LS] `a : S` by auto 
  2054 hence h1: "int(dim LS) = aff_dim S" using assms aff_dim_affine[of S LS] by auto
  2055 have "T ~= {}" using assms by auto
  2056 def LT == "{y. ? x : T. (-a)+x=y}" 
  2057 hence lt: "subspace LT & affine_parallel T LT" using assms parallel_subspace_explicit[of T a LT] `a : T` by auto
  2058 hence "int(dim LT) = aff_dim T" using assms aff_dim_affine[of T LT] `T ~= {}` by auto 
  2059 hence "dim LS = dim LT" using h1 assms by auto
  2060 moreover have "LS <= LT" using LS_def LT_def assms by auto
  2061 ultimately have "LS=LT" using subspace_dim_equal[of LS LT] ls lt by auto
  2062 moreover have "S = {x. ? y : LS. a+y=x}" using LS_def by auto 
  2063 moreover have "T = {x. ? y : LT. a+y=x}" using LT_def by auto
  2064 ultimately show ?thesis by auto 
  2065 qed
  2066 
  2067 lemma affine_hull_univ:
  2068 fixes S :: "('n::euclidean_space) set"
  2069 assumes "aff_dim S = int(DIM('n))"
  2070 shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
  2071 proof-
  2072 have "S ~= {}" using assms aff_dim_empty[of S] by auto
  2073 have h0: "S <= affine hull S" using hull_subset[of S _] by auto
  2074 have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S" using aff_dim_univ assms by auto
  2075 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  
  2076 have h3: "aff_dim S <= aff_dim (affine hull S)" using h0 aff_dim_subset[of S "affine hull S"] assms by auto
  2077 hence h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" using h0 h1 h2 by auto
  2078 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
  2079 qed
  2080 
  2081 lemma aff_dim_convex_hull:
  2082 fixes S :: "('n::euclidean_space) set"
  2083 shows "aff_dim (convex hull S)=aff_dim S"
  2084   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S] 
  2085   hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"] 
  2086   aff_dim_subset[of "convex hull S" "affine hull S"] by auto
  2087 
  2088 lemma aff_dim_cball:
  2089 fixes a :: "'n::euclidean_space" 
  2090 assumes "0<e"
  2091 shows "aff_dim (cball a e) = int (DIM('n))"
  2092 proof-
  2093 have "(%x. a + x) ` (cball 0 e)<=cball a e" unfolding cball_def dist_norm by auto
  2094 hence "aff_dim (cball (0 :: 'n::euclidean_space) e) <= aff_dim (cball a e)"
  2095   using aff_dim_translation_eq[of a "cball 0 e"] 
  2096         aff_dim_subset[of "op + a ` cball 0 e" "cball a e"] by auto
  2097 moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))" 
  2098    using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"] centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms 
  2099    by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
  2100 ultimately show ?thesis using aff_dim_subset_univ[of "cball a e"] by auto 
  2101 qed
  2102 
  2103 lemma aff_dim_open:
  2104 fixes S :: "('n::euclidean_space) set"
  2105 assumes "open S" "S ~= {}"
  2106 shows "aff_dim S = int (DIM('n))"
  2107 proof-
  2108 obtain x where "x:S" using assms by auto
  2109 from this obtain e where e_def: "e>0 & cball x e <= S" using open_contains_cball[of S] assms by auto
  2110 from this have "aff_dim (cball x e) <= aff_dim S" using aff_dim_subset by auto
  2111 from this show ?thesis using aff_dim_cball[of e x] aff_dim_subset_univ[of S] e_def by auto     
  2112 qed
  2113 
  2114 lemma low_dim_interior:
  2115 fixes S :: "('n::euclidean_space) set"
  2116 assumes "~(aff_dim S = int (DIM('n)))"
  2117 shows "interior S = {}"
  2118 proof-
  2119 have "aff_dim(interior S) <= aff_dim S" 
  2120    using interior_subset aff_dim_subset[of "interior S" S] by auto 
  2121 from this show ?thesis using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto   
  2122 qed
  2123 
  2124 subsection{* Relative Interior of a Set *}
  2125 
  2126 definition "rel_interior S = {x. ? T. openin (subtopology euclidean (affine hull S)) T & x : T & T <= S}"
  2127 
  2128 lemma rel_interior: "rel_interior S = {x : S. ? T. open T & x : T & (T Int (affine hull S)) <= S}"
  2129   unfolding rel_interior_def[of S] openin_open[of "affine hull S"] apply auto
  2130 proof-
  2131 fix x T assume a: "x:S" "open T" "x : T" "(T Int (affine hull S)) <= S"
  2132 hence h1: "x : T Int affine hull S" using hull_inc by auto
  2133 show "EX Tb. (EX Ta. open Ta & Tb = affine hull S Int Ta) & x : Tb & Tb <= S"
  2134 apply (rule_tac x="T Int (affine hull S)" in exI)
  2135 using a h1 by auto
  2136 qed
  2137 
  2138 lemma mem_rel_interior: 
  2139      "x : rel_interior S <-> (? T. open T & x : (T Int S) & (T Int (affine hull S)) <= S)" 
  2140      by (auto simp add: rel_interior)
  2141 
  2142 lemma mem_rel_interior_ball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((ball x e) Int (affine hull S)) <= S)"
  2143   apply (simp add: rel_interior, safe)
  2144   apply (force simp add: open_contains_ball)
  2145   apply (rule_tac x="ball x e" in exI)
  2146   apply (simp add: open_ball centre_in_ball)
  2147   done
  2148 
  2149 lemma rel_interior_ball: 
  2150       "rel_interior S = {x : S. ? e. e>0 & ((ball x e) Int (affine hull S)) <= S}" 
  2151       using mem_rel_interior_ball [of _ S] by auto 
  2152 
  2153 lemma mem_rel_interior_cball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((cball x e) Int (affine hull S)) <= S)"
  2154   apply (simp add: rel_interior, safe) 
  2155   apply (force simp add: open_contains_cball)
  2156   apply (rule_tac x="ball x e" in exI)
  2157   apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
  2158   apply auto
  2159   done
  2160 
  2161 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
  2162 
  2163 lemma rel_interior_empty: "rel_interior {} = {}" 
  2164    by (auto simp add: rel_interior_def) 
  2165 
  2166 lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"
  2167 by (metis affine_hull_eq affine_sing)
  2168 
  2169 lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"
  2170    unfolding rel_interior_ball affine_hull_sing apply auto
  2171    apply(rule_tac x="1 :: real" in exI) apply simp
  2172    done
  2173 
  2174 lemma subset_rel_interior:
  2175 fixes S T :: "('n::euclidean_space) set"
  2176 assumes "S<=T" "affine hull S=affine hull T"
  2177 shows "rel_interior S <= rel_interior T"
  2178   using assms by (auto simp add: rel_interior_def)  
  2179 
  2180 lemma rel_interior_subset: "rel_interior S <= S" 
  2181    by (auto simp add: rel_interior_def)
  2182 
  2183 lemma rel_interior_subset_closure: "rel_interior S <= closure S" 
  2184    using rel_interior_subset by (auto simp add: closure_def) 
  2185 
  2186 lemma interior_subset_rel_interior: "interior S <= rel_interior S" 
  2187    by (auto simp add: rel_interior interior_def)
  2188 
  2189 lemma interior_rel_interior:
  2190 fixes S :: "('n::euclidean_space) set"
  2191 assumes "aff_dim S = int(DIM('n))"
  2192 shows "rel_interior S = interior S"
  2193 proof -
  2194 have "affine hull S = UNIV" using assms affine_hull_univ[of S] by auto 
  2195 from this show ?thesis unfolding rel_interior interior_def by auto
  2196 qed
  2197 
  2198 lemma rel_interior_open:
  2199 fixes S :: "('n::euclidean_space) set"
  2200 assumes "open S"
  2201 shows "rel_interior S = S"
  2202 by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
  2203 
  2204 lemma interior_rel_interior_gen:
  2205 fixes S :: "('n::euclidean_space) set"
  2206 shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
  2207 by (metis interior_rel_interior low_dim_interior)
  2208 
  2209 lemma rel_interior_univ: 
  2210 fixes S :: "('n::euclidean_space) set"
  2211 shows "rel_interior (affine hull S) = affine hull S"
  2212 proof-
  2213 have h1: "rel_interior (affine hull S) <= affine hull S" using rel_interior_subset by auto 
  2214 { fix x assume x_def: "x : affine hull S"
  2215   obtain e :: real where "e=1" by auto
  2216   hence "e>0 & ball x e Int affine hull (affine hull S) <= affine hull S" using hull_hull[of _ S] by auto
  2217   hence "x : rel_interior (affine hull S)" using x_def rel_interior_ball[of "affine hull S"] by auto
  2218 } from this show ?thesis using h1 by auto 
  2219 qed
  2220 
  2221 lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
  2222 by (metis open_UNIV rel_interior_open)
  2223 
  2224 lemma rel_interior_convex_shrink:
  2225   fixes S :: "('a::euclidean_space) set"
  2226   assumes "convex S" "c : rel_interior S" "x : S" "0 < e" "e <= 1"
  2227   shows "x - e *\<^sub>R (x - c) : rel_interior S"
  2228 proof- 
  2229 (* Proof is a modified copy of the proof of similar lemma mem_interior_convex_shrink 
  2230 *)
  2231 obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
  2232   using assms(2) unfolding  mem_rel_interior_ball by auto
  2233 {   fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d & y : affine hull S"
  2234     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)
  2235     have "x : affine hull S" using assms hull_subset[of S] by auto
  2236     moreover have "1 / e + - ((1 - e) / e) = 1" 
  2237        using `e>0` mult_left.diff[of "1" "(1-e)" "1/e"] by auto
  2238     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x : affine hull S"
  2239         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)     
  2240     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)"
  2241       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
  2242       by(auto simp add:euclidean_eq[where 'a='a] field_simps) 
  2243     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)
  2244     also have "... < d" using as[unfolded dist_norm] and `e>0`
  2245       by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
  2246     finally have "y : S" apply(subst *) 
  2247 apply(rule assms(1)[unfolded convex_alt,rule_format])
  2248       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) ** by auto
  2249 } hence "ball (x - e *\<^sub>R (x - c)) (e*d) Int affine hull S <= S" by auto
  2250 moreover have "0 < e*d" using `0<e` `0<d` using real_mult_order by auto
  2251 moreover have "c : S" using assms rel_interior_subset by auto
  2252 moreover hence "x - e *\<^sub>R (x - c) : S"
  2253    using mem_convex[of S x c e] apply (simp add: algebra_simps) using assms by auto
  2254 ultimately show ?thesis 
  2255   using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e>0` by auto
  2256 qed
  2257 
  2258 lemma interior_real_semiline:
  2259 fixes a :: real
  2260 shows "interior {a..} = {a<..}"
  2261 proof-
  2262 { fix y assume "a<y" hence "y : interior {a..}"
  2263   apply (simp add: mem_interior) apply (rule_tac x="(y-a)" in exI) apply (auto simp add: dist_norm) 
  2264   done }
  2265 moreover
  2266 { fix y assume "y : interior {a..}" (*hence "a<=y" using interior_subset by auto*)
  2267   from this obtain e where e_def: "e>0 & cball y e \<subseteq> {a..}" 
  2268      using mem_interior_cball[of y "{a..}"] by auto
  2269   moreover hence "y-e : cball y e" by (auto simp add: cball_def dist_norm) 
  2270   ultimately have "a<=y-e" by auto
  2271   hence "a<y" using e_def by auto
  2272 } ultimately show ?thesis by auto
  2273 qed
  2274 
  2275 lemma rel_interior_real_interval:
  2276   fixes a b :: real assumes "a < b" shows "rel_interior {a..b} = {a<..<b}"
  2277 proof-
  2278   have "{a<..<b} \<noteq> {}" using assms unfolding set_eq_iff by (auto intro!: exI[of _ "(a + b) / 2"])
  2279   then show ?thesis
  2280     using interior_rel_interior_gen[of "{a..b}", symmetric]
  2281     by (simp split: split_if_asm add: interior_closed_interval)
  2282 qed
  2283 
  2284 lemma rel_interior_real_semiline:
  2285   fixes a :: real shows "rel_interior {a..} = {a<..}"
  2286 proof-
  2287   have *: "{a<..} \<noteq> {}" unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
  2288   then show ?thesis using interior_real_semiline
  2289      interior_rel_interior_gen[of "{a..}"]
  2290      by (auto split: split_if_asm)
  2291 qed
  2292 
  2293 subsection "Relative open"
  2294 
  2295 definition "rel_open S <-> (rel_interior S) = S"
  2296 
  2297 lemma rel_open: "rel_open S <-> openin (subtopology euclidean (affine hull S)) S"
  2298  unfolding rel_open_def rel_interior_def apply auto
  2299  using openin_subopen[of "subtopology euclidean (affine hull S)" S] by auto
  2300 
  2301 lemma opein_rel_interior: 
  2302   "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
  2303   apply (simp add: rel_interior_def)
  2304   apply (subst openin_subopen) by blast
  2305 
  2306 lemma affine_rel_open: 
  2307   fixes S :: "('n::euclidean_space) set"
  2308   assumes "affine S" shows "rel_open S" 
  2309   unfolding rel_open_def using assms rel_interior_univ[of S] affine_hull_eq[of S] by metis
  2310 
  2311 lemma affine_closed: 
  2312   fixes S :: "('n::euclidean_space) set"
  2313   assumes "affine S" shows "closed S"
  2314 proof-
  2315 { assume "S ~= {}"
  2316   from this obtain L where L_def: "subspace L & affine_parallel S L"
  2317      using assms affine_parallel_subspace[of S] by auto
  2318   from this obtain "a" where a_def: "S=(op + a ` L)" 
  2319      using affine_parallel_def[of L S] affine_parallel_commut by auto 
  2320   have "closed L" using L_def closed_subspace by auto
  2321   hence "closed S" using closed_translation a_def by auto
  2322 } from this show ?thesis by auto
  2323 qed
  2324 
  2325 lemma closure_affine_hull:
  2326   fixes S :: "('n::euclidean_space) set"
  2327   shows "closure S <= affine hull S"
  2328 proof-
  2329 have "closure S <= closure (affine hull S)" using subset_closure by auto
  2330 moreover have "closure (affine hull S) = affine hull S" 
  2331    using affine_affine_hull affine_closed[of "affine hull S"] closure_eq by auto
  2332 ultimately show ?thesis by auto  
  2333 qed
  2334 
  2335 lemma closure_same_affine_hull:
  2336   fixes S :: "('n::euclidean_space) set"
  2337   shows "affine hull (closure S) = affine hull S"
  2338 proof-
  2339 have "affine hull (closure S) <= affine hull S"
  2340    using hull_mono[of "closure S" "affine hull S" "affine"] closure_affine_hull[of S] hull_hull[of "affine" S] by auto
  2341 moreover have "affine hull (closure S) >= affine hull S"  
  2342    using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
  2343 ultimately show ?thesis by auto  
  2344 qed
  2345 
  2346 lemma closure_aff_dim: 
  2347   fixes S :: "('n::euclidean_space) set"
  2348   shows "aff_dim (closure S) = aff_dim S"
  2349 proof-
  2350 have "aff_dim S <= aff_dim (closure S)" using aff_dim_subset closure_subset by auto
  2351 moreover have "aff_dim (closure S) <= aff_dim (affine hull S)" 
  2352   using aff_dim_subset closure_affine_hull by auto
  2353 moreover have "aff_dim (affine hull S) = aff_dim S" using aff_dim_affine_hull by auto
  2354 ultimately show ?thesis by auto
  2355 qed
  2356 
  2357 lemma rel_interior_closure_convex_shrink:
  2358   fixes S :: "(_::euclidean_space) set"
  2359   assumes "convex S" "c : rel_interior S" "x : closure S" "0 < e" "e <= 1"
  2360   shows "x - e *\<^sub>R (x - c) : rel_interior S"
  2361 proof- 
  2362 (* Proof is a modified copy of the proof of similar lemma mem_interior_closure_convex_shrink
  2363 *)
  2364 obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
  2365   using assms(2) unfolding mem_rel_interior_ball by auto
  2366 have "EX y : S. norm (y - x) * (1 - e) < e * d" proof(cases "x : S")
  2367     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  2368     case False hence x:"x islimpt S" using assms(3)[unfolded closure_def] by auto
  2369     show ?thesis proof(cases "e=1")
  2370       case True obtain y where "y : S" "y ~= x" "dist y x < 1"
  2371         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  2372       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  2373       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  2374         using `e<=1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  2375       then obtain y where "y : S" "y ~= x" "dist y x < e * d / (1 - e)"
  2376         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  2377       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  2378   then obtain y where "y : S" and y:"norm (y - x) * (1 - e) < e * d" by auto
  2379   def z == "c + ((1 - e) / e) *\<^sub>R (x - y)"
  2380   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)
  2381   have zball: "z\<in>ball c d"
  2382     using mem_ball z_def dist_norm[of c] using y and assms(4,5) by (auto simp add:field_simps norm_minus_commute)
  2383   have "x : affine hull S" using closure_affine_hull assms by auto
  2384   moreover have "y : affine hull S" using `y : S` hull_subset[of S] by auto
  2385   moreover have "c : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
  2386   ultimately have "z : affine hull S" 
  2387     using z_def affine_affine_hull[of S] 
  2388           mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"] 
  2389           assms by (auto simp add: field_simps)
  2390   hence "z : S" using d zball by auto
  2391   obtain d1 where "d1>0" and d1:"ball z d1 <= ball c d"
  2392     using zball open_ball[of c d] openE[of "ball c d" z] by auto
  2393   hence "(ball z d1) Int (affine hull S) <= (ball c d) Int (affine hull S)" by auto
  2394   hence "(ball z d1) Int (affine hull S) <= S" using d by auto 
  2395   hence "z : rel_interior S" using mem_rel_interior_ball using `d1>0` `z : S` by auto
  2396   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
  2397   thus ?thesis using * by auto 
  2398 qed
  2399 
  2400 subsection{* Relative interior preserves under linear transformations *}
  2401 
  2402 lemma rel_interior_translation_aux:
  2403 fixes a :: "'n::euclidean_space"
  2404 shows "((%x. a + x) ` rel_interior S) <= rel_interior ((%x. a + x) ` S)"
  2405 proof-
  2406 { fix x assume x_def: "x : rel_interior S"
  2407   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 
  2408   from this have "open ((%x. a + x) ` T)" and 
  2409     "(a + x) : (((%x. a + x) ` T) Int ((%x. a + x) ` S))" and 
  2410     "(((%x. a + x) ` T) Int (affine hull ((%x. a + x) ` S))) <= ((%x. a + x) ` S)" 
  2411     using affine_hull_translation[of a S] open_translation[of T a] x_def by auto 
  2412   from this have "(a+x) : rel_interior ((%x. a + x) ` S)" 
  2413     using mem_rel_interior[of "a+x" "((%x. a + x) ` S)"] by auto 
  2414 } from this show ?thesis by auto 
  2415 qed
  2416 
  2417 lemma rel_interior_translation:
  2418 fixes a :: "'n::euclidean_space"
  2419 shows "rel_interior ((%x. a + x) ` S) = ((%x. a + x) ` rel_interior S)"
  2420 proof-
  2421 have "(%x. (-a) + x) ` rel_interior ((%x. a + x) ` S) <= rel_interior S" 
  2422    using rel_interior_translation_aux[of "-a" "(%x. a + x) ` S"] 
  2423          translation_assoc[of "-a" "a"] by auto
  2424 hence "((%x. a + x) ` rel_interior S) >= rel_interior ((%x. a + x) ` S)" 
  2425    using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"] 
  2426    by auto
  2427 from this show ?thesis using  rel_interior_translation_aux[of a S] by auto 
  2428 qed
  2429 
  2430 
  2431 lemma affine_hull_linear_image:
  2432 assumes "bounded_linear f"
  2433 shows "f ` (affine hull s) = affine hull f ` s"
  2434 (* Proof is a modified copy of the proof of similar lemma convex_hull_linear_image
  2435 *)
  2436   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  2437   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  2438   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  2439 proof-
  2440   interpret f: bounded_linear f by fact
  2441   show "affine {x. f x : affine hull f ` s}" 
  2442   unfolding affine_def by(auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format]) next
  2443   interpret f: bounded_linear f by fact
  2444   show "affine {x. x : f ` (affine hull s)}" using affine_affine_hull[unfolded affine_def, of s] 
  2445     unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  2446 qed auto
  2447 
  2448 
  2449 lemma rel_interior_injective_on_span_linear_image:
  2450 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  2451 fixes S :: "('m::euclidean_space) set"
  2452 assumes "bounded_linear f" and "inj_on f (span S)"
  2453 shows "rel_interior (f ` S) = f ` (rel_interior S)"
  2454 proof-
  2455 { fix z assume z_def: "z : rel_interior (f ` S)"
  2456   have "z : f ` S" using z_def rel_interior_subset[of "f ` S"] by auto
  2457   from this obtain x where x_def: "x : S & (f x = z)" by auto
  2458   obtain e2 where e2_def: "e2>0 & cball z e2 Int affine hull (f ` S) <= (f ` S)" 
  2459     using z_def rel_interior_cball[of "f ` S"] by auto
  2460   obtain K where K_def: "K>0 & (! x. norm (f x) <= norm x * K)" 
  2461    using assms RealVector.bounded_linear.pos_bounded[of f] by auto
  2462   def e1 == "1/K" hence e1_def: "e1>0 & (! x. e1 * norm (f x) <= norm x)" 
  2463    using K_def pos_le_divide_eq[of e1] by auto
  2464   def e == "e1 * e2" hence "e>0" using e1_def e2_def real_mult_order by auto 
  2465   { fix y assume y_def: "y : cball x e Int affine hull S"
  2466     from this have h1: "f y : affine hull (f ` S)" 
  2467       using affine_hull_linear_image[of f S] assms by auto 
  2468     from y_def have "norm (x-y)<=e1 * e2" 
  2469       using cball_def[of x e] dist_norm[of x y] e_def by auto
  2470     moreover have "(f x)-(f y)=f (x-y)"
  2471        using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
  2472     moreover have "e1 * norm (f (x-y)) <= norm (x-y)" using e1_def by auto
  2473     ultimately have "e1 * norm ((f x)-(f y)) <= e1 * e2" by auto
  2474     hence "(f y) : (cball z e2)" 
  2475       using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1_def x_def by auto
  2476     hence "f y : (f ` S)" using y_def e2_def h1 by auto
  2477     hence "y : S" using assms y_def hull_subset[of S] affine_hull_subset_span 
  2478          inj_on_image_mem_iff[of f "span S" S y] by auto
  2479   } 
  2480   hence "z : f ` (rel_interior S)" using mem_rel_interior_cball[of x S] `e>0` x_def by auto
  2481 } 
  2482 moreover
  2483 { fix x assume x_def: "x : rel_interior S"
  2484   from this obtain e2 where e2_def: "e2>0 & cball x e2 Int affine hull S <= S" 
  2485     using rel_interior_cball[of S] by auto
  2486   have "x : S" using x_def rel_interior_subset by auto
  2487   hence *: "f x : f ` S" by auto
  2488   have "! x:span S. f x = 0 --> x = 0" 
  2489     using assms subspace_span linear_conv_bounded_linear[of f] 
  2490           linear_injective_on_subspace_0[of f "span S"] by auto
  2491   from this obtain e1 where e1_def: "e1>0 & (! x : span S. e1 * norm x <= norm (f x))" 
  2492    using assms injective_imp_isometric[of "span S" f] 
  2493          subspace_span[of S] closed_subspace[of "span S"] by auto
  2494   def e == "e1 * e2" hence "e>0" using e1_def e2_def real_mult_order by auto 
  2495   { fix y assume y_def: "y : cball (f x) e Int affine hull (f ` S)"
  2496     from this have "y : f ` (affine hull S)" using affine_hull_linear_image[of f S] assms by auto 
  2497     from this obtain xy where xy_def: "xy : affine hull S & (f xy = y)" by auto
  2498     from this y_def have "norm ((f x)-(f xy))<=e1 * e2" 
  2499       using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
  2500     moreover have "(f x)-(f xy)=f (x-xy)"
  2501        using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
  2502     moreover have "x-xy : span S" 
  2503        using subspace_sub[of "span S" x xy] subspace_span `x : S` xy_def 
  2504              affine_hull_subset_span[of S] span_inc by auto
  2505     moreover hence "e1 * norm (x-xy) <= norm (f (x-xy))" using e1_def by auto
  2506     ultimately have "e1 * norm (x-xy) <= e1 * e2" by auto
  2507     hence "xy : (cball x e2)"  using cball_def[of x e2] dist_norm[of x xy] e1_def by auto
  2508     hence "y : (f ` S)" using xy_def e2_def by auto
  2509   } 
  2510   hence "(f x) : rel_interior (f ` S)" 
  2511      using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e>0` by auto
  2512 } 
  2513 ultimately show ?thesis by auto
  2514 qed
  2515 
  2516 lemma rel_interior_injective_linear_image:
  2517 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  2518 assumes "bounded_linear f" and "inj f"
  2519 shows "rel_interior (f ` S) = f ` (rel_interior S)"
  2520 using assms rel_interior_injective_on_span_linear_image[of f S] 
  2521       subset_inj_on[of f "UNIV" "span S"] by auto
  2522 
  2523 subsection{* Some Properties of subset of standard basis *}
  2524 
  2525 lemma affine_hull_substd_basis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  2526   shows "affine hull (insert 0 {basis i | i. i : d}) =
  2527   {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
  2528  (is "affine hull (insert 0 ?A) = ?B")
  2529 proof- have *:"\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A" by auto
  2530   show ?thesis unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,THEN sym] * ..
  2531 qed
  2532 
  2533 lemma affine_hull_convex_hull: "affine hull (convex hull S) = affine hull S"
  2534 by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
  2535 
  2536 subsection {* Openness and compactness are preserved by convex hull operation. *}
  2537 
  2538 lemma open_convex_hull[intro]:
  2539   fixes s :: "'a::real_normed_vector set"
  2540   assumes "open s"
  2541   shows "open(convex hull s)"
  2542   unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
  2543 proof(rule, rule) fix a
  2544   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"
  2545   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
  2546 
  2547   from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
  2548     using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
  2549   have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
  2550 
  2551   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}"
  2552     apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
  2553   proof-
  2554     show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
  2555       using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
  2556   next  fix y assume "y \<in> cball a (Min i)"
  2557     hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
  2558     { fix x assume "x\<in>t"
  2559       hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
  2560       hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
  2561       moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
  2562       ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
  2563     moreover
  2564     have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
  2565     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
  2566       unfolding setsum_reindex[OF *] o_def using obt(4) by auto
  2567     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
  2568       unfolding setsum_reindex[OF *] o_def using obt(4,5)
  2569       by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
  2570     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"
  2571       apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
  2572       using obt(1, 3) by auto
  2573   qed
  2574 qed
  2575 
  2576 lemma compact_convex_combinations:
  2577   fixes s t :: "'a::real_normed_vector set"
  2578   assumes "compact s" "compact t"
  2579   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}"
  2580 proof-
  2581   let ?X = "{0..1} \<times> s \<times> t"
  2582   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  2583   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"
  2584     apply(rule set_eqI) unfolding image_iff mem_Collect_eq
  2585     apply rule apply auto
  2586     apply (rule_tac x=u in rev_bexI, simp)
  2587     apply (erule rev_bexI, erule rev_bexI, simp)
  2588     by auto
  2589   have "continuous_on ({0..1} \<times> s \<times> t)
  2590      (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  2591     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  2592   thus ?thesis unfolding *
  2593     apply (rule compact_continuous_image)
  2594     apply (intro compact_Times compact_interval assms)
  2595     done
  2596 qed
  2597 
  2598 lemma compact_convex_hull: fixes s::"('a::euclidean_space) set"
  2599   assumes "compact s"  shows "compact(convex hull s)"
  2600 proof(cases "s={}")
  2601   case True thus ?thesis using compact_empty by simp
  2602 next
  2603   case False then obtain w where "w\<in>s" by auto
  2604   show ?thesis unfolding caratheodory[of s]
  2605   proof(induct ("DIM('a) + 1"))
  2606     have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
  2607       using compact_empty by auto
  2608     case 0 thus ?case unfolding * by simp
  2609   next
  2610     case (Suc n)
  2611     show ?case proof(cases "n=0")
  2612       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"
  2613         unfolding set_eq_iff and mem_Collect_eq proof(rule, rule)
  2614         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2615         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  2616         show "x\<in>s" proof(cases "card t = 0")
  2617           case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
  2618         next
  2619           case False hence "card t = Suc 0" using t(3) `n=0` by auto
  2620           then obtain a where "t = {a}" unfolding card_Suc_eq by auto
  2621           thus ?thesis using t(2,4) by simp
  2622         qed
  2623       next
  2624         fix x assume "x\<in>s"
  2625         thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2626           apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
  2627       qed thus ?thesis using assms by simp
  2628     next
  2629       case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
  2630         { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
  2631         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}}"
  2632         unfolding set_eq_iff and mem_Collect_eq proof(rule,rule)
  2633         fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  2634           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)"
  2635         then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
  2636           "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
  2637         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
  2638           apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
  2639           using obt(7) and hull_mono[of t "insert u t"] by auto
  2640         ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2641           apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
  2642       next
  2643         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2644         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  2645         let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  2646           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)"
  2647         show ?P proof(cases "card t = Suc n")
  2648           case False hence "card t \<le> n" using t(3) by auto
  2649           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
  2650             by(auto intro!: exI[where x=t])
  2651         next
  2652           case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
  2653           show ?P proof(cases "u={}")
  2654             case True hence "x=a" using t(4)[unfolded au] by auto
  2655             show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
  2656               using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
  2657           next
  2658             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"
  2659               using t(4)[unfolded au convex_hull_insert[OF False]] by auto
  2660             have *:"1 - vx = ux" using obt(3) by auto
  2661             show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
  2662               using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
  2663               by(auto intro!: exI[where x=u])
  2664           qed
  2665         qed
  2666       qed
  2667       thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
  2668     qed
  2669   qed
  2670 qed
  2671 
  2672 lemma finite_imp_compact_convex_hull:
  2673   fixes s :: "('a::euclidean_space) set"
  2674   shows "finite s \<Longrightarrow> compact(convex hull s)"
  2675 by (metis compact_convex_hull finite_imp_compact)
  2676 
  2677 subsection {* Extremal points of a simplex are some vertices. *}
  2678 
  2679 lemma dist_increases_online:
  2680   fixes a b d :: "'a::real_inner"
  2681   assumes "d \<noteq> 0"
  2682   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
  2683 proof(cases "inner a d - inner b d > 0")
  2684   case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
  2685     apply(rule_tac add_pos_pos) using assms by auto
  2686   thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  2687     by (simp add: algebra_simps inner_commute)
  2688 next
  2689   case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
  2690     apply(rule_tac add_pos_nonneg) using assms by auto
  2691   thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  2692     by (simp add: algebra_simps inner_commute)
  2693 qed
  2694 
  2695 lemma norm_increases_online:
  2696   fixes d :: "'a::real_inner"
  2697   shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
  2698   using dist_increases_online[of d a 0] unfolding dist_norm by auto
  2699 
  2700 lemma simplex_furthest_lt:
  2701   fixes s::"'a::real_inner set" assumes "finite s"
  2702   shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
  2703 proof(induct_tac rule: finite_induct[of s])
  2704   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))"
  2705   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))"
  2706   proof(rule,rule,cases "s = {}")
  2707     case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
  2708     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"
  2709       using y(1)[unfolded convex_hull_insert[OF False]] by auto
  2710     show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
  2711     proof(cases "y\<in>convex hull s")
  2712       case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
  2713         using as(3)[THEN bspec[where x=y]] and y(2) by auto
  2714       thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
  2715     next
  2716       case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
  2717         assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
  2718         thus ?thesis using False and obt(4) by auto
  2719       next
  2720         assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
  2721         thus ?thesis using y(2) by auto
  2722       next
  2723         assume "u\<noteq>0" "v\<noteq>0"
  2724         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
  2725         have "x\<noteq>b" proof(rule ccontr) 
  2726           assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
  2727             using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
  2728           thus False using obt(4) and False by simp qed
  2729         hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
  2730         show ?thesis using dist_increases_online[OF *, of a y]
  2731         proof(erule_tac disjE)
  2732           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
  2733           hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
  2734             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  2735           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
  2736             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  2737             apply(rule_tac x="u + w" in exI) apply rule defer 
  2738             apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  2739           ultimately show ?thesis by auto
  2740         next
  2741           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
  2742           hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
  2743             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  2744           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
  2745             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  2746             apply(rule_tac x="u - w" in exI) apply rule defer 
  2747             apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  2748           ultimately show ?thesis by auto
  2749         qed
  2750       qed auto
  2751     qed
  2752   qed auto
  2753 qed (auto simp add: assms)
  2754 
  2755 lemma simplex_furthest_le:
  2756   fixes s :: "('a::euclidean_space) set"
  2757   assumes "finite s" "s \<noteq> {}"
  2758   shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
  2759 proof-
  2760   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  2761   then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
  2762     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
  2763     unfolding dist_commute[of a] unfolding dist_norm by auto
  2764   thus ?thesis proof(cases "x\<in>s")
  2765     case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
  2766       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
  2767     thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
  2768   qed auto
  2769 qed
  2770 
  2771 lemma simplex_furthest_le_exists:
  2772   fixes s :: "('a::euclidean_space) set"
  2773   shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
  2774   using simplex_furthest_le[of s] by (cases "s={}")auto
  2775 
  2776 lemma simplex_extremal_le:
  2777   fixes s :: "('a::euclidean_space) set"
  2778   assumes "finite s" "s \<noteq> {}"
  2779   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)"
  2780 proof-
  2781   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  2782   then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
  2783     "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
  2784     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
  2785   thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
  2786     assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
  2787       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
  2788     thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
  2789   next
  2790     assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
  2791       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
  2792     thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
  2793       by (auto simp add: norm_minus_commute)
  2794   qed auto
  2795 qed 
  2796 
  2797 lemma simplex_extremal_le_exists:
  2798   fixes s :: "('a::euclidean_space) set"
  2799   shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
  2800   \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
  2801   using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
  2802 
  2803 subsection {* Closest point of a convex set is unique, with a continuous projection. *}
  2804 
  2805 definition
  2806   closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a" where
  2807  "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
  2808 
  2809 lemma closest_point_exists:
  2810   assumes "closed s" "s \<noteq> {}"
  2811   shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
  2812   unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
  2813   using distance_attains_inf[OF assms(1,2), of a] by auto
  2814 
  2815 lemma closest_point_in_set:
  2816   "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
  2817   by(meson closest_point_exists)
  2818 
  2819 lemma closest_point_le:
  2820   "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
  2821   using closest_point_exists[of s] by auto
  2822 
  2823 lemma closest_point_self:
  2824   assumes "x \<in> s"  shows "closest_point s x = x"
  2825   unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
  2826   using assms by auto
  2827 
  2828 lemma closest_point_refl:
  2829  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
  2830   using closest_point_in_set[of s x] closest_point_self[of x s] by auto
  2831 
  2832 lemma closer_points_lemma:
  2833   assumes "inner y z > 0"
  2834   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
  2835 proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
  2836   thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
  2837     fix v assume "0<v" "v \<le> inner y z / inner z z"
  2838     thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
  2839       by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
  2840   qed(rule divide_pos_pos, auto) qed
  2841 
  2842 lemma closer_point_lemma:
  2843   assumes "inner (y - x) (z - x) > 0"
  2844   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
  2845 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)"
  2846     using closer_points_lemma[OF assms] by auto
  2847   show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
  2848     unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
  2849 
  2850 lemma any_closest_point_dot:
  2851   assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  2852   shows "inner (a - x) (y - x) \<le> 0"
  2853 proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
  2854   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
  2855   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
  2856   thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
  2857 
  2858 lemma any_closest_point_unique:
  2859   fixes x :: "'a::real_inner"
  2860   assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
  2861   "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
  2862   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)]
  2863   unfolding norm_pths(1) and norm_le_square
  2864   by (auto simp add: algebra_simps)
  2865 
  2866 lemma closest_point_unique:
  2867   assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  2868   shows "x = closest_point s a"
  2869   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
  2870   using closest_point_exists[OF assms(2)] and assms(3) by auto
  2871 
  2872 lemma closest_point_dot:
  2873   assumes "convex s" "closed s" "x \<in> s"
  2874   shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
  2875   apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
  2876   using closest_point_exists[OF assms(2)] and assms(3) by auto
  2877 
  2878 lemma closest_point_lt:
  2879   assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
  2880   shows "dist a (closest_point s a) < dist a x"
  2881   apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
  2882   apply(rule closest_point_unique[OF assms(1-3), of a])
  2883   using closest_point_le[OF assms(2), of _ a] by fastsimp
  2884 
  2885 lemma closest_point_lipschitz:
  2886   assumes "convex s" "closed s" "s \<noteq> {}"
  2887   shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
  2888 proof-
  2889   have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
  2890        "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
  2891     apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
  2892     using closest_point_exists[OF assms(2-3)] by auto
  2893   thus ?thesis unfolding dist_norm and norm_le
  2894     using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
  2895     by (simp add: inner_add inner_diff inner_commute) qed
  2896 
  2897 lemma continuous_at_closest_point:
  2898   assumes "convex s" "closed s" "s \<noteq> {}"
  2899   shows "continuous (at x) (closest_point s)"
  2900   unfolding continuous_at_eps_delta 
  2901   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
  2902 
  2903 lemma continuous_on_closest_point:
  2904   assumes "convex s" "closed s" "s \<noteq> {}"
  2905   shows "continuous_on t (closest_point s)"
  2906 by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
  2907 
  2908 subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
  2909 
  2910 lemma supporting_hyperplane_closed_point:
  2911   fixes z :: "'a::{real_inner,heine_borel}"
  2912   assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
  2913   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)"
  2914 proof-
  2915   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
  2916   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)
  2917     apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
  2918     show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
  2919       unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
  2920   next
  2921     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)"
  2922       using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
  2923     assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
  2924       "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)
  2925     thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
  2926   qed auto
  2927 qed
  2928 
  2929 lemma separating_hyperplane_closed_point:
  2930   fixes z :: "'a::{real_inner,heine_borel}"
  2931   assumes "convex s" "closed s" "z \<notin> s"
  2932   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
  2933 proof(cases "s={}")
  2934   case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
  2935     using less_le_trans[OF _ inner_ge_zero[of z]] by auto
  2936 next
  2937   case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
  2938     using distance_attains_inf[OF assms(2) False] by auto
  2939   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
  2940     apply rule defer apply rule proof-
  2941     fix x assume "x\<in>s"
  2942     have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
  2943       assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
  2944       then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
  2945       thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
  2946         using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
  2947         using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
  2948     moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
  2949     hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
  2950     ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
  2951       unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
  2952   qed(insert `y\<in>s` `z\<notin>s`, auto)
  2953 qed
  2954 
  2955 lemma separating_hyperplane_closed_0:
  2956   assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 \<notin> s"
  2957   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
  2958   proof(cases "s={}")
  2959   case True have "norm ((basis 0)::'a) = 1" by auto
  2960   hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
  2961     apply(subst norm_le_zero_iff[THEN sym]) by auto
  2962   thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
  2963     using True using DIM_positive[where 'a='a] by auto
  2964 next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
  2965     apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
  2966 
  2967 subsection {* Now set-to-set for closed/compact sets. *}
  2968 
  2969 lemma separating_hyperplane_closed_compact:
  2970   assumes "convex (s::('a::euclidean_space) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
  2971   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  2972 proof(cases "s={}")
  2973   case True
  2974   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
  2975   obtain z::"'a" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
  2976   hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
  2977   then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
  2978     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
  2979   thus ?thesis using True by auto
  2980 next
  2981   case False then obtain y where "y\<in>s" by auto
  2982   obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
  2983     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
  2984     using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
  2985   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)
  2986   def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
  2987   show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
  2988     apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
  2989     from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
  2990       apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
  2991     hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
  2992     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
  2993   next
  2994     fix x assume "x\<in>s" 
  2995     hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
  2996       using ab[THEN bspec[where x=x]] by auto
  2997     thus "k + b / 2 < inner a x" using `0 < b` by auto
  2998   qed
  2999 qed
  3000 
  3001 lemma separating_hyperplane_compact_closed:
  3002   fixes s :: "('a::euclidean_space) set"
  3003   assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
  3004   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  3005 proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
  3006     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
  3007   thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
  3008 
  3009 subsection {* General case without assuming closure and getting non-strict separation. *}
  3010 
  3011 lemma separating_hyperplane_set_0:
  3012   assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
  3013   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
  3014 proof- let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
  3015   have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
  3016     apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
  3017     defer apply(rule,rule,erule conjE) proof-
  3018     fix f assume as:"f \<subseteq> ?k ` s" "finite f"
  3019     obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
  3020     then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
  3021       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
  3022       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
  3023       using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
  3024     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)
  3025        using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
  3026        apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
  3027        by(auto simp add: inner_commute elim!: ballE)
  3028     thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
  3029   qed(insert closed_halfspace_ge, auto)
  3030   then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
  3031   thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
  3032 
  3033 lemma separating_hyperplane_sets:
  3034   assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
  3035   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)"
  3036 proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
  3037   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" 
  3038     using assms(3-5) by auto 
  3039   hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
  3040     by (force simp add: inner_diff)
  3041   thus ?thesis
  3042     apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
  3043     apply auto
  3044     apply (rule Sup[THEN isLubD2]) 
  3045     prefer 4
  3046     apply (rule Sup_least) 
  3047      using assms(3-5) apply (auto simp add: setle_def)
  3048     apply metis
  3049     done
  3050 qed
  3051 
  3052 subsection {* More convexity generalities. *}
  3053 
  3054 lemma convex_closure:
  3055   fixes s :: "'a::real_normed_vector set"
  3056   assumes "convex s" shows "convex(closure s)"
  3057   unfolding convex_def Ball_def closure_sequential
  3058   apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
  3059   apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
  3060   apply(rule assms[unfolded convex_def, rule_format]) prefer 6
  3061   apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
  3062 
  3063 lemma convex_interior:
  3064   fixes s :: "'a::real_normed_vector set"
  3065   assumes "convex s" shows "convex(interior s)"
  3066   unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
  3067   fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
  3068   fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
  3069   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)
  3070     apply rule unfolding subset_eq defer apply rule proof-
  3071     fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
  3072     hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
  3073       apply(rule_tac assms[unfolded convex_alt, rule_format])
  3074       using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
  3075     thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
  3076 
  3077 lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
  3078   using hull_subset[of s convex] convex_hull_empty by auto
  3079 
  3080 subsection {* Moving and scaling convex hulls. *}
  3081 
  3082 lemma convex_hull_translation_lemma:
  3083   "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
  3084 by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono mem_def)
  3085 
  3086 lemma convex_hull_bilemma: fixes neg
  3087   assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
  3088   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)
  3089   \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
  3090   using assms by(metis subset_antisym) 
  3091 
  3092 lemma convex_hull_translation:
  3093   "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
  3094   apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
  3095 
  3096 lemma convex_hull_scaling_lemma:
  3097  "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  3098 by (metis convex_convex_hull convex_scaling hull_subset mem_def subset_hull subset_image_iff)
  3099 
  3100 lemma convex_hull_scaling:
  3101   "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  3102   apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
  3103   unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
  3104 
  3105 lemma convex_hull_affinity:
  3106   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
  3107 by(simp only: image_image[THEN sym] convex_hull_scaling convex_hull_translation)
  3108 
  3109 subsection {* Convexity of cone hulls *}
  3110 
  3111 lemma convex_cone_hull:
  3112 fixes S :: "('m::euclidean_space) set"
  3113 assumes "convex S"
  3114 shows "convex (cone hull S)"
  3115 proof-
  3116 { fix x y assume xy_def: "x : cone hull S & y : cone hull S"
  3117   hence "S ~= {}" using cone_hull_empty_iff[of S] by auto
  3118   fix u v assume uv_def: "u>=0 & v>=0 & (u :: real)+v=1"
  3119   hence *: "u *\<^sub>R x : cone hull S & v *\<^sub>R y : cone hull S"
  3120      using cone_cone_hull[of S] xy_def cone_def[of "cone hull S"] by auto
  3121   from * obtain cx xx where x_def: "u *\<^sub>R x = cx *\<^sub>R xx & (cx :: real)>=0 & xx : S"
  3122      using cone_hull_expl[of S] by auto
  3123   from * obtain cy yy where y_def: "v *\<^sub>R y = cy *\<^sub>R yy & (cy :: real)>=0 & yy : S"
  3124      using cone_hull_expl[of S] by auto
  3125   { assume "cx+cy<=0" hence "u *\<^sub>R x=0 & v *\<^sub>R y=0" using x_def y_def by auto
  3126     hence "u *\<^sub>R x+ v *\<^sub>R y = 0" by auto
  3127     hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using cone_hull_contains_0[of S] `S ~= {}` by auto
  3128   }
  3129   moreover
  3130   { assume "cx+cy>0"
  3131     hence "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy : S"
  3132       using assms mem_convex_alt[of S xx yy cx cy] x_def y_def by auto
  3133     hence "cx *\<^sub>R xx + cy *\<^sub>R yy : cone hull S"
  3134       using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"]
  3135       `cx+cy>0` by (auto simp add: scaleR_right_distrib)
  3136     hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using x_def y_def by auto
  3137   }
  3138   moreover have "(cx+cy<=0) | (cx+cy>0)" by auto
  3139   ultimately have "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" by blast
  3140 } from this show ?thesis unfolding convex_def by auto
  3141 qed
  3142 
  3143 lemma cone_convex_hull:
  3144 fixes S :: "('m::euclidean_space) set"
  3145 assumes "cone S"
  3146 shows "cone (convex hull S)"
  3147 proof-
  3148 { assume "S = {}" hence ?thesis by auto }
  3149 moreover
  3150 { assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  3151   { fix c assume "(c :: real)>0"
  3152     hence "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
  3153        using convex_hull_scaling[of _ S] by auto
  3154     also have "...=convex hull S" using * `c>0` by auto
  3155     finally have "op *\<^sub>R c ` (convex hull S) = convex hull S" by auto
  3156   }
  3157   hence "0 : convex hull S & (!c. c>0 --> (op *\<^sub>R c ` (convex hull S)) = (convex hull S))"
  3158      using * hull_subset[of S convex] by auto
  3159   hence ?thesis using `S ~= {}` cone_iff[of "convex hull S"] by auto
  3160 }
  3161 ultimately show ?thesis by blast
  3162 qed
  3163 
  3164 subsection {* Convex set as intersection of halfspaces. *}
  3165 
  3166 lemma convex_halfspace_intersection:
  3167   fixes s :: "('a::euclidean_space) set"
  3168   assumes "closed s" "convex s"
  3169   shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
  3170   apply(rule set_eqI, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
  3171   fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
  3172   hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
  3173   thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
  3174     apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
  3175 qed auto
  3176 
  3177 subsection {* Radon's theorem (from Lars Schewe). *}
  3178 
  3179 lemma radon_ex_lemma:
  3180   assumes "finite c" "affine_dependent c"
  3181   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"
  3182 proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
  3183   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
  3184     and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
  3185 
  3186 lemma radon_s_lemma:
  3187   assumes "finite s" "setsum f s = (0::real)"
  3188   shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
  3189 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
  3190   show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  3191     using assms(2) by assumption qed
  3192 
  3193 lemma radon_v_lemma:
  3194   assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
  3195   shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
  3196 proof-
  3197   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 
  3198   show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  3199     using assms(2) by assumption qed
  3200 
  3201 lemma radon_partition:
  3202   assumes "finite c" "affine_dependent c"
  3203   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
  3204   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
  3205   have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
  3206   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}"
  3207   have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
  3208     case False hence "u v < 0" by auto
  3209     thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
  3210       case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
  3211     next
  3212       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
  3213       thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
  3214   qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
  3215 
  3216   hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
  3217   moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
  3218     "(\<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)"
  3219     using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
  3220   hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
  3221    "(\<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)" 
  3222     unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
  3223   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
  3224     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
  3225 
  3226   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
  3227     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)
  3228     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
  3229     by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
  3230   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
  3231     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
  3232   hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
  3233     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)
  3234     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
  3235     by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
  3236   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
  3237 qed
  3238 
  3239 lemma radon: assumes "affine_dependent c"
  3240   obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
  3241 proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
  3242   hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
  3243   from radon_partition[OF *] guess m .. then guess p ..
  3244   thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
  3245 
  3246 subsection {* Helly's theorem. *}
  3247 
  3248 lemma helly_induct: fixes f::"('a::euclidean_space) set set"
  3249   assumes "card f = n" "n \<ge> DIM('a) + 1"
  3250   "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  3251   shows "\<Inter> f \<noteq> {}"
  3252 using assms proof(induct n arbitrary: f)
  3253 case (Suc n)
  3254 have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
  3255 show "\<Inter> f \<noteq> {}" apply(cases "n = DIM('a)") apply(rule Suc(5)[rule_format])
  3256   unfolding `card f = Suc n` proof-
  3257   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
  3258     apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
  3259     defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
  3260   then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
  3261   show ?thesis proof(cases "inj_on X f")
  3262     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
  3263     hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
  3264     show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
  3265       apply(rule, rule X[rule_format]) using X st by auto
  3266   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> {}"
  3267       using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
  3268       unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
  3269     have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
  3270     then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
  3271     hence "f \<union> (g \<union> h) = f" by auto
  3272     hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
  3273       unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
  3274     have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
  3275     have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
  3276       apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  unfolding mem_def unfolding subset_eq
  3277       apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
  3278       fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
  3279       thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
  3280       fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
  3281       thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
  3282     qed(auto)
  3283     thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
  3284 qed(auto) qed(auto)
  3285 
  3286 lemma helly: fixes f::"('a::euclidean_space) set set"
  3287   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
  3288           "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  3289   shows "\<Inter> f \<noteq>{}"
  3290   apply(rule helly_induct) using assms by auto
  3291 
  3292 subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
  3293 
  3294 lemma compact_frontier_line_lemma:
  3295   fixes s :: "('a::euclidean_space) set"
  3296   assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
  3297   obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
  3298 proof-
  3299   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
  3300   let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
  3301   have A:"?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
  3302     by auto
  3303   have *:"\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
  3304   have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
  3305     apply(rule, rule continuous_vmul)
  3306     apply(rule continuous_at_id) by(rule compact_interval)
  3307   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)])
  3308     unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
  3309   ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
  3310     "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
  3311 
  3312   have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
  3313   { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
  3314     hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
  3315       using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
  3316     hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
  3317       apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
  3318       using as(1) `u\<ge>0` by(auto simp add:field_simps) 
  3319     hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
  3320   } note u_max = this
  3321 
  3322   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]
  3323     prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
  3324     fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
  3325     hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
  3326     thus False using u_max[OF _ as] by auto
  3327   qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
  3328   thus ?thesis by(metis that[of u] u_max obt(1))
  3329 qed
  3330 
  3331 lemma starlike_compact_projective:
  3332   assumes "compact s" "cball (0::'a::euclidean_space) 1 \<subseteq> s "
  3333   "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
  3334   shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
  3335 proof-
  3336   have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
  3337   def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
  3338   have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
  3339     using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
  3340   have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
  3341 
  3342   have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
  3343     apply rule unfolding pi_def
  3344     apply (rule continuous_mul)
  3345     apply (rule continuous_at_inv[unfolded o_def])
  3346     apply (rule continuous_at_norm)
  3347     apply simp
  3348     apply (rule continuous_at_id)
  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(rule_tac continuous_mul, rule_tac continuous_at_norm)
  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 simp add:norm_basis[unfolded One_nat_def])
  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: fixes s::"('a::euclidean_space) set"
  3472   assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
  3473   shows "s homeomorphic (cball (0::'a) 1)"
  3474   apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
  3475   fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
  3476   hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
  3477     apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
  3478     unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
  3479     fix y assume "dist (u *\<^sub>R x) y < 1 - u"
  3480     hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
  3481       using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
  3482       unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
  3483       apply (rule mult_left_le_imp_le[of "1 - u"])
  3484       unfolding mult_assoc[symmetric] using `u<1` by auto
  3485     thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
  3486       using as unfolding scaleR_scaleR by auto qed auto
  3487   thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
  3488 
  3489 lemma homeomorphic_convex_compact_cball: fixes e::real and s::"('a::euclidean_space) set"
  3490   assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
  3491   shows "s homeomorphic (cball (b::'a) e)"
  3492 proof- obtain a where "a\<in>interior s" using assms(3) by auto
  3493   then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
  3494   let ?d = "inverse d" and ?n = "0::'a"
  3495   have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
  3496     apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
  3497     apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
  3498     by(auto simp add: mult_right_le_one_le)
  3499   hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
  3500     using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
  3501     using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
  3502   thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
  3503     apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
  3504     using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
  3505 
  3506 lemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) set"
  3507   assumes "convex s" "compact s" "interior s \<noteq> {}"
  3508           "convex t" "compact t" "interior t \<noteq> {}"
  3509   shows "s homeomorphic t"
  3510   using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
  3511 
  3512 subsection {* Epigraphs of convex functions. *}
  3513 
  3514 definition "epigraph s (f::_ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
  3515 
  3516 lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
  3517 
  3518 (** This might break sooner or later. In fact it did already once. **)
  3519 lemma convex_epigraph: 
  3520   "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
  3521   unfolding convex_def convex_on_def
  3522   unfolding Ball_def split_paired_All epigraph_def
  3523   unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
  3524   apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
  3525   apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
  3526   apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)
  3527 
  3528 lemma convex_epigraphI:
  3529   "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex(epigraph s f)"
  3530 unfolding convex_epigraph by auto
  3531 
  3532 lemma convex_epigraph_convex:
  3533   "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
  3534 by(simp add: convex_epigraph)
  3535 
  3536 subsection {* Use this to derive general bound property of convex function. *}
  3537 
  3538 lemma convex_on:
  3539   assumes "convex s"
  3540   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>
  3541    f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
  3542   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
  3543   unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
  3544   apply safe
  3545   apply (drule_tac x=k in spec)
  3546   apply (drule_tac x=u in spec)
  3547   apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
  3548   apply simp
  3549   using assms[unfolded convex] apply simp
  3550   apply(rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
  3551   defer apply(rule setsum_mono) apply(erule_tac x=i in allE) unfolding real_scaleR_def
  3552   apply(rule mult_left_mono)using assms[unfolded convex] by auto
  3553 
  3554 
  3555 subsection {* Convexity of general and special intervals. *}
  3556 
  3557 lemma convexI: (* TODO: move to Library/Convex.thy *)
  3558   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"
  3559   shows "convex s"
  3560 using assms unfolding convex_def by fast
  3561 
  3562 lemma is_interval_convex:
  3563   fixes s :: "('a::euclidean_space) set"
  3564   assumes "is_interval s" shows "convex s"
  3565 proof (rule convexI)
  3566   fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
  3567   hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
  3568   { fix a b assume "\<not> b \<le> u * a + v * b"
  3569     hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
  3570     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)
  3571     hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
  3572   } moreover
  3573   { fix a b assume "\<not> u * a + v * b \<le> a"
  3574     hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
  3575     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
  3576     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
  3577   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)])
  3578     using as(3-) DIM_positive[where 'a='a] by(auto simp add:euclidean_simps) qed
  3579 
  3580 lemma is_interval_connected:
  3581   fixes s :: "('a::euclidean_space) set"
  3582   shows "is_interval s \<Longrightarrow> connected s"
  3583   using is_interval_convex convex_connected by auto
  3584 
  3585 lemma convex_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"
  3586   apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
  3587 
  3588 (* FIXME: rewrite these lemmas without using vec1
  3589 subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
  3590 
  3591 lemma is_interval_1:
  3592   "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)"
  3593   unfolding is_interval_def forall_1 by auto
  3594 
  3595 lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
  3596   apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
  3597   apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
  3598   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"
  3599   hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
  3600   let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
  3601   { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
  3602     using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
  3603   moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
  3604   hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
  3605   ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
  3606     apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
  3607     apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
  3608     by(auto simp add: field_simps) qed
  3609 
  3610 lemma is_interval_convex_1:
  3611   "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
  3612 by(metis is_interval_convex convex_connected is_interval_connected_1)
  3613 
  3614 lemma convex_connected_1:
  3615   "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
  3616 by(metis is_interval_convex convex_connected is_interval_connected_1)
  3617 *)
  3618 subsection {* Another intermediate value theorem formulation. *}
  3619 
  3620 lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3621   assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
  3622   shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
  3623 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
  3624     using assms(1) by auto
  3625   thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
  3626     using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
  3627     using assms by(auto intro!: imageI) qed
  3628 
  3629 lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3630   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
  3631    \<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
  3632 by(rule ivt_increasing_component_on_1)
  3633   (auto simp add: continuous_at_imp_continuous_on)
  3634 
  3635 lemma ivt_decreasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3636   assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
  3637   shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
  3638   apply(subst neg_equal_iff_equal[THEN sym])
  3639   using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"] using assms using continuous_on_neg
  3640   by (auto simp add:euclidean_simps)
  3641 
  3642 lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3643   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
  3644     \<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
  3645 by(rule ivt_decreasing_component_on_1)
  3646   (auto simp: continuous_at_imp_continuous_on)
  3647 
  3648 subsection {* A bound within a convex hull, and so an interval. *}
  3649 
  3650 lemma convex_on_convex_hull_bound:
  3651   assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
  3652   shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
  3653   fix x assume "x\<in>convex hull s"
  3654   then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
  3655     unfolding convex_hull_indexed mem_Collect_eq by auto
  3656   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
  3657     unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
  3658     using assms(2) obt(1) by auto
  3659   thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
  3660     unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
  3661 
  3662 lemma unit_interval_convex_hull:
  3663   "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
  3664   (is "?int = convex hull ?points")
  3665 proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
  3666   { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n" 
  3667   hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
  3668     case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
  3669     thus "x\<in>convex hull ?points" using 01 by auto
  3670   next
  3671     case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
  3672       case True hence "x = 0" apply(subst euclidean_eq) by auto
  3673       thus "x\<in>convex hull ?points" using 01 by auto
  3674     next
  3675       case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
  3676       have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
  3677       then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
  3678       have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
  3679         unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
  3680         defer apply(rule_tac x=j in bexI) using i' by auto
  3681       have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
  3682         using i'(2-) `x$$i \<noteq> 0` by auto
  3683       show ?thesis proof(cases "x$$i=1")
  3684         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
  3685         proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
  3686           hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
  3687           hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto 
  3688           hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
  3689           thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
  3690         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
  3691           by auto
  3692       next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
  3693         case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
  3694           apply(subst euclidean_eq) by(auto simp add: field_simps euclidean_simps)
  3695         { fix j assume j:"j<DIM('a)"
  3696           have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
  3697             apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
  3698             using Suc(2)[unfolded mem_interval, rule_format, of j] using j
  3699             by(auto simp add:field_simps euclidean_simps)
  3700           hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
  3701         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}"
  3702           using i01 using i'(3) by auto
  3703         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
  3704         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 
  3705           by( auto simp add:euclidean_simps)
  3706         have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
  3707           using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
  3708         ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
  3709           apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
  3710           unfolding mem_interval using i01 Suc(3) by auto
  3711       qed qed qed } note * = this
  3712   have **:"DIM('a) = card {..<DIM('a)}" by auto
  3713   show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
  3714     apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **) 
  3715     apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
  3716     unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
  3717     by(auto simp add: mem_def[of _ convex]) qed
  3718 
  3719 subsection {* And this is a finite set of vertices. *}
  3720 
  3721 lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
  3722   apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
  3723   apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
  3724   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
  3725   fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
  3726   show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
  3727     apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
  3728     using as apply(subst euclidean_eq) by auto qed auto
  3729 
  3730 subsection {* Hence any cube (could do any nonempty interval). *}
  3731 
  3732 lemma cube_convex_hull:
  3733   assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
  3734   "finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
  3735   let ?d = "(\<chi>\<chi> i. d)::'a"
  3736   have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_eqI, rule)
  3737     unfolding image_iff defer apply(erule bexE) proof-
  3738     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
  3739     { fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
  3740         using as[unfolded mem_interval, THEN spec[where x=i]] i
  3741         by(auto simp add:euclidean_simps)
  3742       hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
  3743         apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
  3744         using assms by(auto simp add: field_simps)
  3745       hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
  3746             "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
  3747     hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
  3748       by(auto simp add: euclidean_simps field_simps)
  3749     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) 
  3750       using assms by auto
  3751   next
  3752     fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z" 
  3753     have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
  3754       using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
  3755       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
  3756       using assms by auto
  3757     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
  3758       apply(erule_tac x=i in allE) using assms by(auto simp add: euclidean_simps) qed
  3759   obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
  3760   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
  3761 
  3762 subsection {* Bounded convex function on open set is continuous. *}
  3763 
  3764 lemma convex_on_bounded_continuous:
  3765   fixes s :: "('a::real_normed_vector) set"
  3766   assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
  3767   shows "continuous_on s f"
  3768   apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
  3769   fix x e assume "x\<in>s" "(0::real) < e"
  3770   def B \<equiv> "abs b + 1"
  3771   have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
  3772     unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
  3773   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
  3774   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
  3775     apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
  3776     fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)" 
  3777     show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
  3778       case False def t \<equiv> "k / norm (y - x)"
  3779       have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
  3780       have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
  3781         apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute) 
  3782       { def w \<equiv> "x + t *\<^sub>R (y - x)"
  3783         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  3784           unfolding t_def using `k>0` by auto
  3785         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)
  3786         also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps)
  3787         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)
  3788         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) 
  3789         hence "(f w - f x) / t < e"
  3790           using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps) 
  3791         hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
  3792           using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
  3793           using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
  3794       moreover 
  3795       { def w \<equiv> "x - t *\<^sub>R (y - x)"
  3796         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  3797           unfolding t_def using `k>0` by auto
  3798         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)
  3799         also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
  3800         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)
  3801         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) 
  3802         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) 
  3803         have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y" 
  3804           using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
  3805           using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
  3806         also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding divide_inverse by (auto simp add:field_simps)
  3807         also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
  3808         finally have "f x - f y < e" by auto }
  3809       ultimately show ?thesis by auto 
  3810     qed(insert `0<e`, auto) 
  3811   qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
  3812 
  3813 subsection {* Upper bound on a ball implies upper and lower bounds. *}
  3814 
  3815 lemma convex_bounds_lemma:
  3816   fixes x :: "'a::real_normed_vector"
  3817   assumes "convex_on (cball x e) f"  "\<forall>y \<in> cball x e. f y \<le> b"
  3818   shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
  3819   apply(rule) proof(cases "0 \<le> e") case True
  3820   fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *\<^sub>R x - y"
  3821   have *:"x - (2 *\<^sub>R x - y) = y - x" by (simp add: scaleR_2)
  3822   have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
  3823   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
  3824   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"]
  3825     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
  3826 next case False fix y assume "y\<in>cball x e" 
  3827   hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
  3828   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
  3829 
  3830 subsection {* Hence a convex function on an open set is continuous. *}
  3831 
  3832 lemma convex_on_continuous:
  3833   assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f" 
  3834   (* FIXME: generalize to euclidean_space *)
  3835   shows "continuous_on s f"
  3836   unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
  3837   note dimge1 = DIM_positive[where 'a='a]
  3838   fix x assume "x\<in>s"
  3839   then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
  3840   def d \<equiv> "e / real DIM('a)"
  3841   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
  3842   let ?d = "(\<chi>\<chi> i. d)::'a"
  3843   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
  3844   have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:euclidean_simps)
  3845   hence "c\<noteq>{}" using c by auto
  3846   def k \<equiv> "Max (f ` c)"
  3847   have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
  3848     apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
  3849     fix z assume z:"z\<in>{x - ?d..x + ?d}"
  3850     have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
  3851       unfolding real_eq_of_nat by auto
  3852     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
  3853       using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:euclidean_simps) qed
  3854   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
  3855     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
  3856   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
  3857   hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
  3858   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
  3859   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
  3860     fix y assume y:"y\<in>cball x d"
  3861     { fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i"  "y $$ i \<le> x $$ i + d" 
  3862         using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add:euclidean_simps)  }
  3863     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
  3864       by(auto simp add:euclidean_simps) qed
  3865   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
  3866     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
  3867     apply force
  3868     done
  3869   thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
  3870     using `d>0` by auto 
  3871 qed
  3872 
  3873 subsection {* Line segments, Starlike Sets, etc.*}
  3874 
  3875 (* Use the same overloading tricks as for intervals, so that 
  3876    segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
  3877 
  3878 definition
  3879   midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a" where
  3880   "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
  3881 
  3882 definition
  3883   open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
  3884   "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real.  0 < u \<and> u < 1}"
  3885 
  3886 definition
  3887   closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
  3888   "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
  3889 
  3890 definition "between = (\<lambda> (a,b). closed_segment a b)"
  3891 
  3892 lemmas segment = open_segment_def closed_segment_def
  3893 
  3894 definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
  3895 
  3896 lemma midpoint_refl: "midpoint x x = x"
  3897   unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto
  3898 
  3899 lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
  3900 
  3901 lemma midpoint_eq_iff: "midpoint a b = c \<longleftrightarrow> a + b = c + c"
  3902 proof -
  3903   have "midpoint a b = c \<longleftrightarrow> scaleR 2 (midpoint a b) = scaleR 2 c"
  3904     by simp
  3905   thus ?thesis
  3906     unfolding midpoint_def scaleR_2 [symmetric] by simp
  3907 qed
  3908 
  3909 lemma dist_midpoint:
  3910   fixes a b :: "'a::real_normed_vector" shows
  3911   "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
  3912   "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
  3913   "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
  3914   "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
  3915 proof-
  3916   have *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
  3917   have **:"\<And>x y::'a. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2" by auto
  3918   note scaleR_right_distrib [simp]
  3919   show ?t1 unfolding midpoint_def dist_norm apply (rule **)
  3920     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3921   show ?t2 unfolding midpoint_def dist_norm apply (rule *)
  3922     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3923   show ?t3 unfolding midpoint_def dist_norm apply (rule *)
  3924     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3925   show ?t4 unfolding midpoint_def dist_norm apply (rule **)
  3926     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3927 qed
  3928 
  3929 lemma midpoint_eq_endpoint:
  3930   "midpoint a b = a \<longleftrightarrow> a = b"
  3931   "midpoint a b = b \<longleftrightarrow> a = b"
  3932   unfolding midpoint_eq_iff by auto
  3933 
  3934 lemma convex_contains_segment:
  3935   "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. closed_segment a b \<subseteq> s)"
  3936   unfolding convex_alt closed_segment_def by auto
  3937 
  3938 lemma convex_imp_starlike:
  3939   "convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
  3940   unfolding convex_contains_segment starlike_def by auto
  3941 
  3942 lemma segment_convex_hull:
  3943  "closed_segment a b = convex hull {a,b}" proof-
  3944   have *:"\<And>x. {x} \<noteq> {}" by auto
  3945   have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
  3946   show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_eqI)
  3947     unfolding mem_Collect_eq apply(rule,erule exE) 
  3948     apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
  3949     apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
  3950 
  3951 lemma convex_segment: "convex (closed_segment a b)"
  3952   unfolding segment_convex_hull by(rule convex_convex_hull)
  3953 
  3954 lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
  3955   unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto
  3956 
  3957 lemma segment_furthest_le:
  3958   fixes a b x y :: "'a::euclidean_space"
  3959   assumes "x \<in> closed_segment a b" shows "norm(y - x) \<le> norm(y - a) \<or>  norm(y - x) \<le> norm(y - b)" proof-
  3960   obtain z where "z\<in>{a, b}" "norm (x - y) \<le> norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
  3961     using assms[unfolded segment_convex_hull] by auto
  3962   thus ?thesis by(auto simp add:norm_minus_commute) qed
  3963 
  3964 lemma segment_bound:
  3965   fixes x a b :: "'a::euclidean_space"
  3966   assumes "x \<in> closed_segment a b"
  3967   shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
  3968   using segment_furthest_le[OF assms, of a]
  3969   using segment_furthest_le[OF assms, of b]
  3970   by (auto simp add:norm_minus_commute) 
  3971 
  3972 lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
  3973 
  3974 lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
  3975   unfolding between_def mem_def by auto
  3976 
  3977 lemma between:"between (a,b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
  3978 proof(cases "a = b")
  3979   case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
  3980     by(auto simp add:segment_refl dist_commute) next
  3981   case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto 
  3982   have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
  3983   show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
  3984     apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
  3985       fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" 
  3986       hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
  3987         unfolding as(1) by(auto simp add:algebra_simps)
  3988       show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
  3989         unfolding norm_minus_commute[of x a] * using as(2,3)
  3990         by(auto simp add: field_simps)
  3991     next assume as:"dist a b = dist a x + dist x b"
  3992       have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2]
  3993         unfolding as[unfolded dist_norm] norm_ge_zero by auto 
  3994       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)
  3995         unfolding dist_norm apply(subst euclidean_eq) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
  3996       proof(rule,rule) fix i assume i:"i<DIM('a)"
  3997           have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
  3998             ((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
  3999             using Fal by(auto simp add: field_simps euclidean_simps)
  4000           also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
  4001             unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
  4002             apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps euclidean_simps)
  4003           finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i" 
  4004             by auto
  4005         qed(insert Fal2, auto) qed qed
  4006 
  4007 lemma between_midpoint: fixes a::"'a::euclidean_space" shows
  4008   "between (a,b) (midpoint a b)" (is ?t1) 
  4009   "between (b,a) (midpoint a b)" (is ?t2)
  4010 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
  4011   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
  4012     unfolding euclidean_eq[where 'a='a]
  4013     by(auto simp add:field_simps euclidean_simps) qed
  4014 
  4015 lemma between_mem_convex_hull:
  4016   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
  4017   unfolding between_mem_segment segment_convex_hull ..
  4018 
  4019 subsection {* Shrinking towards the interior of a convex set. *}
  4020 
  4021 lemma mem_interior_convex_shrink:
  4022   fixes s :: "('a::euclidean_space) set"
  4023   assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
  4024   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  4025 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  4026   show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
  4027     apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
  4028     fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
  4029     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)
  4030     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)"
  4031       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
  4032       by(auto simp add: euclidean_simps euclidean_eq[where 'a='a] field_simps) 
  4033     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)
  4034     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
  4035       by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
  4036     finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
  4037       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
  4038   qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
  4039 
  4040 lemma mem_interior_closure_convex_shrink:
  4041   fixes s :: "('a::euclidean_space) set"
  4042   assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
  4043   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  4044 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  4045   have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
  4046     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  4047     case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
  4048     show ?thesis proof(cases "e=1")
  4049       case True obtain y where "y\<in>s" "y \<noteq> x" "dist y x < 1"
  4050         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  4051       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  4052       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  4053         using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  4054       then obtain y where "y\<in>s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
  4055         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  4056       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  4057   then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
  4058   def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
  4059   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)
  4060   have "z\<in>interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
  4061     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
  4062     by(auto simp add:field_simps norm_minus_commute)
  4063   thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink) 
  4064     using assms(1,4-5) `y\<in>s` by auto qed
  4065 
  4066 subsection {* Some obvious but surprisingly hard simplex lemmas. *}
  4067 
  4068 lemma simplex:
  4069   assumes "finite s" "0 \<notin> s"
  4070   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)}"
  4071   unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_eqI, rule) unfolding mem_Collect_eq
  4072   apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
  4073   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)
  4074   unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
  4075 
  4076 lemma substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4077   shows "convex hull (insert 0 { basis i | i. i : d}) =
  4078         {x::'a::euclidean_space . (!i<DIM('a). 0 <= x$$i) & setsum (%i. x$$i) d <= 1 &
  4079   (!i<DIM('a). i ~: d --> x$$i = 0)}" 
  4080   (is "convex hull (insert 0 ?p) = ?s")
  4081 (* Proof is a modified copy of the proof of similar lemma std_simplex in Convex_Euclidean_Space.thy *)
  4082 proof- let ?D = d (*"{..<DIM('a::euclidean_space)}"*)
  4083   have "0 ~: ?p" using assms by (auto simp: image_def)
  4084   have "{(basis i)::'n::euclidean_space |i. i \<in> ?D} = basis ` ?D" by auto
  4085   note sumbas = this setsum_reindex[OF basis_inj_on[of d], unfolded o_def, OF assms]
  4086   show ?thesis unfolding simplex[OF finite_substdbasis `0 ~: ?p`] 
  4087     apply(rule set_eqI) unfolding mem_Collect_eq apply rule
  4088     apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
  4089     fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x"
  4090       "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
  4091     have *:"\<forall>i<DIM('a). i:d --> u (basis i) = x$$i" and "(!i<DIM('a). i ~: d --> x $$ i = 0)" using as(3) 
  4092       unfolding sumbas unfolding substdbasis_expansion_unique[OF assms] by auto
  4093     hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $$ x) ?D" unfolding sumbas 
  4094       apply-apply(rule setsum_cong2) using assms by auto
  4095     have " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1" 
  4096       apply - proof(rule,rule,rule)
  4097       fix i assume i:"i<DIM('a)" have "i : d ==> 0 \<le> x$$i" unfolding *[rule_format,OF i,THEN sym] 
  4098          apply(rule_tac as(1)[rule_format]) by auto
  4099       moreover have "i ~: d ==> 0 \<le> x$$i" 
  4100         using `(!i<DIM('a). i ~: d --> x $$ i = 0)`[rule_format, OF i] by auto
  4101       ultimately show "0 \<le> x$$i" by auto
  4102     qed(insert as(2)[unfolded **], auto)
  4103     from this show " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1 & (!i<DIM('a). i ~: d --> x $$ i = 0)" 
  4104       using `(!i<DIM('a). i ~: d --> x $$ i = 0)` by auto
  4105   next fix x::"'a::euclidean_space" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
  4106       "(!i<DIM('a). i ~: d --> x $$ i = 0)"
  4107     show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and>
  4108       setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
  4109       apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE)
  4110       using as(1) apply(erule_tac x=i in allE) unfolding sumbas apply safe unfolding not_less basis_zero
  4111       unfolding substdbasis_expansion_unique[OF assms] euclidean_component_def[THEN sym]
  4112       using as(2,3) by(auto simp add:dot_basis not_less  basis_zero) 
  4113   qed qed
  4114 
  4115 lemma std_simplex:
  4116   "convex hull (insert 0 { basis i | i. i<DIM('a)}) =
  4117         {x::'a::euclidean_space . (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} \<le> 1 }"
  4118   using substd_simplex[of "{..<DIM('a)}"] by auto
  4119 
  4120 lemma interior_std_simplex:
  4121   "interior (convex hull (insert 0 { basis i| i. i<DIM('a)})) =
  4122   {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 < x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} < 1 }"
  4123   apply(rule set_eqI) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
  4124   unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
  4125   fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x<DIM('a). 0 \<le> xa $$ x) \<and> setsum (op $$ xa) {..<DIM('a)} \<le> 1"
  4126   show "(\<forall>xa<DIM('a). 0 < x $$ xa) \<and> setsum (op $$ x) {..<DIM('a)} < 1" apply(safe) proof-
  4127     fix i assume i:"i<DIM('a)" thus "0 < x $$ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
  4128       unfolding dist_norm  by(auto simp add: inner_simps euclidean_component_def dot_basis elim!:allE[where x=i])
  4129   next have **:"dist x (x + (e / 2) *\<^sub>R basis 0) < e" using  `e>0`
  4130       unfolding dist_norm by(auto intro!: mult_strict_left_mono)
  4131     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)"
  4132       unfolding euclidean_component_def by(auto simp add:inner_simps dot_basis)
  4133     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)}"
  4134       apply(rule_tac setsum_cong) by auto
  4135     have "setsum (op $$ x) {..<DIM('a)} < setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)}" unfolding * setsum_addf
  4136       using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by auto
  4137     also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
  4138     finally show "setsum (op $$ x) {..<DIM('a)} < 1" by auto qed
  4139 next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 < x $$ i" "setsum (op $$ x) {..<DIM('a)} < 1"
  4140   guess a using UNIV_witness[where 'a='b] ..
  4141   let ?d = "(1 - setsum (op $$ x) {..<DIM('a)}) / real (DIM('a))"
  4142   have "Min ((op $$ x) ` {..<DIM('a)}) > 0" apply(rule Min_grI) using as(1) by auto
  4143   moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by(auto simp add: Suc_le_eq)
  4144   ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
  4145     apply(rule_tac x="min (Min ((op $$ x) ` {..<DIM('a)})) ?D" in exI) apply rule defer apply(rule,rule) proof-
  4146     fix y assume y:"dist x y < min (Min (op $$ x ` {..<DIM('a)})) ?d"
  4147     have "setsum (op $$ y) {..<DIM('a)} \<le> setsum (\<lambda>i. x$$i + ?d) {..<DIM('a)}" proof(rule setsum_mono)
  4148       fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
  4149         using component_le_norm[of "y - x" i]
  4150         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
  4151       thus "y $$ i \<le> x $$ i + ?d" by auto qed
  4152     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat by(auto simp add: Suc_le_eq)
  4153     finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1" 
  4154     proof safe fix i assume i:"i<DIM('a)"
  4155       have "norm (x - y) < x$$i" apply(rule less_le_trans) 
  4156         apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
  4157       thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
  4158     qed qed auto qed
  4159 
  4160 lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
  4161   "a \<in> interior(convex hull (insert 0 {basis i | i . i<DIM('a)}))" proof-
  4162   let ?D = "{..<DIM('a)}" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) {(basis i) | i. i<DIM('a)}"
  4163   have *:"{basis i :: 'a | i. i<DIM('a)} = basis ` ?D" by auto
  4164   { fix i assume i:"i<DIM('a)" have "?a $$ i = inverse (2 * real DIM('a))"
  4165       unfolding euclidean_component.setsum * and setsum_reindex[OF basis_inj] and o_def
  4166       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"]) apply(rule setsum_cong2)
  4167       defer apply(subst setsum_delta') unfolding euclidean_component_def using i by(auto simp add:dot_basis) }
  4168   note ** = this
  4169   show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe
  4170     fix i assume i:"i<DIM('a)" show "0 < ?a $$ i" unfolding **[OF i] by(auto simp add: Suc_le_eq)
  4171   next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
  4172     also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat divide_inverse[THEN sym] by (auto simp add:field_simps)
  4173     finally show "setsum (op $$ ?a) ?D < 1" by auto qed qed
  4174 
  4175 lemma rel_interior_substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4176   shows "rel_interior (convex hull (insert 0 { basis i| i. i : d})) =
  4177   {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x$$i) & setsum (%i. x$$i) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)}"
  4178   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
  4179 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex in Convex_Euclidean_Space.thy *)
  4180 proof-
  4181 have "finite d" apply(rule finite_subset) using assms by auto
  4182 { assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq[of _ 0] by auto }
  4183 moreover
  4184 { assume "d~={}"
  4185 have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}" 
  4186    using affine_hull_convex_hull affine_hull_substd_basis assms by auto 
  4187 have aux: "!x::'n::euclidean_space. !i. ((! i:d. 0 <= x$$i) & (!i. i ~: d --> x$$i = 0))--> 0 <= x$$i" by auto
  4188 { fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"
  4189   from this obtain e where e0: "e>0" and 
  4190        "ball x e Int {xa. (!i<DIM('a). i ~: d --> xa$$i = 0)} <= convex hull (insert 0 ?p)" 
  4191        using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto   
  4192   hence as: "ALL xa. (dist x xa < e & (!i<DIM('a). i ~: d --> xa$$i = 0)) -->
  4193     (!i : d. 0 <= xa $$ i) & setsum (op $$ xa) d <= 1"
  4194     unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
  4195   have x0: "(!i<DIM('a). i ~: d --> x$$i = 0)" 
  4196     using x_def rel_interior_subset  substd_simplex[OF assms] by auto
  4197   have "(!i : d. 0 < x $$ i) & setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" apply(rule,rule) 
  4198   proof-
  4199     fix i::nat assume "i:d" 
  4200     hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R basis i) $$ ia" apply-apply(rule as[rule_format,THEN conjunct1])
  4201       unfolding dist_norm using assms `e>0` x0 by auto
  4202     thus "0 < x $$ i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` assms by auto
  4203   next obtain a where a:"a:d" using `d ~= {}` by auto
  4204     have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e"
  4205       using  `e>0` and Euclidean_Space.norm_basis[of a]
  4206       unfolding dist_norm by auto
  4207     have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $$ i = x$$i + (if i = a then e/2 else 0)"
  4208       unfolding euclidean_simps using a assms by auto
  4209     hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d =
  4210       setsum (\<lambda>i. x$$i + (if a = i then e/2 else 0)) d" by(rule_tac setsum_cong, auto)
  4211     have h1: "(ALL i<DIM('a). i ~: d --> (x + (e / 2) *\<^sub>R basis a) $$ i = 0)"
  4212       using as[THEN spec[where x="x + (e / 2) *\<^sub>R basis a"]] `a:d` using x0
  4213       by(auto simp add: norm_basis elim:allE[where x=a]) 
  4214     have "setsum (op $$ x) d < setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d" unfolding * setsum_addf
  4215       using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')
  4216     also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R basis a"] by auto
  4217     finally show "setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" using x0 by auto 
  4218   qed
  4219 }
  4220 moreover
  4221 {
  4222   fix x::"'a::euclidean_space" assume as: "x : ?s"
  4223   have "!i. ((0<x$$i) | (0=x$$i) --> 0<=x$$i)" by auto
  4224   moreover have "!i. (i:d) | (i ~: d)" by auto
  4225   ultimately 
  4226   have "!i. ( (ALL i:d. 0 < x$$i) & (ALL i. i ~: d --> x$$i = 0) ) --> 0 <= x$$i" by metis
  4227   hence h2: "x : convex hull (insert 0 ?p)" using as assms 
  4228     unfolding substd_simplex[OF assms] by fastsimp 
  4229   obtain a where a:"a:d" using `d ~= {}` by auto
  4230   let ?d = "(1 - setsum (op $$ x) d) / real (card d)"
  4231   have "card d >= 1" using `d ~={}` card_ge1[of d] using `finite d` by auto
  4232   have "Min ((op $$ x) ` d) > 0" apply(rule Min_grI) using as `card d >= 1` `finite d` by auto
  4233   moreover have "?d > 0" apply(rule divide_pos_pos) using as using `card d >= 1` by(auto simp add: Suc_le_eq)
  4234   ultimately have h3: "min (Min ((op $$ x) ` d)) ?d > 0" by auto
  4235 
  4236   have "x : rel_interior (convex hull (insert 0 ?p))"
  4237     unfolding rel_interior_ball mem_Collect_eq h0 apply(rule,rule h2)
  4238     unfolding substd_simplex[OF assms]
  4239     apply(rule_tac x="min (Min ((op $$ x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball
  4240   proof- fix y::'a assume y:"dist x y < min (Min (op $$ x ` d)) ?d" and y2:"(!i<DIM('a). i ~: d --> y$$i = 0)"
  4241     have "setsum (op $$ y) d \<le> setsum (\<lambda>i. x$$i + ?d) d" proof(rule setsum_mono)
  4242       fix i assume i:"i\<in>d"
  4243       have "abs (y$$i - x$$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
  4244         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
  4245         by(auto simp add: norm_minus_commute)
  4246       thus "y $$ i \<le> x $$ i + ?d" by auto qed
  4247     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat
  4248       using `card d >= 1` by(auto simp add: Suc_le_eq)
  4249     finally show "setsum (op $$ y) d \<le> 1" .
  4250      
  4251     fix i assume "i<DIM('a)" thus "0 \<le> y$$i" 
  4252     proof(cases "i\<in>d") case True
  4253       have "norm (x - y) < x$$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
  4254         using Min_gr_iff[of "op $$ x ` d" "norm (x - y)"] `card d >= 1` `i:d`
  4255         apply auto by (metis Suc_n_not_le_n True card_eq_0_iff finite_imageI)
  4256       thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format] by auto
  4257     qed(insert y2, auto)
  4258   qed
  4259 } ultimately have
  4260     "!!x :: 'a::euclidean_space. (x : rel_interior (convex hull insert 0 {basis i |i. i : d})) =
  4261     (x : {x. (ALL i:d. 0 < x $$ i) &
  4262     setsum (op $$ x) d < 1 & (ALL i<DIM('a). i ~: d --> x $$ i = 0)})" by blast
  4263 from this have ?thesis by (rule set_eqI)
  4264 } ultimately show ?thesis by blast
  4265 qed
  4266 
  4267 lemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4268   obtains a::"'a::euclidean_space" where
  4269   "a : rel_interior(convex hull (insert 0 {basis i | i . i : d}))" proof-
  4270 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex_nonempty in Convex_Euclidean_Space.thy *)
  4271   let ?D = d let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
  4272   have *:"{basis i :: 'a | i. i \<in> ?D} = basis ` ?D" by auto
  4273   have "finite d" apply(rule finite_subset) using assms(2) by auto
  4274   hence d1: "real(card d) >= 1" using `d ~={}` card_ge1[of d] by auto
  4275   { fix i assume "i:d" have "?a $$ i = inverse (2 * real (card d))"
  4276       unfolding * setsum_reindex[OF basis_inj_on, OF assms(2)] o_def
  4277       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"]) 
  4278       unfolding euclidean_component.setsum
  4279       apply(rule setsum_cong2)
  4280       using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)
  4281       by (auto simp add: Euclidean_Space.basis_component[of i])}
  4282   note ** = this
  4283   show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
  4284   proof safe fix i assume "i:d" 
  4285     have "0 < inverse (2 * real (card d))" using d1 by(auto simp add: Suc_le_eq)
  4286     also have "...=?a $$ i" using **[of i] `i:d` by auto
  4287     finally show "0 < ?a $$ i" by auto
  4288   next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D" 
  4289       by(rule setsum_cong2, rule **) 
  4290     also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat real_divide_def[THEN sym]
  4291       by (auto simp add:field_simps)
  4292     finally show "setsum (op $$ ?a) ?D < 1" by auto
  4293   next fix i assume "i<DIM('a)" and "i~:d"
  4294     have "?a : (span {basis i | i. i : d})" 
  4295       apply (rule span_setsum[of "{basis i |i. i : d}" "(%b. b /\<^sub>R (2 * real (card d)))" "{basis i |i. i : d}"]) 
  4296       using finite_substdbasis[of d] apply blast 
  4297     proof-
  4298       { fix x assume "(x :: 'a::euclidean_space): {basis i |i. i : d}"
  4299         hence "x : span {basis i |i. i : d}" 
  4300           using span_superset[of _ "{basis i |i. i : d}"] by auto
  4301         hence "(x /\<^sub>R (2 * real (card d))) : (span {basis i |i. i : d})"
  4302           using span_mul[of x "{basis i |i. i : d}" "(inverse (real (card d)) / 2)"] by auto
  4303       } 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
  4304     qed
  4305     thus "?a $$ i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i<DIM('a)` by auto
  4306   qed
  4307 qed
  4308 
  4309 subsection{* Relative Interior of Convex Set *}
  4310 
  4311 lemma rel_interior_convex_nonempty_aux: 
  4312 fixes S :: "('n::euclidean_space) set" 
  4313 assumes "convex S" and "0 : S"
  4314 shows "rel_interior S ~= {}"
  4315 proof-
  4316 { assume "S = {0}" hence ?thesis using rel_interior_sing by auto }
  4317 moreover { 
  4318 assume "S ~= {0}"
  4319 obtain B where B_def: "independent B & B<=S & (S <= span B) & card B = dim S" using basis_exists[of S] by auto
  4320 hence "B~={}" using B_def assms `S ~= {0}` span_empty by auto
  4321 have "insert 0 B <= span B" using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
  4322 hence "span (insert 0 B) <= span B" 
  4323     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
  4324 hence "convex hull insert 0 B <= span B" 
  4325     using convex_hull_subset_span[of "insert 0 B"] by auto
  4326 hence "span (convex hull insert 0 B) <= span B"
  4327     using span_span[of B] span_mono[of "convex hull insert 0 B" "span B"] by blast
  4328 hence *: "span (convex hull insert 0 B) = span B" 
  4329     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  4330 hence "span (convex hull insert 0 B) = span S"
  4331     using B_def span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
  4332 moreover have "0 : affine hull (convex hull insert 0 B)"
  4333     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  4334 ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
  4335     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"] 
  4336     assms  hull_subset[of S] by auto
  4337 obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = {basis i |i. i : (d :: nat set)} & 
  4338        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)}"
  4339     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by auto
  4340 hence "bounded_linear f" using linear_conv_bounded_linear by auto
  4341 have "d ~={}" using fd B_def `B ~={}` by auto
  4342 have "(insert 0 {basis i |i. i : d}) = f ` (insert 0 B)" using fd linear_0 by auto
  4343 hence "(convex hull (insert 0 {basis i |i. i : d})) = f ` (convex hull (insert 0 B))"
  4344    using convex_hull_linear_image[of f "(insert 0 {basis i |i. i : d})"] 
  4345    convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by auto
  4346 moreover have "rel_interior (f ` (convex hull insert 0 B)) = 
  4347    f ` rel_interior (convex hull insert 0 B)"
  4348    apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
  4349    using `bounded_linear f` fd * by auto
  4350 ultimately have "rel_interior (convex hull insert 0 B) ~= {}"
  4351    using rel_interior_substd_simplex_nonempty[OF `d~={}` d] apply auto by blast 
  4352 moreover have "convex hull (insert 0 B) <= S"
  4353    using B_def assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
  4354 ultimately have ?thesis using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
  4355 } ultimately show ?thesis by auto
  4356 qed
  4357 
  4358 lemma rel_interior_convex_nonempty:
  4359 fixes S :: "('n::euclidean_space) set"
  4360 assumes "convex S"
  4361 shows "rel_interior S = {} <-> S = {}"
  4362 proof-
  4363 { assume "S ~= {}" from this obtain a where "a : S" by auto
  4364   hence "0 : op + (-a) ` S" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
  4365   hence "rel_interior (op + (-a) ` S) ~= {}"  
  4366     using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"] 
  4367           convex_translation[of S "-a"] assms by auto 
  4368   hence "rel_interior S ~= {}" using rel_interior_translation by auto
  4369 } from this show ?thesis using rel_interior_empty by auto
  4370 qed
  4371 
  4372 lemma convex_rel_interior:
  4373 fixes S :: "(_::euclidean_space) set"
  4374 assumes "convex S"
  4375 shows "convex (rel_interior S)"
  4376 proof-
  4377 { fix "x" "y" "u"
  4378   assume assm: "x:rel_interior S" "y:rel_interior S" "0<=u" "(u :: real) <= 1"
  4379   hence "x:S" using rel_interior_subset by auto
  4380   have "x - u *\<^sub>R (x-y) : rel_interior S"
  4381   proof(cases "0=u")
  4382      case False hence "0<u" using assm by auto 
  4383         thus ?thesis
  4384         using assm rel_interior_convex_shrink[of S y x u] assms `x:S` by auto
  4385      next
  4386      case True thus ?thesis using assm by auto
  4387   qed
  4388   hence "(1-u) *\<^sub>R x + u *\<^sub>R y : rel_interior S" by (simp add: algebra_simps)
  4389 } from this show ?thesis unfolding convex_alt by auto
  4390 qed
  4391 
  4392 lemma convex_closure_rel_interior: 
  4393 fixes S :: "('n::euclidean_space) set" 
  4394 assumes "convex S"
  4395 shows "closure(rel_interior S) = closure S"
  4396 proof-
  4397 have h1: "closure(rel_interior S) <= closure S" 
  4398    using subset_closure[of "rel_interior S" S] rel_interior_subset[of S] by auto
  4399 { assume "S ~= {}" from this obtain a where a_def: "a : rel_interior S" 
  4400     using rel_interior_convex_nonempty assms by auto
  4401   { fix x assume x_def: "x : closure S"
  4402     { assume "x=a" hence "x : closure(rel_interior S)" using a_def unfolding closure_def by auto }
  4403     moreover
  4404     { assume "x ~= a"
  4405        { fix e :: real assume e_def: "e>0" 
  4406          def e1 == "min 1 (e/norm (x - a))" hence e1_def: "e1>0 & e1<=1 & e1*norm(x-a)<=e"
  4407             using `x ~= a` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(x-a)"] by simp 
  4408          hence *: "x - e1 *\<^sub>R (x - a) : rel_interior S"
  4409             using rel_interior_closure_convex_shrink[of S a x e1] assms x_def a_def e1_def by auto
  4410          have "EX y. y:rel_interior S & y ~= x & (dist y x) <= e"
  4411             apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
  4412             using * e1_def dist_norm[of "x - e1 *\<^sub>R (x - a)" x] `x ~= a` by simp
  4413       } hence "x islimpt rel_interior S" unfolding islimpt_approachable_le by auto 
  4414       hence "x : closure(rel_interior S)" unfolding closure_def by auto 
  4415     } ultimately have "x : closure(rel_interior S)" by auto
  4416   } hence ?thesis using h1 by auto
  4417 }
  4418 moreover
  4419 { assume "S = {}" hence "rel_interior S = {}" using rel_interior_empty by auto
  4420   hence "closure(rel_interior S) = {}" using closure_empty by auto 
  4421   hence ?thesis using `S={}` by auto 
  4422 } ultimately show ?thesis by blast
  4423 qed
  4424 
  4425 lemma rel_interior_same_affine_hull:
  4426   fixes S :: "('n::euclidean_space) set"
  4427   assumes "convex S"
  4428   shows "affine hull (rel_interior S) = affine hull S"
  4429 by (metis assms closure_same_affine_hull convex_closure_rel_interior)
  4430 
  4431 lemma rel_interior_aff_dim: 
  4432   fixes S :: "('n::euclidean_space) set"
  4433   assumes "convex S"
  4434   shows "aff_dim (rel_interior S) = aff_dim S"
  4435 by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
  4436 
  4437 lemma rel_interior_rel_interior:
  4438   fixes S :: "('n::euclidean_space) set"
  4439   assumes "convex S"
  4440   shows "rel_interior (rel_interior S) = rel_interior S"
  4441 proof-
  4442 have "openin (subtopology euclidean (affine hull (rel_interior S))) (rel_interior S)"
  4443   using opein_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
  4444 from this show ?thesis using rel_interior_def by auto
  4445 qed
  4446 
  4447 lemma rel_interior_rel_open:
  4448   fixes S :: "('n::euclidean_space) set"
  4449   assumes "convex S"
  4450   shows "rel_open (rel_interior S)"
  4451 unfolding rel_open_def using rel_interior_rel_interior assms by auto
  4452 
  4453 lemma convex_rel_interior_closure_aux:
  4454   fixes x y z :: "_::euclidean_space"
  4455   assumes "0 < a" "0 < b" "(a+b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
  4456   obtains e where "0 < e" "e <= 1" "z = y - e *\<^sub>R (y-x)"
  4457 proof-
  4458 def e == "a/(a+b)"
  4459 have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)" apply auto using assms by simp
  4460 also have "... = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)" using assms
  4461    scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"] by auto
  4462 also have "... = y - e *\<^sub>R (y-x)" using e_def apply (simp add: algebra_simps)
  4463    using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"] by auto
  4464 finally have "z = y - e *\<^sub>R (y-x)" by auto
  4465 moreover have "0<e" using e_def assms divide_pos_pos[of a "a+b"] by auto
  4466 moreover have "e<=1" using e_def assms by auto
  4467 ultimately show ?thesis using that[of e] by auto
  4468 qed
  4469 
  4470 lemma convex_rel_interior_closure: 
  4471   fixes S :: "('n::euclidean_space) set" 
  4472   assumes "convex S"
  4473   shows "rel_interior (closure S) = rel_interior S"
  4474 proof-
  4475 { assume "S={}" hence ?thesis using assms rel_interior_convex_nonempty by auto }
  4476 moreover
  4477 { assume "S ~= {}"
  4478   have "rel_interior (closure S) >= rel_interior S" 
  4479     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset by auto
  4480   moreover
  4481   { fix z assume z_def: "z : rel_interior (closure S)"
  4482     obtain x where x_def: "x : rel_interior S" 
  4483       using `S ~= {}` assms rel_interior_convex_nonempty by auto  
  4484     { assume "x=z" hence "z : rel_interior S" using x_def by auto }
  4485     moreover
  4486     { assume "x ~= z"
  4487       obtain e where e_def: "e > 0 & cball z e Int affine hull closure S <= closure S" 
  4488         using z_def rel_interior_cball[of "closure S"] by auto
  4489       hence *: "0 < e/norm(z-x)" using e_def `x ~= z` divide_pos_pos[of e "norm(z-x)"] by auto 
  4490       def y == "z + (e/norm(z-x)) *\<^sub>R (z-x)"
  4491       have yball: "y : cball z e"
  4492         using mem_cball y_def dist_norm[of z y] e_def by auto 
  4493       have "x : affine hull closure S" 
  4494         using x_def rel_interior_subset_closure hull_inc[of x "closure S"] by auto
  4495       moreover have "z : affine hull closure S" 
  4496         using z_def rel_interior_subset hull_subset[of "closure S"] by auto
  4497       ultimately have "y : affine hull closure S" 
  4498         using y_def affine_affine_hull[of "closure S"] 
  4499           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
  4500       hence "y : closure S" using e_def yball by auto
  4501       have "(1+(e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
  4502         using y_def by (simp add: algebra_simps) 
  4503       from this obtain e1 where "0 < e1 & e1 <= 1 & z = y - e1 *\<^sub>R (y - x)"
  4504         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y] 
  4505           by (auto simp add: algebra_simps)
  4506       hence "z : rel_interior S" 
  4507         using rel_interior_closure_convex_shrink assms x_def `y : closure S` by auto
  4508     } ultimately have "z : rel_interior S" by auto
  4509   } ultimately have ?thesis by auto
  4510 } ultimately show ?thesis by blast
  4511 qed
  4512 
  4513 lemma convex_interior_closure: 
  4514 fixes S :: "('n::euclidean_space) set" 
  4515 assumes "convex S"
  4516 shows "interior (closure S) = interior S"
  4517 using closure_aff_dim[of S] interior_rel_interior_gen[of S] interior_rel_interior_gen[of "closure S"] 
  4518       convex_rel_interior_closure[of S] assms by auto
  4519 
  4520 lemma closure_eq_rel_interior_eq:
  4521 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4522 assumes "convex S1" "convex S2"
  4523 shows "(closure S1 = closure S2) <-> (rel_interior S1 = rel_interior S2)"
  4524  by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
  4525 
  4526 
  4527 lemma closure_eq_between:
  4528 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4529 assumes "convex S1" "convex S2"
  4530 shows "(closure S1 = closure S2) <-> 
  4531       ((rel_interior S1 <= S2) & (S2 <= closure S1))" (is "?A <-> ?B")
  4532 proof-
  4533 have "?A --> ?B" by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
  4534 moreover have "?B --> (closure S1 <= closure S2)" 
  4535      by (metis assms(1) convex_closure_rel_interior subset_closure)
  4536 moreover have "?B --> (closure S1 >= closure S2)" by (metis closed_closure closure_minimal)
  4537 ultimately show ?thesis by blast
  4538 qed
  4539 
  4540 lemma open_inter_closure_rel_interior:
  4541 fixes S A ::  "('n::euclidean_space) set" 
  4542 assumes "convex S" "open A"
  4543 shows "((A Int closure S) = {}) <-> ((A Int rel_interior S) = {})"
  4544 by (metis assms convex_closure_rel_interior open_inter_closure_eq_empty) 
  4545 
  4546 definition "rel_frontier S = closure S - rel_interior S"
  4547 
  4548 lemma closed_affine_hull: "closed (affine hull ((S :: ('n::euclidean_space) set)))"
  4549 by (metis affine_affine_hull affine_closed)
  4550 
  4551 lemma closed_rel_frontier: "closed(rel_frontier (S :: ('n::euclidean_space) set))"
  4552 proof-
  4553 have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)" 
  4554 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] 
  4555   closure_affine_hull[of S] opein_rel_interior[of S] by auto 
  4556 show ?thesis apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"]) 
  4557   unfolding rel_frontier_def using * closed_affine_hull by auto 
  4558 qed
  4559  
  4560 
  4561 lemma convex_rel_frontier_aff_dim:
  4562 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4563 assumes "convex S1" "convex S2" "S2 ~= {}"
  4564 assumes "S1 <= rel_frontier S2"
  4565 shows "aff_dim S1 < aff_dim S2" 
  4566 proof-
  4567 have "S1 <= closure S2" using assms unfolding rel_frontier_def by auto
  4568 hence *: "affine hull S1 <= affine hull S2" 
  4569    using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by auto
  4570 hence "aff_dim S1 <= aff_dim S2" using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] 
  4571     aff_dim_subset[of "affine hull S1" "affine hull S2"] by auto
  4572 moreover
  4573 { assume eq: "aff_dim S1 = aff_dim S2"
  4574   hence "S1 ~= {}" using aff_dim_empty[of S1] aff_dim_empty[of S2] `S2 ~= {}` by auto
  4575   have **: "affine hull S1 = affine hull S2" 
  4576      apply (rule affine_dim_equal) using * affine_affine_hull apply auto
  4577      using `S1 ~= {}` hull_subset[of S1] apply auto
  4578      using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] by auto
  4579   obtain a where a_def: "a : rel_interior S1"
  4580      using  `S1 ~= {}` rel_interior_convex_nonempty assms by auto
  4581   obtain T where T_def: "open T & a : T Int S1 & T Int affine hull S1 <= S1"
  4582      using mem_rel_interior[of a S1] a_def by auto
  4583   hence "a : T Int closure S2" using a_def assms unfolding rel_frontier_def by auto
  4584   from this obtain b where b_def: "b : T Int rel_interior S2" 
  4585      using open_inter_closure_rel_interior[of S2 T] assms T_def by auto
  4586   hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto
  4587   hence "b : S1" using T_def b_def by auto
  4588   hence False using b_def assms unfolding rel_frontier_def by auto
  4589 } ultimately show ?thesis using zless_le by auto
  4590 qed
  4591 
  4592 
  4593 lemma convex_rel_interior_if:
  4594 fixes S ::  "('n::euclidean_space) set" 
  4595 assumes "convex S"
  4596 assumes "z : rel_interior S"
  4597 shows "(!x:affine hull S. EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S ))"
  4598 proof-
  4599 obtain e1 where e1_def: "e1>0 & cball z e1 Int affine hull S <= S" 
  4600     using mem_rel_interior_cball[of z S] assms by auto
  4601 { fix x assume x_def: "x:affine hull S"
  4602   { assume "x ~= z"
  4603     def m == "1+e1/norm(x-z)" 
  4604     hence "m>1" using e1_def `x ~= z` divide_pos_pos[of e1 "norm (x - z)"] by auto 
  4605     { fix e assume e_def: "e>1 & e<=m"
  4606       have "z : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
  4607       hence *: "(1-e)*\<^sub>R x+ e *\<^sub>R z : affine hull S"
  4608          using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x_def by auto
  4609       have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x-z))" by (simp add: algebra_simps)
  4610       also have "...= (e - 1) * norm(x-z)" using norm_scaleR e_def by auto  
  4611       also have "...<=(m - 1) * norm(x-z)" using e_def mult_right_mono[of _ _ "norm(x-z)"] by auto
  4612       also have "...= (e1 / norm (x - z)) * norm (x - z)" using m_def by auto
  4613       also have "...=e1" using `x ~= z` e1_def by simp
  4614       finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) <= e1" by auto
  4615       have "(1-e)*\<^sub>R x+ e *\<^sub>R z : cball z e1"
  4616          using m_def ** unfolding cball_def dist_norm by (auto simp add: algebra_simps)
  4617       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def * e1_def by auto
  4618     } 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
  4619   }
  4620   moreover
  4621   { assume "x=z" def m == "1+e1" hence "m>1" using e1_def by auto
  4622     { fix e assume e_def: "e>1 & e<=m"
  4623       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S"
  4624         using e1_def x_def `x=z` by (auto simp add: algebra_simps)
  4625       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def by auto
  4626     } 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
  4627   } ultimately have "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" by auto
  4628 } from this show ?thesis by auto 
  4629 qed
  4630 
  4631 lemma convex_rel_interior_if2:
  4632 fixes S ::  "('n::euclidean_space) set" 
  4633 assumes "convex S"
  4634 assumes "z : rel_interior S"
  4635 shows "(!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4636 using convex_rel_interior_if[of S z] assms by auto
  4637 
  4638 lemma convex_rel_interior_only_if:
  4639 fixes S ::  "('n::euclidean_space) set" 
  4640 assumes "convex S" "S ~= {}"
  4641 assumes "(!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4642 shows "z : rel_interior S" 
  4643 proof-
  4644 obtain x where x_def: "x : rel_interior S" using rel_interior_convex_nonempty assms by auto
  4645 hence "x:S" using rel_interior_subset by auto
  4646 from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S" using assms by auto
  4647 def y == "(1 - e) *\<^sub>R x + e *\<^sub>R z" hence "y:S" using e_def by auto
  4648 def e1 == "1/e" hence "0<e1 & e1<1" using e_def by auto
  4649 hence "z=y-(1-e1)*\<^sub>R (y-x)" using e1_def y_def by (auto simp add: algebra_simps)
  4650 from this show ?thesis 
  4651     using rel_interior_convex_shrink[of S x y "1-e1"] `0<e1 & e1<1` `y:S` x_def assms by auto
  4652 qed
  4653 
  4654 lemma convex_rel_interior_iff:
  4655 fixes S ::  "('n::euclidean_space) set" 
  4656 assumes "convex S" "S ~= {}"
  4657 shows "z : rel_interior S <-> (!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4658 using assms hull_subset[of S "affine"] 
  4659       convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z] by auto
  4660 
  4661 lemma convex_rel_interior_iff2:
  4662 fixes S ::  "('n::euclidean_space) set" 
  4663 assumes "convex S" "S ~= {}"
  4664 shows "z : rel_interior S <-> (!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4665 using assms hull_subset[of S] 
  4666       convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z] by auto
  4667 
  4668 
  4669 lemma convex_interior_iff:
  4670 fixes S ::  "('n::euclidean_space) set" 
  4671 assumes "convex S"
  4672 shows "z : interior S <-> (!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
  4673 proof-
  4674 { assume a: "~(aff_dim S = int DIM('n))"
  4675   { assume "z : interior S"
  4676     hence False using a interior_rel_interior_gen[of S] by auto
  4677   }
  4678   moreover
  4679   { assume r: "!x. EX e. e>0 & z+ e *\<^sub>R x : S"
  4680     { fix x obtain e1 where e1_def: "e1>0 & z+ e1 *\<^sub>R (x-z) : S" using r by auto
  4681       obtain e2 where e2_def: "e2>0 & z+ e2 *\<^sub>R (z-x) : S" using r by auto
  4682       def x1 == "z+ e1 *\<^sub>R (x-z)"
  4683          hence x1: "x1 : affine hull S" using e1_def hull_subset[of S] by auto
  4684       def x2 == "z+ e2 *\<^sub>R (z-x)"
  4685          hence x2: "x2 : affine hull S" using e2_def hull_subset[of S] by auto
  4686       have *: "e1/(e1+e2) + e2/(e1+e2) = 1" using divide.add[of e1 e2 "e1+e2"] e1_def e2_def by simp
  4687       hence "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
  4688          using x1_def x2_def apply (auto simp add: algebra_simps)
  4689          using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z] by auto
  4690       hence z: "z : affine hull S" 
  4691          using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]  
  4692          x1 x2 affine_affine_hull[of S] * by auto
  4693       have "x1-x2 = (e1+e2) *\<^sub>R (x-z)"
  4694          using x1_def x2_def by (auto simp add: algebra_simps)
  4695       hence "x=z+(1/(e1+e2)) *\<^sub>R (x1-x2)" using e1_def e2_def by simp
  4696       hence "x : affine hull S" using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"] 
  4697           x1 x2 z affine_affine_hull[of S] by auto
  4698     } hence "affine hull S = UNIV" by auto
  4699     hence "aff_dim S = int DIM('n)" using aff_dim_affine_hull[of S] by (simp add: aff_dim_univ)
  4700     hence False using a by auto
  4701   } ultimately have ?thesis by auto
  4702 }
  4703 moreover
  4704 { assume a: "aff_dim S = int DIM('n)"
  4705   hence "S ~= {}" using aff_dim_empty[of S] by auto
  4706   have *: "affine hull S=UNIV" using a affine_hull_univ by auto
  4707   { assume "z : interior S"
  4708     hence "z : rel_interior S" using a interior_rel_interior_gen[of S] by auto
  4709     hence **: "(!x. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4710       using convex_rel_interior_iff2[of S z] assms `S~={}` * by auto
  4711     fix x obtain e1 where e1_def: "e1>1 & (1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z : S"
  4712       using **[rule_format, of "z-x"] by auto
  4713     def e == "e1 - 1"
  4714     hence "(1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z = z+ e *\<^sub>R x" by (simp add: algebra_simps)
  4715     hence "e>0 & z+ e *\<^sub>R x : S" using e1_def e_def by auto
  4716     hence "EX e. e>0 & z+ e *\<^sub>R x : S" by auto
  4717   }
  4718   moreover
  4719   { assume r: "(!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
  4720     { fix x obtain e1 where e1_def: "e1>0 & z + e1*\<^sub>R (z-x) : S"
  4721          using r[rule_format, of "z-x"] by auto
  4722       def e == "e1 + 1"
  4723       hence "z + e1*\<^sub>R (z-x) = (1-e)*\<^sub>R x+ e *\<^sub>R z" by (simp add: algebra_simps)
  4724       hence "e > 1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e1_def e_def by auto
  4725       hence "EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" by auto
  4726     }
  4727     hence "z : rel_interior S" using convex_rel_interior_iff2[of S z] assms `S~={}` by auto
  4728     hence "z : interior S" using a interior_rel_interior_gen[of S] by auto
  4729   } ultimately have ?thesis by auto
  4730 } ultimately show ?thesis by auto
  4731 qed
  4732 
  4733 subsection{* Relative interior and closure under commom operations *}
  4734 
  4735 lemma rel_interior_inter_aux: "Inter {rel_interior S |S. S : I} <= Inter I"
  4736 proof- 
  4737 { fix y assume "y : Inter {rel_interior S |S. S : I}"
  4738   hence y_def: "!S : I. y : rel_interior S" by auto
  4739   { fix S assume "S : I" hence "y : S" using rel_interior_subset y_def by auto }
  4740   hence "y : Inter I" by auto
  4741 } thus ?thesis by auto
  4742 qed
  4743 
  4744 lemma closure_inter: "closure (Inter I) <= Inter {closure S |S. S : I}"
  4745 proof- 
  4746 { fix y assume "y : Inter I" hence y_def: "!S : I. y : S" by auto
  4747   { fix S assume "S : I" hence "y : closure S" using closure_subset y_def by auto }
  4748   hence "y : Inter {closure S |S. S : I}" by auto
  4749 } hence "Inter I <= Inter {closure S |S. S : I}" by auto
  4750 moreover have "Inter {closure S |S. S : I} : closed" 
  4751   unfolding mem_def closed_Inter closed_closure by auto
  4752 ultimately show ?thesis using closure_hull[of "Inter I"]
  4753   hull_minimal[of "Inter I" "Inter {closure S |S. S : I}" "closed"] by auto
  4754 qed
  4755 
  4756 lemma convex_closure_rel_interior_inter: 
  4757 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4758 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4759 shows "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
  4760 proof-
  4761 obtain x where x_def: "!S : I. x : rel_interior S" using assms by auto
  4762 { fix y assume "y : Inter {closure S |S. S : I}" hence y_def: "!S : I. y : closure S" by auto
  4763   { assume "y = x" 
  4764     hence "y : closure (Inter {rel_interior S |S. S : I})"
  4765        using x_def closure_subset[of "Inter {rel_interior S |S. S : I}"] by auto
  4766   }
  4767   moreover
  4768   { assume "y ~= x"
  4769     { fix e :: real assume e_def: "0 < e"
  4770       def e1 == "min 1 (e/norm (y - x))" hence e1_def: "e1>0 & e1<=1 & e1*norm(y-x)<=e"
  4771         using `y ~= x` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(y-x)"] by simp 
  4772       def z == "y - e1 *\<^sub>R (y - x)"
  4773       { fix S assume "S : I" 
  4774         hence "z : rel_interior S" using rel_interior_closure_convex_shrink[of S x y e1] 
  4775            assms x_def y_def e1_def z_def by auto
  4776       } hence *: "z : Inter {rel_interior S |S. S : I}" by auto
  4777       have "EX z. z:Inter {rel_interior S |S. S : I} & z ~= y & (dist z y) <= e"
  4778            apply (rule_tac x="z" in exI) using `y ~= x` z_def * e1_def e_def dist_norm[of z y] by simp
  4779     } hence "y islimpt Inter {rel_interior S |S. S : I}" unfolding islimpt_approachable_le by blast 
  4780     hence "y : closure (Inter {rel_interior S |S. S : I})" unfolding closure_def by auto
  4781   } ultimately have "y : closure (Inter {rel_interior S |S. S : I})" by auto
  4782 } from this show ?thesis by auto
  4783 qed
  4784 
  4785 
  4786 lemma convex_closure_inter: 
  4787 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4788 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4789 shows "closure (Inter I) = Inter {closure S |S. S : I}"
  4790 proof-
  4791 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
  4792   using convex_closure_rel_interior_inter assms by auto
  4793 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
  4794     using rel_interior_inter_aux 
  4795           subset_closure[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
  4796 ultimately show ?thesis using closure_inter[of I] by auto
  4797 qed
  4798 
  4799 lemma convex_inter_rel_interior_same_closure: 
  4800 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4801 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4802 shows "closure (Inter {rel_interior S |S. S : I}) = closure (Inter I)"
  4803 proof-
  4804 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
  4805   using convex_closure_rel_interior_inter assms by auto
  4806 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
  4807     using rel_interior_inter_aux 
  4808           subset_closure[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
  4809 ultimately show ?thesis using closure_inter[of I] by auto
  4810 qed
  4811 
  4812 lemma convex_rel_interior_inter: 
  4813 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4814 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4815 shows "rel_interior (Inter I) <= Inter {rel_interior S |S. S : I}"
  4816 proof-
  4817 have "convex(Inter I)" using assms convex_Inter by auto
  4818 moreover have "convex(Inter {rel_interior S |S. S : I})" apply (rule convex_Inter)
  4819    using assms convex_rel_interior by auto 
  4820 ultimately have "rel_interior (Inter {rel_interior S |S. S : I}) = rel_interior (Inter I)"
  4821    using convex_inter_rel_interior_same_closure assms 
  4822    closure_eq_rel_interior_eq[of "Inter {rel_interior S |S. S : I}" "Inter I"] by blast
  4823 from this show ?thesis using rel_interior_subset[of "Inter {rel_interior S |S. S : I}"] by auto
  4824 qed
  4825 
  4826 lemma convex_rel_interior_finite_inter: 
  4827 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4828 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4829 assumes "finite I"
  4830 shows "rel_interior (Inter I) = Inter {rel_interior S |S. S : I}"
  4831 proof-
  4832 have "Inter I ~= {}" using assms rel_interior_inter_aux[of I] by auto
  4833 have "convex (Inter I)" using convex_Inter assms by auto
  4834 { assume "I={}" hence ?thesis using Inter_empty rel_interior_univ2 by auto }
  4835 moreover
  4836 { assume "I ~= {}"
  4837 { fix z assume z_def: "z : Inter {rel_interior S |S. S : I}"
  4838   { fix x assume x_def: "x : Inter I"
  4839     { fix S assume S_def: "S : I" hence "z : rel_interior S" "x : S" using z_def x_def by auto 
  4840       (*from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S"*)
  4841       hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )"
  4842          using convex_rel_interior_if[of S z] S_def assms hull_subset[of S] by auto
  4843     } from this obtain mS where mS_def: "!S : I. (mS(S) > (1 :: real) & 
  4844          (!e. (e>1 & e<=mS(S)) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S))" by metis
  4845     obtain e where e_def: "e=Min (mS ` I)" by auto 
  4846     have "e : (mS ` I)" using e_def assms `I ~= {}` by (simp add: Min_in) 
  4847     hence "e>(1 :: real)" using mS_def by auto
  4848     moreover have "!S : I. e<=mS(S)" using e_def assms by auto
  4849     ultimately have "EX e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z : Inter I" using mS_def by auto
  4850   } hence "z : rel_interior (Inter I)" using convex_rel_interior_iff[of "Inter I" z]
  4851        `Inter I ~= {}` `convex (Inter I)` by auto
  4852 } from this have ?thesis using convex_rel_interior_inter[of I] assms by auto
  4853 } ultimately show ?thesis by blast
  4854 qed
  4855 
  4856 lemma convex_closure_inter_two: 
  4857 fixes S T :: "('n::euclidean_space) set"
  4858 assumes "convex S" "convex T"
  4859 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
  4860 shows "closure (S Int T) = (closure S) Int (closure T)" 
  4861 using convex_closure_inter[of "{S,T}"] assms by auto
  4862 
  4863 lemma convex_rel_interior_inter_two: 
  4864 fixes S T :: "('n::euclidean_space) set"
  4865 assumes "convex S" "convex T"
  4866 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
  4867 shows "rel_interior (S Int T) = (rel_interior S) Int (rel_interior T)" 
  4868 using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
  4869 
  4870 
  4871 lemma convex_affine_closure_inter: 
  4872 fixes S T :: "('n::euclidean_space) set"
  4873 assumes "convex S" "affine T"
  4874 assumes "(rel_interior S) Int T ~= {}"
  4875 shows "closure (S Int T) = (closure S) Int T"
  4876 proof- 
  4877 have "affine hull T = T" using assms by auto
  4878 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
  4879 moreover have "closure T = T" using assms affine_closed[of T] by auto
  4880 ultimately show ?thesis using convex_closure_inter_two[of S T] assms affine_imp_convex by auto 
  4881 qed
  4882 
  4883 lemma convex_affine_rel_interior_inter: 
  4884 fixes S T :: "('n::euclidean_space) set"
  4885 assumes "convex S" "affine T"
  4886 assumes "(rel_interior S) Int T ~= {}"
  4887 shows "rel_interior (S Int T) = (rel_interior S) Int T"
  4888 proof- 
  4889 have "affine hull T = T" using assms by auto
  4890 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
  4891 moreover have "closure T = T" using assms affine_closed[of T] by auto
  4892 ultimately show ?thesis using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto 
  4893 qed
  4894 
  4895 lemma subset_rel_interior_convex:
  4896 fixes S T :: "('n::euclidean_space) set"
  4897 assumes "convex S" "convex T"
  4898 assumes "S <= closure T"
  4899 assumes "~(S <= rel_frontier T)"
  4900 shows "rel_interior S <= rel_interior T"
  4901 proof-
  4902 have *: "S Int closure T = S" using assms by auto
  4903 have "~(rel_interior S <= rel_frontier T)"
  4904      using subset_closure[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T] 
  4905      closure_closed convex_closure_rel_interior[of S] closure_subset[of S] assms by auto
  4906 hence "(rel_interior S) Int (rel_interior (closure T)) ~= {}" 
  4907      using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T] by auto
  4908 hence "rel_interior S Int rel_interior T = rel_interior (S Int closure T)" using assms convex_closure  
  4909      convex_rel_interior_inter_two[of S "closure T"] convex_rel_interior_closure[of T] by auto
  4910 also have "...=rel_interior (S)" using * by auto
  4911 finally show ?thesis by auto
  4912 qed
  4913 
  4914 
  4915 lemma rel_interior_convex_linear_image:
  4916 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  4917 assumes "linear f"
  4918 assumes "convex S"
  4919 shows "f ` (rel_interior S) = rel_interior (f ` S)"
  4920 proof-
  4921 { assume "S = {}" hence ?thesis using assms rel_interior_empty rel_interior_convex_nonempty by auto }
  4922 moreover
  4923 { assume "S ~= {}"
  4924 have *: "f ` (rel_interior S) <= f ` S" unfolding image_mono using rel_interior_subset by auto
  4925 have "f ` S <= f ` (closure S)" unfolding image_mono using closure_subset by auto
  4926 also have "... = f ` (closure (rel_interior S))" using convex_closure_rel_interior assms by auto  
  4927 also have "... <= closure (f ` (rel_interior S))" using closure_linear_image assms by auto 
  4928 finally have "closure (f ` S) = closure (f ` rel_interior S)"
  4929    using subset_closure[of "f ` S" "closure (f ` rel_interior S)"] closure_closure 
  4930          subset_closure[of "f ` rel_interior S" "f ` S"] * by auto
  4931 hence "rel_interior (f ` S) = rel_interior (f ` rel_interior S)" using assms convex_rel_interior
  4932    linear_conv_bounded_linear[of f] convex_linear_image[of S] convex_linear_image[of "rel_interior S"] 
  4933    closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"] by auto
  4934 hence "rel_interior (f ` S) <= f ` rel_interior S" using rel_interior_subset by auto
  4935 moreover
  4936 { fix z assume z_def: "z : f ` rel_interior S"
  4937   from this obtain z1 where z1_def: "z1 : rel_interior S & (f z1 = z)" by auto
  4938   { fix x assume "x : f ` S"
  4939     from this obtain x1 where x1_def: "x1 : S & (f x1 = x)" by auto
  4940     from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x1 + e *\<^sub>R z1 : S"
  4941        using convex_rel_interior_iff[of S z1] `convex S` x1_def z1_def by auto
  4942     moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
  4943         using x1_def z1_def `linear f` by (simp add: linear_add_cmul)
  4944     ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
  4945         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
  4946     hence "EX e. (e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S)" using e_def by auto
  4947   } from this have "z : rel_interior (f ` S)" using convex_rel_interior_iff[of "f ` S" z] `convex S` 
  4948        `linear f` `S ~= {}` convex_linear_image[of S f]  linear_conv_bounded_linear[of f] by auto
  4949 } ultimately have ?thesis by auto
  4950 } ultimately show ?thesis by blast
  4951 qed
  4952 
  4953 
  4954 lemma convex_linear_preimage:
  4955   assumes c:"convex S" and l:"bounded_linear f"
  4956   shows "convex(f -` S)"
  4957 proof(auto simp add: convex_def)
  4958   interpret f: bounded_linear f by fact
  4959   fix x y assume xy:"f x : S" "f y : S"
  4960   fix u v ::real assume uv:"0 <= u" "0 <= v" "u + v = 1"
  4961   show "f (u *\<^sub>R x + v *\<^sub>R y) : S" unfolding image_iff
  4962     using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR
  4963       c[unfolded convex_def] xy uv by auto
  4964 qed
  4965 
  4966 
  4967 lemma rel_interior_convex_linear_preimage:
  4968 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  4969 assumes "linear f"
  4970 assumes "convex S"
  4971 assumes "f -` (rel_interior S) ~= {}"
  4972 shows "rel_interior (f -` S) = f -` (rel_interior S)"
  4973 proof-
  4974 have "S ~= {}" using assms rel_interior_empty by auto
  4975 have nonemp: "f -` S ~= {}" by (metis assms(3) rel_interior_subset subset_empty vimage_mono) 
  4976 hence "S Int (range f) ~= {}" by auto
  4977 have conv: "convex (f -` S)" using convex_linear_preimage assms linear_conv_bounded_linear by auto
  4978 hence "convex (S Int (range f))"
  4979   by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
  4980 { fix z assume "z : f -` (rel_interior S)"
  4981   hence z_def: "f z : rel_interior S" by auto
  4982   { fix x assume "x : f -` S" from this have x_def: "f x : S" by auto
  4983     from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) : S"
  4984       using convex_rel_interior_iff[of S "f z"] z_def assms `S ~= {}` by auto
  4985     moreover have "(1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R x + e *\<^sub>R z)"
  4986       using `linear f` by (simp add: linear_def)
  4987     ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R z : f -` S" using e_def by auto
  4988   } hence "z : rel_interior (f -` S)" 
  4989        using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
  4990 } 
  4991 moreover
  4992 { fix z assume z_def: "z : rel_interior (f -` S)" 
  4993   { fix x assume x_def: "x: S Int (range f)"
  4994     from this obtain y where y_def: "(f y = x) & (y : f -` S)" by auto
  4995     from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R y+ e *\<^sub>R z : f -` S"
  4996       using convex_rel_interior_iff[of "f -` S" z] z_def conv by auto
  4997     moreover have "(1-e)*\<^sub>R x+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R y + e *\<^sub>R z)"
  4998       using `linear f` y_def by (simp add: linear_def)
  4999     ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R (f z) : S Int (range f)"
  5000       using e_def by auto
  5001   } hence "f z : rel_interior (S Int (range f))" using `convex (S Int (range f))`
  5002     `S Int (range f) ~= {}` convex_rel_interior_iff[of "S Int (range f)" "f z"] by auto
  5003   moreover have "affine (range f)"
  5004     by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
  5005   ultimately have "f z : rel_interior S" 
  5006     using convex_affine_rel_interior_inter[of S "range f"] assms by auto
  5007   hence "z : f -` (rel_interior S)" by auto
  5008 }
  5009 ultimately show ?thesis by auto
  5010 qed
  5011     
  5012 
  5013 lemma convex_direct_sum:
  5014 fixes S :: "('n::euclidean_space) set"
  5015 fixes T :: "('m::euclidean_space) set"
  5016 assumes "convex S" "convex T"
  5017 shows "convex (S <*> T)"
  5018 proof-
  5019 {
  5020 fix x assume "x : S <*> T"
  5021 from this obtain xs xt where xst_def: "xs : S & xt : T & (xs,xt) = x" by auto
  5022 fix y assume "y : S <*> T"
  5023 from this obtain ys yt where yst_def: "ys : S & yt : T & (ys,yt) = y" by auto
  5024 fix u v assume uv_def: "(u :: real)>=0 & (v :: real)>=0 & u+v=1"
  5025 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
  5026 moreover have "u *\<^sub>R xs + v *\<^sub>R ys : S"
  5027    using uv_def xst_def yst_def convex_def[of S] assms by auto
  5028 moreover have "u *\<^sub>R xt + v *\<^sub>R yt : T"
  5029    using uv_def xst_def yst_def convex_def[of T] assms by auto
  5030 ultimately have "u *\<^sub>R x + v *\<^sub>R y : S <*> T" by auto
  5031 } from this show ?thesis unfolding convex_def by auto
  5032 qed
  5033 
  5034 
  5035 lemma convex_hull_direct_sum:
  5036 fixes S :: "('n::euclidean_space) set"
  5037 fixes T :: "('m::euclidean_space) set"
  5038 shows "convex hull (S <*> T) = (convex hull S) <*> (convex hull T)"
  5039 proof-
  5040 { fix x assume "x : (convex hull S) <*> (convex hull T)"
  5041   from this obtain xs xt where xst_def: "xs : convex hull S & xt : convex hull T & (xs,xt) = x" by auto
  5042   from xst_def obtain sI su where s: "finite sI & sI <= S & (ALL x:sI. 0 <= su x) & setsum su sI = 1 
  5043      & (SUM v:sI. su v *\<^sub>R v) = xs" using convex_hull_explicit[of S] by auto
  5044   from xst_def obtain tI tu where t: "finite tI & tI <= T & (ALL x:tI. 0 <= tu x) & setsum tu tI = 1 
  5045      & (SUM v:tI. tu v *\<^sub>R v) = xt" using convex_hull_explicit[of T] by auto
  5046   def I == "(sI <*> tI)"
  5047   def u == "(%i. (su (fst i))*(tu(snd i)))"
  5048   have "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
  5049      (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vs)"
  5050      using fst_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
  5051      by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
  5052   also have "...=(SUM vt:tI. tu vt *\<^sub>R (SUM vs:sI. su vs *\<^sub>R vs))"
  5053      using setsum_commute[of "(%vt vs. (su vs * tu vt) *\<^sub>R vs)" sI tI]
  5054      by (simp add: mult_commute scaleR_right.setsum)
  5055   also have "...=(SUM vt:tI. tu vt *\<^sub>R xs)" using s by auto
  5056   also have "...=(SUM vt:tI. tu vt) *\<^sub>R xs" by (simp add: scaleR_left.setsum)
  5057   also have "...=xs" using t by auto
  5058   finally have h1: "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xs" by auto
  5059   have "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
  5060      (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vt)"
  5061      using snd_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
  5062      by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
  5063   also have "...=(SUM vs:sI. su vs *\<^sub>R (SUM vt:tI. tu vt *\<^sub>R vt))"
  5064      by (simp add: mult_commute scaleR_right.setsum)
  5065   also have "...=(SUM vs:sI. su vs *\<^sub>R xt)" using t by auto
  5066   also have "...=(SUM vs:sI. su vs) *\<^sub>R xt" by (simp add: scaleR_left.setsum)
  5067   also have "...=xt" using s by auto
  5068   finally have h2: "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xt" by auto
  5069   from h1 h2 have "(SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v) = x" using xst_def by auto
  5070 
  5071   moreover have "finite I & (I <= S <*> T)" using s t I_def by auto 
  5072   moreover have "!i:I. 0 <= u i" using s t I_def u_def by (simp add: mult_nonneg_nonneg)
  5073   moreover have "setsum u I = 1" using u_def I_def setsum_cartesian_product[of "(% x y. (su x)*(tu y))"] 
  5074      s t setsum_product[of su sI tu tI] by (auto simp add: split_def)
  5075   ultimately have "x : convex hull (S <*> T)" 
  5076      apply (subst convex_hull_explicit[of "S <*> T"]) apply rule
  5077      apply (rule_tac x="I" in exI) apply (rule_tac x="u" in exI)
  5078      using I_def u_def by auto
  5079 }
  5080 hence "convex hull (S <*> T) >= (convex hull S) <*> (convex hull T)" by auto
  5081 moreover have "(convex hull S) <*> (convex hull T) : convex" 
  5082    unfolding mem_def by (simp add: convex_direct_sum convex_convex_hull)
  5083 ultimately show ?thesis 
  5084    using hull_minimal[of "S <*> T" "(convex hull S) <*> (convex hull T)" "convex"] 
  5085          hull_subset[of S convex] hull_subset[of T convex] by auto
  5086 qed
  5087 
  5088 lemma rel_interior_direct_sum:
  5089 fixes S :: "('n::euclidean_space) set"
  5090 fixes T :: "('m::euclidean_space) set"
  5091 assumes "convex S" "convex T"
  5092 shows "rel_interior (S <*> T) = rel_interior S <*> rel_interior T"
  5093 proof-
  5094 { assume "S={}" hence ?thesis apply auto using rel_interior_empty by auto }
  5095 moreover
  5096 { assume "T={}" hence ?thesis apply auto using rel_interior_empty by auto }
  5097 moreover {
  5098 assume "S ~={}" "T ~={}"
  5099 hence ri: "rel_interior S ~= {}" "rel_interior T ~= {}" using rel_interior_convex_nonempty assms by auto
  5100 hence "fst -` rel_interior S ~= {}" using fst_vimage_eq_Times[of "rel_interior S"] by auto
  5101 hence "rel_interior ((fst :: 'n * 'm => 'n) -` S) = fst -` rel_interior S"
  5102   using fst_linear `convex S` rel_interior_convex_linear_preimage[of fst S] by auto 
  5103 hence s: "rel_interior (S <*> (UNIV :: 'm set)) = rel_interior S <*> UNIV" by (simp add: fst_vimage_eq_Times)
  5104 from ri have "snd -` rel_interior T ~= {}" using snd_vimage_eq_Times[of "rel_interior T"] by auto
  5105 hence "rel_interior ((snd :: 'n * 'm => 'm) -` T) = snd -` rel_interior T"
  5106   using snd_linear `convex T` rel_interior_convex_linear_preimage[of snd T] by auto 
  5107 hence t: "rel_interior ((UNIV :: 'n set) <*> T) = UNIV <*> rel_interior T" by (simp add: snd_vimage_eq_Times)
  5108 from s t have *: "rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T) 
  5109   = rel_interior S <*> rel_interior T" by auto
  5110 have "(S <*> T) = (S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T)" by auto
  5111 hence "rel_interior (S <*> T) = rel_interior ((S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T))" by auto
  5112 also have "...=rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)" 
  5113    apply (subst convex_rel_interior_inter_two[of "S <*> (UNIV :: 'm set)" "(UNIV :: 'n set) <*> T"]) 
  5114    using * ri assms convex_direct_sum by auto
  5115 finally have ?thesis using * by auto
  5116 }
  5117 ultimately show ?thesis by blast
  5118 qed
  5119 
  5120 lemma rel_interior_scaleR: 
  5121 fixes S :: "('n::euclidean_space) set"
  5122 assumes "c ~= 0"
  5123 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  5124 using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
  5125       linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms by auto
  5126 
  5127 lemma rel_interior_convex_scaleR: 
  5128 fixes S :: "('n::euclidean_space) set"
  5129 assumes "convex S"
  5130 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  5131 by (metis assms linear_scaleR rel_interior_convex_linear_image)
  5132 
  5133 lemma convex_rel_open_scaleR: 
  5134 fixes S :: "('n::euclidean_space) set"
  5135 assumes "convex S" "rel_open S"
  5136 shows "convex ((op *\<^sub>R c) ` S) & rel_open ((op *\<^sub>R c) ` S)"
  5137 by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
  5138 
  5139 
  5140 lemma convex_rel_open_finite_inter: 
  5141 assumes "!S : I. (convex (S :: ('n::euclidean_space) set) & rel_open S)"
  5142 assumes "finite I"
  5143 shows "convex (Inter I) & rel_open (Inter I)"
  5144 proof-
  5145 { assume "Inter {rel_interior S |S. S : I} = {}"
  5146   hence "Inter I = {}" using assms unfolding rel_open_def by auto
  5147   hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
  5148 }
  5149 moreover
  5150 { assume "Inter {rel_interior S |S. S : I} ~= {}"
  5151   hence "rel_open (Inter I)" using assms unfolding rel_open_def
  5152     using convex_rel_interior_finite_inter[of I] by auto
  5153   hence ?thesis using convex_Inter assms by auto
  5154 } ultimately show ?thesis by auto
  5155 qed
  5156 
  5157 lemma convex_rel_open_linear_image:
  5158 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  5159 assumes "linear f"
  5160 assumes "convex S" "rel_open S"
  5161 shows "convex (f ` S) & rel_open (f ` S)"
  5162 by (metis assms convex_linear_image rel_interior_convex_linear_image 
  5163    linear_conv_bounded_linear rel_open_def)
  5164 
  5165 lemma convex_rel_open_linear_preimage:
  5166 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  5167 assumes "linear f"
  5168 assumes "convex S" "rel_open S"
  5169 shows "convex (f -` S) & rel_open (f -` S)" 
  5170 proof-
  5171 { assume "f -` (rel_interior S) = {}"
  5172   hence "f -` S = {}" using assms unfolding rel_open_def by auto
  5173   hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
  5174 }
  5175 moreover
  5176 { assume "f -` (rel_interior S) ~= {}"
  5177   hence "rel_open (f -` S)" using assms unfolding rel_open_def
  5178     using rel_interior_convex_linear_preimage[of f S] by auto
  5179   hence ?thesis using convex_linear_preimage assms linear_conv_bounded_linear by auto
  5180 } ultimately show ?thesis by auto
  5181 qed
  5182 
  5183 lemma rel_interior_projection:
  5184 fixes S :: "('m::euclidean_space*'n::euclidean_space) set"
  5185 fixes f :: "'m::euclidean_space => ('n::euclidean_space) set"
  5186 assumes "convex S"
  5187 assumes "f = (%y. {z. (y,z) : S})"
  5188 shows "(y,z) : rel_interior S <-> (y : rel_interior {y. (f y ~= {})} & z : rel_interior (f y))"
  5189 proof-
  5190 { fix y assume "y : {y. (f y ~= {})}" from this obtain z where "(y,z) : S" using assms by auto
  5191   hence "EX x. x : S & y = fst x" apply (rule_tac x="(y,z)" in exI) by auto
  5192   from this obtain x where "x : S & y = fst x" by blast
  5193   hence "y : fst ` S" unfolding image_def by auto
  5194 }
  5195 hence "fst ` S = {y. (f y ~= {})}" unfolding fst_def using assms by auto
  5196 hence h1: "fst ` rel_interior S = rel_interior {y. (f y ~= {})}"
  5197    using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
  5198 { fix y assume "y : rel_interior {y. (f y ~= {})}"
  5199   hence "y : fst ` rel_interior S" using h1 by auto
  5200   hence *: "rel_interior S Int fst -` {y} ~= {}" by auto
  5201   moreover have aff: "affine (fst -` {y})" unfolding affine_alt by (simp add: algebra_simps)
  5202   ultimately have **: "rel_interior (S Int fst -` {y}) = rel_interior S Int fst -` {y}"
  5203     using convex_affine_rel_interior_inter[of S "fst -` {y}"] assms by auto
  5204   have conv: "convex (S Int fst -` {y})" using convex_Int assms aff affine_imp_convex by auto
  5205   { fix x assume "x : f y"
  5206     hence "(y,x) : S Int (fst -` {y})" using assms by auto
  5207     moreover have "x = snd (y,x)" by auto
  5208     ultimately have "x : snd ` (S Int fst -` {y})" by blast
  5209   }
  5210   hence "snd ` (S Int fst -` {y}) = f y" using assms by auto
  5211   hence ***: "rel_interior (f y) = snd ` rel_interior (S Int fst -` {y})"
  5212     using rel_interior_convex_linear_image[of snd "S Int fst -` {y}"] snd_linear conv by auto 
  5213   { fix z assume "z : rel_interior (f y)"
  5214     hence "z : snd ` rel_interior (S Int fst -` {y})" using *** by auto
  5215     moreover have "{y} = fst ` rel_interior (S Int fst -` {y})" using * ** rel_interior_subset by auto   
  5216     ultimately have "(y,z) : rel_interior (S Int fst -` {y})" by force
  5217     hence "(y,z) : rel_interior S" using ** by auto
  5218   }
  5219   moreover
  5220   { fix z assume "(y,z) : rel_interior S"
  5221     hence "(y,z) : rel_interior (S Int fst -` {y})" using ** by auto
  5222     hence "z : snd ` rel_interior (S Int fst -` {y})" by (metis Range_iff snd_eq_Range) 
  5223     hence "z : rel_interior (f y)" using *** by auto
  5224   }
  5225   ultimately have "!!z. (y,z) : rel_interior S <-> z : rel_interior (f y)" by auto
  5226 } 
  5227 hence h2: "!!y z. y : rel_interior {t. f t ~= {}} ==> ((y, z) : rel_interior S) = (z : rel_interior (f y))"
  5228   by auto
  5229 { fix y z assume asm: "(y, z) : rel_interior S"
  5230   hence "y : fst ` rel_interior S" by (metis Domain_iff fst_eq_Domain)
  5231   hence "y : rel_interior {t. f t ~= {}}" using h1 by auto
  5232   hence "y : rel_interior {t. f t ~= {}} & (z : rel_interior (f y))" using h2 asm by auto
  5233 } from this show ?thesis using h2 by blast
  5234 qed
  5235 
  5236 subsection{* Relative interior of convex cone *}
  5237 
  5238 lemma cone_rel_interior:
  5239 fixes S :: "('m::euclidean_space) set"
  5240 assumes "cone S"
  5241 shows "cone ({0} Un (rel_interior S))"
  5242 proof-
  5243 { assume "S = {}" hence ?thesis by (simp add: rel_interior_empty cone_0) }
  5244 moreover
  5245 { assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  5246   hence *: "0:({0} Un (rel_interior S)) &
  5247            (!c. c>0 --> op *\<^sub>R c ` ({0} Un rel_interior S) = ({0} Un rel_interior S))"
  5248            by (auto simp add: rel_interior_scaleR)
  5249   hence ?thesis using cone_iff[of "{0} Un rel_interior S"] by auto
  5250 }
  5251 ultimately show ?thesis by blast
  5252 qed
  5253 
  5254 lemma rel_interior_convex_cone_aux:
  5255 fixes S :: "('m::euclidean_space) set"
  5256 assumes "convex S"
  5257 shows "(c,x) : rel_interior (cone hull ({(1 :: real)} <*> S)) <-> 
  5258        c>0 & x : ((op *\<^sub>R c) ` (rel_interior S))"
  5259 proof-
  5260 { assume "S={}" hence ?thesis by (simp add: rel_interior_empty cone_hull_empty) } 
  5261 moreover
  5262 { assume "S ~= {}" from this obtain s where "s : S" by auto
  5263 have conv: "convex ({(1 :: real)} <*> S)" using convex_direct_sum[of "{(1 :: real)}" S] 
  5264    assms convex_singleton[of "1 :: real"] by auto
  5265 def f == "(%y. {z. (y,z) : cone hull ({(1 :: real)} <*> S)})"
  5266 hence *: "(c, x) : rel_interior (cone hull ({(1 :: real)} <*> S)) =
  5267       (c : rel_interior {y. f y ~= {}} & x : rel_interior (f c))"
  5268   apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} <*> S)" f c x])
  5269   using convex_cone_hull[of "{(1 :: real)} <*> S"] conv by auto
  5270 { fix y assume "(y :: real)>=0"
  5271   hence "y *\<^sub>R (1,s) : cone hull ({(1 :: real)} <*> S)"
  5272      using cone_hull_expl[of "{(1 :: real)} <*> S"] `s:S` by auto
  5273   hence "f y ~= {}" using f_def by auto
  5274 }
  5275 hence "{y. f y ~= {}} = {0..}" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
  5276 hence **: "rel_interior {y. f y ~= {}} = {0<..}" using rel_interior_real_semiline by auto
  5277 { fix c assume "c>(0 :: real)"
  5278   hence "f c = (op *\<^sub>R c ` S)" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
  5279   hence "rel_interior (f c)= (op *\<^sub>R c ` rel_interior S)"
  5280      using rel_interior_convex_scaleR[of S c] assms by auto
  5281 }
  5282 hence ?thesis using * ** by auto
  5283 } ultimately show ?thesis by blast
  5284 qed
  5285 
  5286 
  5287 lemma rel_interior_convex_cone:
  5288 fixes S :: "('m::euclidean_space) set"
  5289 assumes "convex S"
  5290 shows "rel_interior (cone hull ({(1 :: real)} <*> S)) = 
  5291        {(c,c *\<^sub>R x) |c x. c>0 & x : (rel_interior S)}"
  5292 (is "?lhs=?rhs")
  5293 proof-
  5294 { fix z assume "z:?lhs" 
  5295   have *: "z=(fst z,snd z)" by auto 
  5296   have "z:?rhs" using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms `z:?lhs` apply auto
  5297      apply (rule_tac x="fst z" in exI) apply (rule_tac x="x" in exI) using * by auto
  5298 }
  5299 moreover
  5300 { fix z assume "z:?rhs" hence "z:?lhs" 
  5301   using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms by auto
  5302 }
  5303 ultimately show ?thesis by blast
  5304 qed
  5305 
  5306 lemma convex_hull_finite_union:
  5307 assumes "finite I"
  5308 assumes "!i:I. (convex (S i) & (S i) ~= {})"
  5309 shows "convex hull (Union (S ` I)) = 
  5310        {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)}"
  5311   (is "?lhs = ?rhs")
  5312 proof-
  5313 { fix x assume "x : ?rhs" 
  5314   from this obtain c s 
  5315     where *: "setsum (%i. c i *\<^sub>R s i) I=x" "(setsum c I = 1)"
  5316      "(!i:I. c i >= 0) & (!i:I. s i : S i)" by auto
  5317   hence "!i:I. s i : convex hull (Union (S ` I))" using hull_subset[of "Union (S ` I)" convex] by auto
  5318   hence "x : ?lhs" unfolding *(1)[THEN sym]
  5319      apply (subst convex_setsum[of I "convex hull Union (S ` I)" c s])
  5320      using * assms convex_convex_hull by auto
  5321 } hence "?lhs >= ?rhs" by auto
  5322 
  5323 { fix i assume "i:I"
  5324     from this assms have "EX p. p : S i" by auto
  5325 } 
  5326 from this obtain p where p_def: "!i:I. p i : S i" by metis
  5327 
  5328 { fix i assume "i:I"
  5329   { fix x assume "x : S i"
  5330     def c == "(%j. if (j=i) then (1::real) else 0)"
  5331     hence *: "setsum c I = 1" using `finite I` `i:I` setsum_delta[of I i "(%(j::'a). (1::real))"] by auto
  5332     def s == "(%j. if (j=i) then x else p j)"
  5333     hence "!j. c j *\<^sub>R s j = (if (j=i) then x else 0)" using c_def by (auto simp add: algebra_simps)
  5334     hence "x = setsum (%i. c i *\<^sub>R s i) I"
  5335        using s_def c_def `finite I` `i:I` setsum_delta[of I i "(%(j::'a). x)"] by auto 
  5336     hence "x : ?rhs" apply auto
  5337       apply (rule_tac x="c" in exI) 
  5338       apply (rule_tac x="s" in exI) using * c_def s_def p_def `x : S i` by auto 
  5339   } hence "?rhs >= S i" by auto
  5340 } hence *: "?rhs >= Union (S ` I)" by auto
  5341 
  5342 { fix u v assume uv: "(u :: real)>=0 & v>=0 & u+v=1"
  5343   fix x y assume xy: "(x : ?rhs) & (y : ?rhs)"
  5344   from xy obtain c s where xc: "x=setsum (%i. c i *\<^sub>R s i) I &
  5345      (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)" by auto
  5346   from xy obtain d t where yc: "y=setsum (%i. d i *\<^sub>R t i) I &
  5347      (!i:I. d i >= 0) & (setsum d I = 1) & (!i:I. t i : S i)" by auto
  5348   def e == "(%i. u * (c i)+v * (d i))"
  5349   have ge0: "!i:I. e i >= 0"  using e_def xc yc uv by (simp add: mult_nonneg_nonneg)
  5350   have "setsum (%i. u * c i) I = u * setsum c I" by (simp add: setsum_right_distrib)
  5351   moreover have "setsum (%i. v * d i) I = v * setsum d I" by (simp add: setsum_right_distrib)
  5352   ultimately have sum1: "setsum e I = 1" using e_def xc yc uv by (simp add: setsum_addf)
  5353   def q == "(%i. if (e i = 0) then (p i) 
  5354                  else (u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))"
  5355   { fix i assume "i:I"
  5356     { assume "e i = 0" hence "q i : S i" using `i:I` p_def q_def by auto }
  5357     moreover
  5358     { assume "e i ~= 0" 
  5359       hence "q i : S i" using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"] 
  5360          mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
  5361          assms q_def e_def `i:I` `e i ~= 0` xc yc uv by auto
  5362     } ultimately have "q i : S i" by auto
  5363   } hence qs: "!i:I. q i : S i" by auto
  5364   { fix i assume "i:I"
  5365     { assume "e i = 0" 
  5366       have ge: "u * (c i) >= 0 & v * (d i) >= 0" using xc yc uv `i:I` by (simp add: mult_nonneg_nonneg)
  5367       moreover hence "u * (c i) <= 0 & v * (d i) <= 0" using `e i = 0` e_def `i:I` by simp 
  5368       ultimately have "u * (c i) = 0 & v * (d i) = 0" by auto
  5369       hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
  5370          using `e i = 0` by auto
  5371     }
  5372     moreover
  5373     { assume "e i ~= 0"
  5374       hence "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
  5375          using q_def by auto
  5376       hence "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
  5377              = (e i) *\<^sub>R (q i)" by auto
  5378       hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
  5379          using `e i ~= 0` by (simp add: algebra_simps)
  5380     } ultimately have 
  5381       "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by blast
  5382   } hence *: "!i:I.
  5383     (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by auto
  5384   have "u *\<^sub>R x + v *\<^sub>R y =
  5385        setsum (%i. (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i)) I"
  5386           using xc yc by (simp add: algebra_simps scaleR_right.setsum setsum_addf)
  5387   also have "...=setsum (%i. (e i) *\<^sub>R (q i)) I" using * by auto
  5388   finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (%i. (e i) *\<^sub>R (q i)) I" by auto
  5389   hence "u *\<^sub>R x + v *\<^sub>R y : ?rhs" using ge0 sum1 qs by auto
  5390 } hence "convex ?rhs" unfolding convex_def by auto
  5391 hence "?rhs : convex" unfolding mem_def by auto
  5392 from this show ?thesis using `?lhs >= ?rhs` * 
  5393    hull_minimal[of "Union (S ` I)" "?rhs" "convex"] by blast
  5394 qed
  5395 
  5396 lemma convex_hull_union_two:
  5397 fixes S T :: "('m::euclidean_space) set"
  5398 assumes "convex S" "S ~= {}" "convex T" "T ~= {}"
  5399 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}"
  5400   (is "?lhs = ?rhs")
  5401 proof-
  5402 def I == "{(1::nat),2}"
  5403 def s == "(%i. (if i=(1::nat) then S else T))"
  5404 have "Union (s ` I) = S Un T" using s_def I_def by auto
  5405 hence "convex hull (Union (s ` I)) = convex hull (S Un T)" by auto
  5406 moreover have "convex hull Union (s ` I) =
  5407     {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)}"
  5408     apply (subst convex_hull_finite_union[of I s]) using assms s_def I_def by auto
  5409 moreover have 
  5410   "{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)} <=
  5411   ?rhs"
  5412   using s_def I_def by auto
  5413 ultimately have "?lhs<=?rhs" by auto 
  5414 { fix x assume "x : ?rhs" 
  5415   from this obtain u v s t 
  5416     where *: "x=u *\<^sub>R s + v *\<^sub>R t & u>=0 & v>=0 & u+v=1 & s:S & t:T" by auto
  5417   hence "x : convex hull {s,t}" using convex_hull_2[of s t] by auto
  5418   hence "x : convex hull (S Un T)" using * hull_mono[of "{s, t}" "S Un T"] by auto
  5419 } hence "?lhs >= ?rhs" by blast
  5420 from this show ?thesis using `?lhs<=?rhs` by auto
  5421 qed
  5422 
  5423 subsection {* Convexity on direct sums *}
  5424 
  5425 lemma closure_sum:
  5426   fixes S T :: "('n::euclidean_space) set"
  5427   shows "closure S \<oplus> closure T \<subseteq> closure (S \<oplus> T)"
  5428 proof-
  5429   have "(closure S) \<oplus> (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
  5430     by (simp add: set_plus_image)
  5431   also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
  5432     using closure_direct_sum by auto
  5433   also have "... \<subseteq> closure (S \<oplus> T)"
  5434     using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
  5435     by (auto simp: set_plus_image)
  5436   finally show ?thesis
  5437     by auto
  5438 qed
  5439 
  5440 lemma convex_oplus:
  5441 fixes S T :: "('n::euclidean_space) set"
  5442 assumes "convex S" "convex T"
  5443 shows "convex (S \<oplus> T)"
  5444 proof-
  5445 have "{x + y |x y. x : S & y : T} = {c. EX a:S. EX b:T. c = a + b}" by auto
  5446 thus ?thesis unfolding set_plus_def using convex_sums[of S T] assms by auto