src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author huffman
Wed Aug 10 18:02:16 2011 -0700 (2011-08-10)
changeset 44142 8e27e0177518
parent 44132 0f35a870ecf1
child 44170 510ac30f44c0
permissions -rw-r--r--
avoid warnings about duplicate rules
     1 (*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
     2     Author:     Robert Himmelmann, TU Muenchen
     3     Author:     Bogdan Grechuk, University of Edinburgh
     4 *)
     5 
     6 header {* Convex sets, functions and related things. *}
     7 
     8 theory Convex_Euclidean_Space
     9 imports
    10   Topology_Euclidean_Space
    11   "~~/src/HOL/Library/Convex"
    12   "~~/src/HOL/Library/Set_Algebras"
    13 begin
    14 
    15 
    16 (* ------------------------------------------------------------------------- *)
    17 (* To be moved elsewhere                                                     *)
    18 (* ------------------------------------------------------------------------- *)
    19 
    20 lemma linear_scaleR: "linear (%(x :: 'n::euclidean_space). scaleR c x)"
    21   by (metis linear_conv_bounded_linear scaleR.bounded_linear_right)
    22 
    23 lemma injective_scaleR: 
    24 assumes "(c :: real) ~= 0"
    25 shows "inj (%(x :: 'n::euclidean_space). scaleR c x)"
    26   by (metis assms injI real_vector.scale_cancel_left)
    27 
    28 lemma linear_add_cmul:
    29 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)" 
    30 assumes "linear f"
    31 shows "f(a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
    32 using linear_add[of f] linear_cmul[of f] assms by (simp) 
    33 
    34 lemma mem_convex_2:
    35   assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v=1"
    36   shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
    37   using assms convex_def[of S] by auto
    38 
    39 lemma mem_convex_alt:
    40   assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
    41   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
    42 apply (subst mem_convex_2) 
    43 using assms apply (auto simp add: algebra_simps zero_le_divide_iff)
    44 using add_divide_distrib[of u v "u+v"] by auto
    45 
    46 lemma card_ge1: assumes "d ~= {}" "finite d" shows "card d >= 1"
    47 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)
    48 
    49 lemma inj_on_image_mem_iff: "inj_on f B ==> (A <= B) ==> (f a : f`A) ==> (a : B) ==> (a : A)" 
    50 by (blast dest: inj_onD)
    51 
    52 lemma independent_injective_on_span_image:
    53   assumes iS: "independent (S::(_::euclidean_space) set)" 
    54      and lf: "linear f" and fi: "inj_on f (span S)"
    55   shows "independent (f ` S)"
    56 proof-
    57   {fix a assume a: "a : S" "f a : span (f ` S - {f a})"
    58     have eq: "f ` S - {f a} = f ` (S - {a})" using fi a span_inc
    59       by (auto simp add: inj_on_def)
    60     from a have "f a : f ` span (S -{a})"
    61       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
    62     moreover have "span (S -{a}) <= span S" using span_mono[of "S-{a}" S] by auto
    63     ultimately have "a : span (S -{a})" using fi a span_inc by (auto simp add: inj_on_def)
    64     with a(1) iS  have False by (simp add: dependent_def) }
    65   then show ?thesis unfolding dependent_def by blast
    66 qed
    67 
    68 lemma dim_image_eq:
    69 fixes f :: "'n::euclidean_space => 'm::euclidean_space"
    70 assumes lf: "linear f" and fi: "inj_on f (span S)" 
    71 shows "dim (f ` S) = dim (S:: ('n::euclidean_space) set)"
    72 proof-
    73 obtain B where B_def: "B<=S & independent B & S <= span B & card B = dim S" 
    74   using basis_exists[of S] by auto
    75 hence "span S = span B" using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
    76 hence "independent (f ` B)" using independent_injective_on_span_image[of B f] B_def assms by auto
    77 moreover have "card (f ` B) = card B" using assms card_image[of f B] subset_inj_on[of f "span S" B] 
    78    B_def span_inc by auto
    79 moreover have "(f ` B) <= (f ` S)" using B_def by auto
    80 ultimately have "dim (f ` S) >= dim S" 
    81   using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
    82 from this show ?thesis using dim_image_le[of f S] assms by auto
    83 qed
    84 
    85 lemma linear_injective_on_subspace_0:
    86 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
    87 assumes lf: "linear f" and "subspace S"
    88   shows "inj_on f S <-> (!x : S. f x = 0 --> x = 0)"
    89 proof-
    90   have "inj_on f S <-> (!x : S. !y : S. f x = f y --> x = y)" by (simp add: inj_on_def)
    91   also have "... <-> (!x : S. !y : S. f x - f y = 0 --> x - y = 0)" by simp
    92   also have "... <-> (!x : S. !y : S. f (x - y) = 0 --> x - y = 0)"
    93     by (simp add: linear_sub[OF lf])
    94   also have "... <-> (! x : S. f x = 0 --> x = 0)" 
    95     using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
    96   finally show ?thesis .
    97 qed
    98 
    99 lemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"
   100   unfolding subspace_def by auto 
   101 
   102 lemma span_eq[simp]: "(span s = s) <-> subspace s"
   103 proof-
   104   { fix f assume "f <= subspace"
   105     hence "subspace (Inter f)" using subspace_Inter[of f] unfolding subset_eq mem_def by auto  }
   106   thus ?thesis using hull_eq[unfolded mem_def, of subspace s] span_def by auto
   107 qed
   108 
   109 lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
   110   by(auto simp add: inj_on_def euclidean_eq[where 'a='n])
   111   
   112 lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
   113 proof-
   114   have eq: "?S = basis ` d" by blast
   115   show ?thesis unfolding eq apply(rule finite_subset[OF _ range_basis_finite]) by auto
   116 qed
   117 
   118 lemma card_substdbasis: assumes "d \<subseteq> {..<DIM('n::euclidean_space)}"
   119   shows "card {basis i ::'n::euclidean_space | i. i : d} = card d" (is "card ?S = _")
   120 proof-
   121   have eq: "?S = basis ` d" by blast
   122   show ?thesis unfolding eq using card_image[OF basis_inj_on[of d]] assms by auto
   123 qed
   124 
   125 lemma substdbasis_expansion_unique: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
   126   shows "setsum (%i. f i *\<^sub>R basis i) d = (x::'a::euclidean_space)
   127       <-> (!i<DIM('a). (i:d --> f i = x$$i) & (i ~: d --> x $$ i = 0))"
   128 proof- have *:"\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto
   129   have **:"finite d" apply(rule finite_subset[OF assms]) by fastsimp
   130   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)"
   131     unfolding euclidean_component.setsum euclidean_scaleR basis_component *
   132     apply(rule setsum_cong2) using assms by auto
   133   show ?thesis unfolding euclidean_eq[where 'a='a] *** setsum_delta[OF **] using assms by auto
   134 qed
   135 
   136 lemma independent_substdbasis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
   137   shows "independent {basis i ::'a::euclidean_space |i. i : (d :: nat set)}" (is "independent ?A")
   138 proof -
   139   have *: "{basis i |i. i < DIM('a)} = basis ` {..<DIM('a)}" by auto
   140   show ?thesis
   141     apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
   142     using independent_basis[where 'a='a] assms by (auto simp: *)
   143 qed
   144 
   145 lemma dim_cball: 
   146 assumes "0<e"
   147 shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
   148 proof-
   149 { fix x :: "'n::euclidean_space" def y == "(e/norm x) *\<^sub>R x"
   150   hence "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto
   151   moreover have "x = (norm x/e) *\<^sub>R y" using y_def assms by simp
   152   moreover hence "x = (norm x/e) *\<^sub>R y"  by auto
   153   ultimately have "x : span (cball 0 e)"
   154      using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
   155 } hence "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto 
   156 from this show ?thesis using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
   157 qed
   158 
   159 lemma indep_card_eq_dim_span:
   160 fixes B :: "('n::euclidean_space) set"
   161 assumes "independent B"
   162 shows "finite B & card B = dim (span B)" 
   163   using assms basis_card_eq_dim[of B "span B"] span_inc by auto
   164 
   165 lemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"
   166   apply(rule ccontr) by auto
   167 
   168 lemma translate_inj_on: 
   169 fixes A :: "('n::euclidean_space) set"
   170 shows "inj_on (%x. a+x) A" unfolding inj_on_def by auto
   171 
   172 lemma translation_assoc:
   173   fixes a b :: "'a::ab_group_add"
   174   shows "(\<lambda>x. b+x) ` ((\<lambda>x. a+x) ` S) = (\<lambda>x. (a+b)+x) ` S" by auto
   175 
   176 lemma translation_invert:
   177   fixes a :: "'a::ab_group_add"
   178   assumes "(\<lambda>x. a+x) ` A = (\<lambda>x. a+x) ` B"
   179   shows "A=B"
   180 proof-
   181   have "(%x. -a+x) ` ((%x. a+x) ` A) = (%x. -a+x) ` ((%x. a+x) ` B)" using assms by auto
   182   from this show ?thesis using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto 
   183 qed
   184 
   185 lemma translation_galois:
   186   fixes a :: "'a::ab_group_add"
   187   shows "T=((\<lambda>x. a+x) ` S) <-> S=((\<lambda>x. (-a)+x) ` T)"
   188   using translation_assoc[of "-a" a S] apply auto
   189   using translation_assoc[of a "-a" T] by auto
   190 
   191 lemma translation_inverse_subset:
   192   assumes "((%x. -a+x) ` V) <= (S :: 'n::ab_group_add set)" 
   193   shows "V <= ((%x. a+x) ` S)"
   194 proof-
   195 { fix x assume "x:V" hence "x-a : S" using assms by auto 
   196   hence "x : {a + v |v. v : S}" apply auto apply (rule exI[of _ "x-a"]) apply simp done 
   197   hence "x : ((%x. a+x) ` S)" by auto }
   198   from this show ?thesis by auto
   199 qed
   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 subsection {* Some explicit formulations (from Lars Schewe). *}
   401 
   402 lemma affine: fixes V::"'a::real_vector set"
   403   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)"
   404 unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
   405 defer apply(rule, rule, rule, rule, rule) proof-
   406   fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
   407     "\<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"
   408   thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
   409     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) 
   410     by(auto simp add: scaleR_left_distrib[THEN sym])
   411 next
   412   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"
   413     "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
   414   def n \<equiv> "card s"
   415   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   416   thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
   417     assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
   418     then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
   419     thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
   420       by(auto simp add: setsum_clauses(2))
   421   next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
   422       case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
   423       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;
   424                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
   425         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"
   426            "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
   427       have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
   428         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
   429         thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
   430           less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
   431       then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
   432 
   433       have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
   434       have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
   435       have **:"setsum u (s - {x}) = 1 - u x"
   436         using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
   437       have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
   438       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
   439         case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
   440           assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
   441           thus False using True by auto qed auto
   442         thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
   443         unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
   444       next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
   445         then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
   446         thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
   447           using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
   448       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"
   449         apply-apply(rule as(3)[rule_format]) 
   450         unfolding  RealVector.scaleR_right.setsum using x(1) as(6) by auto
   451       thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
   452          apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
   453          using `u x \<noteq> 1` by auto 
   454     qed auto
   455   next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
   456     thus ?thesis using as(4,5) by simp
   457   qed(insert `s\<noteq>{}` `finite s`, auto)
   458 qed
   459 
   460 lemma affine_hull_explicit:
   461   "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}"
   462   apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
   463   apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
   464   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"
   465     apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
   466 next
   467   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" 
   468   thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
   469 next
   470   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
   471     apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
   472     fix u v ::real assume uv:"u + v = 1"
   473     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"
   474     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
   475     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"
   476     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
   477     have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
   478     have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
   479     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"
   480       apply(rule_tac x="sx \<union> sy" in exI)
   481       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)
   482       unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
   483       unfolding scaleR_scaleR[THEN sym] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
   484       unfolding x y using x(1-3) y(1-3) uv by simp qed qed
   485 
   486 lemma affine_hull_finite:
   487   assumes "finite s"
   488   shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
   489   unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq apply (rule,rule)
   490   apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
   491   fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   492   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"
   493     apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
   494 next
   495   fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
   496   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"
   497   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)
   498     unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
   499 
   500 subsection {* Stepping theorems and hence small special cases. *}
   501 
   502 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   503   apply(rule hull_unique) unfolding mem_def by auto
   504 
   505 lemma affine_hull_finite_step:
   506   fixes y :: "'a::real_vector"
   507   shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
   508   "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>
   509                 (\<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)")
   510 proof-
   511   show ?th1 by simp
   512   assume ?as 
   513   { assume ?lhs
   514     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
   515     have ?rhs proof(cases "a\<in>s")
   516       case True hence *:"insert a s = s" by auto
   517       show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
   518     next
   519       case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
   520     qed  } moreover
   521   { assume ?rhs
   522     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
   523     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
   524     have ?lhs proof(cases "a\<in>s")
   525       case True thus ?thesis
   526         apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
   527         unfolding setsum_clauses(2)[OF `?as`]  apply simp
   528         unfolding scaleR_left_distrib and setsum_addf 
   529         unfolding vu and * and scaleR_zero_left
   530         by (auto simp add: setsum_delta[OF `?as`])
   531     next
   532       case False 
   533       hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
   534                "\<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
   535       from False show ?thesis
   536         apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
   537         unfolding setsum_clauses(2)[OF `?as`] and * using vu
   538         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)]
   539         using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
   540     qed }
   541   ultimately show "?lhs = ?rhs" by blast
   542 qed
   543 
   544 lemma affine_hull_2:
   545   fixes a b :: "'a::real_vector"
   546   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
   547 proof-
   548   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   549          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   550   have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
   551     using affine_hull_finite[of "{a,b}"] by auto
   552   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
   553     by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
   554   also have "\<dots> = ?rhs" unfolding * by auto
   555   finally show ?thesis by auto
   556 qed
   557 
   558 lemma affine_hull_3:
   559   fixes a b c :: "'a::real_vector"
   560   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")
   561 proof-
   562   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   563          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   564   show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
   565     unfolding * apply auto
   566     apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
   567     apply(rule_tac x=u in exI) by(auto intro!: exI)
   568 qed
   569 
   570 lemma mem_affine:
   571   assumes "affine S" "x : S" "y : S" "u+v=1"
   572   shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
   573   using assms affine_def[of S] by auto
   574 
   575 lemma mem_affine_3:
   576   assumes "affine S" "x : S" "y : S" "z : S" "u+v+w=1"
   577   shows "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : S"
   578 proof-
   579 have "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : affine hull {x, y, z}"
   580   using affine_hull_3[of x y z] assms by auto
   581 moreover have " affine hull {x, y, z} <= affine hull S" 
   582   using hull_mono[of "{x, y, z}" "S"] assms by auto
   583 moreover have "affine hull S = S" 
   584   using assms affine_hull_eq[of S] by auto
   585 ultimately show ?thesis by auto 
   586 qed
   587 
   588 lemma mem_affine_3_minus:
   589   assumes "affine S" "x : S" "y : S" "z : S"
   590   shows "x + v *\<^sub>R (y-z) : S"
   591 using mem_affine_3[of S x y z 1 v "-v"] assms by (simp add: algebra_simps)
   592 
   593 
   594 subsection {* Some relations between affine hull and subspaces. *}
   595 
   596 lemma affine_hull_insert_subset_span:
   597   fixes a :: "'a::euclidean_space"
   598   shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
   599   unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
   600   apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
   601   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"
   602   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
   603   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)"
   604     apply(rule_tac x="x - a" in exI)
   605     apply (rule conjI, simp)
   606     apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
   607     apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
   608     apply (rule conjI) using as(1) apply simp
   609     apply (erule conjI)
   610     using as(1)
   611     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)
   612     unfolding as by simp qed
   613 
   614 lemma affine_hull_insert_span:
   615   fixes a :: "'a::euclidean_space"
   616   assumes "a \<notin> s"
   617   shows "affine hull (insert a s) =
   618             {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
   619   apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
   620   unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
   621   fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
   622   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
   623   def f \<equiv> "(\<lambda>x. x + a) ` t"
   624   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 
   625     by(auto simp add: setsum_reindex[unfolded inj_on_def])
   626   have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
   627   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"
   628     apply(rule_tac x="insert a f" in exI)
   629     apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
   630     using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
   631     unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
   632     by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
   633 
   634 lemma affine_hull_span:
   635   fixes a :: "'a::euclidean_space"
   636   assumes "a \<in> s"
   637   shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
   638   using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
   639 
   640 subsection{* Parallel Affine Sets *}
   641 
   642 definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
   643 where "affine_parallel S T = (? a. T = ((%x. a + x) ` S))"
   644 
   645 lemma affine_parallel_expl_aux:
   646    fixes S T :: "'a::real_vector set"
   647    assumes "!x. (x : S <-> (a+x) : T)" 
   648    shows "T = ((%x. a + x) ` S)"
   649 proof-
   650 { fix x assume "x : T" hence "(-a)+x : S" using assms by auto
   651   hence " x : ((%x. a + x) ` S)" using imageI[of "-a+x" S "(%x. a+x)"] by auto}
   652 moreover have "T >= ((%x. a + x) ` S)" using assms by auto 
   653 ultimately show ?thesis by auto
   654 qed
   655 
   656 lemma affine_parallel_expl: 
   657    "affine_parallel S T = (? a. !x. (x : S <-> (a+x) : T))" 
   658    unfolding affine_parallel_def using affine_parallel_expl_aux[of S _ T] by auto
   659 
   660 lemma affine_parallel_reflex: "affine_parallel S S" unfolding affine_parallel_def apply (rule exI[of _ "0"]) by auto
   661 
   662 lemma affine_parallel_commut:
   663 assumes "affine_parallel A B" shows "affine_parallel B A" 
   664 proof-
   665 from assms obtain a where "B=((%x. a + x) ` A)" unfolding affine_parallel_def by auto 
   666 from this show ?thesis using translation_galois[of B a A] unfolding affine_parallel_def by auto
   667 qed
   668 
   669 lemma affine_parallel_assoc:
   670 assumes "affine_parallel A B" "affine_parallel B C"
   671 shows "affine_parallel A C" 
   672 proof-
   673 from assms obtain ab where "B=((%x. ab + x) ` A)" unfolding affine_parallel_def by auto 
   674 moreover 
   675 from assms obtain bc where "C=((%x. bc + x) ` B)" unfolding affine_parallel_def by auto
   676 ultimately show ?thesis using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto 
   677 qed
   678 
   679 lemma affine_translation_aux:
   680   fixes a :: "'a::real_vector"
   681   assumes "affine ((%x. a + x) ` S)" shows "affine S"
   682 proof-
   683 { fix x y u v
   684   assume xy: "x : S" "y : S" "(u :: real)+v=1"
   685   hence "(a+x):((%x. a + x) ` S)" "(a+y):((%x. a + x) ` S)" by auto
   686   hence h1: "u *\<^sub>R  (a+x) + v *\<^sub>R (a+y) : ((%x. a + x) ` S)" using xy assms unfolding affine_def by auto
   687   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)
   688   also have "...= a + (u *\<^sub>R x + v *\<^sub>R y)" using `u+v=1` by auto
   689   ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) : ((%x. a + x) ` S)" using h1 by auto
   690   hence "u *\<^sub>R x + v *\<^sub>R y : S" by auto
   691 } from this show ?thesis unfolding affine_def by auto
   692 qed
   693 
   694 lemma affine_translation:
   695   fixes a :: "'a::real_vector"
   696   shows "affine S <-> affine ((%x. a + x) ` S)"
   697 proof-
   698 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
   699 from this show ?thesis using affine_translation_aux by auto
   700 qed
   701 
   702 lemma parallel_is_affine:
   703 fixes S T :: "'a::real_vector set"
   704 assumes "affine S" "affine_parallel S T"
   705 shows "affine T"
   706 proof-
   707   from assms obtain a where "T=((%x. a + x) ` S)" unfolding affine_parallel_def by auto 
   708   from this show ?thesis using affine_translation assms by auto
   709 qed
   710 
   711 lemma subspace_imp_affine:
   712   fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> affine s"
   713   unfolding subspace_def affine_def by auto
   714 
   715 subsection{* Subspace Parallel to an Affine Set *}
   716 
   717 lemma subspace_affine:
   718   fixes S :: "('n::euclidean_space) set"
   719   shows "subspace S <-> (affine S & 0 : S)"
   720 proof-
   721 have h0: "subspace S ==> (affine S & 0 : S)" using subspace_imp_affine[of S] subspace_0 by auto
   722 { assume assm: "affine S & 0 : S"
   723   { fix c :: real 
   724     fix x assume x_def: "x : S"
   725     have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
   726     moreover have "(1-c) *\<^sub>R 0 + c *\<^sub>R x : S" using affine_alt[of S] assm x_def by auto
   727     ultimately have "c *\<^sub>R x : S" by auto
   728   } hence h1: "!c. !x : S. c *\<^sub>R x : S" by auto
   729   { fix x y assume xy_def: "x : S" "y : S"
   730     def u == "(1 :: real)/2"
   731     have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)" by auto
   732     moreover have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y" by (simp add: algebra_simps)
   733     moreover have "(1-u) *\<^sub>R x + u *\<^sub>R y : S" using affine_alt[of S] assm xy_def by auto
   734     ultimately have "(1/2) *\<^sub>R (x+y) : S" using u_def by auto
   735     moreover have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" by auto
   736     ultimately have "(x+y) : S" using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
   737   } hence "!x : S. !y : S. (x+y) : S" by auto 
   738   hence "subspace S" using h1 assm unfolding subspace_def by auto
   739 } from this show ?thesis using h0 by metis
   740 qed
   741 
   742 lemma affine_diffs_subspace:
   743   fixes S :: "('n::euclidean_space) set"
   744   assumes "affine S" "a : S"
   745   shows "subspace ((%x. (-a)+x) ` S)"
   746 proof-
   747 have "affine ((%x. (-a)+x) ` S)" using  affine_translation assms by auto  
   748 moreover have "0 : ((%x. (-a)+x) ` S)" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
   749 ultimately show ?thesis using subspace_affine by auto 
   750 qed
   751 
   752 lemma parallel_subspace_explicit:
   753 fixes a :: "'n::euclidean_space"
   754 assumes "affine S" "a : S"
   755 assumes "L == {y. ? x : S. (-a)+x=y}" 
   756 shows "subspace L & affine_parallel S L" 
   757 proof-
   758 have par: "affine_parallel S L" unfolding affine_parallel_def using assms by auto
   759 hence "affine L" using assms parallel_is_affine by auto  
   760 moreover have "0 : L" using assms apply auto using exI[of "(%x. x:S & -a+x=0)" a] by auto
   761 ultimately show ?thesis using subspace_affine par by auto 
   762 qed
   763 
   764 lemma parallel_subspace_aux:
   765 fixes A B :: "('n::euclidean_space) set"
   766 assumes "subspace A" "subspace B" "affine_parallel A B"
   767 shows "A>=B"
   768 proof-
   769 from assms obtain a where a_def: "!x. (x : A <-> (a+x) : B)" using affine_parallel_expl[of A B] by auto
   770 hence "-a : A" using assms subspace_0[of B] by auto
   771 hence "a : A" using assms subspace_neg[of A "-a"] by auto
   772 from this show ?thesis using assms a_def unfolding subspace_def by auto
   773 qed
   774 
   775 lemma parallel_subspace:
   776 fixes A B :: "('n::euclidean_space) set"
   777 assumes "subspace A" "subspace B" "affine_parallel A B"
   778 shows "A=B"
   779 proof-
   780 have "A>=B" using assms parallel_subspace_aux by auto
   781 moreover have "A<=B" using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
   782 ultimately show ?thesis by auto  
   783 qed
   784 
   785 lemma affine_parallel_subspace:
   786 fixes S :: "('n::euclidean_space) set"
   787 assumes "affine S" "S ~= {}"
   788 shows "?!L. subspace L & affine_parallel S L" 
   789 proof-
   790 have ex: "? L. subspace L & affine_parallel S L" using assms  parallel_subspace_explicit by auto 
   791 { fix L1 L2 assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"
   792   hence "affine_parallel L1 L2" using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
   793   hence "L1=L2" using ass parallel_subspace by auto
   794 } from this show ?thesis using ex by auto
   795 qed
   796 
   797 subsection {* Cones. *}
   798 
   799 definition
   800   cone :: "'a::real_vector set \<Rightarrow> bool" where
   801   "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   802 
   803 lemma cone_empty[intro, simp]: "cone {}"
   804   unfolding cone_def by auto
   805 
   806 lemma cone_univ[intro, simp]: "cone UNIV"
   807   unfolding cone_def by auto
   808 
   809 lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
   810   unfolding cone_def by auto
   811 
   812 subsection {* Conic hull. *}
   813 
   814 lemma cone_cone_hull: "cone (cone hull s)"
   815   unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
   816   by (auto simp add: mem_def)
   817 
   818 lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
   819   apply(rule hull_eq[unfolded mem_def])
   820   using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
   821 
   822 lemma mem_cone:
   823   assumes "cone S" "x : S" "c>=0"
   824   shows "c *\<^sub>R x : S"
   825   using assms cone_def[of S] by auto
   826 
   827 lemma cone_contains_0:
   828 fixes S :: "('m::euclidean_space) set"
   829 assumes "cone S"
   830 shows "(S ~= {}) <-> (0 : S)"
   831 proof-
   832 { assume "S ~= {}" from this obtain a where "a:S" by auto
   833   hence "0 : S" using assms mem_cone[of S a 0] by auto
   834 } from this show ?thesis by auto
   835 qed
   836 
   837 lemma cone_0:
   838 shows "cone {(0 :: 'm::euclidean_space)}"
   839 unfolding cone_def by auto
   840 
   841 lemma cone_Union[intro]: "(!s:f. cone s) --> (cone (Union f))"
   842   unfolding cone_def by blast
   843 
   844 lemma cone_iff:
   845 fixes S :: "('m::euclidean_space) set"
   846 assumes "S ~= {}"
   847 shows "cone S <-> 0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
   848 proof-
   849 { assume "cone S"
   850   { fix c assume "(c :: real)>0"
   851     { fix x assume "x : S" hence "x : (op *\<^sub>R c) ` S" unfolding image_def
   852         using `cone S` `c>0` mem_cone[of S x "1/c"]
   853         exI[of "(%t. t:S & x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"] by auto
   854     }
   855     moreover
   856     { fix x assume "x : (op *\<^sub>R c) ` S"
   857       (*from this obtain t where "t:S & x = c *\<^sub>R t" by auto*)
   858       hence "x:S" using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto
   859     }
   860     ultimately have "((op *\<^sub>R c) ` S) = S" by auto
   861   } hence "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)" using `cone S` cone_contains_0[of S] assms by auto
   862 }
   863 moreover
   864 { assume a: "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
   865   { fix x assume "x:S"
   866     fix c1 assume "(c1 :: real)>=0"
   867     hence "(c1=0) | (c1>0)" by auto
   868     hence "c1 *\<^sub>R x : S" using a `x:S` by auto
   869   }
   870  hence "cone S" unfolding cone_def by auto
   871 } ultimately show ?thesis by blast
   872 qed
   873 
   874 lemma cone_hull_empty:
   875 "cone hull {} = {}"
   876 by (metis cone_empty cone_hull_eq)
   877 
   878 lemma cone_hull_empty_iff:
   879 fixes S :: "('m::euclidean_space) set"
   880 shows "(S = {}) <-> (cone hull S = {})"
   881 by (metis bot_least cone_hull_empty hull_subset xtrans(5))
   882 
   883 lemma cone_hull_contains_0: 
   884 fixes S :: "('m::euclidean_space) set"
   885 shows "(S ~= {}) <-> (0 : cone hull S)"
   886 using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] by auto
   887 
   888 lemma mem_cone_hull:
   889   assumes "x : S" "c>=0"
   890   shows "c *\<^sub>R x : cone hull S"
   891 by (metis assms cone_cone_hull hull_inc mem_cone mem_def)
   892 
   893 lemma cone_hull_expl:
   894 fixes S :: "('m::euclidean_space) set"
   895 shows "cone hull S = {c *\<^sub>R x | c x. c>=0 & x : S}" (is "?lhs = ?rhs")
   896 proof-
   897 { fix x assume "x : ?rhs"
   898   from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
   899   fix c assume c_def: "(c :: real)>=0"
   900   hence "c *\<^sub>R x = (c*cx) *\<^sub>R xx" using x_def by (simp add: algebra_simps)
   901   moreover have "(c*cx) >= 0" using c_def x_def using mult_nonneg_nonneg by auto
   902   ultimately have "c *\<^sub>R x : ?rhs" using x_def by auto
   903 } hence "cone ?rhs" unfolding cone_def by auto
   904   hence "?rhs : cone" unfolding mem_def by auto
   905 { fix x assume "x : S" hence "1 *\<^sub>R x : ?rhs" apply auto apply(rule_tac x="1" in exI) by auto
   906   hence "x : ?rhs" by auto
   907 } hence "S <= ?rhs" by auto
   908 hence "?lhs <= ?rhs" using `?rhs : cone` hull_minimal[of S "?rhs" "cone"] by auto
   909 moreover
   910 { fix x assume "x : ?rhs"
   911   from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
   912   hence "xx : cone hull S" using hull_subset[of S] by auto
   913   hence "x : ?lhs" using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
   914 } ultimately show ?thesis by auto
   915 qed
   916 
   917 lemma cone_closure:
   918 fixes S :: "('m::euclidean_space) set"
   919 assumes "cone S"
   920 shows "cone (closure S)"
   921 proof-
   922 { assume "S = {}" hence ?thesis by auto }
   923 moreover
   924 { assume "S ~= {}" hence "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
   925   hence "0:(closure S) & (!c. c>0 --> op *\<^sub>R c ` (closure S) = (closure S))"
   926      using closure_subset by (auto simp add: closure_scaleR)
   927   hence ?thesis using cone_iff[of "closure S"] by auto
   928 }
   929 ultimately show ?thesis by blast
   930 qed
   931 
   932 subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
   933 
   934 definition
   935   affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
   936   "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
   937 
   938 lemma affine_dependent_explicit:
   939   "affine_dependent p \<longleftrightarrow>
   940     (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
   941     (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   942   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
   943   apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
   944 proof-
   945   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"
   946   have "x\<notin>s" using as(1,4) by auto
   947   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"
   948     apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
   949     unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
   950 next
   951   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"
   952   have "s \<noteq> {v}" using as(3,6) by auto
   953   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" 
   954     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)
   955     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
   956 qed
   957 
   958 lemma affine_dependent_explicit_finite:
   959   fixes s :: "'a::real_vector set" assumes "finite s"
   960   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)"
   961   (is "?lhs = ?rhs")
   962 proof
   963   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
   964   assume ?lhs
   965   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"
   966     unfolding affine_dependent_explicit by auto
   967   thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   968     apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
   969     unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
   970 next
   971   assume ?rhs
   972   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
   973   thus ?lhs unfolding affine_dependent_explicit using assms by auto
   974 qed
   975 
   976 subsection {* A general lemma. *}
   977 
   978 lemma convex_connected:
   979   fixes s :: "'a::real_normed_vector set"
   980   assumes "convex s" shows "connected s"
   981 proof-
   982   { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
   983     assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
   984     then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
   985     hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
   986 
   987     { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
   988       { 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"
   989           by (simp add: algebra_simps)
   990         assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
   991         hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
   992           unfolding * and scaleR_right_diff_distrib[THEN sym]
   993           unfolding less_divide_eq using n by auto  }
   994       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"
   995         apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
   996         apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
   997 
   998     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"
   999       apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
  1000       using * apply(simp add: dist_norm)
  1001       using as(1,2)[unfolded open_dist] apply simp
  1002       using as(1,2)[unfolded open_dist] apply simp
  1003       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
  1004       using as(3) by auto
  1005     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
  1006     hence False using as(4) 
  1007       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
  1008       using x1(2) x2(2) by auto  }
  1009   thus ?thesis unfolding connected_def by auto
  1010 qed
  1011 
  1012 subsection {* One rather trivial consequence. *}
  1013 
  1014 lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
  1015   by(simp add: convex_connected convex_UNIV)
  1016 
  1017 subsection {* Balls, being convex, are connected. *}
  1018 
  1019 lemma convex_box: fixes a::"'a::euclidean_space"
  1020   assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
  1021   shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
  1022   using assms unfolding convex_def by(auto simp add:euclidean_simps)
  1023 
  1024 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
  1025   by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
  1026 
  1027 lemma convex_local_global_minimum:
  1028   fixes s :: "'a::real_normed_vector set"
  1029   assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
  1030   shows "\<forall>y\<in>s. f x \<le> f y"
  1031 proof(rule ccontr)
  1032   have "x\<in>s" using assms(1,3) by auto
  1033   assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
  1034   then obtain y where "y\<in>s" and y:"f x > f y" by auto
  1035   hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
  1036 
  1037   then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
  1038     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
  1039   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`
  1040     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
  1041   moreover
  1042   have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
  1043   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]
  1044     using u unfolding pos_less_divide_eq[OF xy] by auto
  1045   hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
  1046   ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
  1047 qed
  1048 
  1049 lemma convex_ball:
  1050   fixes x :: "'a::real_normed_vector"
  1051   shows "convex (ball x e)" 
  1052 proof(auto simp add: convex_def)
  1053   fix y z assume yz:"dist x y < e" "dist x z < e"
  1054   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
  1055   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
  1056     using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
  1057   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using convex_bound_lt[OF yz uv] by auto
  1058 qed
  1059 
  1060 lemma convex_cball:
  1061   fixes x :: "'a::real_normed_vector"
  1062   shows "convex(cball x e)"
  1063 proof(auto simp add: convex_def Ball_def)
  1064   fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
  1065   fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
  1066   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
  1067     using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
  1068   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto 
  1069 qed
  1070 
  1071 lemma connected_ball:
  1072   fixes x :: "'a::real_normed_vector"
  1073   shows "connected (ball x e)"
  1074   using convex_connected convex_ball by auto
  1075 
  1076 lemma connected_cball:
  1077   fixes x :: "'a::real_normed_vector"
  1078   shows "connected(cball x e)"
  1079   using convex_connected convex_cball by auto
  1080 
  1081 subsection {* Convex hull. *}
  1082 
  1083 lemma convex_convex_hull: "convex(convex hull s)"
  1084   unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
  1085   unfolding mem_def by auto
  1086 
  1087 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
  1088 by (metis convex_convex_hull hull_same mem_def)
  1089 
  1090 lemma bounded_convex_hull:
  1091   fixes s :: "'a::real_normed_vector set"
  1092   assumes "bounded s" shows "bounded(convex hull s)"
  1093 proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
  1094   show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
  1095     unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
  1096     unfolding subset_eq mem_cball dist_norm using B by auto qed
  1097 
  1098 lemma finite_imp_bounded_convex_hull:
  1099   fixes s :: "'a::real_normed_vector set"
  1100   shows "finite s \<Longrightarrow> bounded(convex hull s)"
  1101   using bounded_convex_hull finite_imp_bounded by auto
  1102 
  1103 subsection {* Convex hull is "preserved" by a linear function. *}
  1104 
  1105 lemma convex_hull_linear_image:
  1106   assumes "bounded_linear f"
  1107   shows "f ` (convex hull s) = convex hull (f ` s)"
  1108   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  1109   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  1110   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  1111 proof-
  1112   interpret f: bounded_linear f by fact
  1113   show "convex {x. f x \<in> convex hull f ` s}" 
  1114   unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
  1115   interpret f: bounded_linear f by fact
  1116   show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
  1117     unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  1118 qed auto
  1119 
  1120 lemma in_convex_hull_linear_image:
  1121   assumes "bounded_linear f" "x \<in> convex hull s"
  1122   shows "(f x) \<in> convex hull (f ` s)"
  1123 using convex_hull_linear_image[OF assms(1)] assms(2) by auto
  1124 
  1125 subsection {* Stepping theorems for convex hulls of finite sets. *}
  1126 
  1127 lemma convex_hull_empty[simp]: "convex hull {} = {}"
  1128   apply(rule hull_unique) unfolding mem_def by auto
  1129 
  1130 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
  1131   apply(rule hull_unique) unfolding mem_def by auto
  1132 
  1133 lemma convex_hull_insert:
  1134   fixes s :: "'a::real_vector set"
  1135   assumes "s \<noteq> {}"
  1136   shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
  1137                                     b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
  1138  apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
  1139  fix x assume x:"x = a \<or> x \<in> s"
  1140  thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
  1141    apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
  1142 next
  1143   fix x assume "x\<in>?hull"
  1144   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
  1145   have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
  1146     using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
  1147   thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
  1148     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
  1149 next
  1150   show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
  1151     fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
  1152     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
  1153     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
  1154     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)
  1155     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)"
  1156     proof(cases "u * v1 + v * v2 = 0")
  1157       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)
  1158       case True hence **:"u * v1 = 0" "v * v2 = 0"
  1159         using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by arith+
  1160       hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
  1161       thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
  1162     next
  1163       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)
  1164       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
  1165       also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
  1166       case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
  1167         apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
  1168         using as(1,2) obt1(1,2) obt2(1,2) by auto 
  1169       thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
  1170         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
  1171         apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
  1172         unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
  1173         by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
  1174     qed note * = this
  1175     have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
  1176     have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
  1177     have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
  1178       apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
  1179     also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
  1180     finally 
  1181     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)
  1182       apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
  1183       using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
  1184   qed
  1185 qed
  1186 
  1187 
  1188 subsection {* Explicit expression for convex hull. *}
  1189 
  1190 lemma convex_hull_indexed:
  1191   fixes s :: "'a::real_vector set"
  1192   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>
  1193                             (setsum u {1..k} = 1) \<and>
  1194                             (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
  1195   apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
  1196   apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
  1197 proof-
  1198   fix x assume "x\<in>s"
  1199   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
  1200 next
  1201   fix t assume as:"s \<subseteq> t" "convex t"
  1202   show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
  1203     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"
  1204     show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
  1205       using assm(1,2) as(1) by auto qed
  1206 next
  1207   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"
  1208   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
  1209   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
  1210   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)"
  1211     "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
  1212     prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
  1213   have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
  1214   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
  1215     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)
  1216     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)
  1217     unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
  1218     unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
  1219     fix i assume i:"i \<in> {1..k1+k2}"
  1220     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"
  1221     proof(cases "i\<in>{1..k1}")
  1222       case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
  1223     next def j \<equiv> "i - k1"
  1224       case False with i have "j \<in> {1..k2}" unfolding j_def by auto
  1225       thus ?thesis unfolding j_def[symmetric] using False
  1226         using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
  1227   qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
  1228 qed
  1229 
  1230 lemma convex_hull_finite:
  1231   fixes s :: "'a::real_vector set"
  1232   assumes "finite s"
  1233   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
  1234          setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
  1235 proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
  1236   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" 
  1237     apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
  1238     unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
  1239 next
  1240   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
  1241   fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
  1242   fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
  1243   { fix x assume "x\<in>s"
  1244     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)
  1245       by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
  1246   moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
  1247     unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
  1248   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)"
  1249     unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
  1250   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)"
  1251     apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
  1252 next
  1253   fix t assume t:"s \<subseteq> t" "convex t" 
  1254   fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
  1255   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]]
  1256     using assms and t(1) by auto
  1257 qed
  1258 
  1259 subsection {* Another formulation from Lars Schewe. *}
  1260 
  1261 lemma setsum_constant_scaleR:
  1262   fixes y :: "'a::real_vector"
  1263   shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
  1264 apply (cases "finite A")
  1265 apply (induct set: finite)
  1266 apply (simp_all add: algebra_simps)
  1267 done
  1268 
  1269 lemma convex_hull_explicit:
  1270   fixes p :: "'a::real_vector set"
  1271   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
  1272              (\<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")
  1273 proof-
  1274   { fix x assume "x\<in>?lhs"
  1275     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"
  1276       unfolding convex_hull_indexed by auto
  1277 
  1278     have fin:"finite {1..k}" by auto
  1279     have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
  1280     { fix j assume "j\<in>{1..k}"
  1281       hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
  1282         using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
  1283         apply(rule setsum_nonneg) using obt(1) by auto } 
  1284     moreover
  1285     have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
  1286       unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
  1287     moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
  1288       using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
  1289       unfolding scaleR_left.setsum using obt(3) by auto
  1290     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"
  1291       apply(rule_tac x="y ` {1..k}" in exI)
  1292       apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
  1293     hence "x\<in>?rhs" by auto  }
  1294   moreover
  1295   { fix y assume "y\<in>?rhs"
  1296     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
  1297 
  1298     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
  1299     
  1300     { fix i::nat assume "i\<in>{1..card s}"
  1301       hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
  1302       hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
  1303     moreover have *:"finite {1..card s}" by auto
  1304     { fix y assume "y\<in>s"
  1305       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
  1306       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
  1307       hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
  1308       hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
  1309             "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
  1310         by (auto simp add: setsum_constant_scaleR)   }
  1311 
  1312     hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
  1313       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] 
  1314       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"]
  1315       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
  1316     
  1317     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"
  1318       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
  1319     hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
  1320   ultimately show ?thesis unfolding set_eq_iff by blast
  1321 qed
  1322 
  1323 subsection {* A stepping theorem for that expansion. *}
  1324 
  1325 lemma convex_hull_finite_step:
  1326   fixes s :: "'a::real_vector set" assumes "finite s"
  1327   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)
  1328      \<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")
  1329 proof(rule, case_tac[!] "a\<in>s")
  1330   assume "a\<in>s" hence *:"insert a s = s" by auto
  1331   assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
  1332 next
  1333   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
  1334   assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
  1335     apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
  1336 next
  1337   assume "a\<in>s" hence *:"insert a s = s" by auto
  1338   have fin:"finite (insert a s)" using assms by auto
  1339   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
  1340   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]
  1341     unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
  1342 next
  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   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)"
  1345     apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
  1346   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
  1347 qed
  1348 
  1349 subsection {* Hence some special cases. *}
  1350 
  1351 lemma convex_hull_2:
  1352   "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}"
  1353 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
  1354 show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
  1355   apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
  1356   apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
  1357 
  1358 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
  1359   unfolding convex_hull_2 unfolding Collect_def 
  1360 proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
  1361   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)"
  1362     unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
  1363 
  1364 lemma convex_hull_3:
  1365   "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}"
  1366 proof-
  1367   have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
  1368   have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
  1369          "\<And>x y z ::_::euclidean_space. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: field_simps)
  1370   show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
  1371     unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
  1372     apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
  1373     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
  1374 
  1375 lemma convex_hull_3_alt:
  1376   "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}"
  1377 proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
  1378   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)
  1379     apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
  1380 
  1381 subsection {* Relations among closure notions and corresponding hulls. *}
  1382 
  1383 text {* TODO: Generalize linear algebra concepts defined in @{text
  1384 Euclidean_Space.thy} so that we can generalize these lemmas. *}
  1385 
  1386 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
  1387   unfolding affine_def convex_def by auto
  1388 
  1389 lemma subspace_imp_convex:
  1390   fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> convex s"
  1391   using subspace_imp_affine affine_imp_convex by auto
  1392 
  1393 lemma affine_hull_subset_span:
  1394   fixes s :: "(_::euclidean_space) set" shows "(affine hull s) \<subseteq> (span s)"
  1395 by (metis hull_minimal mem_def span_inc subspace_imp_affine subspace_span)
  1396 
  1397 lemma convex_hull_subset_span:
  1398   fixes s :: "(_::euclidean_space) set" shows "(convex hull s) \<subseteq> (span s)"
  1399 by (metis hull_minimal mem_def span_inc subspace_imp_convex subspace_span)
  1400 
  1401 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
  1402 by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset mem_def)
  1403 
  1404 
  1405 lemma affine_dependent_imp_dependent:
  1406   fixes s :: "(_::euclidean_space) set" shows "affine_dependent s \<Longrightarrow> dependent s"
  1407   unfolding affine_dependent_def dependent_def 
  1408   using affine_hull_subset_span by auto
  1409 
  1410 lemma dependent_imp_affine_dependent:
  1411   fixes s :: "(_::euclidean_space) set"
  1412   assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
  1413   shows "affine_dependent (insert a s)"
  1414 proof-
  1415   from assms(1)[unfolded dependent_explicit] obtain S u v 
  1416     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
  1417   def t \<equiv> "(\<lambda>x. x + a) ` S"
  1418 
  1419   have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
  1420   have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
  1421   have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
  1422 
  1423   hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
  1424   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
  1425     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1426   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
  1427     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
  1428   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"
  1429     apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
  1430   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)"
  1431     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1432   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
  1433     unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
  1434     using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
  1435   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"
  1436     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *)
  1437   ultimately show ?thesis unfolding affine_dependent_explicit
  1438     apply(rule_tac x="insert a t" in exI) by auto 
  1439 qed
  1440 
  1441 lemma convex_cone:
  1442   "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")
  1443 proof-
  1444   { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
  1445     hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
  1446     hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
  1447       apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
  1448       apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
  1449   thus ?thesis unfolding convex_def cone_def by blast
  1450 qed
  1451 
  1452 lemma affine_dependent_biggerset: fixes s::"('a::euclidean_space) set"
  1453   assumes "finite s" "card s \<ge> DIM('a) + 2"
  1454   shows "affine_dependent s"
  1455 proof-
  1456   have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
  1457   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1458   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1459     apply(rule card_image) unfolding inj_on_def by auto
  1460   also have "\<dots> > DIM('a)" using assms(2)
  1461     unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1462   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1463     apply(rule dependent_imp_affine_dependent)
  1464     apply(rule dependent_biggerset) by auto qed
  1465 
  1466 lemma affine_dependent_biggerset_general:
  1467   assumes "finite (s::('a::euclidean_space) set)" "card s \<ge> dim s + 2"
  1468   shows "affine_dependent s"
  1469 proof-
  1470   from assms(2) have "s \<noteq> {}" by auto
  1471   then obtain a where "a\<in>s" by auto
  1472   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1473   have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1474     apply(rule card_image) unfolding inj_on_def by auto
  1475   have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
  1476     apply(rule subset_le_dim) unfolding subset_eq
  1477     using `a\<in>s` by (auto simp add:span_superset span_sub)
  1478   also have "\<dots> < dim s + 1" by auto
  1479   also have "\<dots> \<le> card (s - {a})" using assms
  1480     using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1481   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1482     apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
  1483 
  1484 subsection {* Caratheodory's theorem. *}
  1485 
  1486 lemma convex_hull_caratheodory: fixes p::"('a::euclidean_space) set"
  1487   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
  1488   (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  1489   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
  1490 proof(rule,rule)
  1491   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"
  1492   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"
  1493   then obtain N where "?P N" by auto
  1494   hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
  1495   then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
  1496   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
  1497 
  1498   have "card s \<le> DIM('a) + 1" proof(rule ccontr, simp only: not_le)
  1499     assume "DIM('a) + 1 < card s"
  1500     hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
  1501     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"
  1502       using affine_dependent_explicit_finite[OF obt(1)] by auto
  1503     def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
  1504     have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
  1505       assume as:"\<forall>x\<in>s. 0 \<le> w x"
  1506       hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
  1507       hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
  1508         using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
  1509       thus False using wv(1) by auto
  1510     qed hence "i\<noteq>{}" unfolding i_def by auto
  1511 
  1512     hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
  1513       using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
  1514     have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
  1515       fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
  1516       show"0 \<le> u v + t * w v" proof(cases "w v < 0")
  1517         case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
  1518           using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
  1519         case True hence "t \<le> u v / (- w v)" using `v\<in>s`
  1520           unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
  1521         thus ?thesis unfolding real_0_le_add_iff
  1522           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
  1523       qed qed
  1524 
  1525     obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
  1526       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
  1527     hence a:"a\<in>s" "u a + t * w a = 0" by auto
  1528     have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
  1529       unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
  1530     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
  1531       unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
  1532     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" 
  1533       unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
  1534       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
  1535     ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
  1536       apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a
  1537       by (auto simp add: * scaleR_left_distrib)
  1538     thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
  1539   thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1
  1540     \<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
  1541 qed auto
  1542 
  1543 lemma caratheodory:
  1544  "convex hull p = {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
  1545       card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
  1546   unfolding set_eq_iff apply(rule, rule) unfolding mem_Collect_eq proof-
  1547   fix x assume "x \<in> convex hull p"
  1548   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
  1549      "\<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
  1550   thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  1551     apply(rule_tac x=s in exI) using hull_subset[of s convex]
  1552   using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
  1553 next
  1554   fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  1555   then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s" by auto
  1556   thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
  1557 qed
  1558 
  1559 
  1560 subsection {* Some Properties of Affine Dependent Sets *}
  1561 
  1562 lemma affine_independent_empty: "~(affine_dependent {})"
  1563   by (simp add: affine_dependent_def)
  1564 
  1565 lemma affine_independent_sing:
  1566 fixes a :: "'n::euclidean_space" 
  1567 shows "~(affine_dependent {a})"
  1568  by (simp add: affine_dependent_def)
  1569 
  1570 lemma affine_hull_translation:
  1571 "affine hull ((%x. a + x) `  S) = (%x. a + x) ` (affine hull S)"
  1572 proof-
  1573 have "affine ((%x. a + x) ` (affine hull S))" using affine_translation affine_affine_hull by auto
  1574 moreover have "(%x. a + x) `  S <= (%x. a + x) ` (affine hull S)" using hull_subset[of S] by auto
  1575 ultimately have h1: "affine hull ((%x. a + x) `  S) <= (%x. a + x) ` (affine hull S)" by (metis hull_minimal mem_def)
  1576 have "affine((%x. -a + x) ` (affine hull ((%x. a + x) `  S)))"  using affine_translation affine_affine_hull by auto
  1577 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 
  1578 moreover have "S=(%x. -a + x) ` (%x. a + x) `  S" using  translation_assoc[of "-a" a] by auto
  1579 ultimately have "(%x. -a + x) ` (affine hull ((%x. a + x) `  S)) >= (affine hull S)" by (metis hull_minimal mem_def)
  1580 hence "affine hull ((%x. a + x) `  S) >= (%x. a + x) ` (affine hull S)" by auto
  1581 from this show ?thesis using h1 by auto
  1582 qed
  1583 
  1584 lemma affine_dependent_translation:
  1585   assumes "affine_dependent S"
  1586   shows "affine_dependent ((%x. a + x) ` S)"
  1587 proof-
  1588 obtain x where x_def: "x : S & x : affine hull (S - {x})" using assms affine_dependent_def by auto
  1589 have "op + a ` (S - {x}) = op + a ` S - {a + x}" by auto
  1590 hence "a+x : affine hull ((%x. a + x) ` S - {a+x})" using  affine_hull_translation[of a "S-{x}"] x_def by auto
  1591 moreover have "a+x : (%x. a + x) ` S" using x_def by auto  
  1592 ultimately show ?thesis unfolding affine_dependent_def by auto 
  1593 qed
  1594 
  1595 lemma affine_dependent_translation_eq:
  1596   "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"
  1597 proof-
  1598 { assume "affine_dependent ((%x. a + x) ` S)" 
  1599   hence "affine_dependent S" using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] by auto  
  1600 } from this show ?thesis using affine_dependent_translation by auto
  1601 qed
  1602 
  1603 lemma affine_hull_0_dependent:
  1604   fixes S ::  "('n::euclidean_space) set"
  1605   assumes "0 : affine hull S"
  1606   shows "dependent S"
  1607 proof-
  1608 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
  1609 hence "EX v:s. u v ~= 0" using setsum_not_0[of "u" "s"] by auto 
  1610 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
  1611 from this show ?thesis unfolding dependent_explicit[of S] by auto
  1612 qed
  1613 
  1614 lemma affine_dependent_imp_dependent2:
  1615   fixes S :: "('n::euclidean_space) set" 
  1616   assumes "affine_dependent (insert 0 S)"
  1617   shows "dependent S"
  1618 proof-
  1619 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
  1620 hence "x : span (insert 0 S - {x})" using affine_hull_subset_span by auto
  1621 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
  1622 ultimately have "x : span (S - {x})" by auto
  1623 hence "(x~=0) ==> dependent S" using x_def dependent_def by auto
  1624 moreover
  1625 { assume "x=0" hence "0 : affine hull S" using x_def hull_mono[of "S - {0}" S] by auto
  1626                hence "dependent S" using affine_hull_0_dependent by auto  
  1627 } ultimately show ?thesis by auto
  1628 qed
  1629 
  1630 lemma affine_dependent_iff_dependent:
  1631   fixes S :: "('n::euclidean_space) set" 
  1632   assumes "a ~: S"
  1633   shows "affine_dependent (insert a S) <-> dependent ((%x. -a + x) ` S)" 
  1634 proof-
  1635 have "(op + (- a) ` S)={x - a| x . x : S}" by auto
  1636 from this show ?thesis using affine_dependent_translation_eq[of "(insert a S)" "-a"] 
  1637       affine_dependent_imp_dependent2 assms 
  1638       dependent_imp_affine_dependent[of a S] by auto
  1639 qed
  1640 
  1641 lemma affine_dependent_iff_dependent2:
  1642   fixes S :: "('n::euclidean_space) set" 
  1643   assumes "a : S"
  1644   shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"
  1645 proof-
  1646 have "insert a (S - {a})=S" using assms by auto
  1647 from this show ?thesis using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto 
  1648 qed
  1649 
  1650 lemma affine_hull_insert_span_gen:
  1651   fixes a :: "_::euclidean_space"
  1652   shows "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)" 
  1653 proof-
  1654 have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)" by auto
  1655 { assume "a ~: s" hence ?thesis using affine_hull_insert_span[of a s] h1 by auto}  
  1656 moreover
  1657 { assume a1: "a : s"
  1658   have "EX x. x:s & -a+x=0" apply (rule exI[of _ a]) using a1 by auto
  1659   hence "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s" by auto
  1660   hence "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)" 
  1661     using span_insert_0[of "op + (- a) ` (s - {a})"] by auto
  1662   moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))" by auto 
  1663   moreover have "insert a (s - {a})=(insert a s)" using assms by auto
  1664   ultimately have ?thesis using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
  1665 } 
  1666 ultimately show ?thesis by auto  
  1667 qed
  1668 
  1669 lemma affine_hull_span2:
  1670   fixes a :: "_::euclidean_space"
  1671   assumes "a : s"
  1672   shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` (s-{a}))"
  1673   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
  1674 
  1675 lemma affine_hull_span_gen:
  1676   fixes a :: "_::euclidean_space"
  1677   assumes "a : affine hull s"
  1678   shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` s)"
  1679 proof-
  1680 have "affine hull (insert a s) = affine hull s" using hull_redundant[of a affine s] assms by auto
  1681 from this show ?thesis using affine_hull_insert_span_gen[of a "s"] by auto
  1682 qed
  1683 
  1684 lemma affine_hull_span_0:
  1685   assumes "(0 :: _::euclidean_space) : affine hull S"
  1686   shows "affine hull S = span S"
  1687 using affine_hull_span_gen[of "0" S] assms by auto
  1688 
  1689 
  1690 lemma extend_to_affine_basis:
  1691 fixes S V :: "('n::euclidean_space) set"
  1692 assumes "~(affine_dependent S)" "S <= V" "S~={}"
  1693 shows "? T. ~(affine_dependent T) & S<=T & T<=V & affine hull T = affine hull V"
  1694 proof-
  1695 obtain a where a_def: "a : S" using assms by auto
  1696 hence h0: "independent  ((%x. -a + x) ` (S-{a}))" using affine_dependent_iff_dependent2 assms by auto
  1697 from this obtain B 
  1698    where B_def: "(%x. -a+x) ` (S - {a}) <= B & B <= (%x. -a+x) ` V & independent B & (%x. -a+x) ` V <= span B" 
  1699    using maximal_independent_subset_extend[of "(%x. -a+x) ` (S-{a})" "(%x. -a + x) ` V"] assms by blast
  1700 def T == "(%x. a+x) ` (insert 0 B)" hence "T=insert a ((%x. a+x) ` B)" by auto
  1701 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
  1702 hence "V <= affine hull T" using B_def assms translation_inverse_subset[of a V "span B"] by auto
  1703 moreover have "T<=V" using T_def B_def a_def assms by auto
  1704 ultimately have "affine hull T = affine hull V" 
  1705     by (metis Int_absorb1 Int_absorb2 Int_commute Int_lower2 assms hull_hull hull_mono) 
  1706 moreover have "S<=T" using T_def B_def translation_inverse_subset[of a "S-{a}" B] by auto
  1707 moreover have "~(affine_dependent T)" using T_def affine_dependent_translation_eq[of "insert 0 B"] affine_dependent_imp_dependent2 B_def by auto
  1708 ultimately show ?thesis using `T<=V` by auto
  1709 qed
  1710 
  1711 lemma affine_basis_exists: 
  1712 fixes V :: "('n::euclidean_space) set"
  1713 shows "? B. B <= V & ~(affine_dependent B) & affine hull V = affine hull B"
  1714 proof-
  1715 { assume empt: "V={}" have "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)" using empt affine_independent_empty by auto
  1716 }
  1717 moreover
  1718 { assume nonempt: "V~={}" obtain x where "x:V" using nonempt by auto
  1719   hence "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)"
  1720   using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}:: ('n::euclidean_space) set" V] by auto
  1721 }
  1722 ultimately show ?thesis by auto
  1723 qed
  1724 
  1725 subsection {* Affine Dimension of a Set *}
  1726 
  1727 definition "aff_dim V = (SOME d :: int. ? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1))"
  1728 
  1729 lemma aff_dim_basis_exists:
  1730   fixes V :: "('n::euclidean_space) set" 
  1731   shows "? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
  1732 proof-
  1733 obtain B where B_def: "~(affine_dependent B) & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
  1734 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
  1735 qed
  1736 
  1737 lemma affine_hull_nonempty: "(S ~= {}) <-> affine hull S ~= {}"
  1738 proof-
  1739 fix S have "(S = {}) ==> affine hull S = {}"using affine_hull_empty by auto 
  1740 moreover have "affine hull S = {} ==> S = {}" unfolding hull_def by auto
  1741 ultimately show "(S ~= {}) <-> affine hull S ~= {}" by blast
  1742 qed
  1743 
  1744 lemma aff_dim_parallel_subspace_aux:
  1745 fixes B :: "('n::euclidean_space) set"
  1746 assumes "~(affine_dependent B)" "a:B"
  1747 shows "finite B & ((card B) - 1 = dim (span ((%x. -a+x) ` (B-{a}))))" 
  1748 proof-
  1749 have "independent ((%x. -a + x) ` (B-{a}))" using affine_dependent_iff_dependent2 assms by auto
  1750 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
  1751 { assume emp: "(%x. -a + x) ` (B - {a}) = {}" 
  1752   have "B=insert a ((%x. a + x) ` (%x. -a + x) ` (B - {a}))" using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
  1753   hence "B={a}" using emp by auto
  1754   hence ?thesis using assms fin by auto  
  1755 }
  1756 moreover
  1757 { assume "(%x. -a + x) ` (B - {a}) ~= {}"
  1758   hence "card ((%x. -a + x) ` (B - {a}))>0" using fin by auto
  1759   moreover have h1: "card ((%x. -a + x) ` (B-{a})) = card (B-{a})"  
  1760      apply (rule card_image) using translate_inj_on by auto
  1761   ultimately have "card (B-{a})>0" by auto
  1762   hence "finite(B-{a})" using card_gt_0_iff[of "(B - {a})"] by auto
  1763   moreover hence "(card (B-{a})= (card B) - 1)" using card_Diff_singleton assms by auto
  1764   ultimately have ?thesis using fin h1 by auto
  1765 } ultimately show ?thesis by auto
  1766 qed
  1767 
  1768 lemma aff_dim_parallel_subspace:
  1769 fixes V L :: "('n::euclidean_space) set"
  1770 assumes "V ~= {}"
  1771 assumes "subspace L" "affine_parallel (affine hull V) L"
  1772 shows "aff_dim V=int(dim L)"
  1773 proof-
  1774 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
  1775 hence "B~={}" using assms B_def  affine_hull_nonempty[of V] affine_hull_nonempty[of B] by auto 
  1776 from this obtain a where a_def: "a : B" by auto
  1777 def Lb == "span ((%x. -a+x) ` (B-{a}))"
  1778   moreover have "affine_parallel (affine hull B) Lb"
  1779      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
  1780   moreover have "subspace Lb" using Lb_def subspace_span by auto
  1781   moreover have "affine hull B ~= {}" using assms B_def affine_hull_nonempty[of V] by auto
  1782   ultimately have "L=Lb" using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B_def by auto 
  1783   hence "dim L=dim Lb" by auto 
  1784   moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def B_def by auto
  1785 (*  hence "card B=dim Lb+1" using `B~={}` card_gt_0_iff[of B] by auto *)
  1786   ultimately show ?thesis using B_def `B~={}` card_gt_0_iff[of B] by auto
  1787 qed
  1788 
  1789 lemma aff_independent_finite:
  1790 fixes B :: "('n::euclidean_space) set"
  1791 assumes "~(affine_dependent B)"
  1792 shows "finite B"
  1793 proof-
  1794 { assume "B~={}" from this obtain a where "a:B" by auto 
  1795   hence ?thesis using aff_dim_parallel_subspace_aux assms by auto 
  1796 } from this show ?thesis by auto
  1797 qed
  1798 
  1799 lemma independent_finite:
  1800 fixes B :: "('n::euclidean_space) set"
  1801 assumes "independent B" 
  1802 shows "finite B"
  1803 using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms by auto
  1804 
  1805 lemma subspace_dim_equal:
  1806 assumes "subspace (S :: ('n::euclidean_space) set)" "subspace T" "S <= T" "dim S >= dim T"
  1807 shows "S=T"
  1808 proof- 
  1809 obtain B where B_def: "B <= S & independent B & S <= span B & (card B = dim S)" using basis_exists[of S] by auto
  1810 hence "span B <= S" using span_mono[of B S] span_eq[of S] assms by metis 
  1811 hence "span B = S" using B_def by auto
  1812 have "dim S = dim T" using assms dim_subset[of S T] by auto
  1813 hence "T <= span B" using card_eq_dim[of B T] B_def independent_finite assms by auto
  1814 from this show ?thesis using assms `span B=S` by auto
  1815 qed
  1816 
  1817 lemma span_substd_basis:  assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  1818   shows "(span {basis i | i. i : d}) = {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
  1819   (is "span ?A = ?B")
  1820 proof-
  1821 have "?A <= ?B" by auto
  1822 moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .
  1823 ultimately have "span ?A <= ?B" using span_mono[of "?A" "?B"] span_eq[of "?B"] by blast
  1824 moreover have "card d <= dim (span ?A)" using independent_card_le_dim[of "?A" "span ?A"] 
  1825    independent_substdbasis[OF assms] card_substdbasis[OF assms] span_inc[of "?A"] by auto
  1826 moreover hence "dim ?B <= dim (span ?A)" using dim_substandard[OF assms] by auto
  1827 ultimately show ?thesis using s subspace_dim_equal[of "span ?A" "?B"] 
  1828   subspace_span[of "?A"] by auto
  1829 qed
  1830 
  1831 lemma basis_to_substdbasis_subspace_isomorphism:
  1832 fixes B :: "('a::euclidean_space) set" 
  1833 assumes "independent B"
  1834 shows "EX f d. card d = card B & linear f & f ` B = {basis i::'a |i. i : (d :: nat set)} & 
  1835        f ` span B = {x. ALL i<DIM('a). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B) \<and> d\<subseteq>{..<DIM('a)}" 
  1836 proof-
  1837   have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto
  1838   def d \<equiv> "{..<dim B}" have t:"card d = dim B" unfolding d_def by auto
  1839   have "dim B <= DIM('a)" using dim_subset_UNIV[of B] by auto
  1840   hence d:"d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
  1841   let ?t = "{x::'a::euclidean_space. !i<DIM('a). i ~: d --> x$$i = 0}"
  1842   have "EX f. linear f & f ` B = {basis i |i. i : d} &
  1843     f ` span B = ?t & inj_on f (span B)"
  1844     apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "{basis i |i. i : d}"])
  1845     apply(rule subspace_span) apply(rule subspace_substandard) defer
  1846     apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)
  1847     unfolding span_substd_basis[OF d,THEN sym] card_substdbasis[OF d] apply(rule span_inc)
  1848     apply(rule independent_substdbasis[OF d]) apply(rule,assumption)
  1849     unfolding t[THEN sym] span_substd_basis[OF d] dim_substandard[OF d] by auto
  1850   from this t `card B=dim B` show ?thesis using d by auto 
  1851 qed
  1852 
  1853 lemma aff_dim_empty:
  1854 fixes S :: "('n::euclidean_space) set" 
  1855 shows "S = {} <-> aff_dim S = -1"
  1856 proof-
  1857 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
  1858 moreover hence "S={} <-> B={}" using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
  1859 ultimately show ?thesis using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
  1860 qed
  1861 
  1862 lemma aff_dim_affine_hull:
  1863 fixes S :: "('n::euclidean_space) set"
  1864 shows "aff_dim (affine hull S)=aff_dim S" 
  1865 unfolding aff_dim_def using hull_hull[of _ S] by auto 
  1866 
  1867 lemma aff_dim_affine_hull2:
  1868 fixes S T :: "('n::euclidean_space) set"
  1869 assumes "affine hull S=affine hull T"
  1870 shows "aff_dim S=aff_dim T" unfolding aff_dim_def using assms by auto
  1871 
  1872 lemma aff_dim_unique: 
  1873 fixes B V :: "('n::euclidean_space) set" 
  1874 assumes "(affine hull B=affine hull V) & ~(affine_dependent B)"
  1875 shows "of_nat(card B) = aff_dim V+1"
  1876 proof-
  1877 { assume "B={}" hence "V={}" using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms by auto
  1878   hence "aff_dim V = (-1::int)"  using aff_dim_empty by auto  
  1879   hence ?thesis using `B={}` by auto
  1880 }
  1881 moreover
  1882 { assume "B~={}" from this obtain a where a_def: "a:B" by auto 
  1883   def Lb == "span ((%x. -a+x) ` (B-{a}))"
  1884   have "affine_parallel (affine hull B) Lb"
  1885      using Lb_def affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"] 
  1886      unfolding affine_parallel_def by auto
  1887   moreover have "subspace Lb" using Lb_def subspace_span by auto
  1888   ultimately have "aff_dim B=int(dim Lb)" using aff_dim_parallel_subspace[of B Lb] `B~={}` by auto 
  1889   moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def assms by auto
  1890   ultimately have "(of_nat(card B) = aff_dim B+1)" using  `B~={}` card_gt_0_iff[of B] by auto
  1891   hence ?thesis using aff_dim_affine_hull2 assms by auto
  1892 } ultimately show ?thesis by blast
  1893 qed
  1894 
  1895 lemma aff_dim_affine_independent: 
  1896 fixes B :: "('n::euclidean_space) set" 
  1897 assumes "~(affine_dependent B)"
  1898 shows "of_nat(card B) = aff_dim B+1"
  1899   using aff_dim_unique[of B B] assms by auto
  1900 
  1901 lemma aff_dim_sing: 
  1902 fixes a :: "'n::euclidean_space" 
  1903 shows "aff_dim {a}=0"
  1904   using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
  1905 
  1906 lemma aff_dim_inner_basis_exists:
  1907   fixes V :: "('n::euclidean_space) set" 
  1908   shows "? B. B<=V & (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
  1909 proof-
  1910 obtain B where B_def: "~(affine_dependent B) & B<=V & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
  1911 moreover hence "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
  1912 ultimately show ?thesis by auto
  1913 qed
  1914 
  1915 lemma aff_dim_le_card:
  1916 fixes V :: "('n::euclidean_space) set" 
  1917 assumes "finite V"
  1918 shows "aff_dim V <= of_nat(card V) - 1"
  1919  proof-
  1920  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 
  1921  moreover hence "card B <= card V" using assms card_mono by auto
  1922  ultimately show ?thesis by auto
  1923 qed
  1924 
  1925 lemma aff_dim_parallel_eq:
  1926 fixes S T :: "('n::euclidean_space) set"
  1927 assumes "affine_parallel (affine hull S) (affine hull T)"
  1928 shows "aff_dim S=aff_dim T"
  1929 proof-
  1930 { assume "T~={}" "S~={}" 
  1931   from this obtain L where L_def: "subspace L & affine_parallel (affine hull T) L" 
  1932        using affine_parallel_subspace[of "affine hull T"] affine_affine_hull[of T] affine_hull_nonempty by auto
  1933   hence "aff_dim T = int(dim L)" using aff_dim_parallel_subspace `T~={}` by auto
  1934   moreover have "subspace L & affine_parallel (affine hull S) L" 
  1935      using L_def affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
  1936   moreover hence "aff_dim S = int(dim L)" using aff_dim_parallel_subspace `S~={}` by auto 
  1937   ultimately have ?thesis by auto
  1938 }
  1939 moreover
  1940 { assume "S={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
  1941   hence ?thesis using aff_dim_empty by auto
  1942 }
  1943 moreover
  1944 { assume "T={}" 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 ultimately show ?thesis by blast
  1948 qed
  1949 
  1950 lemma aff_dim_translation_eq:
  1951 fixes a :: "'n::euclidean_space"
  1952 shows "aff_dim ((%x. a + x) ` S)=aff_dim S" 
  1953 proof-
  1954 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
  1955 from this show ?thesis using  aff_dim_parallel_eq[of S "(%x. a + x) ` S"] by auto 
  1956 qed
  1957 
  1958 lemma aff_dim_affine:
  1959 fixes S L :: "('n::euclidean_space) set"
  1960 assumes "S ~= {}" "affine S"
  1961 assumes "subspace L" "affine_parallel S L"
  1962 shows "aff_dim S=int(dim L)" 
  1963 proof-
  1964 have 1: "(affine hull S) = S" using assms affine_hull_eq[of S] by auto 
  1965 hence "affine_parallel (affine hull S) L" using assms by (simp add:1)
  1966 from this show ?thesis using assms aff_dim_parallel_subspace[of S L] by blast 
  1967 qed
  1968 
  1969 lemma dim_affine_hull:
  1970 fixes S :: "('n::euclidean_space) set"
  1971 shows "dim (affine hull S)=dim S"
  1972 proof-
  1973 have "dim (affine hull S)>=dim S" using dim_subset by auto
  1974 moreover have "dim(span S) >= dim (affine hull S)" using dim_subset affine_hull_subset_span by auto
  1975 moreover have "dim(span S)=dim S" using dim_span by auto
  1976 ultimately show ?thesis by auto
  1977 qed
  1978 
  1979 lemma aff_dim_subspace:
  1980 fixes S :: "('n::euclidean_space) set"
  1981 assumes "S ~= {}" "subspace S"
  1982 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 
  1983 
  1984 lemma aff_dim_zero:
  1985 fixes S :: "('n::euclidean_space) set"
  1986 assumes "0 : affine hull S"
  1987 shows "aff_dim S=int(dim S)"
  1988 proof-
  1989 have "subspace(affine hull S)" using subspace_affine[of "affine hull S"] affine_affine_hull assms by auto
  1990 hence "aff_dim (affine hull S) =int(dim (affine hull S))" using assms aff_dim_subspace[of "affine hull S"] by auto  
  1991 from this show ?thesis using aff_dim_affine_hull[of S] dim_affine_hull[of S] by auto
  1992 qed
  1993 
  1994 lemma aff_dim_univ: "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))"
  1995   using aff_dim_subspace[of "(UNIV :: ('n::euclidean_space) set)"]
  1996     dim_UNIV[where 'a="'n::euclidean_space"] by auto
  1997 
  1998 lemma aff_dim_geq:
  1999   fixes V :: "('n::euclidean_space) set"
  2000   shows "aff_dim V >= -1"
  2001 proof-
  2002 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
  2003 from this show ?thesis by auto
  2004 qed
  2005 
  2006 lemma independent_card_le_aff_dim: 
  2007   assumes "(B::('n::euclidean_space) set) <= V"
  2008   assumes "~(affine_dependent B)" 
  2009   shows "int(card B) <= aff_dim V+1"
  2010 proof-
  2011 { assume "B~={}" 
  2012   from this obtain T where T_def: "~(affine_dependent T) & B<=T & T<=V & affine hull T = affine hull V" 
  2013   using assms extend_to_affine_basis[of B V] by auto
  2014   hence "of_nat(card T) = aff_dim V+1" using aff_dim_unique by auto
  2015   hence ?thesis using T_def card_mono[of T B] aff_independent_finite[of T] by auto
  2016 }
  2017 moreover
  2018 { assume "B={}"
  2019   moreover have "-1<= aff_dim V" using aff_dim_geq by auto
  2020   ultimately have ?thesis by auto
  2021 }  ultimately show ?thesis by blast
  2022 qed
  2023 
  2024 lemma aff_dim_subset:
  2025   fixes S T :: "('n::euclidean_space) set"
  2026   assumes "S <= T"
  2027   shows "aff_dim S <= aff_dim T"
  2028 proof-
  2029 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
  2030 moreover hence "int (card B) <= aff_dim T + 1" using assms independent_card_le_aff_dim[of B T] by auto
  2031 ultimately show ?thesis by auto
  2032 qed
  2033 
  2034 lemma aff_dim_subset_univ:
  2035 fixes S :: "('n::euclidean_space) set"
  2036 shows "aff_dim S <= int(DIM('n))"
  2037 proof - 
  2038   have "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))" using aff_dim_univ by auto
  2039   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
  2040 qed
  2041 
  2042 lemma affine_dim_equal:
  2043 assumes "affine (S :: ('n::euclidean_space) set)" "affine T" "S ~= {}" "S <= T" "aff_dim S = aff_dim T"
  2044 shows "S=T"
  2045 proof-
  2046 obtain a where "a : S" using assms by auto 
  2047 hence "a : T" using assms by auto
  2048 def LS == "{y. ? x : S. (-a)+x=y}"
  2049 hence ls: "subspace LS & affine_parallel S LS" using assms parallel_subspace_explicit[of S a LS] `a : S` by auto 
  2050 hence h1: "int(dim LS) = aff_dim S" using assms aff_dim_affine[of S LS] by auto
  2051 have "T ~= {}" using assms by auto
  2052 def LT == "{y. ? x : T. (-a)+x=y}" 
  2053 hence lt: "subspace LT & affine_parallel T LT" using assms parallel_subspace_explicit[of T a LT] `a : T` by auto
  2054 hence "int(dim LT) = aff_dim T" using assms aff_dim_affine[of T LT] `T ~= {}` by auto 
  2055 hence "dim LS = dim LT" using h1 assms by auto
  2056 moreover have "LS <= LT" using LS_def LT_def assms by auto
  2057 ultimately have "LS=LT" using subspace_dim_equal[of LS LT] ls lt by auto
  2058 moreover have "S = {x. ? y : LS. a+y=x}" using LS_def by auto 
  2059 moreover have "T = {x. ? y : LT. a+y=x}" using LT_def by auto
  2060 ultimately show ?thesis by auto 
  2061 qed
  2062 
  2063 lemma affine_hull_univ:
  2064 fixes S :: "('n::euclidean_space) set"
  2065 assumes "aff_dim S = int(DIM('n))"
  2066 shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
  2067 proof-
  2068 have "S ~= {}" using assms aff_dim_empty[of S] by auto
  2069 have h0: "S <= affine hull S" using hull_subset[of S _] by auto
  2070 have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S" using aff_dim_univ assms by auto
  2071 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  
  2072 have h3: "aff_dim S <= aff_dim (affine hull S)" using h0 aff_dim_subset[of S "affine hull S"] assms by auto
  2073 hence h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" using h0 h1 h2 by auto
  2074 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
  2075 qed
  2076 
  2077 lemma aff_dim_convex_hull:
  2078 fixes S :: "('n::euclidean_space) set"
  2079 shows "aff_dim (convex hull S)=aff_dim S"
  2080   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S] 
  2081   hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"] 
  2082   aff_dim_subset[of "convex hull S" "affine hull S"] by auto
  2083 
  2084 lemma aff_dim_cball:
  2085 fixes a :: "'n::euclidean_space" 
  2086 assumes "0<e"
  2087 shows "aff_dim (cball a e) = int (DIM('n))"
  2088 proof-
  2089 have "(%x. a + x) ` (cball 0 e)<=cball a e" unfolding cball_def dist_norm by auto
  2090 hence "aff_dim (cball (0 :: 'n::euclidean_space) e) <= aff_dim (cball a e)"
  2091   using aff_dim_translation_eq[of a "cball 0 e"] 
  2092         aff_dim_subset[of "op + a ` cball 0 e" "cball a e"] by auto
  2093 moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))" 
  2094    using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"] centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms 
  2095    by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
  2096 ultimately show ?thesis using aff_dim_subset_univ[of "cball a e"] by auto 
  2097 qed
  2098 
  2099 lemma aff_dim_open:
  2100 fixes S :: "('n::euclidean_space) set"
  2101 assumes "open S" "S ~= {}"
  2102 shows "aff_dim S = int (DIM('n))"
  2103 proof-
  2104 obtain x where "x:S" using assms by auto
  2105 from this obtain e where e_def: "e>0 & cball x e <= S" using open_contains_cball[of S] assms by auto
  2106 from this have "aff_dim (cball x e) <= aff_dim S" using aff_dim_subset by auto
  2107 from this show ?thesis using aff_dim_cball[of e x] aff_dim_subset_univ[of S] e_def by auto     
  2108 qed
  2109 
  2110 lemma low_dim_interior:
  2111 fixes S :: "('n::euclidean_space) set"
  2112 assumes "~(aff_dim S = int (DIM('n)))"
  2113 shows "interior S = {}"
  2114 proof-
  2115 have "aff_dim(interior S) <= aff_dim S" 
  2116    using interior_subset aff_dim_subset[of "interior S" S] by auto 
  2117 from this show ?thesis using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto   
  2118 qed
  2119 
  2120 subsection{* Relative Interior of a Set *}
  2121 
  2122 definition "rel_interior S = {x. ? T. openin (subtopology euclidean (affine hull S)) T & x : T & T <= S}"
  2123 
  2124 lemma rel_interior: "rel_interior S = {x : S. ? T. open T & x : T & (T Int (affine hull S)) <= S}"
  2125   unfolding rel_interior_def[of S] openin_open[of "affine hull S"] apply auto
  2126 proof-
  2127 fix x T assume a: "x:S" "open T" "x : T" "(T Int (affine hull S)) <= S"
  2128 hence h1: "x : T Int affine hull S" using hull_inc by auto
  2129 show "EX Tb. (EX Ta. open Ta & Tb = affine hull S Int Ta) & x : Tb & Tb <= S"
  2130 apply (rule_tac x="T Int (affine hull S)" in exI)
  2131 using a h1 by auto
  2132 qed
  2133 
  2134 lemma mem_rel_interior: 
  2135      "x : rel_interior S <-> (? T. open T & x : (T Int S) & (T Int (affine hull S)) <= S)" 
  2136      by (auto simp add: rel_interior)
  2137 
  2138 lemma mem_rel_interior_ball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((ball x e) Int (affine hull S)) <= S)"
  2139   apply (simp add: rel_interior, safe)
  2140   apply (force simp add: open_contains_ball)
  2141   apply (rule_tac x="ball x e" in exI)
  2142   apply (simp add: centre_in_ball)
  2143   done
  2144 
  2145 lemma rel_interior_ball: 
  2146       "rel_interior S = {x : S. ? e. e>0 & ((ball x e) Int (affine hull S)) <= S}" 
  2147       using mem_rel_interior_ball [of _ S] by auto 
  2148 
  2149 lemma mem_rel_interior_cball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((cball x e) Int (affine hull S)) <= S)"
  2150   apply (simp add: rel_interior, safe) 
  2151   apply (force simp add: open_contains_cball)
  2152   apply (rule_tac x="ball x e" in exI)
  2153   apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
  2154   apply auto
  2155   done
  2156 
  2157 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
  2158 
  2159 lemma rel_interior_empty: "rel_interior {} = {}" 
  2160    by (auto simp add: rel_interior_def) 
  2161 
  2162 lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"
  2163 by (metis affine_hull_eq affine_sing)
  2164 
  2165 lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"
  2166    unfolding rel_interior_ball affine_hull_sing apply auto
  2167    apply(rule_tac x="1 :: real" in exI) apply simp
  2168    done
  2169 
  2170 lemma subset_rel_interior:
  2171 fixes S T :: "('n::euclidean_space) set"
  2172 assumes "S<=T" "affine hull S=affine hull T"
  2173 shows "rel_interior S <= rel_interior T"
  2174   using assms by (auto simp add: rel_interior_def)  
  2175 
  2176 lemma rel_interior_subset: "rel_interior S <= S" 
  2177    by (auto simp add: rel_interior_def)
  2178 
  2179 lemma rel_interior_subset_closure: "rel_interior S <= closure S" 
  2180    using rel_interior_subset by (auto simp add: closure_def) 
  2181 
  2182 lemma interior_subset_rel_interior: "interior S <= rel_interior S" 
  2183    by (auto simp add: rel_interior interior_def)
  2184 
  2185 lemma interior_rel_interior:
  2186 fixes S :: "('n::euclidean_space) set"
  2187 assumes "aff_dim S = int(DIM('n))"
  2188 shows "rel_interior S = interior S"
  2189 proof -
  2190 have "affine hull S = UNIV" using assms affine_hull_univ[of S] by auto 
  2191 from this show ?thesis unfolding rel_interior interior_def by auto
  2192 qed
  2193 
  2194 lemma rel_interior_open:
  2195 fixes S :: "('n::euclidean_space) set"
  2196 assumes "open S"
  2197 shows "rel_interior S = S"
  2198 by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
  2199 
  2200 lemma interior_rel_interior_gen:
  2201 fixes S :: "('n::euclidean_space) set"
  2202 shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
  2203 by (metis interior_rel_interior low_dim_interior)
  2204 
  2205 lemma rel_interior_univ: 
  2206 fixes S :: "('n::euclidean_space) set"
  2207 shows "rel_interior (affine hull S) = affine hull S"
  2208 proof-
  2209 have h1: "rel_interior (affine hull S) <= affine hull S" using rel_interior_subset by auto 
  2210 { fix x assume x_def: "x : affine hull S"
  2211   obtain e :: real where "e=1" by auto
  2212   hence "e>0 & ball x e Int affine hull (affine hull S) <= affine hull S" using hull_hull[of _ S] by auto
  2213   hence "x : rel_interior (affine hull S)" using x_def rel_interior_ball[of "affine hull S"] by auto
  2214 } from this show ?thesis using h1 by auto 
  2215 qed
  2216 
  2217 lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
  2218 by (metis open_UNIV rel_interior_open)
  2219 
  2220 lemma rel_interior_convex_shrink:
  2221   fixes S :: "('a::euclidean_space) set"
  2222   assumes "convex S" "c : rel_interior S" "x : S" "0 < e" "e <= 1"
  2223   shows "x - e *\<^sub>R (x - c) : rel_interior S"
  2224 proof- 
  2225 (* Proof is a modified copy of the proof of similar lemma mem_interior_convex_shrink 
  2226 *)
  2227 obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
  2228   using assms(2) unfolding  mem_rel_interior_ball by auto
  2229 {   fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d & y : affine hull S"
  2230     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)
  2231     have "x : affine hull S" using assms hull_subset[of S] by auto
  2232     moreover have "1 / e + - ((1 - e) / e) = 1" 
  2233        using `e>0` mult_left.diff[of "1" "(1-e)" "1/e"] by auto
  2234     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x : affine hull S"
  2235         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)     
  2236     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)"
  2237       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
  2238       by(auto simp add:euclidean_eq[where 'a='a] field_simps) 
  2239     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)
  2240     also have "... < d" using as[unfolded dist_norm] and `e>0`
  2241       by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
  2242     finally have "y : S" apply(subst *) 
  2243 apply(rule assms(1)[unfolded convex_alt,rule_format])
  2244       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) ** by auto
  2245 } hence "ball (x - e *\<^sub>R (x - c)) (e*d) Int affine hull S <= S" by auto
  2246 moreover have "0 < e*d" using `0<e` `0<d` using real_mult_order by auto
  2247 moreover have "c : S" using assms rel_interior_subset by auto
  2248 moreover hence "x - e *\<^sub>R (x - c) : S"
  2249    using mem_convex[of S x c e] apply (simp add: algebra_simps) using assms by auto
  2250 ultimately show ?thesis 
  2251   using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e>0` by auto
  2252 qed
  2253 
  2254 lemma interior_real_semiline:
  2255 fixes a :: real
  2256 shows "interior {a..} = {a<..}"
  2257 proof-
  2258 { fix y assume "a<y" hence "y : interior {a..}"
  2259   apply (simp add: mem_interior) apply (rule_tac x="(y-a)" in exI) apply (auto simp add: dist_norm) 
  2260   done }
  2261 moreover
  2262 { fix y assume "y : interior {a..}" (*hence "a<=y" using interior_subset by auto*)
  2263   from this obtain e where e_def: "e>0 & cball y e \<subseteq> {a..}" 
  2264      using mem_interior_cball[of y "{a..}"] by auto
  2265   moreover hence "y-e : cball y e" by (auto simp add: cball_def dist_norm) 
  2266   ultimately have "a<=y-e" by auto
  2267   hence "a<y" using e_def by auto
  2268 } ultimately show ?thesis by auto
  2269 qed
  2270 
  2271 lemma rel_interior_real_interval:
  2272   fixes a b :: real assumes "a < b" shows "rel_interior {a..b} = {a<..<b}"
  2273 proof-
  2274   have "{a<..<b} \<noteq> {}" using assms unfolding set_eq_iff by (auto intro!: exI[of _ "(a + b) / 2"])
  2275   then show ?thesis
  2276     using interior_rel_interior_gen[of "{a..b}", symmetric]
  2277     by (simp split: split_if_asm add: interior_closed_interval)
  2278 qed
  2279 
  2280 lemma rel_interior_real_semiline:
  2281   fixes a :: real shows "rel_interior {a..} = {a<..}"
  2282 proof-
  2283   have *: "{a<..} \<noteq> {}" unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
  2284   then show ?thesis using interior_real_semiline
  2285      interior_rel_interior_gen[of "{a..}"]
  2286      by (auto split: split_if_asm)
  2287 qed
  2288 
  2289 subsection "Relative open"
  2290 
  2291 definition "rel_open S <-> (rel_interior S) = S"
  2292 
  2293 lemma rel_open: "rel_open S <-> openin (subtopology euclidean (affine hull S)) S"
  2294  unfolding rel_open_def rel_interior_def apply auto
  2295  using openin_subopen[of "subtopology euclidean (affine hull S)" S] by auto
  2296 
  2297 lemma opein_rel_interior: 
  2298   "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
  2299   apply (simp add: rel_interior_def)
  2300   apply (subst openin_subopen) by blast
  2301 
  2302 lemma affine_rel_open: 
  2303   fixes S :: "('n::euclidean_space) set"
  2304   assumes "affine S" shows "rel_open S" 
  2305   unfolding rel_open_def using assms rel_interior_univ[of S] affine_hull_eq[of S] by metis
  2306 
  2307 lemma affine_closed: 
  2308   fixes S :: "('n::euclidean_space) set"
  2309   assumes "affine S" shows "closed S"
  2310 proof-
  2311 { assume "S ~= {}"
  2312   from this obtain L where L_def: "subspace L & affine_parallel S L"
  2313      using assms affine_parallel_subspace[of S] by auto
  2314   from this obtain "a" where a_def: "S=(op + a ` L)" 
  2315      using affine_parallel_def[of L S] affine_parallel_commut by auto 
  2316   have "closed L" using L_def closed_subspace by auto
  2317   hence "closed S" using closed_translation a_def by auto
  2318 } from this show ?thesis by auto
  2319 qed
  2320 
  2321 lemma closure_affine_hull:
  2322   fixes S :: "('n::euclidean_space) set"
  2323   shows "closure S <= affine hull S"
  2324 proof-
  2325 have "closure S <= closure (affine hull S)" using subset_closure by auto
  2326 moreover have "closure (affine hull S) = affine hull S" 
  2327    using affine_affine_hull affine_closed[of "affine hull S"] closure_eq by auto
  2328 ultimately show ?thesis by auto  
  2329 qed
  2330 
  2331 lemma closure_same_affine_hull:
  2332   fixes S :: "('n::euclidean_space) set"
  2333   shows "affine hull (closure S) = affine hull S"
  2334 proof-
  2335 have "affine hull (closure S) <= affine hull S"
  2336    using hull_mono[of "closure S" "affine hull S" "affine"] closure_affine_hull[of S] hull_hull[of "affine" S] by auto
  2337 moreover have "affine hull (closure S) >= affine hull S"  
  2338    using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
  2339 ultimately show ?thesis by auto  
  2340 qed
  2341 
  2342 lemma closure_aff_dim: 
  2343   fixes S :: "('n::euclidean_space) set"
  2344   shows "aff_dim (closure S) = aff_dim S"
  2345 proof-
  2346 have "aff_dim S <= aff_dim (closure S)" using aff_dim_subset closure_subset by auto
  2347 moreover have "aff_dim (closure S) <= aff_dim (affine hull S)" 
  2348   using aff_dim_subset closure_affine_hull by auto
  2349 moreover have "aff_dim (affine hull S) = aff_dim S" using aff_dim_affine_hull by auto
  2350 ultimately show ?thesis by auto
  2351 qed
  2352 
  2353 lemma rel_interior_closure_convex_shrink:
  2354   fixes S :: "(_::euclidean_space) set"
  2355   assumes "convex S" "c : rel_interior S" "x : closure S" "0 < e" "e <= 1"
  2356   shows "x - e *\<^sub>R (x - c) : rel_interior S"
  2357 proof- 
  2358 (* Proof is a modified copy of the proof of similar lemma mem_interior_closure_convex_shrink
  2359 *)
  2360 obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
  2361   using assms(2) unfolding mem_rel_interior_ball by auto
  2362 have "EX y : S. norm (y - x) * (1 - e) < e * d" proof(cases "x : S")
  2363     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  2364     case False hence x:"x islimpt S" using assms(3)[unfolded closure_def] by auto
  2365     show ?thesis proof(cases "e=1")
  2366       case True obtain y where "y : S" "y ~= x" "dist y x < 1"
  2367         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  2368       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  2369       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  2370         using `e<=1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  2371       then obtain y where "y : S" "y ~= x" "dist y x < e * d / (1 - e)"
  2372         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  2373       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  2374   then obtain y where "y : S" and y:"norm (y - x) * (1 - e) < e * d" by auto
  2375   def z == "c + ((1 - e) / e) *\<^sub>R (x - y)"
  2376   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)
  2377   have zball: "z\<in>ball c d"
  2378     using mem_ball z_def dist_norm[of c] using y and assms(4,5) by (auto simp add:field_simps norm_minus_commute)
  2379   have "x : affine hull S" using closure_affine_hull assms by auto
  2380   moreover have "y : affine hull S" using `y : S` hull_subset[of S] by auto
  2381   moreover have "c : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
  2382   ultimately have "z : affine hull S" 
  2383     using z_def affine_affine_hull[of S] 
  2384           mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"] 
  2385           assms by (auto simp add: field_simps)
  2386   hence "z : S" using d zball by auto
  2387   obtain d1 where "d1>0" and d1:"ball z d1 <= ball c d"
  2388     using zball open_ball[of c d] openE[of "ball c d" z] by auto
  2389   hence "(ball z d1) Int (affine hull S) <= (ball c d) Int (affine hull S)" by auto
  2390   hence "(ball z d1) Int (affine hull S) <= S" using d by auto 
  2391   hence "z : rel_interior S" using mem_rel_interior_ball using `d1>0` `z : S` by auto
  2392   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
  2393   thus ?thesis using * by auto 
  2394 qed
  2395 
  2396 subsection{* Relative interior preserves under linear transformations *}
  2397 
  2398 lemma rel_interior_translation_aux:
  2399 fixes a :: "'n::euclidean_space"
  2400 shows "((%x. a + x) ` rel_interior S) <= rel_interior ((%x. a + x) ` S)"
  2401 proof-
  2402 { fix x assume x_def: "x : rel_interior S"
  2403   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 
  2404   from this have "open ((%x. a + x) ` T)" and 
  2405     "(a + x) : (((%x. a + x) ` T) Int ((%x. a + x) ` S))" and 
  2406     "(((%x. a + x) ` T) Int (affine hull ((%x. a + x) ` S))) <= ((%x. a + x) ` S)" 
  2407     using affine_hull_translation[of a S] open_translation[of T a] x_def by auto 
  2408   from this have "(a+x) : rel_interior ((%x. a + x) ` S)" 
  2409     using mem_rel_interior[of "a+x" "((%x. a + x) ` S)"] by auto 
  2410 } from this show ?thesis by auto 
  2411 qed
  2412 
  2413 lemma rel_interior_translation:
  2414 fixes a :: "'n::euclidean_space"
  2415 shows "rel_interior ((%x. a + x) ` S) = ((%x. a + x) ` rel_interior S)"
  2416 proof-
  2417 have "(%x. (-a) + x) ` rel_interior ((%x. a + x) ` S) <= rel_interior S" 
  2418    using rel_interior_translation_aux[of "-a" "(%x. a + x) ` S"] 
  2419          translation_assoc[of "-a" "a"] by auto
  2420 hence "((%x. a + x) ` rel_interior S) >= rel_interior ((%x. a + x) ` S)" 
  2421    using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"] 
  2422    by auto
  2423 from this show ?thesis using  rel_interior_translation_aux[of a S] by auto 
  2424 qed
  2425 
  2426 
  2427 lemma affine_hull_linear_image:
  2428 assumes "bounded_linear f"
  2429 shows "f ` (affine hull s) = affine hull f ` s"
  2430 (* Proof is a modified copy of the proof of similar lemma convex_hull_linear_image
  2431 *)
  2432   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  2433   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  2434   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  2435 proof-
  2436   interpret f: bounded_linear f by fact
  2437   show "affine {x. f x : affine hull f ` s}" 
  2438   unfolding affine_def by(auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format]) next
  2439   interpret f: bounded_linear f by fact
  2440   show "affine {x. x : f ` (affine hull s)}" using affine_affine_hull[unfolded affine_def, of s] 
  2441     unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  2442 qed auto
  2443 
  2444 
  2445 lemma rel_interior_injective_on_span_linear_image:
  2446 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  2447 fixes S :: "('m::euclidean_space) set"
  2448 assumes "bounded_linear f" and "inj_on f (span S)"
  2449 shows "rel_interior (f ` S) = f ` (rel_interior S)"
  2450 proof-
  2451 { fix z assume z_def: "z : rel_interior (f ` S)"
  2452   have "z : f ` S" using z_def rel_interior_subset[of "f ` S"] by auto
  2453   from this obtain x where x_def: "x : S & (f x = z)" by auto
  2454   obtain e2 where e2_def: "e2>0 & cball z e2 Int affine hull (f ` S) <= (f ` S)" 
  2455     using z_def rel_interior_cball[of "f ` S"] by auto
  2456   obtain K where K_def: "K>0 & (! x. norm (f x) <= norm x * K)" 
  2457    using assms RealVector.bounded_linear.pos_bounded[of f] by auto
  2458   def e1 == "1/K" hence e1_def: "e1>0 & (! x. e1 * norm (f x) <= norm x)" 
  2459    using K_def pos_le_divide_eq[of e1] by auto
  2460   def e == "e1 * e2" hence "e>0" using e1_def e2_def real_mult_order by auto 
  2461   { fix y assume y_def: "y : cball x e Int affine hull S"
  2462     from this have h1: "f y : affine hull (f ` S)" 
  2463       using affine_hull_linear_image[of f S] assms by auto 
  2464     from y_def have "norm (x-y)<=e1 * e2" 
  2465       using cball_def[of x e] dist_norm[of x y] e_def by auto
  2466     moreover have "(f x)-(f y)=f (x-y)"
  2467        using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
  2468     moreover have "e1 * norm (f (x-y)) <= norm (x-y)" using e1_def by auto
  2469     ultimately have "e1 * norm ((f x)-(f y)) <= e1 * e2" by auto
  2470     hence "(f y) : (cball z e2)" 
  2471       using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1_def x_def by auto
  2472     hence "f y : (f ` S)" using y_def e2_def h1 by auto
  2473     hence "y : S" using assms y_def hull_subset[of S] affine_hull_subset_span 
  2474          inj_on_image_mem_iff[of f "span S" S y] by auto
  2475   } 
  2476   hence "z : f ` (rel_interior S)" using mem_rel_interior_cball[of x S] `e>0` x_def by auto
  2477 } 
  2478 moreover
  2479 { fix x assume x_def: "x : rel_interior S"
  2480   from this obtain e2 where e2_def: "e2>0 & cball x e2 Int affine hull S <= S" 
  2481     using rel_interior_cball[of S] by auto
  2482   have "x : S" using x_def rel_interior_subset by auto
  2483   hence *: "f x : f ` S" by auto
  2484   have "! x:span S. f x = 0 --> x = 0" 
  2485     using assms subspace_span linear_conv_bounded_linear[of f] 
  2486           linear_injective_on_subspace_0[of f "span S"] by auto
  2487   from this obtain e1 where e1_def: "e1>0 & (! x : span S. e1 * norm x <= norm (f x))" 
  2488    using assms injective_imp_isometric[of "span S" f] 
  2489          subspace_span[of S] closed_subspace[of "span S"] by auto
  2490   def e == "e1 * e2" hence "e>0" using e1_def e2_def real_mult_order by auto 
  2491   { fix y assume y_def: "y : cball (f x) e Int affine hull (f ` S)"
  2492     from this have "y : f ` (affine hull S)" using affine_hull_linear_image[of f S] assms by auto 
  2493     from this obtain xy where xy_def: "xy : affine hull S & (f xy = y)" by auto
  2494     from this y_def have "norm ((f x)-(f xy))<=e1 * e2" 
  2495       using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
  2496     moreover have "(f x)-(f xy)=f (x-xy)"
  2497        using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
  2498     moreover have "x-xy : span S" 
  2499        using subspace_sub[of "span S" x xy] subspace_span `x : S` xy_def 
  2500              affine_hull_subset_span[of S] span_inc by auto
  2501     moreover hence "e1 * norm (x-xy) <= norm (f (x-xy))" using e1_def by auto
  2502     ultimately have "e1 * norm (x-xy) <= e1 * e2" by auto
  2503     hence "xy : (cball x e2)"  using cball_def[of x e2] dist_norm[of x xy] e1_def by auto
  2504     hence "y : (f ` S)" using xy_def e2_def by auto
  2505   } 
  2506   hence "(f x) : rel_interior (f ` S)" 
  2507      using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e>0` by auto
  2508 } 
  2509 ultimately show ?thesis by auto
  2510 qed
  2511 
  2512 lemma rel_interior_injective_linear_image:
  2513 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  2514 assumes "bounded_linear f" and "inj f"
  2515 shows "rel_interior (f ` S) = f ` (rel_interior S)"
  2516 using assms rel_interior_injective_on_span_linear_image[of f S] 
  2517       subset_inj_on[of f "UNIV" "span S"] by auto
  2518 
  2519 subsection{* Some Properties of subset of standard basis *}
  2520 
  2521 lemma affine_hull_substd_basis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  2522   shows "affine hull (insert 0 {basis i | i. i : d}) =
  2523   {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
  2524  (is "affine hull (insert 0 ?A) = ?B")
  2525 proof- have *:"\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A" by auto
  2526   show ?thesis unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,THEN sym] * ..
  2527 qed
  2528 
  2529 lemma affine_hull_convex_hull: "affine hull (convex hull S) = affine hull S"
  2530 by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
  2531 
  2532 subsection {* Openness and compactness are preserved by convex hull operation. *}
  2533 
  2534 lemma open_convex_hull[intro]:
  2535   fixes s :: "'a::real_normed_vector set"
  2536   assumes "open s"
  2537   shows "open(convex hull s)"
  2538   unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(8)
  2539 proof(rule, rule) fix a
  2540   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"
  2541   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
  2542 
  2543   from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
  2544     using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
  2545   have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
  2546 
  2547   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}"
  2548     apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
  2549   proof-
  2550     show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
  2551       using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
  2552   next  fix y assume "y \<in> cball a (Min i)"
  2553     hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
  2554     { fix x assume "x\<in>t"
  2555       hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
  2556       hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
  2557       moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
  2558       ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
  2559     moreover
  2560     have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
  2561     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
  2562       unfolding setsum_reindex[OF *] o_def using obt(4) by auto
  2563     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
  2564       unfolding setsum_reindex[OF *] o_def using obt(4,5)
  2565       by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
  2566     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"
  2567       apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
  2568       using obt(1, 3) by auto
  2569   qed
  2570 qed
  2571 
  2572 lemma compact_convex_combinations:
  2573   fixes s t :: "'a::real_normed_vector set"
  2574   assumes "compact s" "compact t"
  2575   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}"
  2576 proof-
  2577   let ?X = "{0..1} \<times> s \<times> t"
  2578   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  2579   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"
  2580     apply(rule set_eqI) unfolding image_iff mem_Collect_eq
  2581     apply rule apply auto
  2582     apply (rule_tac x=u in rev_bexI, simp)
  2583     apply (erule rev_bexI, erule rev_bexI, simp)
  2584     by auto
  2585   have "continuous_on ({0..1} \<times> s \<times> t)
  2586      (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  2587     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  2588   thus ?thesis unfolding *
  2589     apply (rule compact_continuous_image)
  2590     apply (intro compact_Times compact_interval assms)
  2591     done
  2592 qed
  2593 
  2594 lemma compact_convex_hull: fixes s::"('a::euclidean_space) set"
  2595   assumes "compact s"  shows "compact(convex hull s)"
  2596 proof(cases "s={}")
  2597   case True thus ?thesis using compact_empty by simp
  2598 next
  2599   case False then obtain w where "w\<in>s" by auto
  2600   show ?thesis unfolding caratheodory[of s]
  2601   proof(induct ("DIM('a) + 1"))
  2602     have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
  2603       using compact_empty by auto
  2604     case 0 thus ?case unfolding * by simp
  2605   next
  2606     case (Suc n)
  2607     show ?case proof(cases "n=0")
  2608       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"
  2609         unfolding set_eq_iff and mem_Collect_eq proof(rule, rule)
  2610         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2611         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  2612         show "x\<in>s" proof(cases "card t = 0")
  2613           case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
  2614         next
  2615           case False hence "card t = Suc 0" using t(3) `n=0` by auto
  2616           then obtain a where "t = {a}" unfolding card_Suc_eq by auto
  2617           thus ?thesis using t(2,4) by simp
  2618         qed
  2619       next
  2620         fix x assume "x\<in>s"
  2621         thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2622           apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
  2623       qed thus ?thesis using assms by simp
  2624     next
  2625       case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
  2626         { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
  2627         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}}"
  2628         unfolding set_eq_iff and mem_Collect_eq proof(rule,rule)
  2629         fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  2630           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)"
  2631         then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
  2632           "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
  2633         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
  2634           apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
  2635           using obt(7) and hull_mono[of t "insert u t"] by auto
  2636         ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2637           apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
  2638       next
  2639         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  2640         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  2641         let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  2642           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)"
  2643         show ?P proof(cases "card t = Suc n")
  2644           case False hence "card t \<le> n" using t(3) by auto
  2645           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
  2646             by(auto intro!: exI[where x=t])
  2647         next
  2648           case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
  2649           show ?P proof(cases "u={}")
  2650             case True hence "x=a" using t(4)[unfolded au] by auto
  2651             show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
  2652               using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
  2653           next
  2654             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"
  2655               using t(4)[unfolded au convex_hull_insert[OF False]] by auto
  2656             have *:"1 - vx = ux" using obt(3) by auto
  2657             show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
  2658               using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
  2659               by(auto intro!: exI[where x=u])
  2660           qed
  2661         qed
  2662       qed
  2663       thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
  2664     qed
  2665   qed
  2666 qed
  2667 
  2668 lemma finite_imp_compact_convex_hull:
  2669   fixes s :: "('a::euclidean_space) set"
  2670   shows "finite s \<Longrightarrow> compact(convex hull s)"
  2671 by (metis compact_convex_hull finite_imp_compact)
  2672 
  2673 subsection {* Extremal points of a simplex are some vertices. *}
  2674 
  2675 lemma dist_increases_online:
  2676   fixes a b d :: "'a::real_inner"
  2677   assumes "d \<noteq> 0"
  2678   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
  2679 proof(cases "inner a d - inner b d > 0")
  2680   case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
  2681     apply(rule_tac add_pos_pos) using assms by auto
  2682   thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  2683     by (simp add: algebra_simps inner_commute)
  2684 next
  2685   case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
  2686     apply(rule_tac add_pos_nonneg) using assms by auto
  2687   thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  2688     by (simp add: algebra_simps inner_commute)
  2689 qed
  2690 
  2691 lemma norm_increases_online:
  2692   fixes d :: "'a::real_inner"
  2693   shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
  2694   using dist_increases_online[of d a 0] unfolding dist_norm by auto
  2695 
  2696 lemma simplex_furthest_lt:
  2697   fixes s::"'a::real_inner set" assumes "finite s"
  2698   shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
  2699 proof(induct_tac rule: finite_induct[of s])
  2700   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))"
  2701   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))"
  2702   proof(rule,rule,cases "s = {}")
  2703     case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
  2704     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"
  2705       using y(1)[unfolded convex_hull_insert[OF False]] by auto
  2706     show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
  2707     proof(cases "y\<in>convex hull s")
  2708       case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
  2709         using as(3)[THEN bspec[where x=y]] and y(2) by auto
  2710       thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
  2711     next
  2712       case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
  2713         assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
  2714         thus ?thesis using False and obt(4) by auto
  2715       next
  2716         assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
  2717         thus ?thesis using y(2) by auto
  2718       next
  2719         assume "u\<noteq>0" "v\<noteq>0"
  2720         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
  2721         have "x\<noteq>b" proof(rule ccontr) 
  2722           assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
  2723             using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
  2724           thus False using obt(4) and False by simp qed
  2725         hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
  2726         show ?thesis using dist_increases_online[OF *, of a y]
  2727         proof(erule_tac disjE)
  2728           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
  2729           hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
  2730             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  2731           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
  2732             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  2733             apply(rule_tac x="u + w" in exI) apply rule defer 
  2734             apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  2735           ultimately show ?thesis by auto
  2736         next
  2737           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
  2738           hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
  2739             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  2740           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
  2741             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  2742             apply(rule_tac x="u - w" in exI) apply rule defer 
  2743             apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  2744           ultimately show ?thesis by auto
  2745         qed
  2746       qed auto
  2747     qed
  2748   qed auto
  2749 qed (auto simp add: assms)
  2750 
  2751 lemma simplex_furthest_le:
  2752   fixes s :: "('a::euclidean_space) set"
  2753   assumes "finite s" "s \<noteq> {}"
  2754   shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
  2755 proof-
  2756   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  2757   then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
  2758     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
  2759     unfolding dist_commute[of a] unfolding dist_norm by auto
  2760   thus ?thesis proof(cases "x\<in>s")
  2761     case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
  2762       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
  2763     thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
  2764   qed auto
  2765 qed
  2766 
  2767 lemma simplex_furthest_le_exists:
  2768   fixes s :: "('a::euclidean_space) set"
  2769   shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
  2770   using simplex_furthest_le[of s] by (cases "s={}")auto
  2771 
  2772 lemma simplex_extremal_le:
  2773   fixes s :: "('a::euclidean_space) set"
  2774   assumes "finite s" "s \<noteq> {}"
  2775   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)"
  2776 proof-
  2777   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  2778   then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
  2779     "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
  2780     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
  2781   thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
  2782     assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
  2783       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
  2784     thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
  2785   next
  2786     assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
  2787       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
  2788     thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
  2789       by (auto simp add: norm_minus_commute)
  2790   qed auto
  2791 qed 
  2792 
  2793 lemma simplex_extremal_le_exists:
  2794   fixes s :: "('a::euclidean_space) set"
  2795   shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
  2796   \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
  2797   using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
  2798 
  2799 subsection {* Closest point of a convex set is unique, with a continuous projection. *}
  2800 
  2801 definition
  2802   closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a" where
  2803  "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
  2804 
  2805 lemma closest_point_exists:
  2806   assumes "closed s" "s \<noteq> {}"
  2807   shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
  2808   unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
  2809   using distance_attains_inf[OF assms(1,2), of a] by auto
  2810 
  2811 lemma closest_point_in_set:
  2812   "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
  2813   by(meson closest_point_exists)
  2814 
  2815 lemma closest_point_le:
  2816   "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
  2817   using closest_point_exists[of s] by auto
  2818 
  2819 lemma closest_point_self:
  2820   assumes "x \<in> s"  shows "closest_point s x = x"
  2821   unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
  2822   using assms by auto
  2823 
  2824 lemma closest_point_refl:
  2825  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
  2826   using closest_point_in_set[of s x] closest_point_self[of x s] by auto
  2827 
  2828 lemma closer_points_lemma:
  2829   assumes "inner y z > 0"
  2830   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
  2831 proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
  2832   thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
  2833     fix v assume "0<v" "v \<le> inner y z / inner z z"
  2834     thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
  2835       by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
  2836   qed(rule divide_pos_pos, auto) qed
  2837 
  2838 lemma closer_point_lemma:
  2839   assumes "inner (y - x) (z - x) > 0"
  2840   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
  2841 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)"
  2842     using closer_points_lemma[OF assms] by auto
  2843   show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
  2844     unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
  2845 
  2846 lemma any_closest_point_dot:
  2847   assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  2848   shows "inner (a - x) (y - x) \<le> 0"
  2849 proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
  2850   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
  2851   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
  2852   thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
  2853 
  2854 lemma any_closest_point_unique:
  2855   fixes x :: "'a::real_inner"
  2856   assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
  2857   "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
  2858   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)]
  2859   unfolding norm_pths(1) and norm_le_square
  2860   by (auto simp add: algebra_simps)
  2861 
  2862 lemma closest_point_unique:
  2863   assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  2864   shows "x = closest_point s a"
  2865   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
  2866   using closest_point_exists[OF assms(2)] and assms(3) by auto
  2867 
  2868 lemma closest_point_dot:
  2869   assumes "convex s" "closed s" "x \<in> s"
  2870   shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
  2871   apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
  2872   using closest_point_exists[OF assms(2)] and assms(3) by auto
  2873 
  2874 lemma closest_point_lt:
  2875   assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
  2876   shows "dist a (closest_point s a) < dist a x"
  2877   apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
  2878   apply(rule closest_point_unique[OF assms(1-3), of a])
  2879   using closest_point_le[OF assms(2), of _ a] by fastsimp
  2880 
  2881 lemma closest_point_lipschitz:
  2882   assumes "convex s" "closed s" "s \<noteq> {}"
  2883   shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
  2884 proof-
  2885   have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
  2886        "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
  2887     apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
  2888     using closest_point_exists[OF assms(2-3)] by auto
  2889   thus ?thesis unfolding dist_norm and norm_le
  2890     using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
  2891     by (simp add: inner_add inner_diff inner_commute) qed
  2892 
  2893 lemma continuous_at_closest_point:
  2894   assumes "convex s" "closed s" "s \<noteq> {}"
  2895   shows "continuous (at x) (closest_point s)"
  2896   unfolding continuous_at_eps_delta 
  2897   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
  2898 
  2899 lemma continuous_on_closest_point:
  2900   assumes "convex s" "closed s" "s \<noteq> {}"
  2901   shows "continuous_on t (closest_point s)"
  2902 by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
  2903 
  2904 subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
  2905 
  2906 lemma supporting_hyperplane_closed_point:
  2907   fixes z :: "'a::{real_inner,heine_borel}"
  2908   assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
  2909   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)"
  2910 proof-
  2911   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
  2912   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)
  2913     apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
  2914     show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
  2915       unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
  2916   next
  2917     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)"
  2918       using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
  2919     assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
  2920       "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)
  2921     thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
  2922   qed auto
  2923 qed
  2924 
  2925 lemma separating_hyperplane_closed_point:
  2926   fixes z :: "'a::{real_inner,heine_borel}"
  2927   assumes "convex s" "closed s" "z \<notin> s"
  2928   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
  2929 proof(cases "s={}")
  2930   case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
  2931     using less_le_trans[OF _ inner_ge_zero[of z]] by auto
  2932 next
  2933   case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
  2934     using distance_attains_inf[OF assms(2) False] by auto
  2935   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
  2936     apply rule defer apply rule proof-
  2937     fix x assume "x\<in>s"
  2938     have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
  2939       assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
  2940       then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
  2941       thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
  2942         using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
  2943         using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
  2944     moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
  2945     hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
  2946     ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
  2947       unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
  2948   qed(insert `y\<in>s` `z\<notin>s`, auto)
  2949 qed
  2950 
  2951 lemma separating_hyperplane_closed_0:
  2952   assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 \<notin> s"
  2953   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
  2954   proof(cases "s={}")
  2955   case True have "norm ((basis 0)::'a) = 1" by auto
  2956   hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
  2957     apply(subst norm_le_zero_iff[THEN sym]) by auto
  2958   thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
  2959     using True using DIM_positive[where 'a='a] by auto
  2960 next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
  2961     apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
  2962 
  2963 subsection {* Now set-to-set for closed/compact sets. *}
  2964 
  2965 lemma separating_hyperplane_closed_compact:
  2966   assumes "convex (s::('a::euclidean_space) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
  2967   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  2968 proof(cases "s={}")
  2969   case True
  2970   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
  2971   obtain z::"'a" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
  2972   hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
  2973   then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
  2974     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
  2975   thus ?thesis using True by auto
  2976 next
  2977   case False then obtain y where "y\<in>s" by auto
  2978   obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
  2979     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
  2980     using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
  2981   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)
  2982   def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
  2983   show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
  2984     apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
  2985     from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
  2986       apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
  2987     hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
  2988     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
  2989   next
  2990     fix x assume "x\<in>s" 
  2991     hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
  2992       using ab[THEN bspec[where x=x]] by auto
  2993     thus "k + b / 2 < inner a x" using `0 < b` by auto
  2994   qed
  2995 qed
  2996 
  2997 lemma separating_hyperplane_compact_closed:
  2998   fixes s :: "('a::euclidean_space) set"
  2999   assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
  3000   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  3001 proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
  3002     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
  3003   thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
  3004 
  3005 subsection {* General case without assuming closure and getting non-strict separation. *}
  3006 
  3007 lemma separating_hyperplane_set_0:
  3008   assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
  3009   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
  3010 proof- let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
  3011   have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
  3012     apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
  3013     defer apply(rule,rule,erule conjE) proof-
  3014     fix f assume as:"f \<subseteq> ?k ` s" "finite f"
  3015     obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
  3016     then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
  3017       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
  3018       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
  3019       using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
  3020     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)
  3021        using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
  3022        apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
  3023        by(auto simp add: inner_commute elim!: ballE)
  3024     thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
  3025   qed(insert closed_halfspace_ge, auto)
  3026   then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
  3027   thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
  3028 
  3029 lemma separating_hyperplane_sets:
  3030   assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
  3031   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)"
  3032 proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
  3033   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" 
  3034     using assms(3-5) by auto 
  3035   hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
  3036     by (force simp add: inner_diff)
  3037   thus ?thesis
  3038     apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
  3039     apply auto
  3040     apply (rule Sup[THEN isLubD2]) 
  3041     prefer 4
  3042     apply (rule Sup_least) 
  3043      using assms(3-5) apply (auto simp add: setle_def)
  3044     apply metis
  3045     done
  3046 qed
  3047 
  3048 subsection {* More convexity generalities. *}
  3049 
  3050 lemma convex_closure:
  3051   fixes s :: "'a::real_normed_vector set"
  3052   assumes "convex s" shows "convex(closure s)"
  3053   unfolding convex_def Ball_def closure_sequential
  3054   apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
  3055   apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
  3056   apply(rule assms[unfolded convex_def, rule_format]) prefer 6
  3057   by (auto intro: tendsto_intros)
  3058 
  3059 lemma convex_interior:
  3060   fixes s :: "'a::real_normed_vector set"
  3061   assumes "convex s" shows "convex(interior s)"
  3062   unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
  3063   fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
  3064   fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
  3065   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)
  3066     apply rule unfolding subset_eq defer apply rule proof-
  3067     fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
  3068     hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
  3069       apply(rule_tac assms[unfolded convex_alt, rule_format])
  3070       using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
  3071     thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
  3072 
  3073 lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
  3074   using hull_subset[of s convex] convex_hull_empty by auto
  3075 
  3076 subsection {* Moving and scaling convex hulls. *}
  3077 
  3078 lemma convex_hull_translation_lemma:
  3079   "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
  3080 by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono mem_def)
  3081 
  3082 lemma convex_hull_bilemma: fixes neg
  3083   assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
  3084   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)
  3085   \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
  3086   using assms by(metis subset_antisym) 
  3087 
  3088 lemma convex_hull_translation:
  3089   "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
  3090   apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
  3091 
  3092 lemma convex_hull_scaling_lemma:
  3093  "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  3094 by (metis convex_convex_hull convex_scaling hull_subset mem_def subset_hull subset_image_iff)
  3095 
  3096 lemma convex_hull_scaling:
  3097   "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  3098   apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
  3099   unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
  3100 
  3101 lemma convex_hull_affinity:
  3102   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
  3103 by(simp only: image_image[THEN sym] convex_hull_scaling convex_hull_translation)
  3104 
  3105 subsection {* Convexity of cone hulls *}
  3106 
  3107 lemma convex_cone_hull:
  3108 fixes S :: "('m::euclidean_space) set"
  3109 assumes "convex S"
  3110 shows "convex (cone hull S)"
  3111 proof-
  3112 { fix x y assume xy_def: "x : cone hull S & y : cone hull S"
  3113   hence "S ~= {}" using cone_hull_empty_iff[of S] by auto
  3114   fix u v assume uv_def: "u>=0 & v>=0 & (u :: real)+v=1"
  3115   hence *: "u *\<^sub>R x : cone hull S & v *\<^sub>R y : cone hull S"
  3116      using cone_cone_hull[of S] xy_def cone_def[of "cone hull S"] by auto
  3117   from * obtain cx xx where x_def: "u *\<^sub>R x = cx *\<^sub>R xx & (cx :: real)>=0 & xx : S"
  3118      using cone_hull_expl[of S] by auto
  3119   from * obtain cy yy where y_def: "v *\<^sub>R y = cy *\<^sub>R yy & (cy :: real)>=0 & yy : S"
  3120      using cone_hull_expl[of S] by auto
  3121   { assume "cx+cy<=0" hence "u *\<^sub>R x=0 & v *\<^sub>R y=0" using x_def y_def by auto
  3122     hence "u *\<^sub>R x+ v *\<^sub>R y = 0" by auto
  3123     hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using cone_hull_contains_0[of S] `S ~= {}` by auto
  3124   }
  3125   moreover
  3126   { assume "cx+cy>0"
  3127     hence "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy : S"
  3128       using assms mem_convex_alt[of S xx yy cx cy] x_def y_def by auto
  3129     hence "cx *\<^sub>R xx + cy *\<^sub>R yy : cone hull S"
  3130       using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"]
  3131       `cx+cy>0` by (auto simp add: scaleR_right_distrib)
  3132     hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using x_def y_def by auto
  3133   }
  3134   moreover have "(cx+cy<=0) | (cx+cy>0)" by auto
  3135   ultimately have "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" by blast
  3136 } from this show ?thesis unfolding convex_def by auto
  3137 qed
  3138 
  3139 lemma cone_convex_hull:
  3140 fixes S :: "('m::euclidean_space) set"
  3141 assumes "cone S"
  3142 shows "cone (convex hull S)"
  3143 proof-
  3144 { assume "S = {}" hence ?thesis by auto }
  3145 moreover
  3146 { assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  3147   { fix c assume "(c :: real)>0"
  3148     hence "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
  3149        using convex_hull_scaling[of _ S] by auto
  3150     also have "...=convex hull S" using * `c>0` by auto
  3151     finally have "op *\<^sub>R c ` (convex hull S) = convex hull S" by auto
  3152   }
  3153   hence "0 : convex hull S & (!c. c>0 --> (op *\<^sub>R c ` (convex hull S)) = (convex hull S))"
  3154      using * hull_subset[of S convex] by auto
  3155   hence ?thesis using `S ~= {}` cone_iff[of "convex hull S"] by auto
  3156 }
  3157 ultimately show ?thesis by blast
  3158 qed
  3159 
  3160 subsection {* Convex set as intersection of halfspaces. *}
  3161 
  3162 lemma convex_halfspace_intersection:
  3163   fixes s :: "('a::euclidean_space) set"
  3164   assumes "closed s" "convex s"
  3165   shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
  3166   apply(rule set_eqI, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
  3167   fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
  3168   hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
  3169   thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
  3170     apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
  3171 qed auto
  3172 
  3173 subsection {* Radon's theorem (from Lars Schewe). *}
  3174 
  3175 lemma radon_ex_lemma:
  3176   assumes "finite c" "affine_dependent c"
  3177   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"
  3178 proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
  3179   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
  3180     and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
  3181 
  3182 lemma radon_s_lemma:
  3183   assumes "finite s" "setsum f s = (0::real)"
  3184   shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
  3185 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
  3186   show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  3187     using assms(2) by assumption qed
  3188 
  3189 lemma radon_v_lemma:
  3190   assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
  3191   shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
  3192 proof-
  3193   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 
  3194   show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  3195     using assms(2) by assumption qed
  3196 
  3197 lemma radon_partition:
  3198   assumes "finite c" "affine_dependent c"
  3199   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
  3200   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
  3201   have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
  3202   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}"
  3203   have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
  3204     case False hence "u v < 0" by auto
  3205     thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
  3206       case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
  3207     next
  3208       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
  3209       thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
  3210   qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
  3211 
  3212   hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
  3213   moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
  3214     "(\<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)"
  3215     using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
  3216   hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
  3217    "(\<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)" 
  3218     unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
  3219   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
  3220     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
  3221 
  3222   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
  3223     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)
  3224     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
  3225     by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
  3226   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
  3227     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
  3228   hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
  3229     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)
  3230     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
  3231     by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
  3232   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
  3233 qed
  3234 
  3235 lemma radon: assumes "affine_dependent c"
  3236   obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
  3237 proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
  3238   hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
  3239   from radon_partition[OF *] guess m .. then guess p ..
  3240   thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
  3241 
  3242 subsection {* Helly's theorem. *}
  3243 
  3244 lemma helly_induct: fixes f::"('a::euclidean_space) set set"
  3245   assumes "card f = n" "n \<ge> DIM('a) + 1"
  3246   "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  3247   shows "\<Inter> f \<noteq> {}"
  3248 using assms proof(induct n arbitrary: f)
  3249 case (Suc n)
  3250 have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
  3251 show "\<Inter> f \<noteq> {}" apply(cases "n = DIM('a)") apply(rule Suc(5)[rule_format])
  3252   unfolding `card f = Suc n` proof-
  3253   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
  3254     apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
  3255     defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
  3256   then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
  3257   show ?thesis proof(cases "inj_on X f")
  3258     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
  3259     hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
  3260     show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
  3261       apply(rule, rule X[rule_format]) using X st by auto
  3262   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> {}"
  3263       using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
  3264       unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
  3265     have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
  3266     then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
  3267     hence "f \<union> (g \<union> h) = f" by auto
  3268     hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
  3269       unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
  3270     have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
  3271     have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
  3272       apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  unfolding mem_def unfolding subset_eq
  3273       apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
  3274       fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
  3275       thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
  3276       fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
  3277       thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
  3278     qed(auto)
  3279     thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
  3280 qed(auto) qed(auto)
  3281 
  3282 lemma helly: fixes f::"('a::euclidean_space) set set"
  3283   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
  3284           "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  3285   shows "\<Inter> f \<noteq>{}"
  3286   apply(rule helly_induct) using assms by auto
  3287 
  3288 subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
  3289 
  3290 lemma compact_frontier_line_lemma:
  3291   fixes s :: "('a::euclidean_space) set"
  3292   assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
  3293   obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
  3294 proof-
  3295   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
  3296   let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
  3297   have A:"?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
  3298     by auto
  3299   have *:"\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
  3300   have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
  3301     apply(rule, rule continuous_vmul)
  3302     apply(rule continuous_at_id) by(rule compact_interval)
  3303   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)])
  3304     unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
  3305   ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
  3306     "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
  3307 
  3308   have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
  3309   { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
  3310     hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
  3311       using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
  3312     hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
  3313       apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
  3314       using as(1) `u\<ge>0` by(auto simp add:field_simps) 
  3315     hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
  3316   } note u_max = this
  3317 
  3318   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]
  3319     prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
  3320     fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
  3321     hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
  3322     thus False using u_max[OF _ as] by auto
  3323   qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
  3324   thus ?thesis by(metis that[of u] u_max obt(1))
  3325 qed
  3326 
  3327 lemma starlike_compact_projective:
  3328   assumes "compact s" "cball (0::'a::euclidean_space) 1 \<subseteq> s "
  3329   "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
  3330   shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
  3331 proof-
  3332   have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
  3333   def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
  3334   have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
  3335     using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
  3336   have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
  3337 
  3338   have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
  3339     apply rule unfolding pi_def
  3340     apply (rule continuous_mul)
  3341     apply (rule continuous_at_inv[unfolded o_def])
  3342     apply (rule continuous_at_norm)
  3343     apply simp
  3344     apply (rule continuous_at_id)
  3345     done
  3346   def sphere \<equiv> "{x::'a. norm x = 1}"
  3347   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
  3348 
  3349   have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
  3350   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)
  3351     fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
  3352     hence "x\<noteq>0" using `0\<notin>frontier s` by auto
  3353     obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
  3354       using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
  3355     have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
  3356       assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
  3357       assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
  3358         using v and x and fs unfolding inverse_less_1_iff by auto qed
  3359     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-
  3360       assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
  3361         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
  3362 
  3363   have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
  3364     apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
  3365     apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_eqI,rule) 
  3366     unfolding inj_on_def prefer 3 apply(rule,rule,rule)
  3367   proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
  3368     thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
  3369   next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
  3370     then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
  3371       using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
  3372     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
  3373   next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
  3374     hence xys:"x\<in>s" "y\<in>s" using fs by auto
  3375     from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
  3376     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 
  3377     from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto 
  3378     have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
  3379       unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
  3380     hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
  3381       using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
  3382       using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
  3383       using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
  3384     thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
  3385   qed(insert `0 \<notin> frontier s`, auto)
  3386   then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
  3387     "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
  3388   
  3389   have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
  3390     apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
  3391 
  3392   { fix x assume as:"x \<in> cball (0::'a) 1"
  3393     have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
  3394       case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
  3395       thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
  3396         apply(rule_tac fs[unfolded subset_eq, rule_format])
  3397         unfolding surf(5)[THEN sym] by auto
  3398     next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
  3399         unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
  3400 
  3401   { fix x assume "x\<in>s"
  3402     hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
  3403       case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
  3404     next let ?a = "inverse (norm (surf (pi x)))"
  3405       case False hence invn:"inverse (norm x) \<noteq> 0" by auto
  3406       from False have pix:"pi x\<in>sphere" using pi(1) by auto
  3407       hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
  3408       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
  3409       hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
  3410         apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
  3411       have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
  3412       hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
  3413         unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
  3414       moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))" 
  3415         unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
  3416       moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
  3417       hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
  3418         using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
  3419         using False `x\<in>s` by(auto simp add:field_simps)
  3420       ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
  3421         apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
  3422         unfolding pi(2)[OF `?a > 0`] by auto
  3423     qed } note hom2 = this
  3424 
  3425   show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
  3426     apply(rule compact_cball) defer apply(rule set_eqI, rule, erule imageE, drule hom)
  3427     prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
  3428     fix x::"'a" assume as:"x \<in> cball 0 1"
  3429     thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
  3430       case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
  3431         using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
  3432     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
  3433       hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer 
  3434         apply(erule_tac x="basis 0" in ballE)
  3435         unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
  3436         by(auto simp add:norm_basis[unfolded One_nat_def])
  3437       case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
  3438         apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
  3439         unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
  3440         fix e and x::"'a" assume as:"norm x < e / B" "0 < norm x" "0<e"
  3441         hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
  3442         hence "norm (surf (pi x)) \<le> B" using B fs by auto
  3443         hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
  3444         also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
  3445         also have "\<dots> = e" using `B>0` by auto
  3446         finally show "norm x * norm (surf (pi x)) < e" by assumption
  3447       qed(insert `B>0`, auto) qed
  3448   next { fix x assume as:"surf (pi x) = 0"
  3449       have "x = 0" proof(rule ccontr)
  3450         assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
  3451         hence "surf (pi x) \<in> frontier s" using surf(5) by auto
  3452         thus False using `0\<notin>frontier s` unfolding as by simp qed
  3453     } note surf_0 = this
  3454     show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
  3455       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)"
  3456       thus "x=y" proof(cases "x=0 \<or> y=0") 
  3457         case True thus ?thesis using as by(auto elim: surf_0) next
  3458         case False
  3459         hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
  3460           using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
  3461         moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
  3462         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 
  3463         moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
  3464         ultimately show ?thesis using injpi by auto qed qed
  3465   qed auto qed
  3466 
  3467 lemma homeomorphic_convex_compact_lemma: fixes s::"('a::euclidean_space) set"
  3468   assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
  3469   shows "s homeomorphic (cball (0::'a) 1)"
  3470   apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
  3471   fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
  3472   hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
  3473     apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
  3474     unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
  3475     fix y assume "dist (u *\<^sub>R x) y < 1 - u"
  3476     hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
  3477       using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
  3478       unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
  3479       apply (rule mult_left_le_imp_le[of "1 - u"])
  3480       unfolding mult_assoc[symmetric] using `u<1` by auto
  3481     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]
  3482       using as unfolding scaleR_scaleR by auto qed auto
  3483   thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
  3484 
  3485 lemma homeomorphic_convex_compact_cball: fixes e::real and s::"('a::euclidean_space) set"
  3486   assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
  3487   shows "s homeomorphic (cball (b::'a) e)"
  3488 proof- obtain a where "a\<in>interior s" using assms(3) by auto
  3489   then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
  3490   let ?d = "inverse d" and ?n = "0::'a"
  3491   have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
  3492     apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
  3493     apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
  3494     by(auto simp add: mult_right_le_one_le)
  3495   hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
  3496     using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
  3497     using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
  3498   thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
  3499     apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
  3500     using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
  3501 
  3502 lemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) set"
  3503   assumes "convex s" "compact s" "interior s \<noteq> {}"
  3504           "convex t" "compact t" "interior t \<noteq> {}"
  3505   shows "s homeomorphic t"
  3506   using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
  3507 
  3508 subsection {* Epigraphs of convex functions. *}
  3509 
  3510 definition "epigraph s (f::_ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
  3511 
  3512 lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
  3513 
  3514 (** This might break sooner or later. In fact it did already once. **)
  3515 lemma convex_epigraph: 
  3516   "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
  3517   unfolding convex_def convex_on_def
  3518   unfolding Ball_def split_paired_All epigraph_def
  3519   unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
  3520   apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
  3521   apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
  3522   apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)
  3523 
  3524 lemma convex_epigraphI:
  3525   "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex(epigraph s f)"
  3526 unfolding convex_epigraph by auto
  3527 
  3528 lemma convex_epigraph_convex:
  3529   "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
  3530 by(simp add: convex_epigraph)
  3531 
  3532 subsection {* Use this to derive general bound property of convex function. *}
  3533 
  3534 lemma convex_on:
  3535   assumes "convex s"
  3536   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>
  3537    f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
  3538   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
  3539   unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
  3540   apply safe
  3541   apply (drule_tac x=k in spec)
  3542   apply (drule_tac x=u in spec)
  3543   apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
  3544   apply simp
  3545   using assms[unfolded convex] apply simp
  3546   apply(rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
  3547   defer apply(rule setsum_mono) apply(erule_tac x=i in allE) unfolding real_scaleR_def
  3548   apply(rule mult_left_mono)using assms[unfolded convex] by auto
  3549 
  3550 
  3551 subsection {* Convexity of general and special intervals. *}
  3552 
  3553 lemma convexI: (* TODO: move to Library/Convex.thy *)
  3554   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"
  3555   shows "convex s"
  3556 using assms unfolding convex_def by fast
  3557 
  3558 lemma is_interval_convex:
  3559   fixes s :: "('a::euclidean_space) set"
  3560   assumes "is_interval s" shows "convex s"
  3561 proof (rule convexI)
  3562   fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
  3563   hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
  3564   { fix a b assume "\<not> b \<le> u * a + v * b"
  3565     hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
  3566     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)
  3567     hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
  3568   } moreover
  3569   { fix a b assume "\<not> u * a + v * b \<le> a"
  3570     hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
  3571     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
  3572     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
  3573   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)])
  3574     using as(3-) DIM_positive[where 'a='a] by(auto simp add:euclidean_simps) qed
  3575 
  3576 lemma is_interval_connected:
  3577   fixes s :: "('a::euclidean_space) set"
  3578   shows "is_interval s \<Longrightarrow> connected s"
  3579   using is_interval_convex convex_connected by auto
  3580 
  3581 lemma convex_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"
  3582   apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
  3583 
  3584 (* FIXME: rewrite these lemmas without using vec1
  3585 subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
  3586 
  3587 lemma is_interval_1:
  3588   "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)"
  3589   unfolding is_interval_def forall_1 by auto
  3590 
  3591 lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
  3592   apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
  3593   apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
  3594   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"
  3595   hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
  3596   let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
  3597   { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
  3598     using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
  3599   moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
  3600   hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
  3601   ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
  3602     apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
  3603     apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
  3604     by(auto simp add: field_simps) qed
  3605 
  3606 lemma is_interval_convex_1:
  3607   "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
  3608 by(metis is_interval_convex convex_connected is_interval_connected_1)
  3609 
  3610 lemma convex_connected_1:
  3611   "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
  3612 by(metis is_interval_convex convex_connected is_interval_connected_1)
  3613 *)
  3614 subsection {* Another intermediate value theorem formulation. *}
  3615 
  3616 lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3617   assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
  3618   shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
  3619 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
  3620     using assms(1) by auto
  3621   thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
  3622     using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
  3623     using assms by(auto intro!: imageI) qed
  3624 
  3625 lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3626   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
  3627    \<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
  3628 by(rule ivt_increasing_component_on_1)
  3629   (auto simp add: continuous_at_imp_continuous_on)
  3630 
  3631 lemma ivt_decreasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3632   assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
  3633   shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
  3634   apply(subst neg_equal_iff_equal[THEN sym])
  3635   using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"] using assms using continuous_on_neg
  3636   by (auto simp add:euclidean_simps)
  3637 
  3638 lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
  3639   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
  3640     \<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
  3641 by(rule ivt_decreasing_component_on_1)
  3642   (auto simp: continuous_at_imp_continuous_on)
  3643 
  3644 subsection {* A bound within a convex hull, and so an interval. *}
  3645 
  3646 lemma convex_on_convex_hull_bound:
  3647   assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
  3648   shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
  3649   fix x assume "x\<in>convex hull s"
  3650   then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
  3651     unfolding convex_hull_indexed mem_Collect_eq by auto
  3652   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
  3653     unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
  3654     using assms(2) obt(1) by auto
  3655   thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
  3656     unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
  3657 
  3658 lemma unit_interval_convex_hull:
  3659   "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
  3660   (is "?int = convex hull ?points")
  3661 proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
  3662   { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n" 
  3663   hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
  3664     case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
  3665     thus "x\<in>convex hull ?points" using 01 by auto
  3666   next
  3667     case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
  3668       case True hence "x = 0" apply(subst euclidean_eq) by auto
  3669       thus "x\<in>convex hull ?points" using 01 by auto
  3670     next
  3671       case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
  3672       have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
  3673       then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
  3674       have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
  3675         unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
  3676         defer apply(rule_tac x=j in bexI) using i' by auto
  3677       have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
  3678         using i'(2-) `x$$i \<noteq> 0` by auto
  3679       show ?thesis proof(cases "x$$i=1")
  3680         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
  3681         proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
  3682           hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
  3683           hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto 
  3684           hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
  3685           thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
  3686         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
  3687           by auto
  3688       next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
  3689         case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
  3690           apply(subst euclidean_eq) by(auto simp add: field_simps euclidean_simps)
  3691         { fix j assume j:"j<DIM('a)"
  3692           have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
  3693             apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
  3694             using Suc(2)[unfolded mem_interval, rule_format, of j] using j
  3695             by(auto simp add:field_simps euclidean_simps)
  3696           hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
  3697         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}"
  3698           using i01 using i'(3) by auto
  3699         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
  3700         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 
  3701           by( auto simp add:euclidean_simps)
  3702         have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
  3703           using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
  3704         ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
  3705           apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
  3706           unfolding mem_interval using i01 Suc(3) by auto
  3707       qed qed qed } note * = this
  3708   have **:"DIM('a) = card {..<DIM('a)}" by auto
  3709   show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
  3710     apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **) 
  3711     apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
  3712     unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
  3713     by(auto simp add: mem_def[of _ convex]) qed
  3714 
  3715 subsection {* And this is a finite set of vertices. *}
  3716 
  3717 lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
  3718   apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
  3719   apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
  3720   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
  3721   fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
  3722   show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
  3723     apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
  3724     using as apply(subst euclidean_eq) by auto qed auto
  3725 
  3726 subsection {* Hence any cube (could do any nonempty interval). *}
  3727 
  3728 lemma cube_convex_hull:
  3729   assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
  3730   "finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
  3731   let ?d = "(\<chi>\<chi> i. d)::'a"
  3732   have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_eqI, rule)
  3733     unfolding image_iff defer apply(erule bexE) proof-
  3734     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
  3735     { fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
  3736         using as[unfolded mem_interval, THEN spec[where x=i]] i
  3737         by(auto simp add:euclidean_simps)
  3738       hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
  3739         apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
  3740         using assms by(auto simp add: field_simps)
  3741       hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
  3742             "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
  3743     hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
  3744       by(auto simp add: euclidean_simps field_simps)
  3745     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) 
  3746       using assms by auto
  3747   next
  3748     fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z" 
  3749     have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
  3750       using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
  3751       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
  3752       using assms by auto
  3753     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
  3754       apply(erule_tac x=i in allE) using assms by(auto simp add: euclidean_simps) qed
  3755   obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
  3756   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
  3757 
  3758 subsection {* Bounded convex function on open set is continuous. *}
  3759 
  3760 lemma convex_on_bounded_continuous:
  3761   fixes s :: "('a::real_normed_vector) set"
  3762   assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
  3763   shows "continuous_on s f"
  3764   apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
  3765   fix x e assume "x\<in>s" "(0::real) < e"
  3766   def B \<equiv> "abs b + 1"
  3767   have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
  3768     unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
  3769   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
  3770   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
  3771     apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
  3772     fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)" 
  3773     show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
  3774       case False def t \<equiv> "k / norm (y - x)"
  3775       have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
  3776       have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
  3777         apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute) 
  3778       { def w \<equiv> "x + t *\<^sub>R (y - x)"
  3779         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  3780           unfolding t_def using `k>0` by auto
  3781         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)
  3782         also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps)
  3783         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)
  3784         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) 
  3785         hence "(f w - f x) / t < e"
  3786           using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps) 
  3787         hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
  3788           using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
  3789           using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
  3790       moreover 
  3791       { def w \<equiv> "x - t *\<^sub>R (y - x)"
  3792         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  3793           unfolding t_def using `k>0` by auto
  3794         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)
  3795         also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
  3796         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)
  3797         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) 
  3798         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) 
  3799         have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y" 
  3800           using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
  3801           using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
  3802         also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding divide_inverse by (auto simp add:field_simps)
  3803         also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
  3804         finally have "f x - f y < e" by auto }
  3805       ultimately show ?thesis by auto 
  3806     qed(insert `0<e`, auto) 
  3807   qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
  3808 
  3809 subsection {* Upper bound on a ball implies upper and lower bounds. *}
  3810 
  3811 lemma convex_bounds_lemma:
  3812   fixes x :: "'a::real_normed_vector"
  3813   assumes "convex_on (cball x e) f"  "\<forall>y \<in> cball x e. f y \<le> b"
  3814   shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
  3815   apply(rule) proof(cases "0 \<le> e") case True
  3816   fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *\<^sub>R x - y"
  3817   have *:"x - (2 *\<^sub>R x - y) = y - x" by (simp add: scaleR_2)
  3818   have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
  3819   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
  3820   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"]
  3821     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
  3822 next case False fix y assume "y\<in>cball x e" 
  3823   hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
  3824   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
  3825 
  3826 subsection {* Hence a convex function on an open set is continuous. *}
  3827 
  3828 lemma convex_on_continuous:
  3829   assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f" 
  3830   (* FIXME: generalize to euclidean_space *)
  3831   shows "continuous_on s f"
  3832   unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
  3833   note dimge1 = DIM_positive[where 'a='a]
  3834   fix x assume "x\<in>s"
  3835   then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
  3836   def d \<equiv> "e / real DIM('a)"
  3837   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
  3838   let ?d = "(\<chi>\<chi> i. d)::'a"
  3839   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
  3840   have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:euclidean_simps)
  3841   hence "c\<noteq>{}" using c by auto
  3842   def k \<equiv> "Max (f ` c)"
  3843   have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
  3844     apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
  3845     fix z assume z:"z\<in>{x - ?d..x + ?d}"
  3846     have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
  3847       unfolding real_eq_of_nat by auto
  3848     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
  3849       using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:euclidean_simps) qed
  3850   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
  3851     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
  3852   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
  3853   hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
  3854   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
  3855   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
  3856     fix y assume y:"y\<in>cball x d"
  3857     { fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i"  "y $$ i \<le> x $$ i + d" 
  3858         using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add:euclidean_simps)  }
  3859     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
  3860       by(auto simp add:euclidean_simps) qed
  3861   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
  3862     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
  3863     apply force
  3864     done
  3865   thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
  3866     using `d>0` by auto 
  3867 qed
  3868 
  3869 subsection {* Line segments, Starlike Sets, etc.*}
  3870 
  3871 (* Use the same overloading tricks as for intervals, so that 
  3872    segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
  3873 
  3874 definition
  3875   midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a" where
  3876   "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
  3877 
  3878 definition
  3879   open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
  3880   "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real.  0 < u \<and> u < 1}"
  3881 
  3882 definition
  3883   closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
  3884   "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
  3885 
  3886 definition "between = (\<lambda> (a,b). closed_segment a b)"
  3887 
  3888 lemmas segment = open_segment_def closed_segment_def
  3889 
  3890 definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
  3891 
  3892 lemma midpoint_refl: "midpoint x x = x"
  3893   unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto
  3894 
  3895 lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
  3896 
  3897 lemma midpoint_eq_iff: "midpoint a b = c \<longleftrightarrow> a + b = c + c"
  3898 proof -
  3899   have "midpoint a b = c \<longleftrightarrow> scaleR 2 (midpoint a b) = scaleR 2 c"
  3900     by simp
  3901   thus ?thesis
  3902     unfolding midpoint_def scaleR_2 [symmetric] by simp
  3903 qed
  3904 
  3905 lemma dist_midpoint:
  3906   fixes a b :: "'a::real_normed_vector" shows
  3907   "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
  3908   "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
  3909   "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
  3910   "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
  3911 proof-
  3912   have *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
  3913   have **:"\<And>x y::'a. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2" by auto
  3914   note scaleR_right_distrib [simp]
  3915   show ?t1 unfolding midpoint_def dist_norm apply (rule **)
  3916     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3917   show ?t2 unfolding midpoint_def dist_norm apply (rule *)
  3918     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3919   show ?t3 unfolding midpoint_def dist_norm apply (rule *)
  3920     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3921   show ?t4 unfolding midpoint_def dist_norm apply (rule **)
  3922     by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
  3923 qed
  3924 
  3925 lemma midpoint_eq_endpoint:
  3926   "midpoint a b = a \<longleftrightarrow> a = b"
  3927   "midpoint a b = b \<longleftrightarrow> a = b"
  3928   unfolding midpoint_eq_iff by auto
  3929 
  3930 lemma convex_contains_segment:
  3931   "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. closed_segment a b \<subseteq> s)"
  3932   unfolding convex_alt closed_segment_def by auto
  3933 
  3934 lemma convex_imp_starlike:
  3935   "convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
  3936   unfolding convex_contains_segment starlike_def by auto
  3937 
  3938 lemma segment_convex_hull:
  3939  "closed_segment a b = convex hull {a,b}" proof-
  3940   have *:"\<And>x. {x} \<noteq> {}" by auto
  3941   have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
  3942   show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_eqI)
  3943     unfolding mem_Collect_eq apply(rule,erule exE) 
  3944     apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
  3945     apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
  3946 
  3947 lemma convex_segment: "convex (closed_segment a b)"
  3948   unfolding segment_convex_hull by(rule convex_convex_hull)
  3949 
  3950 lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
  3951   unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto
  3952 
  3953 lemma segment_furthest_le:
  3954   fixes a b x y :: "'a::euclidean_space"
  3955   assumes "x \<in> closed_segment a b" shows "norm(y - x) \<le> norm(y - a) \<or>  norm(y - x) \<le> norm(y - b)" proof-
  3956   obtain z where "z\<in>{a, b}" "norm (x - y) \<le> norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
  3957     using assms[unfolded segment_convex_hull] by auto
  3958   thus ?thesis by(auto simp add:norm_minus_commute) qed
  3959 
  3960 lemma segment_bound:
  3961   fixes x a b :: "'a::euclidean_space"
  3962   assumes "x \<in> closed_segment a b"
  3963   shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
  3964   using segment_furthest_le[OF assms, of a]
  3965   using segment_furthest_le[OF assms, of b]
  3966   by (auto simp add:norm_minus_commute) 
  3967 
  3968 lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
  3969 
  3970 lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
  3971   unfolding between_def mem_def by auto
  3972 
  3973 lemma between:"between (a,b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
  3974 proof(cases "a = b")
  3975   case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
  3976     by(auto simp add:segment_refl dist_commute) next
  3977   case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto 
  3978   have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
  3979   show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
  3980     apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
  3981       fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" 
  3982       hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
  3983         unfolding as(1) by(auto simp add:algebra_simps)
  3984       show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
  3985         unfolding norm_minus_commute[of x a] * using as(2,3)
  3986         by(auto simp add: field_simps)
  3987     next assume as:"dist a b = dist a x + dist x b"
  3988       have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2]
  3989         unfolding as[unfolded dist_norm] norm_ge_zero by auto 
  3990       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)
  3991         unfolding dist_norm apply(subst euclidean_eq) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
  3992       proof(rule,rule) fix i assume i:"i<DIM('a)"
  3993           have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
  3994             ((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
  3995             using Fal by(auto simp add: field_simps euclidean_simps)
  3996           also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
  3997             unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
  3998             apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps euclidean_simps)
  3999           finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i" 
  4000             by auto
  4001         qed(insert Fal2, auto) qed qed
  4002 
  4003 lemma between_midpoint: fixes a::"'a::euclidean_space" shows
  4004   "between (a,b) (midpoint a b)" (is ?t1) 
  4005   "between (b,a) (midpoint a b)" (is ?t2)
  4006 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
  4007   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
  4008     unfolding euclidean_eq[where 'a='a]
  4009     by(auto simp add:field_simps euclidean_simps) qed
  4010 
  4011 lemma between_mem_convex_hull:
  4012   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
  4013   unfolding between_mem_segment segment_convex_hull ..
  4014 
  4015 subsection {* Shrinking towards the interior of a convex set. *}
  4016 
  4017 lemma mem_interior_convex_shrink:
  4018   fixes s :: "('a::euclidean_space) set"
  4019   assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
  4020   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  4021 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  4022   show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
  4023     apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
  4024     fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
  4025     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)
  4026     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)"
  4027       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
  4028       by(auto simp add: euclidean_simps euclidean_eq[where 'a='a] field_simps) 
  4029     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)
  4030     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
  4031       by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
  4032     finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
  4033       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
  4034   qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
  4035 
  4036 lemma mem_interior_closure_convex_shrink:
  4037   fixes s :: "('a::euclidean_space) set"
  4038   assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
  4039   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  4040 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  4041   have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
  4042     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  4043     case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
  4044     show ?thesis proof(cases "e=1")
  4045       case True obtain y where "y\<in>s" "y \<noteq> x" "dist y x < 1"
  4046         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  4047       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  4048       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  4049         using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  4050       then obtain y where "y\<in>s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
  4051         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  4052       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  4053   then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
  4054   def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
  4055   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)
  4056   have "z\<in>interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
  4057     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
  4058     by(auto simp add:field_simps norm_minus_commute)
  4059   thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink) 
  4060     using assms(1,4-5) `y\<in>s` by auto qed
  4061 
  4062 subsection {* Some obvious but surprisingly hard simplex lemmas. *}
  4063 
  4064 lemma simplex:
  4065   assumes "finite s" "0 \<notin> s"
  4066   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)}"
  4067   unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_eqI, rule) unfolding mem_Collect_eq
  4068   apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
  4069   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)
  4070   unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
  4071 
  4072 lemma substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4073   shows "convex hull (insert 0 { basis i | i. i : d}) =
  4074         {x::'a::euclidean_space . (!i<DIM('a). 0 <= x$$i) & setsum (%i. x$$i) d <= 1 &
  4075   (!i<DIM('a). i ~: d --> x$$i = 0)}" 
  4076   (is "convex hull (insert 0 ?p) = ?s")
  4077 (* Proof is a modified copy of the proof of similar lemma std_simplex in Convex_Euclidean_Space.thy *)
  4078 proof- let ?D = d (*"{..<DIM('a::euclidean_space)}"*)
  4079   have "0 ~: ?p" using assms by (auto simp: image_def)
  4080   have "{(basis i)::'n::euclidean_space |i. i \<in> ?D} = basis ` ?D" by auto
  4081   note sumbas = this setsum_reindex[OF basis_inj_on[of d], unfolded o_def, OF assms]
  4082   show ?thesis unfolding simplex[OF finite_substdbasis `0 ~: ?p`] 
  4083     apply(rule set_eqI) unfolding mem_Collect_eq apply rule
  4084     apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
  4085     fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x"
  4086       "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
  4087     have *:"\<forall>i<DIM('a). i:d --> u (basis i) = x$$i" and "(!i<DIM('a). i ~: d --> x $$ i = 0)" using as(3) 
  4088       unfolding sumbas unfolding substdbasis_expansion_unique[OF assms] by auto
  4089     hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $$ x) ?D" unfolding sumbas 
  4090       apply-apply(rule setsum_cong2) using assms by auto
  4091     have " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1" 
  4092       apply - proof(rule,rule,rule)
  4093       fix i assume i:"i<DIM('a)" have "i : d ==> 0 \<le> x$$i" unfolding *[rule_format,OF i,THEN sym] 
  4094          apply(rule_tac as(1)[rule_format]) by auto
  4095       moreover have "i ~: d ==> 0 \<le> x$$i" 
  4096         using `(!i<DIM('a). i ~: d --> x $$ i = 0)`[rule_format, OF i] by auto
  4097       ultimately show "0 \<le> x$$i" by auto
  4098     qed(insert as(2)[unfolded **], auto)
  4099     from this show " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1 & (!i<DIM('a). i ~: d --> x $$ i = 0)" 
  4100       using `(!i<DIM('a). i ~: d --> x $$ i = 0)` by auto
  4101   next fix x::"'a::euclidean_space" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
  4102       "(!i<DIM('a). i ~: d --> x $$ i = 0)"
  4103     show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and>
  4104       setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
  4105       apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE)
  4106       using as(1) apply(erule_tac x=i in allE) unfolding sumbas apply safe unfolding not_less basis_zero
  4107       unfolding substdbasis_expansion_unique[OF assms] euclidean_component_def[THEN sym]
  4108       using as(2,3) by(auto simp add:dot_basis not_less  basis_zero) 
  4109   qed qed
  4110 
  4111 lemma std_simplex:
  4112   "convex hull (insert 0 { basis i | i. i<DIM('a)}) =
  4113         {x::'a::euclidean_space . (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} \<le> 1 }"
  4114   using substd_simplex[of "{..<DIM('a)}"] by auto
  4115 
  4116 lemma interior_std_simplex:
  4117   "interior (convex hull (insert 0 { basis i| i. i<DIM('a)})) =
  4118   {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 < x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} < 1 }"
  4119   apply(rule set_eqI) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
  4120   unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
  4121   fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x<DIM('a). 0 \<le> xa $$ x) \<and> setsum (op $$ xa) {..<DIM('a)} \<le> 1"
  4122   show "(\<forall>xa<DIM('a). 0 < x $$ xa) \<and> setsum (op $$ x) {..<DIM('a)} < 1" apply(safe) proof-
  4123     fix i assume i:"i<DIM('a)" thus "0 < x $$ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
  4124       unfolding dist_norm  by(auto simp add: inner_simps euclidean_component_def dot_basis elim!:allE[where x=i])
  4125   next have **:"dist x (x + (e / 2) *\<^sub>R basis 0) < e" using  `e>0`
  4126       unfolding dist_norm by(auto intro!: mult_strict_left_mono)
  4127     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)"
  4128       unfolding euclidean_component_def by(auto simp add:inner_simps dot_basis)
  4129     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)}"
  4130       apply(rule_tac setsum_cong) by auto
  4131     have "setsum (op $$ x) {..<DIM('a)} < setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)}" unfolding * setsum_addf
  4132       using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by auto
  4133     also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
  4134     finally show "setsum (op $$ x) {..<DIM('a)} < 1" by auto qed
  4135 next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 < x $$ i" "setsum (op $$ x) {..<DIM('a)} < 1"
  4136   guess a using UNIV_witness[where 'a='b] ..
  4137   let ?d = "(1 - setsum (op $$ x) {..<DIM('a)}) / real (DIM('a))"
  4138   have "Min ((op $$ x) ` {..<DIM('a)}) > 0" apply(rule Min_grI) using as(1) by auto
  4139   moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by(auto simp add: Suc_le_eq)
  4140   ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
  4141     apply(rule_tac x="min (Min ((op $$ x) ` {..<DIM('a)})) ?D" in exI) apply rule defer apply(rule,rule) proof-
  4142     fix y assume y:"dist x y < min (Min (op $$ x ` {..<DIM('a)})) ?d"
  4143     have "setsum (op $$ y) {..<DIM('a)} \<le> setsum (\<lambda>i. x$$i + ?d) {..<DIM('a)}" proof(rule setsum_mono)
  4144       fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
  4145         using component_le_norm[of "y - x" i]
  4146         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
  4147       thus "y $$ i \<le> x $$ i + ?d" by auto qed
  4148     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat by(auto simp add: Suc_le_eq)
  4149     finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1" 
  4150     proof safe fix i assume i:"i<DIM('a)"
  4151       have "norm (x - y) < x$$i" apply(rule less_le_trans) 
  4152         apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
  4153       thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
  4154     qed qed auto qed
  4155 
  4156 lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
  4157   "a \<in> interior(convex hull (insert 0 {basis i | i . i<DIM('a)}))" proof-
  4158   let ?D = "{..<DIM('a)}" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) {(basis i) | i. i<DIM('a)}"
  4159   have *:"{basis i :: 'a | i. i<DIM('a)} = basis ` ?D" by auto
  4160   { fix i assume i:"i<DIM('a)" have "?a $$ i = inverse (2 * real DIM('a))"
  4161       unfolding euclidean_component.setsum * and setsum_reindex[OF basis_inj] and o_def
  4162       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"]) apply(rule setsum_cong2)
  4163       defer apply(subst setsum_delta') unfolding euclidean_component_def using i by(auto simp add:dot_basis) }
  4164   note ** = this
  4165   show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe
  4166     fix i assume i:"i<DIM('a)" show "0 < ?a $$ i" unfolding **[OF i] by(auto simp add: Suc_le_eq)
  4167   next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
  4168     also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat divide_inverse[THEN sym] by (auto simp add:field_simps)
  4169     finally show "setsum (op $$ ?a) ?D < 1" by auto qed qed
  4170 
  4171 lemma rel_interior_substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4172   shows "rel_interior (convex hull (insert 0 { basis i| i. i : d})) =
  4173   {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x$$i) & setsum (%i. x$$i) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)}"
  4174   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
  4175 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex in Convex_Euclidean_Space.thy *)
  4176 proof-
  4177 have "finite d" apply(rule finite_subset) using assms by auto
  4178 { assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq[of _ 0] by auto }
  4179 moreover
  4180 { assume "d~={}"
  4181 have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}" 
  4182    using affine_hull_convex_hull affine_hull_substd_basis assms by auto 
  4183 have aux: "!x::'n::euclidean_space. !i. ((! i:d. 0 <= x$$i) & (!i. i ~: d --> x$$i = 0))--> 0 <= x$$i" by auto
  4184 { fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"
  4185   from this obtain e where e0: "e>0" and 
  4186        "ball x e Int {xa. (!i<DIM('a). i ~: d --> xa$$i = 0)} <= convex hull (insert 0 ?p)" 
  4187        using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto   
  4188   hence as: "ALL xa. (dist x xa < e & (!i<DIM('a). i ~: d --> xa$$i = 0)) -->
  4189     (!i : d. 0 <= xa $$ i) & setsum (op $$ xa) d <= 1"
  4190     unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
  4191   have x0: "(!i<DIM('a). i ~: d --> x$$i = 0)" 
  4192     using x_def rel_interior_subset  substd_simplex[OF assms] by auto
  4193   have "(!i : d. 0 < x $$ i) & setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" apply(rule,rule) 
  4194   proof-
  4195     fix i::nat assume "i:d" 
  4196     hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R basis i) $$ ia" apply-apply(rule as[rule_format,THEN conjunct1])
  4197       unfolding dist_norm using assms `e>0` x0 by auto
  4198     thus "0 < x $$ i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` assms by auto
  4199   next obtain a where a:"a:d" using `d ~= {}` by auto
  4200     have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e"
  4201       using  `e>0` and Euclidean_Space.norm_basis[of a]
  4202       unfolding dist_norm by auto
  4203     have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $$ i = x$$i + (if i = a then e/2 else 0)"
  4204       unfolding euclidean_simps using a assms by auto
  4205     hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d =
  4206       setsum (\<lambda>i. x$$i + (if a = i then e/2 else 0)) d" by(rule_tac setsum_cong, auto)
  4207     have h1: "(ALL i<DIM('a). i ~: d --> (x + (e / 2) *\<^sub>R basis a) $$ i = 0)"
  4208       using as[THEN spec[where x="x + (e / 2) *\<^sub>R basis a"]] `a:d` using x0
  4209       by(auto simp add: norm_basis elim:allE[where x=a]) 
  4210     have "setsum (op $$ x) d < setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d" unfolding * setsum_addf
  4211       using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')
  4212     also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R basis a"] by auto
  4213     finally show "setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" using x0 by auto 
  4214   qed
  4215 }
  4216 moreover
  4217 {
  4218   fix x::"'a::euclidean_space" assume as: "x : ?s"
  4219   have "!i. ((0<x$$i) | (0=x$$i) --> 0<=x$$i)" by auto
  4220   moreover have "!i. (i:d) | (i ~: d)" by auto
  4221   ultimately 
  4222   have "!i. ( (ALL i:d. 0 < x$$i) & (ALL i. i ~: d --> x$$i = 0) ) --> 0 <= x$$i" by metis
  4223   hence h2: "x : convex hull (insert 0 ?p)" using as assms 
  4224     unfolding substd_simplex[OF assms] by fastsimp 
  4225   obtain a where a:"a:d" using `d ~= {}` by auto
  4226   let ?d = "(1 - setsum (op $$ x) d) / real (card d)"
  4227   have "card d >= 1" using `d ~={}` card_ge1[of d] using `finite d` by auto
  4228   have "Min ((op $$ x) ` d) > 0" apply(rule Min_grI) using as `card d >= 1` `finite d` by auto
  4229   moreover have "?d > 0" apply(rule divide_pos_pos) using as using `card d >= 1` by(auto simp add: Suc_le_eq)
  4230   ultimately have h3: "min (Min ((op $$ x) ` d)) ?d > 0" by auto
  4231 
  4232   have "x : rel_interior (convex hull (insert 0 ?p))"
  4233     unfolding rel_interior_ball mem_Collect_eq h0 apply(rule,rule h2)
  4234     unfolding substd_simplex[OF assms]
  4235     apply(rule_tac x="min (Min ((op $$ x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball
  4236   proof- fix y::'a assume y:"dist x y < min (Min (op $$ x ` d)) ?d" and y2:"(!i<DIM('a). i ~: d --> y$$i = 0)"
  4237     have "setsum (op $$ y) d \<le> setsum (\<lambda>i. x$$i + ?d) d" proof(rule setsum_mono)
  4238       fix i assume i:"i\<in>d"
  4239       have "abs (y$$i - x$$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
  4240         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
  4241         by(auto simp add: norm_minus_commute)
  4242       thus "y $$ i \<le> x $$ i + ?d" by auto qed
  4243     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat
  4244       using `card d >= 1` by(auto simp add: Suc_le_eq)
  4245     finally show "setsum (op $$ y) d \<le> 1" .
  4246      
  4247     fix i assume "i<DIM('a)" thus "0 \<le> y$$i" 
  4248     proof(cases "i\<in>d") case True
  4249       have "norm (x - y) < x$$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
  4250         using Min_gr_iff[of "op $$ x ` d" "norm (x - y)"] `card d >= 1` `i:d`
  4251         apply auto by (metis Suc_n_not_le_n True card_eq_0_iff finite_imageI)
  4252       thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format] by auto
  4253     qed(insert y2, auto)
  4254   qed
  4255 } ultimately have
  4256     "!!x :: 'a::euclidean_space. (x : rel_interior (convex hull insert 0 {basis i |i. i : d})) =
  4257     (x : {x. (ALL i:d. 0 < x $$ i) &
  4258     setsum (op $$ x) d < 1 & (ALL i<DIM('a). i ~: d --> x $$ i = 0)})" by blast
  4259 from this have ?thesis by (rule set_eqI)
  4260 } ultimately show ?thesis by blast
  4261 qed
  4262 
  4263 lemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d\<subseteq>{..<DIM('a::euclidean_space)}"
  4264   obtains a::"'a::euclidean_space" where
  4265   "a : rel_interior(convex hull (insert 0 {basis i | i . i : d}))" proof-
  4266 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex_nonempty in Convex_Euclidean_Space.thy *)
  4267   let ?D = d let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
  4268   have *:"{basis i :: 'a | i. i \<in> ?D} = basis ` ?D" by auto
  4269   have "finite d" apply(rule finite_subset) using assms(2) by auto
  4270   hence d1: "real(card d) >= 1" using `d ~={}` card_ge1[of d] by auto
  4271   { fix i assume "i:d" have "?a $$ i = inverse (2 * real (card d))"
  4272       unfolding * setsum_reindex[OF basis_inj_on, OF assms(2)] o_def
  4273       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"]) 
  4274       unfolding euclidean_component.setsum
  4275       apply(rule setsum_cong2)
  4276       using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)
  4277       by (auto simp add: Euclidean_Space.basis_component[of i])}
  4278   note ** = this
  4279   show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
  4280   proof safe fix i assume "i:d" 
  4281     have "0 < inverse (2 * real (card d))" using d1 by(auto simp add: Suc_le_eq)
  4282     also have "...=?a $$ i" using **[of i] `i:d` by auto
  4283     finally show "0 < ?a $$ i" by auto
  4284   next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D" 
  4285       by(rule setsum_cong2, rule **) 
  4286     also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat real_divide_def[THEN sym]
  4287       by (auto simp add:field_simps)
  4288     finally show "setsum (op $$ ?a) ?D < 1" by auto
  4289   next fix i assume "i<DIM('a)" and "i~:d"
  4290     have "?a : (span {basis i | i. i : d})" 
  4291       apply (rule span_setsum[of "{basis i |i. i : d}" "(%b. b /\<^sub>R (2 * real (card d)))" "{basis i |i. i : d}"]) 
  4292       using finite_substdbasis[of d] apply blast 
  4293     proof-
  4294       { fix x assume "(x :: 'a::euclidean_space): {basis i |i. i : d}"
  4295         hence "x : span {basis i |i. i : d}" 
  4296           using span_superset[of _ "{basis i |i. i : d}"] by auto
  4297         hence "(x /\<^sub>R (2 * real (card d))) : (span {basis i |i. i : d})"
  4298           using span_mul[of x "{basis i |i. i : d}" "(inverse (real (card d)) / 2)"] by auto
  4299       } 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
  4300     qed
  4301     thus "?a $$ i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i<DIM('a)` by auto
  4302   qed
  4303 qed
  4304 
  4305 subsection{* Relative Interior of Convex Set *}
  4306 
  4307 lemma rel_interior_convex_nonempty_aux: 
  4308 fixes S :: "('n::euclidean_space) set" 
  4309 assumes "convex S" and "0 : S"
  4310 shows "rel_interior S ~= {}"
  4311 proof-
  4312 { assume "S = {0}" hence ?thesis using rel_interior_sing by auto }
  4313 moreover { 
  4314 assume "S ~= {0}"
  4315 obtain B where B_def: "independent B & B<=S & (S <= span B) & card B = dim S" using basis_exists[of S] by auto
  4316 hence "B~={}" using B_def assms `S ~= {0}` span_empty by auto
  4317 have "insert 0 B <= span B" using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
  4318 hence "span (insert 0 B) <= span B" 
  4319     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
  4320 hence "convex hull insert 0 B <= span B" 
  4321     using convex_hull_subset_span[of "insert 0 B"] by auto
  4322 hence "span (convex hull insert 0 B) <= span B"
  4323     using span_span[of B] span_mono[of "convex hull insert 0 B" "span B"] by blast
  4324 hence *: "span (convex hull insert 0 B) = span B" 
  4325     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  4326 hence "span (convex hull insert 0 B) = span S"
  4327     using B_def span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
  4328 moreover have "0 : affine hull (convex hull insert 0 B)"
  4329     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  4330 ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
  4331     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"] 
  4332     assms  hull_subset[of S] by auto
  4333 obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = {basis i |i. i : (d :: nat set)} & 
  4334        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)}"
  4335     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by auto
  4336 hence "bounded_linear f" using linear_conv_bounded_linear by auto
  4337 have "d ~={}" using fd B_def `B ~={}` by auto
  4338 have "(insert 0 {basis i |i. i : d}) = f ` (insert 0 B)" using fd linear_0 by auto
  4339 hence "(convex hull (insert 0 {basis i |i. i : d})) = f ` (convex hull (insert 0 B))"
  4340    using convex_hull_linear_image[of f "(insert 0 {basis i |i. i : d})"] 
  4341    convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by auto
  4342 moreover have "rel_interior (f ` (convex hull insert 0 B)) = 
  4343    f ` rel_interior (convex hull insert 0 B)"
  4344    apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
  4345    using `bounded_linear f` fd * by auto
  4346 ultimately have "rel_interior (convex hull insert 0 B) ~= {}"
  4347    using rel_interior_substd_simplex_nonempty[OF `d~={}` d] apply auto by blast 
  4348 moreover have "convex hull (insert 0 B) <= S"
  4349    using B_def assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
  4350 ultimately have ?thesis using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
  4351 } ultimately show ?thesis by auto
  4352 qed
  4353 
  4354 lemma rel_interior_convex_nonempty:
  4355 fixes S :: "('n::euclidean_space) set"
  4356 assumes "convex S"
  4357 shows "rel_interior S = {} <-> S = {}"
  4358 proof-
  4359 { assume "S ~= {}" from this obtain a where "a : S" by auto
  4360   hence "0 : op + (-a) ` S" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
  4361   hence "rel_interior (op + (-a) ` S) ~= {}"  
  4362     using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"] 
  4363           convex_translation[of S "-a"] assms by auto 
  4364   hence "rel_interior S ~= {}" using rel_interior_translation by auto
  4365 } from this show ?thesis using rel_interior_empty by auto
  4366 qed
  4367 
  4368 lemma convex_rel_interior:
  4369 fixes S :: "(_::euclidean_space) set"
  4370 assumes "convex S"
  4371 shows "convex (rel_interior S)"
  4372 proof-
  4373 { fix "x" "y" "u"
  4374   assume assm: "x:rel_interior S" "y:rel_interior S" "0<=u" "(u :: real) <= 1"
  4375   hence "x:S" using rel_interior_subset by auto
  4376   have "x - u *\<^sub>R (x-y) : rel_interior S"
  4377   proof(cases "0=u")
  4378      case False hence "0<u" using assm by auto 
  4379         thus ?thesis
  4380         using assm rel_interior_convex_shrink[of S y x u] assms `x:S` by auto
  4381      next
  4382      case True thus ?thesis using assm by auto
  4383   qed
  4384   hence "(1-u) *\<^sub>R x + u *\<^sub>R y : rel_interior S" by (simp add: algebra_simps)
  4385 } from this show ?thesis unfolding convex_alt by auto
  4386 qed
  4387 
  4388 lemma convex_closure_rel_interior: 
  4389 fixes S :: "('n::euclidean_space) set" 
  4390 assumes "convex S"
  4391 shows "closure(rel_interior S) = closure S"
  4392 proof-
  4393 have h1: "closure(rel_interior S) <= closure S" 
  4394    using subset_closure[of "rel_interior S" S] rel_interior_subset[of S] by auto
  4395 { assume "S ~= {}" from this obtain a where a_def: "a : rel_interior S" 
  4396     using rel_interior_convex_nonempty assms by auto
  4397   { fix x assume x_def: "x : closure S"
  4398     { assume "x=a" hence "x : closure(rel_interior S)" using a_def unfolding closure_def by auto }
  4399     moreover
  4400     { assume "x ~= a"
  4401        { fix e :: real assume e_def: "e>0" 
  4402          def e1 == "min 1 (e/norm (x - a))" hence e1_def: "e1>0 & e1<=1 & e1*norm(x-a)<=e"
  4403             using `x ~= a` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(x-a)"] by simp 
  4404          hence *: "x - e1 *\<^sub>R (x - a) : rel_interior S"
  4405             using rel_interior_closure_convex_shrink[of S a x e1] assms x_def a_def e1_def by auto
  4406          have "EX y. y:rel_interior S & y ~= x & (dist y x) <= e"
  4407             apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
  4408             using * e1_def dist_norm[of "x - e1 *\<^sub>R (x - a)" x] `x ~= a` by simp
  4409       } hence "x islimpt rel_interior S" unfolding islimpt_approachable_le by auto 
  4410       hence "x : closure(rel_interior S)" unfolding closure_def by auto 
  4411     } ultimately have "x : closure(rel_interior S)" by auto
  4412   } hence ?thesis using h1 by auto
  4413 }
  4414 moreover
  4415 { assume "S = {}" hence "rel_interior S = {}" using rel_interior_empty by auto
  4416   hence "closure(rel_interior S) = {}" using closure_empty by auto 
  4417   hence ?thesis using `S={}` by auto 
  4418 } ultimately show ?thesis by blast
  4419 qed
  4420 
  4421 lemma rel_interior_same_affine_hull:
  4422   fixes S :: "('n::euclidean_space) set"
  4423   assumes "convex S"
  4424   shows "affine hull (rel_interior S) = affine hull S"
  4425 by (metis assms closure_same_affine_hull convex_closure_rel_interior)
  4426 
  4427 lemma rel_interior_aff_dim: 
  4428   fixes S :: "('n::euclidean_space) set"
  4429   assumes "convex S"
  4430   shows "aff_dim (rel_interior S) = aff_dim S"
  4431 by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
  4432 
  4433 lemma rel_interior_rel_interior:
  4434   fixes S :: "('n::euclidean_space) set"
  4435   assumes "convex S"
  4436   shows "rel_interior (rel_interior S) = rel_interior S"
  4437 proof-
  4438 have "openin (subtopology euclidean (affine hull (rel_interior S))) (rel_interior S)"
  4439   using opein_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
  4440 from this show ?thesis using rel_interior_def by auto
  4441 qed
  4442 
  4443 lemma rel_interior_rel_open:
  4444   fixes S :: "('n::euclidean_space) set"
  4445   assumes "convex S"
  4446   shows "rel_open (rel_interior S)"
  4447 unfolding rel_open_def using rel_interior_rel_interior assms by auto
  4448 
  4449 lemma convex_rel_interior_closure_aux:
  4450   fixes x y z :: "_::euclidean_space"
  4451   assumes "0 < a" "0 < b" "(a+b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
  4452   obtains e where "0 < e" "e <= 1" "z = y - e *\<^sub>R (y-x)"
  4453 proof-
  4454 def e == "a/(a+b)"
  4455 have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)" apply auto using assms by simp
  4456 also have "... = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)" using assms
  4457    scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"] by auto
  4458 also have "... = y - e *\<^sub>R (y-x)" using e_def apply (simp add: algebra_simps)
  4459    using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"] by auto
  4460 finally have "z = y - e *\<^sub>R (y-x)" by auto
  4461 moreover have "0<e" using e_def assms divide_pos_pos[of a "a+b"] by auto
  4462 moreover have "e<=1" using e_def assms by auto
  4463 ultimately show ?thesis using that[of e] by auto
  4464 qed
  4465 
  4466 lemma convex_rel_interior_closure: 
  4467   fixes S :: "('n::euclidean_space) set" 
  4468   assumes "convex S"
  4469   shows "rel_interior (closure S) = rel_interior S"
  4470 proof-
  4471 { assume "S={}" hence ?thesis using assms rel_interior_convex_nonempty by auto }
  4472 moreover
  4473 { assume "S ~= {}"
  4474   have "rel_interior (closure S) >= rel_interior S" 
  4475     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset by auto
  4476   moreover
  4477   { fix z assume z_def: "z : rel_interior (closure S)"
  4478     obtain x where x_def: "x : rel_interior S" 
  4479       using `S ~= {}` assms rel_interior_convex_nonempty by auto  
  4480     { assume "x=z" hence "z : rel_interior S" using x_def by auto }
  4481     moreover
  4482     { assume "x ~= z"
  4483       obtain e where e_def: "e > 0 & cball z e Int affine hull closure S <= closure S" 
  4484         using z_def rel_interior_cball[of "closure S"] by auto
  4485       hence *: "0 < e/norm(z-x)" using e_def `x ~= z` divide_pos_pos[of e "norm(z-x)"] by auto 
  4486       def y == "z + (e/norm(z-x)) *\<^sub>R (z-x)"
  4487       have yball: "y : cball z e"
  4488         using mem_cball y_def dist_norm[of z y] e_def by auto 
  4489       have "x : affine hull closure S" 
  4490         using x_def rel_interior_subset_closure hull_inc[of x "closure S"] by auto
  4491       moreover have "z : affine hull closure S" 
  4492         using z_def rel_interior_subset hull_subset[of "closure S"] by auto
  4493       ultimately have "y : affine hull closure S" 
  4494         using y_def affine_affine_hull[of "closure S"] 
  4495           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
  4496       hence "y : closure S" using e_def yball by auto
  4497       have "(1+(e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
  4498         using y_def by (simp add: algebra_simps) 
  4499       from this obtain e1 where "0 < e1 & e1 <= 1 & z = y - e1 *\<^sub>R (y - x)"
  4500         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y] 
  4501           by (auto simp add: algebra_simps)
  4502       hence "z : rel_interior S" 
  4503         using rel_interior_closure_convex_shrink assms x_def `y : closure S` by auto
  4504     } ultimately have "z : rel_interior S" by auto
  4505   } ultimately have ?thesis by auto
  4506 } ultimately show ?thesis by blast
  4507 qed
  4508 
  4509 lemma convex_interior_closure: 
  4510 fixes S :: "('n::euclidean_space) set" 
  4511 assumes "convex S"
  4512 shows "interior (closure S) = interior S"
  4513 using closure_aff_dim[of S] interior_rel_interior_gen[of S] interior_rel_interior_gen[of "closure S"] 
  4514       convex_rel_interior_closure[of S] assms by auto
  4515 
  4516 lemma closure_eq_rel_interior_eq:
  4517 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4518 assumes "convex S1" "convex S2"
  4519 shows "(closure S1 = closure S2) <-> (rel_interior S1 = rel_interior S2)"
  4520  by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
  4521 
  4522 
  4523 lemma closure_eq_between:
  4524 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4525 assumes "convex S1" "convex S2"
  4526 shows "(closure S1 = closure S2) <-> 
  4527       ((rel_interior S1 <= S2) & (S2 <= closure S1))" (is "?A <-> ?B")
  4528 proof-
  4529 have "?A --> ?B" by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
  4530 moreover have "?B --> (closure S1 <= closure S2)" 
  4531      by (metis assms(1) convex_closure_rel_interior subset_closure)
  4532 moreover have "?B --> (closure S1 >= closure S2)" by (metis closed_closure closure_minimal)
  4533 ultimately show ?thesis by blast
  4534 qed
  4535 
  4536 lemma open_inter_closure_rel_interior:
  4537 fixes S A ::  "('n::euclidean_space) set" 
  4538 assumes "convex S" "open A"
  4539 shows "((A Int closure S) = {}) <-> ((A Int rel_interior S) = {})"
  4540 by (metis assms convex_closure_rel_interior open_inter_closure_eq_empty) 
  4541 
  4542 definition "rel_frontier S = closure S - rel_interior S"
  4543 
  4544 lemma closed_affine_hull: "closed (affine hull ((S :: ('n::euclidean_space) set)))"
  4545 by (metis affine_affine_hull affine_closed)
  4546 
  4547 lemma closed_rel_frontier: "closed(rel_frontier (S :: ('n::euclidean_space) set))"
  4548 proof-
  4549 have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)" 
  4550 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] 
  4551   closure_affine_hull[of S] opein_rel_interior[of S] by auto 
  4552 show ?thesis apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"]) 
  4553   unfolding rel_frontier_def using * closed_affine_hull by auto 
  4554 qed
  4555  
  4556 
  4557 lemma convex_rel_frontier_aff_dim:
  4558 fixes S1 S2 ::  "('n::euclidean_space) set" 
  4559 assumes "convex S1" "convex S2" "S2 ~= {}"
  4560 assumes "S1 <= rel_frontier S2"
  4561 shows "aff_dim S1 < aff_dim S2" 
  4562 proof-
  4563 have "S1 <= closure S2" using assms unfolding rel_frontier_def by auto
  4564 hence *: "affine hull S1 <= affine hull S2" 
  4565    using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by auto
  4566 hence "aff_dim S1 <= aff_dim S2" using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] 
  4567     aff_dim_subset[of "affine hull S1" "affine hull S2"] by auto
  4568 moreover
  4569 { assume eq: "aff_dim S1 = aff_dim S2"
  4570   hence "S1 ~= {}" using aff_dim_empty[of S1] aff_dim_empty[of S2] `S2 ~= {}` by auto
  4571   have **: "affine hull S1 = affine hull S2" 
  4572      apply (rule affine_dim_equal) using * affine_affine_hull apply auto
  4573      using `S1 ~= {}` hull_subset[of S1] apply auto
  4574      using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] by auto
  4575   obtain a where a_def: "a : rel_interior S1"
  4576      using  `S1 ~= {}` rel_interior_convex_nonempty assms by auto
  4577   obtain T where T_def: "open T & a : T Int S1 & T Int affine hull S1 <= S1"
  4578      using mem_rel_interior[of a S1] a_def by auto
  4579   hence "a : T Int closure S2" using a_def assms unfolding rel_frontier_def by auto
  4580   from this obtain b where b_def: "b : T Int rel_interior S2" 
  4581      using open_inter_closure_rel_interior[of S2 T] assms T_def by auto
  4582   hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto
  4583   hence "b : S1" using T_def b_def by auto
  4584   hence False using b_def assms unfolding rel_frontier_def by auto
  4585 } ultimately show ?thesis using zless_le by auto
  4586 qed
  4587 
  4588 
  4589 lemma convex_rel_interior_if:
  4590 fixes S ::  "('n::euclidean_space) set" 
  4591 assumes "convex S"
  4592 assumes "z : rel_interior S"
  4593 shows "(!x:affine hull S. EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S ))"
  4594 proof-
  4595 obtain e1 where e1_def: "e1>0 & cball z e1 Int affine hull S <= S" 
  4596     using mem_rel_interior_cball[of z S] assms by auto
  4597 { fix x assume x_def: "x:affine hull S"
  4598   { assume "x ~= z"
  4599     def m == "1+e1/norm(x-z)" 
  4600     hence "m>1" using e1_def `x ~= z` divide_pos_pos[of e1 "norm (x - z)"] by auto 
  4601     { fix e assume e_def: "e>1 & e<=m"
  4602       have "z : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
  4603       hence *: "(1-e)*\<^sub>R x+ e *\<^sub>R z : affine hull S"
  4604          using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x_def by auto
  4605       have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x-z))" by (simp add: algebra_simps)
  4606       also have "...= (e - 1) * norm(x-z)" using norm_scaleR e_def by auto  
  4607       also have "...<=(m - 1) * norm(x-z)" using e_def mult_right_mono[of _ _ "norm(x-z)"] by auto
  4608       also have "...= (e1 / norm (x - z)) * norm (x - z)" using m_def by auto
  4609       also have "...=e1" using `x ~= z` e1_def by simp
  4610       finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) <= e1" by auto
  4611       have "(1-e)*\<^sub>R x+ e *\<^sub>R z : cball z e1"
  4612          using m_def ** unfolding cball_def dist_norm by (auto simp add: algebra_simps)
  4613       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def * e1_def by auto
  4614     } 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
  4615   }
  4616   moreover
  4617   { assume "x=z" def m == "1+e1" hence "m>1" using e1_def by auto
  4618     { fix e assume e_def: "e>1 & e<=m"
  4619       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S"
  4620         using e1_def x_def `x=z` by (auto simp add: algebra_simps)
  4621       hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def by auto
  4622     } 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
  4623   } ultimately have "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" by auto
  4624 } from this show ?thesis by auto 
  4625 qed
  4626 
  4627 lemma convex_rel_interior_if2:
  4628 fixes S ::  "('n::euclidean_space) set" 
  4629 assumes "convex S"
  4630 assumes "z : rel_interior S"
  4631 shows "(!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4632 using convex_rel_interior_if[of S z] assms by auto
  4633 
  4634 lemma convex_rel_interior_only_if:
  4635 fixes S ::  "('n::euclidean_space) set" 
  4636 assumes "convex S" "S ~= {}"
  4637 assumes "(!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4638 shows "z : rel_interior S" 
  4639 proof-
  4640 obtain x where x_def: "x : rel_interior S" using rel_interior_convex_nonempty assms by auto
  4641 hence "x:S" using rel_interior_subset by auto
  4642 from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S" using assms by auto
  4643 def y == "(1 - e) *\<^sub>R x + e *\<^sub>R z" hence "y:S" using e_def by auto
  4644 def e1 == "1/e" hence "0<e1 & e1<1" using e_def by auto
  4645 hence "z=y-(1-e1)*\<^sub>R (y-x)" using e1_def y_def by (auto simp add: algebra_simps)
  4646 from this show ?thesis 
  4647     using rel_interior_convex_shrink[of S x y "1-e1"] `0<e1 & e1<1` `y:S` x_def assms by auto
  4648 qed
  4649 
  4650 lemma convex_rel_interior_iff:
  4651 fixes S ::  "('n::euclidean_space) set" 
  4652 assumes "convex S" "S ~= {}"
  4653 shows "z : rel_interior S <-> (!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4654 using assms hull_subset[of S "affine"] 
  4655       convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z] by auto
  4656 
  4657 lemma convex_rel_interior_iff2:
  4658 fixes S ::  "('n::euclidean_space) set" 
  4659 assumes "convex S" "S ~= {}"
  4660 shows "z : rel_interior S <-> (!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4661 using assms hull_subset[of S] 
  4662       convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z] by auto
  4663 
  4664 
  4665 lemma convex_interior_iff:
  4666 fixes S ::  "('n::euclidean_space) set" 
  4667 assumes "convex S"
  4668 shows "z : interior S <-> (!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
  4669 proof-
  4670 { assume a: "~(aff_dim S = int DIM('n))"
  4671   { assume "z : interior S"
  4672     hence False using a interior_rel_interior_gen[of S] by auto
  4673   }
  4674   moreover
  4675   { assume r: "!x. EX e. e>0 & z+ e *\<^sub>R x : S"
  4676     { fix x obtain e1 where e1_def: "e1>0 & z+ e1 *\<^sub>R (x-z) : S" using r by auto
  4677       obtain e2 where e2_def: "e2>0 & z+ e2 *\<^sub>R (z-x) : S" using r by auto
  4678       def x1 == "z+ e1 *\<^sub>R (x-z)"
  4679          hence x1: "x1 : affine hull S" using e1_def hull_subset[of S] by auto
  4680       def x2 == "z+ e2 *\<^sub>R (z-x)"
  4681          hence x2: "x2 : affine hull S" using e2_def hull_subset[of S] by auto
  4682       have *: "e1/(e1+e2) + e2/(e1+e2) = 1" using divide.add[of e1 e2 "e1+e2"] e1_def e2_def by simp
  4683       hence "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
  4684          using x1_def x2_def apply (auto simp add: algebra_simps)
  4685          using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z] by auto
  4686       hence z: "z : affine hull S" 
  4687          using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]  
  4688          x1 x2 affine_affine_hull[of S] * by auto
  4689       have "x1-x2 = (e1+e2) *\<^sub>R (x-z)"
  4690          using x1_def x2_def by (auto simp add: algebra_simps)
  4691       hence "x=z+(1/(e1+e2)) *\<^sub>R (x1-x2)" using e1_def e2_def by simp
  4692       hence "x : affine hull S" using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"] 
  4693           x1 x2 z affine_affine_hull[of S] by auto
  4694     } hence "affine hull S = UNIV" by auto
  4695     hence "aff_dim S = int DIM('n)" using aff_dim_affine_hull[of S] by (simp add: aff_dim_univ)
  4696     hence False using a by auto
  4697   } ultimately have ?thesis by auto
  4698 }
  4699 moreover
  4700 { assume a: "aff_dim S = int DIM('n)"
  4701   hence "S ~= {}" using aff_dim_empty[of S] by auto
  4702   have *: "affine hull S=UNIV" using a affine_hull_univ by auto
  4703   { assume "z : interior S"
  4704     hence "z : rel_interior S" using a interior_rel_interior_gen[of S] by auto
  4705     hence **: "(!x. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
  4706       using convex_rel_interior_iff2[of S z] assms `S~={}` * by auto
  4707     fix x obtain e1 where e1_def: "e1>1 & (1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z : S"
  4708       using **[rule_format, of "z-x"] by auto
  4709     def e == "e1 - 1"
  4710     hence "(1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z = z+ e *\<^sub>R x" by (simp add: algebra_simps)
  4711     hence "e>0 & z+ e *\<^sub>R x : S" using e1_def e_def by auto
  4712     hence "EX e. e>0 & z+ e *\<^sub>R x : S" by auto
  4713   }
  4714   moreover
  4715   { assume r: "(!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
  4716     { fix x obtain e1 where e1_def: "e1>0 & z + e1*\<^sub>R (z-x) : S"
  4717          using r[rule_format, of "z-x"] by auto
  4718       def e == "e1 + 1"
  4719       hence "z + e1*\<^sub>R (z-x) = (1-e)*\<^sub>R x+ e *\<^sub>R z" by (simp add: algebra_simps)
  4720       hence "e > 1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e1_def e_def by auto
  4721       hence "EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" by auto
  4722     }
  4723     hence "z : rel_interior S" using convex_rel_interior_iff2[of S z] assms `S~={}` by auto
  4724     hence "z : interior S" using a interior_rel_interior_gen[of S] by auto
  4725   } ultimately have ?thesis by auto
  4726 } ultimately show ?thesis by auto
  4727 qed
  4728 
  4729 subsection{* Relative interior and closure under commom operations *}
  4730 
  4731 lemma rel_interior_inter_aux: "Inter {rel_interior S |S. S : I} <= Inter I"
  4732 proof- 
  4733 { fix y assume "y : Inter {rel_interior S |S. S : I}"
  4734   hence y_def: "!S : I. y : rel_interior S" by auto
  4735   { fix S assume "S : I" hence "y : S" using rel_interior_subset y_def by auto }
  4736   hence "y : Inter I" by auto
  4737 } thus ?thesis by auto
  4738 qed
  4739 
  4740 lemma closure_inter: "closure (Inter I) <= Inter {closure S |S. S : I}"
  4741 proof- 
  4742 { fix y assume "y : Inter I" hence y_def: "!S : I. y : S" by auto
  4743   { fix S assume "S : I" hence "y : closure S" using closure_subset y_def by auto }
  4744   hence "y : Inter {closure S |S. S : I}" by auto
  4745 } hence "Inter I <= Inter {closure S |S. S : I}" by auto
  4746 moreover have "Inter {closure S |S. S : I} : closed" 
  4747   unfolding mem_def closed_Inter closed_closure by auto
  4748 ultimately show ?thesis using closure_hull[of "Inter I"]
  4749   hull_minimal[of "Inter I" "Inter {closure S |S. S : I}" "closed"] by auto
  4750 qed
  4751 
  4752 lemma convex_closure_rel_interior_inter: 
  4753 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4754 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4755 shows "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
  4756 proof-
  4757 obtain x where x_def: "!S : I. x : rel_interior S" using assms by auto
  4758 { fix y assume "y : Inter {closure S |S. S : I}" hence y_def: "!S : I. y : closure S" by auto
  4759   { assume "y = x" 
  4760     hence "y : closure (Inter {rel_interior S |S. S : I})"
  4761        using x_def closure_subset[of "Inter {rel_interior S |S. S : I}"] by auto
  4762   }
  4763   moreover
  4764   { assume "y ~= x"
  4765     { fix e :: real assume e_def: "0 < e"
  4766       def e1 == "min 1 (e/norm (y - x))" hence e1_def: "e1>0 & e1<=1 & e1*norm(y-x)<=e"
  4767         using `y ~= x` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(y-x)"] by simp 
  4768       def z == "y - e1 *\<^sub>R (y - x)"
  4769       { fix S assume "S : I" 
  4770         hence "z : rel_interior S" using rel_interior_closure_convex_shrink[of S x y e1] 
  4771            assms x_def y_def e1_def z_def by auto
  4772       } hence *: "z : Inter {rel_interior S |S. S : I}" by auto
  4773       have "EX z. z:Inter {rel_interior S |S. S : I} & z ~= y & (dist z y) <= e"
  4774            apply (rule_tac x="z" in exI) using `y ~= x` z_def * e1_def e_def dist_norm[of z y] by simp
  4775     } hence "y islimpt Inter {rel_interior S |S. S : I}" unfolding islimpt_approachable_le by blast 
  4776     hence "y : closure (Inter {rel_interior S |S. S : I})" unfolding closure_def by auto
  4777   } ultimately have "y : closure (Inter {rel_interior S |S. S : I})" by auto
  4778 } from this show ?thesis by auto
  4779 qed
  4780 
  4781 
  4782 lemma convex_closure_inter: 
  4783 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4784 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4785 shows "closure (Inter I) = Inter {closure S |S. S : I}"
  4786 proof-
  4787 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
  4788   using convex_closure_rel_interior_inter assms by auto
  4789 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
  4790     using rel_interior_inter_aux 
  4791           subset_closure[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
  4792 ultimately show ?thesis using closure_inter[of I] by auto
  4793 qed
  4794 
  4795 lemma convex_inter_rel_interior_same_closure: 
  4796 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4797 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4798 shows "closure (Inter {rel_interior S |S. S : I}) = closure (Inter I)"
  4799 proof-
  4800 have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
  4801   using convex_closure_rel_interior_inter assms by auto
  4802 moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
  4803     using rel_interior_inter_aux 
  4804           subset_closure[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
  4805 ultimately show ?thesis using closure_inter[of I] by auto
  4806 qed
  4807 
  4808 lemma convex_rel_interior_inter: 
  4809 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4810 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4811 shows "rel_interior (Inter I) <= Inter {rel_interior S |S. S : I}"
  4812 proof-
  4813 have "convex(Inter I)" using assms convex_Inter by auto
  4814 moreover have "convex(Inter {rel_interior S |S. S : I})" apply (rule convex_Inter)
  4815    using assms convex_rel_interior by auto 
  4816 ultimately have "rel_interior (Inter {rel_interior S |S. S : I}) = rel_interior (Inter I)"
  4817    using convex_inter_rel_interior_same_closure assms 
  4818    closure_eq_rel_interior_eq[of "Inter {rel_interior S |S. S : I}" "Inter I"] by blast
  4819 from this show ?thesis using rel_interior_subset[of "Inter {rel_interior S |S. S : I}"] by auto
  4820 qed
  4821 
  4822 lemma convex_rel_interior_finite_inter: 
  4823 assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
  4824 assumes "Inter {rel_interior S |S. S : I} ~= {}"
  4825 assumes "finite I"
  4826 shows "rel_interior (Inter I) = Inter {rel_interior S |S. S : I}"
  4827 proof-
  4828 have "Inter I ~= {}" using assms rel_interior_inter_aux[of I] by auto
  4829 have "convex (Inter I)" using convex_Inter assms by auto
  4830 { assume "I={}" hence ?thesis using Inter_empty rel_interior_univ2 by auto }
  4831 moreover
  4832 { assume "I ~= {}"
  4833 { fix z assume z_def: "z : Inter {rel_interior S |S. S : I}"
  4834   { fix x assume x_def: "x : Inter I"
  4835     { fix S assume S_def: "S : I" hence "z : rel_interior S" "x : S" using z_def x_def by auto 
  4836       (*from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S"*)
  4837       hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )"
  4838          using convex_rel_interior_if[of S z] S_def assms hull_subset[of S] by auto
  4839     } from this obtain mS where mS_def: "!S : I. (mS(S) > (1 :: real) & 
  4840          (!e. (e>1 & e<=mS(S)) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S))" by metis
  4841     obtain e where e_def: "e=Min (mS ` I)" by auto 
  4842     have "e : (mS ` I)" using e_def assms `I ~= {}` by (simp add: Min_in) 
  4843     hence "e>(1 :: real)" using mS_def by auto
  4844     moreover have "!S : I. e<=mS(S)" using e_def assms by auto
  4845     ultimately have "EX e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z : Inter I" using mS_def by auto
  4846   } hence "z : rel_interior (Inter I)" using convex_rel_interior_iff[of "Inter I" z]
  4847        `Inter I ~= {}` `convex (Inter I)` by auto
  4848 } from this have ?thesis using convex_rel_interior_inter[of I] assms by auto
  4849 } ultimately show ?thesis by blast
  4850 qed
  4851 
  4852 lemma convex_closure_inter_two: 
  4853 fixes S T :: "('n::euclidean_space) set"
  4854 assumes "convex S" "convex T"
  4855 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
  4856 shows "closure (S Int T) = (closure S) Int (closure T)" 
  4857 using convex_closure_inter[of "{S,T}"] assms by auto
  4858 
  4859 lemma convex_rel_interior_inter_two: 
  4860 fixes S T :: "('n::euclidean_space) set"
  4861 assumes "convex S" "convex T"
  4862 assumes "(rel_interior S) Int (rel_interior T) ~= {}"
  4863 shows "rel_interior (S Int T) = (rel_interior S) Int (rel_interior T)" 
  4864 using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
  4865 
  4866 
  4867 lemma convex_affine_closure_inter: 
  4868 fixes S T :: "('n::euclidean_space) set"
  4869 assumes "convex S" "affine T"
  4870 assumes "(rel_interior S) Int T ~= {}"
  4871 shows "closure (S Int T) = (closure S) Int T"
  4872 proof- 
  4873 have "affine hull T = T" using assms by auto
  4874 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
  4875 moreover have "closure T = T" using assms affine_closed[of T] by auto
  4876 ultimately show ?thesis using convex_closure_inter_two[of S T] assms affine_imp_convex by auto 
  4877 qed
  4878 
  4879 lemma convex_affine_rel_interior_inter: 
  4880 fixes S T :: "('n::euclidean_space) set"
  4881 assumes "convex S" "affine T"
  4882 assumes "(rel_interior S) Int T ~= {}"
  4883 shows "rel_interior (S Int T) = (rel_interior S) Int T"
  4884 proof- 
  4885 have "affine hull T = T" using assms by auto
  4886 hence "rel_interior T = T" using rel_interior_univ[of T] by metis
  4887 moreover have "closure T = T" using assms affine_closed[of T] by auto
  4888 ultimately show ?thesis using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto 
  4889 qed
  4890 
  4891 lemma subset_rel_interior_convex:
  4892 fixes S T :: "('n::euclidean_space) set"
  4893 assumes "convex S" "convex T"
  4894 assumes "S <= closure T"
  4895 assumes "~(S <= rel_frontier T)"
  4896 shows "rel_interior S <= rel_interior T"
  4897 proof-
  4898 have *: "S Int closure T = S" using assms by auto
  4899 have "~(rel_interior S <= rel_frontier T)"
  4900      using subset_closure[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T] 
  4901      closure_closed convex_closure_rel_interior[of S] closure_subset[of S] assms by auto
  4902 hence "(rel_interior S) Int (rel_interior (closure T)) ~= {}" 
  4903      using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T] by auto
  4904 hence "rel_interior S Int rel_interior T = rel_interior (S Int closure T)" using assms convex_closure  
  4905      convex_rel_interior_inter_two[of S "closure T"] convex_rel_interior_closure[of T] by auto
  4906 also have "...=rel_interior (S)" using * by auto
  4907 finally show ?thesis by auto
  4908 qed
  4909 
  4910 
  4911 lemma rel_interior_convex_linear_image:
  4912 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  4913 assumes "linear f"
  4914 assumes "convex S"
  4915 shows "f ` (rel_interior S) = rel_interior (f ` S)"
  4916 proof-
  4917 { assume "S = {}" hence ?thesis using assms rel_interior_empty rel_interior_convex_nonempty by auto }
  4918 moreover
  4919 { assume "S ~= {}"
  4920 have *: "f ` (rel_interior S) <= f ` S" unfolding image_mono using rel_interior_subset by auto
  4921 have "f ` S <= f ` (closure S)" unfolding image_mono using closure_subset by auto
  4922 also have "... = f ` (closure (rel_interior S))" using convex_closure_rel_interior assms by auto  
  4923 also have "... <= closure (f ` (rel_interior S))" using closure_linear_image assms by auto 
  4924 finally have "closure (f ` S) = closure (f ` rel_interior S)"
  4925    using subset_closure[of "f ` S" "closure (f ` rel_interior S)"] closure_closure 
  4926          subset_closure[of "f ` rel_interior S" "f ` S"] * by auto
  4927 hence "rel_interior (f ` S) = rel_interior (f ` rel_interior S)" using assms convex_rel_interior
  4928    linear_conv_bounded_linear[of f] convex_linear_image[of S] convex_linear_image[of "rel_interior S"] 
  4929    closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"] by auto
  4930 hence "rel_interior (f ` S) <= f ` rel_interior S" using rel_interior_subset by auto
  4931 moreover
  4932 { fix z assume z_def: "z : f ` rel_interior S"
  4933   from this obtain z1 where z1_def: "z1 : rel_interior S & (f z1 = z)" by auto
  4934   { fix x assume "x : f ` S"
  4935     from this obtain x1 where x1_def: "x1 : S & (f x1 = x)" by auto
  4936     from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x1 + e *\<^sub>R z1 : S"
  4937        using convex_rel_interior_iff[of S z1] `convex S` x1_def z1_def by auto
  4938     moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
  4939         using x1_def z1_def `linear f` by (simp add: linear_add_cmul)
  4940     ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
  4941         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
  4942     hence "EX e. (e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S)" using e_def by auto
  4943   } from this have "z : rel_interior (f ` S)" using convex_rel_interior_iff[of "f ` S" z] `convex S` 
  4944        `linear f` `S ~= {}` convex_linear_image[of S f]  linear_conv_bounded_linear[of f] by auto
  4945 } ultimately have ?thesis by auto
  4946 } ultimately show ?thesis by blast
  4947 qed
  4948 
  4949 
  4950 lemma convex_linear_preimage:
  4951   assumes c:"convex S" and l:"bounded_linear f"
  4952   shows "convex(f -` S)"
  4953 proof(auto simp add: convex_def)
  4954   interpret f: bounded_linear f by fact
  4955   fix x y assume xy:"f x : S" "f y : S"
  4956   fix u v ::real assume uv:"0 <= u" "0 <= v" "u + v = 1"
  4957   show "f (u *\<^sub>R x + v *\<^sub>R y) : S" unfolding image_iff
  4958     using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR
  4959       c[unfolded convex_def] xy uv by auto
  4960 qed
  4961 
  4962 
  4963 lemma rel_interior_convex_linear_preimage:
  4964 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  4965 assumes "linear f"
  4966 assumes "convex S"
  4967 assumes "f -` (rel_interior S) ~= {}"
  4968 shows "rel_interior (f -` S) = f -` (rel_interior S)"
  4969 proof-
  4970 have "S ~= {}" using assms rel_interior_empty by auto
  4971 have nonemp: "f -` S ~= {}" by (metis assms(3) rel_interior_subset subset_empty vimage_mono) 
  4972 hence "S Int (range f) ~= {}" by auto
  4973 have conv: "convex (f -` S)" using convex_linear_preimage assms linear_conv_bounded_linear by auto
  4974 hence "convex (S Int (range f))"
  4975   by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
  4976 { fix z assume "z : f -` (rel_interior S)"
  4977   hence z_def: "f z : rel_interior S" by auto
  4978   { fix x assume "x : f -` S" from this have x_def: "f x : S" by auto
  4979     from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) : S"
  4980       using convex_rel_interior_iff[of S "f z"] z_def assms `S ~= {}` by auto
  4981     moreover have "(1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R x + e *\<^sub>R z)"
  4982       using `linear f` by (simp add: linear_def)
  4983     ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R z : f -` S" using e_def by auto
  4984   } hence "z : rel_interior (f -` S)" 
  4985        using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
  4986 } 
  4987 moreover
  4988 { fix z assume z_def: "z : rel_interior (f -` S)" 
  4989   { fix x assume x_def: "x: S Int (range f)"
  4990     from this obtain y where y_def: "(f y = x) & (y : f -` S)" by auto
  4991     from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R y+ e *\<^sub>R z : f -` S"
  4992       using convex_rel_interior_iff[of "f -` S" z] z_def conv by auto
  4993     moreover have "(1-e)*\<^sub>R x+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R y + e *\<^sub>R z)"
  4994       using `linear f` y_def by (simp add: linear_def)
  4995     ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R (f z) : S Int (range f)"
  4996       using e_def by auto
  4997   } hence "f z : rel_interior (S Int (range f))" using `convex (S Int (range f))`
  4998     `S Int (range f) ~= {}` convex_rel_interior_iff[of "S Int (range f)" "f z"] by auto
  4999   moreover have "affine (range f)"
  5000     by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
  5001   ultimately have "f z : rel_interior S" 
  5002     using convex_affine_rel_interior_inter[of S "range f"] assms by auto
  5003   hence "z : f -` (rel_interior S)" by auto
  5004 }
  5005 ultimately show ?thesis by auto
  5006 qed
  5007     
  5008 
  5009 lemma convex_direct_sum:
  5010 fixes S :: "('n::euclidean_space) set"
  5011 fixes T :: "('m::euclidean_space) set"
  5012 assumes "convex S" "convex T"
  5013 shows "convex (S <*> T)"
  5014 proof-
  5015 {
  5016 fix x assume "x : S <*> T"
  5017 from this obtain xs xt where xst_def: "xs : S & xt : T & (xs,xt) = x" by auto
  5018 fix y assume "y : S <*> T"
  5019 from this obtain ys yt where yst_def: "ys : S & yt : T & (ys,yt) = y" by auto
  5020 fix u v assume uv_def: "(u :: real)>=0 & (v :: real)>=0 & u+v=1"
  5021 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
  5022 moreover have "u *\<^sub>R xs + v *\<^sub>R ys : S"
  5023    using uv_def xst_def yst_def convex_def[of S] assms by auto
  5024 moreover have "u *\<^sub>R xt + v *\<^sub>R yt : T"
  5025    using uv_def xst_def yst_def convex_def[of T] assms by auto
  5026 ultimately have "u *\<^sub>R x + v *\<^sub>R y : S <*> T" by auto
  5027 } from this show ?thesis unfolding convex_def by auto
  5028 qed
  5029 
  5030 
  5031 lemma convex_hull_direct_sum:
  5032 fixes S :: "('n::euclidean_space) set"
  5033 fixes T :: "('m::euclidean_space) set"
  5034 shows "convex hull (S <*> T) = (convex hull S) <*> (convex hull T)"
  5035 proof-
  5036 { fix x assume "x : (convex hull S) <*> (convex hull T)"
  5037   from this obtain xs xt where xst_def: "xs : convex hull S & xt : convex hull T & (xs,xt) = x" by auto
  5038   from xst_def obtain sI su where s: "finite sI & sI <= S & (ALL x:sI. 0 <= su x) & setsum su sI = 1 
  5039      & (SUM v:sI. su v *\<^sub>R v) = xs" using convex_hull_explicit[of S] by auto
  5040   from xst_def obtain tI tu where t: "finite tI & tI <= T & (ALL x:tI. 0 <= tu x) & setsum tu tI = 1 
  5041      & (SUM v:tI. tu v *\<^sub>R v) = xt" using convex_hull_explicit[of T] by auto
  5042   def I == "(sI <*> tI)"
  5043   def u == "(%i. (su (fst i))*(tu(snd i)))"
  5044   have "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
  5045      (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vs)"
  5046      using fst_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
  5047      by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
  5048   also have "...=(SUM vt:tI. tu vt *\<^sub>R (SUM vs:sI. su vs *\<^sub>R vs))"
  5049      using setsum_commute[of "(%vt vs. (su vs * tu vt) *\<^sub>R vs)" sI tI]
  5050      by (simp add: mult_commute scaleR_right.setsum)
  5051   also have "...=(SUM vt:tI. tu vt *\<^sub>R xs)" using s by auto
  5052   also have "...=(SUM vt:tI. tu vt) *\<^sub>R xs" by (simp add: scaleR_left.setsum)
  5053   also have "...=xs" using t by auto
  5054   finally have h1: "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xs" by auto
  5055   have "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
  5056      (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vt)"
  5057      using snd_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
  5058      by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
  5059   also have "...=(SUM vs:sI. su vs *\<^sub>R (SUM vt:tI. tu vt *\<^sub>R vt))"
  5060      by (simp add: mult_commute scaleR_right.setsum)
  5061   also have "...=(SUM vs:sI. su vs *\<^sub>R xt)" using t by auto
  5062   also have "...=(SUM vs:sI. su vs) *\<^sub>R xt" by (simp add: scaleR_left.setsum)
  5063   also have "...=xt" using s by auto
  5064   finally have h2: "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xt" by auto
  5065   from h1 h2 have "(SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v) = x" using xst_def by auto
  5066 
  5067   moreover have "finite I & (I <= S <*> T)" using s t I_def by auto 
  5068   moreover have "!i:I. 0 <= u i" using s t I_def u_def by (simp add: mult_nonneg_nonneg)
  5069   moreover have "setsum u I = 1" using u_def I_def setsum_cartesian_product[of "(% x y. (su x)*(tu y))"] 
  5070      s t setsum_product[of su sI tu tI] by (auto simp add: split_def)
  5071   ultimately have "x : convex hull (S <*> T)" 
  5072      apply (subst convex_hull_explicit[of "S <*> T"]) apply rule
  5073      apply (rule_tac x="I" in exI) apply (rule_tac x="u" in exI)
  5074      using I_def u_def by auto
  5075 }
  5076 hence "convex hull (S <*> T) >= (convex hull S) <*> (convex hull T)" by auto
  5077 moreover have "(convex hull S) <*> (convex hull T) : convex" 
  5078    unfolding mem_def by (simp add: convex_direct_sum convex_convex_hull)
  5079 ultimately show ?thesis 
  5080    using hull_minimal[of "S <*> T" "(convex hull S) <*> (convex hull T)" "convex"] 
  5081          hull_subset[of S convex] hull_subset[of T convex] by auto
  5082 qed
  5083 
  5084 lemma rel_interior_direct_sum:
  5085 fixes S :: "('n::euclidean_space) set"
  5086 fixes T :: "('m::euclidean_space) set"
  5087 assumes "convex S" "convex T"
  5088 shows "rel_interior (S <*> T) = rel_interior S <*> rel_interior T"
  5089 proof-
  5090 { assume "S={}" hence ?thesis apply auto using rel_interior_empty by auto }
  5091 moreover
  5092 { assume "T={}" hence ?thesis apply auto using rel_interior_empty by auto }
  5093 moreover {
  5094 assume "S ~={}" "T ~={}"
  5095 hence ri: "rel_interior S ~= {}" "rel_interior T ~= {}" using rel_interior_convex_nonempty assms by auto
  5096 hence "fst -` rel_interior S ~= {}" using fst_vimage_eq_Times[of "rel_interior S"] by auto
  5097 hence "rel_interior ((fst :: 'n * 'm => 'n) -` S) = fst -` rel_interior S"
  5098   using fst_linear `convex S` rel_interior_convex_linear_preimage[of fst S] by auto 
  5099 hence s: "rel_interior (S <*> (UNIV :: 'm set)) = rel_interior S <*> UNIV" by (simp add: fst_vimage_eq_Times)
  5100 from ri have "snd -` rel_interior T ~= {}" using snd_vimage_eq_Times[of "rel_interior T"] by auto
  5101 hence "rel_interior ((snd :: 'n * 'm => 'm) -` T) = snd -` rel_interior T"
  5102   using snd_linear `convex T` rel_interior_convex_linear_preimage[of snd T] by auto 
  5103 hence t: "rel_interior ((UNIV :: 'n set) <*> T) = UNIV <*> rel_interior T" by (simp add: snd_vimage_eq_Times)
  5104 from s t have *: "rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T) 
  5105   = rel_interior S <*> rel_interior T" by auto
  5106 have "(S <*> T) = (S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T)" by auto
  5107 hence "rel_interior (S <*> T) = rel_interior ((S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T))" by auto
  5108 also have "...=rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)" 
  5109    apply (subst convex_rel_interior_inter_two[of "S <*> (UNIV :: 'm set)" "(UNIV :: 'n set) <*> T"]) 
  5110    using * ri assms convex_direct_sum by auto
  5111 finally have ?thesis using * by auto
  5112 }
  5113 ultimately show ?thesis by blast
  5114 qed
  5115 
  5116 lemma rel_interior_scaleR: 
  5117 fixes S :: "('n::euclidean_space) set"
  5118 assumes "c ~= 0"
  5119 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  5120 using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
  5121       linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms by auto
  5122 
  5123 lemma rel_interior_convex_scaleR: 
  5124 fixes S :: "('n::euclidean_space) set"
  5125 assumes "convex S"
  5126 shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  5127 by (metis assms linear_scaleR rel_interior_convex_linear_image)
  5128 
  5129 lemma convex_rel_open_scaleR: 
  5130 fixes S :: "('n::euclidean_space) set"
  5131 assumes "convex S" "rel_open S"
  5132 shows "convex ((op *\<^sub>R c) ` S) & rel_open ((op *\<^sub>R c) ` S)"
  5133 by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
  5134 
  5135 
  5136 lemma convex_rel_open_finite_inter: 
  5137 assumes "!S : I. (convex (S :: ('n::euclidean_space) set) & rel_open S)"
  5138 assumes "finite I"
  5139 shows "convex (Inter I) & rel_open (Inter I)"
  5140 proof-
  5141 { assume "Inter {rel_interior S |S. S : I} = {}"
  5142   hence "Inter I = {}" using assms unfolding rel_open_def by auto
  5143   hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
  5144 }
  5145 moreover
  5146 { assume "Inter {rel_interior S |S. S : I} ~= {}"
  5147   hence "rel_open (Inter I)" using assms unfolding rel_open_def
  5148     using convex_rel_interior_finite_inter[of I] by auto
  5149   hence ?thesis using convex_Inter assms by auto
  5150 } ultimately show ?thesis by auto
  5151 qed
  5152 
  5153 lemma convex_rel_open_linear_image:
  5154 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  5155 assumes "linear f"
  5156 assumes "convex S" "rel_open S"
  5157 shows "convex (f ` S) & rel_open (f ` S)"
  5158 by (metis assms convex_linear_image rel_interior_convex_linear_image 
  5159    linear_conv_bounded_linear rel_open_def)
  5160 
  5161 lemma convex_rel_open_linear_preimage:
  5162 fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
  5163 assumes "linear f"
  5164 assumes "convex S" "rel_open S"
  5165 shows "convex (f -` S) & rel_open (f -` S)" 
  5166 proof-
  5167 { assume "f -` (rel_interior S) = {}"
  5168   hence "f -` S = {}" using assms unfolding rel_open_def by auto
  5169   hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
  5170 }
  5171 moreover
  5172 { assume "f -` (rel_interior S) ~= {}"
  5173   hence "rel_open (f -` S)" using assms unfolding rel_open_def
  5174     using rel_interior_convex_linear_preimage[of f S] by auto
  5175   hence ?thesis using convex_linear_preimage assms linear_conv_bounded_linear by auto
  5176 } ultimately show ?thesis by auto
  5177 qed
  5178 
  5179 lemma rel_interior_projection:
  5180 fixes S :: "('m::euclidean_space*'n::euclidean_space) set"
  5181 fixes f :: "'m::euclidean_space => ('n::euclidean_space) set"
  5182 assumes "convex S"
  5183 assumes "f = (%y. {z. (y,z) : S})"
  5184 shows "(y,z) : rel_interior S <-> (y : rel_interior {y. (f y ~= {})} & z : rel_interior (f y))"
  5185 proof-
  5186 { fix y assume "y : {y. (f y ~= {})}" from this obtain z where "(y,z) : S" using assms by auto
  5187   hence "EX x. x : S & y = fst x" apply (rule_tac x="(y,z)" in exI) by auto
  5188   from this obtain x where "x : S & y = fst x" by blast
  5189   hence "y : fst ` S" unfolding image_def by auto
  5190 }
  5191 hence "fst ` S = {y. (f y ~= {})}" unfolding fst_def using assms by auto
  5192 hence h1: "fst ` rel_interior S = rel_interior {y. (f y ~= {})}"
  5193    using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
  5194 { fix y assume "y : rel_interior {y. (f y ~= {})}"
  5195   hence "y : fst ` rel_interior S" using h1 by auto
  5196   hence *: "rel_interior S Int fst -` {y} ~= {}" by auto
  5197   moreover have aff: "affine (fst -` {y})" unfolding affine_alt by (simp add: algebra_simps)
  5198   ultimately have **: "rel_interior (S Int fst -` {y}) = rel_interior S Int fst -` {y}"
  5199     using convex_affine_rel_interior_inter[of S "fst -` {y}"] assms by auto
  5200   have conv: "convex (S Int fst -` {y})" using convex_Int assms aff affine_imp_convex by auto
  5201   { fix x assume "x : f y"
  5202     hence "(y,x) : S Int (fst -` {y})" using assms by auto
  5203     moreover have "x = snd (y,x)" by auto
  5204     ultimately have "x : snd ` (S Int fst -` {y})" by blast
  5205   }
  5206   hence "snd ` (S Int fst -` {y}) = f y" using assms by auto
  5207   hence ***: "rel_interior (f y) = snd ` rel_interior (S Int fst -` {y})"
  5208     using rel_interior_convex_linear_image[of snd "S Int fst -` {y}"] snd_linear conv by auto 
  5209   { fix z assume "z : rel_interior (f y)"
  5210     hence "z : snd ` rel_interior (S Int fst -` {y})" using *** by auto
  5211     moreover have "{y} = fst ` rel_interior (S Int fst -` {y})" using * ** rel_interior_subset by auto   
  5212     ultimately have "(y,z) : rel_interior (S Int fst -` {y})" by force
  5213     hence "(y,z) : rel_interior S" using ** by auto
  5214   }
  5215   moreover
  5216   { fix z assume "(y,z) : rel_interior S"
  5217     hence "(y,z) : rel_interior (S Int fst -` {y})" using ** by auto
  5218     hence "z : snd ` rel_interior (S Int fst -` {y})" by (metis Range_iff snd_eq_Range) 
  5219     hence "z : rel_interior (f y)" using *** by auto
  5220   }
  5221   ultimately have "!!z. (y,z) : rel_interior S <-> z : rel_interior (f y)" by auto
  5222 } 
  5223 hence h2: "!!y z. y : rel_interior {t. f t ~= {}} ==> ((y, z) : rel_interior S) = (z : rel_interior (f y))"
  5224   by auto
  5225 { fix y z assume asm: "(y, z) : rel_interior S"
  5226   hence "y : fst ` rel_interior S" by (metis Domain_iff fst_eq_Domain)
  5227   hence "y : rel_interior {t. f t ~= {}}" using h1 by auto
  5228   hence "y : rel_interior {t. f t ~= {}} & (z : rel_interior (f y))" using h2 asm by auto
  5229 } from this show ?thesis using h2 by blast
  5230 qed
  5231 
  5232 subsection{* Relative interior of convex cone *}
  5233 
  5234 lemma cone_rel_interior:
  5235 fixes S :: "('m::euclidean_space) set"
  5236 assumes "cone S"
  5237 shows "cone ({0} Un (rel_interior S))"
  5238 proof-
  5239 { assume "S = {}" hence ?thesis by (simp add: rel_interior_empty cone_0) }
  5240 moreover
  5241 { assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
  5242   hence *: "0:({0} Un (rel_interior S)) &
  5243            (!c. c>0 --> op *\<^sub>R c ` ({0} Un rel_interior S) = ({0} Un rel_interior S))"
  5244            by (auto simp add: rel_interior_scaleR)
  5245   hence ?thesis using cone_iff[of "{0} Un rel_interior S"] by auto
  5246 }
  5247 ultimately show ?thesis by blast
  5248 qed
  5249 
  5250 lemma rel_interior_convex_cone_aux:
  5251 fixes S :: "('m::euclidean_space) set"
  5252 assumes "convex S"
  5253 shows "(c,x) : rel_interior (cone hull ({(1 :: real)} <*> S)) <-> 
  5254        c>0 & x : ((op *\<^sub>R c) ` (rel_interior S))"
  5255 proof-
  5256 { assume "S={}" hence ?thesis by (simp add: rel_interior_empty cone_hull_empty) } 
  5257 moreover
  5258 { assume "S ~= {}" from this obtain s where "s : S" by auto
  5259 have conv: "convex ({(1 :: real)} <*> S)" using convex_direct_sum[of "{(1 :: real)}" S] 
  5260    assms convex_singleton[of "1 :: real"] by auto
  5261 def f == "(%y. {z. (y,z) : cone hull ({(1 :: real)} <*> S)})"
  5262 hence *: "(c, x) : rel_interior (cone hull ({(1 :: real)} <*> S)) =
  5263       (c : rel_interior {y. f y ~= {}} & x : rel_interior (f c))"
  5264   apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} <*> S)" f c x])
  5265   using convex_cone_hull[of "{(1 :: real)} <*> S"] conv by auto
  5266 { fix y assume "(y :: real)>=0"
  5267   hence "y *\<^sub>R (1,s) : cone hull ({(1 :: real)} <*> S)"
  5268      using cone_hull_expl[of "{(1 :: real)} <*> S"] `s:S` by auto
  5269   hence "f y ~= {}" using f_def by auto
  5270 }
  5271 hence "{y. f y ~= {}} = {0..}" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
  5272 hence **: "rel_interior {y. f y ~= {}} = {0<..}" using rel_interior_real_semiline by auto
  5273 { fix c assume "c>(0 :: real)"
  5274   hence "f c = (op *\<^sub>R c ` S)" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
  5275   hence "rel_interior (f c)= (op *\<^sub>R c ` rel_interior S)"
  5276      using rel_interior_convex_scaleR[of S c] assms by auto
  5277 }
  5278 hence ?thesis using * ** by auto
  5279 } ultimately show ?thesis by blast
  5280 qed
  5281 
  5282 
  5283 lemma rel_interior_convex_cone:
  5284 fixes S :: "('m::euclidean_space) set"
  5285 assumes "convex S"
  5286 shows "rel_interior (cone hull ({(1 :: real)} <*> S)) = 
  5287        {(c,c *\<^sub>R x) |c x. c>0 & x : (rel_interior S)}"
  5288 (is "?lhs=?rhs")
  5289 proof-
  5290 { fix z assume "z:?lhs" 
  5291   have *: "z=(fst z,snd z)" by auto 
  5292   have "z:?rhs" using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms `z:?lhs` apply auto
  5293      apply (rule_tac x="fst z" in exI) apply (rule_tac x="x" in exI) using * by auto
  5294 }
  5295 moreover
  5296 { fix z assume "z:?rhs" hence "z:?lhs" 
  5297   using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms by auto
  5298 }
  5299 ultimately show ?thesis by blast
  5300 qed
  5301 
  5302 lemma convex_hull_finite_union:
  5303 assumes "finite I"
  5304 assumes "!i:I. (convex (S i) & (S i) ~= {})"
  5305 shows "convex hull (Union (S ` I)) = 
  5306        {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)}"
  5307   (is "?lhs = ?rhs")
  5308 proof-
  5309 { fix x assume "x : ?rhs" 
  5310   from this obtain c s 
  5311     where *: "setsum (%i. c i *\<^sub>R s i) I=x" "(setsum c I = 1)"
  5312      "(!i:I. c i >= 0) & (!i:I. s i : S i)" by auto
  5313   hence "!i:I. s i : convex hull (Union (S ` I))" using hull_subset[of "Union (S ` I)" convex] by auto
  5314   hence "x : ?lhs" unfolding *(1)[THEN sym]
  5315      apply (subst convex_setsum[of I "convex hull Union (S ` I)" c s])
  5316      using * assms convex_convex_hull by auto
  5317 } hence "?lhs >= ?rhs" by auto
  5318 
  5319 { fix i assume "i:I"
  5320     from this assms have "EX p. p : S i" by auto
  5321 } 
  5322 from this obtain p where p_def: "!i:I. p i : S i" by metis
  5323 
  5324 { fix i assume "i:I"
  5325   { fix x assume "x : S i"
  5326     def c == "(%j. if (j=i) then (1::real) else 0)"
  5327     hence *: "setsum c I = 1" using `finite I` `i:I` setsum_delta[of I i "(%(j::'a). (1::real))"] by auto
  5328     def s == "(%j. if (j=i) then x else p j)"
  5329     hence "!j. c j *\<^sub>R s j = (if (j=i) then x else 0)" using c_def by (auto simp add: algebra_simps)
  5330     hence "x = setsum (%i. c i *\<^sub>R s i) I"
  5331        using s_def c_def `finite I` `i:I` setsum_delta[of I i "(%(j::'a). x)"] by auto 
  5332     hence "x : ?rhs" apply auto
  5333       apply (rule_tac x="c" in exI) 
  5334       apply (rule_tac x="s" in exI) using * c_def s_def p_def `x : S i` by auto 
  5335   } hence "?rhs >= S i" by auto
  5336 } hence *: "?rhs >= Union (S ` I)" by auto
  5337 
  5338 { fix u v assume uv: "(u :: real)>=0 & v>=0 & u+v=1"
  5339   fix x y assume xy: "(x : ?rhs) & (y : ?rhs)"
  5340   from xy obtain c s where xc: "x=setsum (%i. c i *\<^sub>R s i) I &
  5341      (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)" by auto
  5342   from xy obtain d t where yc: "y=setsum (%i. d i *\<^sub>R t i) I &
  5343      (!i:I. d i >= 0) & (setsum d I = 1) & (!i:I. t i : S i)" by auto
  5344   def e == "(%i. u * (c i)+v * (d i))"
  5345   have ge0: "!i:I. e i >= 0"  using e_def xc yc uv by (simp add: mult_nonneg_nonneg)
  5346   have "setsum (%i. u * c i) I = u * setsum c I" by (simp add: setsum_right_distrib)
  5347   moreover have "setsum (%i. v * d i) I = v * setsum d I" by (simp add: setsum_right_distrib)
  5348   ultimately have sum1: "setsum e I = 1" using e_def xc yc uv by (simp add: setsum_addf)
  5349   def q == "(%i. if (e i = 0) then (p i) 
  5350                  else (u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))"
  5351   { fix i assume "i:I"
  5352     { assume "e i = 0" hence "q i : S i" using `i:I` p_def q_def by auto }
  5353     moreover
  5354     { assume "e i ~= 0" 
  5355       hence "q i : S i" using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"] 
  5356          mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
  5357          assms q_def e_def `i:I` `e i ~= 0` xc yc uv by auto
  5358     } ultimately have "q i : S i" by auto
  5359   } hence qs: "!i:I. q i : S i" by auto
  5360   { fix i assume "i:I"
  5361     { assume "e i = 0" 
  5362       have ge: "u * (c i) >= 0 & v * (d i) >= 0" using xc yc uv `i:I` by (simp add: mult_nonneg_nonneg)
  5363       moreover hence "u * (c i) <= 0 & v * (d i) <= 0" using `e i = 0` e_def `i:I` by simp 
  5364       ultimately have "u * (c i) = 0 & v * (d i) = 0" by auto
  5365       hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
  5366          using `e i = 0` by auto
  5367     }
  5368     moreover
  5369     { assume "e i ~= 0"
  5370       hence "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
  5371          using q_def by auto
  5372       hence "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
  5373              = (e i) *\<^sub>R (q i)" by auto
  5374       hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
  5375          using `e i ~= 0` by (simp add: algebra_simps)
  5376     } ultimately have 
  5377       "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by blast
  5378   } hence *: "!i:I.
  5379     (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by auto
  5380   have "u *\<^sub>R x + v *\<^sub>R y =
  5381        setsum (%i. (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i)) I"
  5382           using xc yc by (simp add: algebra_simps scaleR_right.setsum setsum_addf)
  5383   also have "...=setsum (%i. (e i) *\<^sub>R (q i)) I" using * by auto
  5384   finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (%i. (e i) *\<^sub>R (q i)) I" by auto
  5385   hence "u *\<^sub>R x + v *\<^sub>R y : ?rhs" using ge0 sum1 qs by auto
  5386 } hence "convex ?rhs" unfolding convex_def by auto
  5387 hence "?rhs : convex" unfolding mem_def by auto
  5388 from this show ?thesis using `?lhs >= ?rhs` * 
  5389    hull_minimal[of "Union (S ` I)" "?rhs" "convex"] by blast
  5390 qed
  5391 
  5392 lemma convex_hull_union_two:
  5393 fixes S T :: "('m::euclidean_space) set"
  5394 assumes "convex S" "S ~= {}" "convex T" "T ~= {}"
  5395 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}"
  5396   (is "?lhs = ?rhs")
  5397 proof-
  5398 def I == "{(1::nat),2}"
  5399 def s == "(%i. (if i=(1::nat) then S else T))"
  5400 have "Union (s ` I) = S Un T" using s_def I_def by auto
  5401 hence "convex hull (Union (s ` I)) = convex hull (S Un T)" by auto
  5402 moreover have "convex hull Union (s ` I) =
  5403     {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)}"
  5404     apply (subst convex_hull_finite_union[of I s]) using assms s_def I_def by auto
  5405 moreover have 
  5406   "{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)} <=
  5407   ?rhs"
  5408   using s_def I_def by auto
  5409 ultimately have "?lhs<=?rhs" by auto 
  5410 { fix x assume "x : ?rhs" 
  5411   from this obtain u v s t 
  5412     where *: "x=u *\<^sub>R s + v *\<^sub>R t & u>=0 & v>=0 & u+v=1 & s:S & t:T" by auto
  5413   hence "x : convex hull {s,t}" using convex_hull_2[of s t] by auto
  5414   hence "x : convex hull (S Un T)" using * hull_mono[of "{s, t}" "S Un T"] by auto
  5415 } hence "?lhs >= ?rhs" by blast
  5416 from this show ?thesis using `?lhs<=?rhs` by auto
  5417 qed
  5418 
  5419 subsection {* Convexity on direct sums *}
  5420 
  5421 lemma closure_sum:
  5422   fixes S T :: "('n::euclidean_space) set"
  5423   shows "closure S \<oplus> closure T \<subseteq> closure (S \<oplus> T)"
  5424 proof-
  5425   have "(closure S) \<oplus> (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
  5426     by (simp add: set_plus_image)
  5427   also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
  5428     using closure_direct_sum by auto
  5429   also have "... \<subseteq> closure (S \<oplus> T)"
  5430     using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
  5431     by (auto simp: set_plus_image)
  5432   finally show ?thesis
  5433     by auto
  5434 qed
  5435 
  5436 lemma convex_oplus:
  5437 fixes S T :: "('n::euclidean_space) set"
  5438 assumes "convex S" "convex T"
  5439 shows "convex (S \<oplus> T)"
  5440 proof-
  5441 have "{x + y |x y. x : S & y : T} = {c. EX a:S. EX b:T. c = a + b}" by auto
  5442 thus ?thesis unfolding set_plus_def using convex_sums[of S T] assms by auto
  5443 qed
  5444 
  5445 lemma convex_hull_sum:
  5446 fixes S T :: "('n::euclidean_space) set"
  5447 shows "convex hull (S \<oplus> T) = (convex hull S) \<oplus> (convex hull T)"
  5448 proof-