src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author hoelzl
Fri Oct 24 15:07:51 2014 +0200 (2014-10-24)
changeset 58776 95e58e04e534
parent 57865 dcfb33c26f50
child 58877 262572d90bc6
permissions -rw-r--r--
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
     1 (*  Title:      HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
     2     Author:     Robert Himmelmann, TU Muenchen
     3     Author:     Bogdan Grechuk, University of Edinburgh
     4 *)
     5 
     6 header {* Convex sets, functions and related things. *}
     7 
     8 theory Convex_Euclidean_Space
     9 imports
    10   Topology_Euclidean_Space
    11   "~~/src/HOL/Library/Convex"
    12   "~~/src/HOL/Library/Set_Algebras"
    13 begin
    14 
    15 
    16 (* ------------------------------------------------------------------------- *)
    17 (* To be moved elsewhere                                                     *)
    18 (* ------------------------------------------------------------------------- *)
    19 
    20 lemma linear_scaleR: "linear (\<lambda>x. scaleR c x)"
    21   by (simp add: linear_iff scaleR_add_right)
    22 
    23 lemma linear_scaleR_left: "linear (\<lambda>r. scaleR r x)"
    24   by (simp add: linear_iff scaleR_add_left)
    25 
    26 lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
    27   by (simp add: inj_on_def)
    28 
    29 lemma linear_add_cmul:
    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_alt:
    35   assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
    36   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
    37   apply (rule convexD)
    38   using assms
    39   apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
    40   done
    41 
    42 lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
    43   by (blast dest: inj_onD)
    44 
    45 lemma independent_injective_on_span_image:
    46   assumes iS: "independent S"
    47     and lf: "linear f"
    48     and fi: "inj_on f (span S)"
    49   shows "independent (f ` S)"
    50 proof -
    51   {
    52     fix a
    53     assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
    54     have eq: "f ` S - {f a} = f ` (S - {a})"
    55       using fi a span_inc by (auto simp add: inj_on_def)
    56     from a have "f a \<in> f ` span (S -{a})"
    57       unfolding eq span_linear_image [OF lf, of "S - {a}"] by blast
    58     moreover have "span (S - {a}) \<subseteq> span S"
    59       using span_mono[of "S - {a}" S] by auto
    60     ultimately have "a \<in> span (S - {a})"
    61       using fi a span_inc by (auto simp add: inj_on_def)
    62     with a(1) iS have False
    63       by (simp add: dependent_def)
    64   }
    65   then show ?thesis
    66     unfolding dependent_def by blast
    67 qed
    68 
    69 lemma dim_image_eq:
    70   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
    71   assumes lf: "linear f"
    72     and fi: "inj_on f (span S)"
    73   shows "dim (f ` S) = dim (S::'n::euclidean_space set)"
    74 proof -
    75   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
    76     using basis_exists[of S] by auto
    77   then have "span S = span B"
    78     using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
    79   then have "independent (f ` B)"
    80     using independent_injective_on_span_image[of B f] B assms by auto
    81   moreover have "card (f ` B) = card B"
    82     using assms card_image[of f B] subset_inj_on[of f "span S" B] B span_inc by auto
    83   moreover have "(f ` B) \<subseteq> (f ` S)"
    84     using B by auto
    85   ultimately have "dim (f ` S) \<ge> dim S"
    86     using independent_card_le_dim[of "f ` B" "f ` S"] B by auto
    87   then show ?thesis
    88     using dim_image_le[of f S] assms by auto
    89 qed
    90 
    91 lemma linear_injective_on_subspace_0:
    92   assumes lf: "linear f"
    93     and "subspace S"
    94   shows "inj_on f S \<longleftrightarrow> (\<forall>x \<in> S. f x = 0 \<longrightarrow> x = 0)"
    95 proof -
    96   have "inj_on f S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. f x = f y \<longrightarrow> x = y)"
    97     by (simp add: inj_on_def)
    98   also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. f x - f y = 0 \<longrightarrow> x - y = 0)"
    99     by simp
   100   also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. f (x - y) = 0 \<longrightarrow> x - y = 0)"
   101     by (simp add: linear_sub[OF lf])
   102   also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. f x = 0 \<longrightarrow> x = 0)"
   103     using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
   104   finally show ?thesis .
   105 qed
   106 
   107 lemma subspace_Inter: "\<forall>s \<in> f. subspace s \<Longrightarrow> subspace (Inter f)"
   108   unfolding subspace_def by auto
   109 
   110 lemma span_eq[simp]: "span s = s \<longleftrightarrow> subspace s"
   111   unfolding span_def by (rule hull_eq) (rule subspace_Inter)
   112 
   113 lemma substdbasis_expansion_unique:
   114   assumes d: "d \<subseteq> Basis"
   115   shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
   116     (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
   117 proof -
   118   have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
   119     by auto
   120   have **: "finite d"
   121     by (auto intro: finite_subset[OF assms])
   122   have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
   123     using d
   124     by (auto intro!: setsum.cong simp: inner_Basis inner_setsum_left)
   125   show ?thesis
   126     unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum.delta[OF **] ***)
   127 qed
   128 
   129 lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
   130   by (rule independent_mono[OF independent_Basis])
   131 
   132 lemma dim_cball:
   133   assumes "e > 0"
   134   shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
   135 proof -
   136   {
   137     fix x :: "'n::euclidean_space"
   138     def y \<equiv> "(e / norm x) *\<^sub>R x"
   139     then have "y \<in> cball 0 e"
   140       using cball_def dist_norm[of 0 y] assms by auto
   141     moreover have *: "x = (norm x / e) *\<^sub>R y"
   142       using y_def assms by simp
   143     moreover from * have "x = (norm x/e) *\<^sub>R y"
   144       by auto
   145     ultimately have "x \<in> span (cball 0 e)"
   146       using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
   147   }
   148   then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
   149     by auto
   150   then show ?thesis
   151     using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
   152 qed
   153 
   154 lemma indep_card_eq_dim_span:
   155   fixes B :: "'n::euclidean_space set"
   156   assumes "independent B"
   157   shows "finite B \<and> card B = dim (span B)"
   158   using assms basis_card_eq_dim[of B "span B"] span_inc by auto
   159 
   160 lemma setsum_not_0: "setsum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
   161   by (rule ccontr) auto
   162 
   163 lemma translate_inj_on:
   164   fixes A :: "'a::ab_group_add set"
   165   shows "inj_on (\<lambda>x. a + x) A"
   166   unfolding inj_on_def by auto
   167 
   168 lemma translation_assoc:
   169   fixes a b :: "'a::ab_group_add"
   170   shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
   171   by auto
   172 
   173 lemma translation_invert:
   174   fixes a :: "'a::ab_group_add"
   175   assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
   176   shows "A = B"
   177 proof -
   178   have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
   179     using assms by auto
   180   then show ?thesis
   181     using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
   182 qed
   183 
   184 lemma translation_galois:
   185   fixes a :: "'a::ab_group_add"
   186   shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
   187   using translation_assoc[of "-a" a S]
   188   apply auto
   189   using translation_assoc[of a "-a" T]
   190   apply auto
   191   done
   192 
   193 lemma translation_inverse_subset:
   194   assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
   195   shows "V \<le> ((\<lambda>x. a + x) ` S)"
   196 proof -
   197   {
   198     fix x
   199     assume "x \<in> V"
   200     then have "x-a \<in> S" using assms by auto
   201     then have "x \<in> {a + v |v. v \<in> S}"
   202       apply auto
   203       apply (rule exI[of _ "x-a"])
   204       apply simp
   205       done
   206     then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
   207   }
   208   then show ?thesis by auto
   209 qed
   210 
   211 lemma basis_to_basis_subspace_isomorphism:
   212   assumes s: "subspace (S:: ('n::euclidean_space) set)"
   213     and t: "subspace (T :: ('m::euclidean_space) set)"
   214     and d: "dim S = dim T"
   215     and B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
   216     and C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T"
   217   shows "\<exists>f. linear f \<and> f ` B = C \<and> f ` S = T \<and> inj_on f S"
   218 proof -
   219   from B independent_bound have fB: "finite B"
   220     by blast
   221   from C independent_bound have fC: "finite C"
   222     by blast
   223   from B(4) C(4) card_le_inj[of B C] d obtain f where
   224     f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
   225   from linear_independent_extend[OF B(2)] obtain g where
   226     g: "linear g" "\<forall>x \<in> B. g x = f x" by blast
   227   from inj_on_iff_eq_card[OF fB, of f] f(2)
   228   have "card (f ` B) = card B" by simp
   229   with B(4) C(4) have ceq: "card (f ` B) = card C" using d
   230     by simp
   231   have "g ` B = f ` B" using g(2)
   232     by (auto simp add: image_iff)
   233   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
   234   finally have gBC: "g ` B = C" .
   235   have gi: "inj_on g B" using f(2) g(2)
   236     by (auto simp add: inj_on_def)
   237   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
   238   {
   239     fix x y
   240     assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
   241     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
   242       by blast+
   243     from gxy have th0: "g (x - y) = 0"
   244       by (simp add: linear_sub[OF g(1)])
   245     have th1: "x - y \<in> span B" using x' y'
   246       by (metis span_sub)
   247     have "x = y" using g0[OF th1 th0] by simp
   248   }
   249   then have giS: "inj_on g S" unfolding inj_on_def by blast
   250   from span_subspace[OF B(1,3) s]
   251   have "g ` S = span (g ` B)"
   252     by (simp add: span_linear_image[OF g(1)])
   253   also have "\<dots> = span C"
   254     unfolding gBC ..
   255   also have "\<dots> = T"
   256     using span_subspace[OF C(1,3) t] .
   257   finally have gS: "g ` S = T" .
   258   from g(1) gS giS gBC show ?thesis
   259     by blast
   260 qed
   261 
   262 lemma closure_bounded_linear_image:
   263   assumes f: "bounded_linear f"
   264   shows "f ` closure S \<subseteq> closure (f ` S)"
   265   using linear_continuous_on [OF f] closed_closure closure_subset
   266   by (rule image_closure_subset)
   267 
   268 lemma closure_linear_image:
   269   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::real_normed_vector"
   270   assumes "linear f"
   271   shows "f ` (closure S) \<le> closure (f ` S)"
   272   using assms unfolding linear_conv_bounded_linear
   273   by (rule closure_bounded_linear_image)
   274 
   275 lemma closure_injective_linear_image:
   276   fixes f :: "'n::euclidean_space \<Rightarrow> 'n::euclidean_space"
   277   assumes "linear f" "inj f"
   278   shows "f ` (closure S) = closure (f ` S)"
   279 proof -
   280   obtain f' where f': "linear f' \<and> f \<circ> f' = id \<and> f' \<circ> f = id"
   281     using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
   282   then have "f' ` closure (f ` S) \<le> closure (S)"
   283     using closure_linear_image[of f' "f ` S"] image_comp[of f' f] by auto
   284   then have "f ` f' ` closure (f ` S) \<le> f ` closure S" by auto
   285   then have "closure (f ` S) \<le> f ` closure S"
   286     using image_comp[of f f' "closure (f ` S)"] f' by auto
   287   then show ?thesis using closure_linear_image[of f S] assms by auto
   288 qed
   289 
   290 lemma closure_scaleR:
   291   fixes S :: "'a::real_normed_vector set"
   292   shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
   293 proof
   294   show "(op *\<^sub>R c) ` (closure S) \<subseteq> closure ((op *\<^sub>R c) ` S)"
   295     using bounded_linear_scaleR_right
   296     by (rule closure_bounded_linear_image)
   297   show "closure ((op *\<^sub>R c) ` S) \<subseteq> (op *\<^sub>R c) ` (closure S)"
   298     by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
   299 qed
   300 
   301 lemma fst_linear: "linear fst"
   302   unfolding linear_iff by (simp add: algebra_simps)
   303 
   304 lemma snd_linear: "linear snd"
   305   unfolding linear_iff by (simp add: algebra_simps)
   306 
   307 lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
   308   unfolding linear_iff by (simp add: algebra_simps)
   309 
   310 lemma scaleR_2:
   311   fixes x :: "'a::real_vector"
   312   shows "scaleR 2 x = x + x"
   313   unfolding one_add_one [symmetric] scaleR_left_distrib by simp
   314 
   315 lemma vector_choose_size:
   316   "0 \<le> c \<Longrightarrow> \<exists>x::'a::euclidean_space. norm x = c"
   317   apply (rule exI [where x="c *\<^sub>R (SOME i. i \<in> Basis)"])
   318   apply (auto simp: SOME_Basis)
   319   done
   320 
   321 lemma setsum_delta_notmem:
   322   assumes "x \<notin> s"
   323   shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
   324     and "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
   325     and "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
   326     and "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
   327   apply (rule_tac [!] setsum.cong)
   328   using assms
   329   apply auto
   330   done
   331 
   332 lemma setsum_delta'':
   333   fixes s::"'a::real_vector set"
   334   assumes "finite s"
   335   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)"
   336 proof -
   337   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)"
   338     by auto
   339   show ?thesis
   340     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)"
   344   by (fact if_distrib)
   345 
   346 lemma dist_triangle_eq:
   347   fixes x y z :: "'a::real_inner"
   348   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
   349     norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
   350 proof -
   351   have *: "x - y + (y - z) = x - z" by auto
   352   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
   353     by (auto simp add:norm_minus_commute)
   354 qed
   355 
   356 lemma norm_minus_eqI: "x = - y \<Longrightarrow> norm x = norm y" by auto
   357 
   358 lemma Min_grI:
   359   assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
   360   shows "x < Min A"
   361   unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
   362 
   363 lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
   364   unfolding norm_eq_sqrt_inner by simp
   365 
   366 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
   367   unfolding norm_eq_sqrt_inner by simp
   368 
   369 
   370 subsection {* Affine set and affine hull *}
   371 
   372 definition affine :: "'a::real_vector set \<Rightarrow> bool"
   373   where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
   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
   395   using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
   396 
   397 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   398   by (metis affine_affine_hull hull_same)
   399 
   400 
   401 subsubsection {* Some explicit formulations (from Lars Schewe) *}
   402 
   403 lemma affine:
   404   fixes V::"'a::real_vector set"
   405   shows "affine V \<longleftrightarrow>
   406     (\<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)"
   407   unfolding affine_def
   408   apply rule
   409   apply(rule, rule, rule)
   410   apply(erule conjE)+
   411   defer
   412   apply (rule, rule, rule, rule, rule)
   413 proof -
   414   fix x y u v
   415   assume as: "x \<in> V" "y \<in> V" "u + v = (1::real)"
   416     "\<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"
   417   then show "u *\<^sub>R x + v *\<^sub>R y \<in> V"
   418     apply (cases "x = y")
   419     using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]]
   420       and as(1-3)
   421     apply (auto simp add: scaleR_left_distrib[symmetric])
   422     done
   423 next
   424   fix s u
   425   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"
   426     "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
   427   def n \<equiv> "card s"
   428   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   429   then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
   430   proof (auto simp only: disjE)
   431     assume "card s = 2"
   432     then have "card s = Suc (Suc 0)"
   433       by auto
   434     then obtain a b where "s = {a, b}"
   435       unfolding card_Suc_eq by auto
   436     then show ?thesis
   437       using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
   438       by (auto simp add: setsum_clauses(2))
   439   next
   440     assume "card s > 2"
   441     then show ?thesis using as and n_def
   442     proof (induct n arbitrary: u s)
   443       case 0
   444       then show ?case by auto
   445     next
   446       case (Suc n)
   447       fix s :: "'a set" and u :: "'a \<Rightarrow> real"
   448       assume IA:
   449         "\<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;
   450           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"
   451         and as:
   452           "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"
   453            "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
   454       have "\<exists>x\<in>s. u x \<noteq> 1"
   455       proof (rule ccontr)
   456         assume "\<not> ?thesis"
   457         then have "setsum u s = real_of_nat (card s)"
   458           unfolding card_eq_setsum by auto
   459         then show False
   460           using as(7) and `card s > 2`
   461           by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
   462       qed
   463       then obtain x where x:"x \<in> s" "u x \<noteq> 1" by auto
   464 
   465       have c: "card (s - {x}) = card s - 1"
   466         apply (rule card_Diff_singleton)
   467         using `x\<in>s` as(4)
   468         apply auto
   469         done
   470       have *: "s = insert x (s - {x})" "finite (s - {x})"
   471         using `x\<in>s` and as(4) by auto
   472       have **: "setsum u (s - {x}) = 1 - u x"
   473         using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
   474       have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"
   475         unfolding ** using `u x \<noteq> 1` by auto
   476       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V"
   477       proof (cases "card (s - {x}) > 2")
   478         case True
   479         then have "s - {x} \<noteq> {}" "card (s - {x}) = n"
   480           unfolding c and as(1)[symmetric]
   481         proof (rule_tac ccontr)
   482           assume "\<not> s - {x} \<noteq> {}"
   483           then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
   484           then show False using True by auto
   485         qed auto
   486         then show ?thesis
   487           apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
   488           unfolding setsum_right_distrib[symmetric]
   489           using as and *** and True
   490           apply auto
   491           done
   492       next
   493         case False
   494         then have "card (s - {x}) = Suc (Suc 0)"
   495           using as(2) and c by auto
   496         then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b"
   497           unfolding card_Suc_eq by auto
   498         then show ?thesis
   499           using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
   500           using *** *(2) and `s \<subseteq> V`
   501           unfolding setsum_right_distrib
   502           by (auto simp add: setsum_clauses(2))
   503       qed
   504       then have "u x + (1 - u x) = 1 \<Longrightarrow>
   505           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"
   506         apply -
   507         apply (rule as(3)[rule_format])
   508         unfolding  Real_Vector_Spaces.scaleR_right.setsum
   509         using x(1) as(6)
   510         apply auto
   511         done
   512       then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
   513         unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
   514         apply (subst *)
   515         unfolding setsum_clauses(2)[OF *(2)]
   516         using `u x \<noteq> 1`
   517         apply auto
   518         done
   519     qed
   520   next
   521     assume "card s = 1"
   522     then obtain a where "s={a}"
   523       by (auto simp add: card_Suc_eq)
   524     then show ?thesis
   525       using as(4,5) by simp
   526   qed (insert `s\<noteq>{}` `finite s`, auto)
   527 qed
   528 
   529 lemma affine_hull_explicit:
   530   "affine hull p =
   531     {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}"
   532   apply (rule hull_unique)
   533   apply (subst subset_eq)
   534   prefer 3
   535   apply rule
   536   unfolding mem_Collect_eq
   537   apply (erule exE)+
   538   apply (erule conjE)+
   539   prefer 2
   540   apply rule
   541 proof -
   542   fix x
   543   assume "x\<in>p"
   544   then show "\<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"
   545     apply (rule_tac x="{x}" in exI)
   546     apply (rule_tac x="\<lambda>x. 1" in exI)
   547     apply auto
   548     done
   549 next
   550   fix t x s u
   551   assume as: "p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}"
   552     "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   553   then show "x \<in> t"
   554     using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]]
   555     by auto
   556 next
   557   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}"
   558     unfolding affine_def
   559     apply (rule, rule, rule, rule, rule)
   560     unfolding mem_Collect_eq
   561   proof -
   562     fix u v :: real
   563     assume uv: "u + v = 1"
   564     fix x
   565     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"
   566     then obtain sx ux where
   567       x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x"
   568       by auto
   569     fix y
   570     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"
   571     then obtain sy uy where
   572       y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
   573     have xy: "finite (sx \<union> sy)"
   574       using x(1) y(1) by auto
   575     have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
   576       by auto
   577     show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and>
   578         setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
   579       apply (rule_tac x="sx \<union> sy" in exI)
   580       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)
   581       unfolding scaleR_left_distrib setsum.distrib if_smult scaleR_zero_left
   582         ** setsum.inter_restrict[OF xy, symmetric]
   583       unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
   584         and setsum_right_distrib[symmetric]
   585       unfolding x y
   586       using x(1-3) y(1-3) uv
   587       apply simp
   588       done
   589   qed
   590 qed
   591 
   592 lemma affine_hull_finite:
   593   assumes "finite s"
   594   shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
   595   unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq
   596   apply (rule, rule)
   597   apply (erule exE)+
   598   apply (erule conjE)+
   599   defer
   600   apply (erule exE)
   601   apply (erule conjE)
   602 proof -
   603   fix x u
   604   assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   605   then show "\<exists>sa u. finite sa \<and>
   606       \<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"
   607     apply (rule_tac x=s in exI, rule_tac x=u in exI)
   608     using assms
   609     apply auto
   610     done
   611 next
   612   fix x t u
   613   assume "t \<subseteq> s"
   614   then have *: "s \<inter> t = t"
   615     by auto
   616   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"
   617   then show "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   618     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   619     unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms, symmetric] and *
   620     apply auto
   621     done
   622 qed
   623 
   624 
   625 subsubsection {* Stepping theorems and hence small special cases *}
   626 
   627 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   628   by (rule hull_unique) auto
   629 
   630 lemma affine_hull_finite_step:
   631   fixes y :: "'a::real_vector"
   632   shows
   633     "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
   634     and
   635     "finite s \<Longrightarrow>
   636       (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
   637       (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
   638 proof -
   639   show ?th1 by simp
   640   assume fin: "finite s"
   641   show "?lhs = ?rhs"
   642   proof
   643     assume ?lhs
   644     then obtain u where u: "setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
   645       by auto
   646     show ?rhs
   647     proof (cases "a \<in> s")
   648       case True
   649       then have *: "insert a s = s" by auto
   650       show ?thesis
   651         using u[unfolded *]
   652         apply(rule_tac x=0 in exI)
   653         apply auto
   654         done
   655     next
   656       case False
   657       then show ?thesis
   658         apply (rule_tac x="u a" in exI)
   659         using u and fin
   660         apply auto
   661         done
   662     qed
   663   next
   664     assume ?rhs
   665     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"
   666       by auto
   667     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)"
   668       by auto
   669     show ?lhs
   670     proof (cases "a \<in> s")
   671       case True
   672       then show ?thesis
   673         apply (rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
   674         unfolding setsum_clauses(2)[OF fin]
   675         apply simp
   676         unfolding scaleR_left_distrib and setsum.distrib
   677         unfolding vu and * and scaleR_zero_left
   678         apply (auto simp add: setsum.delta[OF fin])
   679         done
   680     next
   681       case False
   682       then have **:
   683         "\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
   684         "\<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
   685       from False show ?thesis
   686         apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
   687         unfolding setsum_clauses(2)[OF fin] and * using vu
   688         using setsum.cong [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)]
   689         using setsum.cong [of s _ u "\<lambda>x. if x = a then v else u x", OF _ **(1)]
   690         apply auto
   691         done
   692     qed
   693   qed
   694 qed
   695 
   696 lemma affine_hull_2:
   697   fixes a b :: "'a::real_vector"
   698   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
   699   (is "?lhs = ?rhs")
   700 proof -
   701   have *:
   702     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
   703     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   704   have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
   705     using affine_hull_finite[of "{a,b}"] by auto
   706   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
   707     by (simp add: affine_hull_finite_step(2)[of "{b}" a])
   708   also have "\<dots> = ?rhs" unfolding * by auto
   709   finally show ?thesis by auto
   710 qed
   711 
   712 lemma affine_hull_3:
   713   fixes a b c :: "'a::real_vector"
   714   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}"
   715 proof -
   716   have *:
   717     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
   718     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   719   show ?thesis
   720     apply (simp add: affine_hull_finite affine_hull_finite_step)
   721     unfolding *
   722     apply auto
   723     apply (rule_tac x=v in exI)
   724     apply (rule_tac x=va in exI)
   725     apply auto
   726     apply (rule_tac x=u in exI)
   727     apply force
   728     done
   729 qed
   730 
   731 lemma mem_affine:
   732   assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
   733   shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
   734   using assms affine_def[of S] by auto
   735 
   736 lemma mem_affine_3:
   737   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
   738   shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
   739 proof -
   740   have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
   741     using affine_hull_3[of x y z] assms by auto
   742   moreover
   743   have "affine hull {x, y, z} \<subseteq> affine hull S"
   744     using hull_mono[of "{x, y, z}" "S"] assms by auto
   745   moreover
   746   have "affine hull S = S"
   747     using assms affine_hull_eq[of S] by auto
   748   ultimately show ?thesis by auto
   749 qed
   750 
   751 lemma mem_affine_3_minus:
   752   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
   753   shows "x + v *\<^sub>R (y-z) \<in> S"
   754   using mem_affine_3[of S x y z 1 v "-v"] assms
   755   by (simp add: algebra_simps)
   756 
   757 
   758 subsubsection {* Some relations between affine hull and subspaces *}
   759 
   760 lemma affine_hull_insert_subset_span:
   761   "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
   762   unfolding subset_eq Ball_def
   763   unfolding affine_hull_explicit span_explicit mem_Collect_eq
   764   apply (rule, rule)
   765   apply (erule exE)+
   766   apply (erule conjE)+
   767 proof -
   768   fix x t u
   769   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"
   770   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}"
   771     using as(3) by auto
   772   then show "\<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)"
   773     apply (rule_tac x="x - a" in exI)
   774     apply (rule conjI, simp)
   775     apply (rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
   776     apply (rule_tac x="\<lambda>x. u (x + a)" in exI)
   777     apply (rule conjI) using as(1) apply simp
   778     apply (erule conjI)
   779     using as(1)
   780     apply (simp add: setsum.reindex[unfolded inj_on_def] scaleR_right_diff_distrib
   781       setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)
   782     unfolding as
   783     apply simp
   784     done
   785 qed
   786 
   787 lemma affine_hull_insert_span:
   788   assumes "a \<notin> s"
   789   shows "affine hull (insert a s) = {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
   790   apply (rule, rule affine_hull_insert_subset_span)
   791   unfolding subset_eq Ball_def
   792   unfolding affine_hull_explicit and mem_Collect_eq
   793 proof (rule, rule, erule exE, erule conjE)
   794   fix y v
   795   assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
   796   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"
   797     unfolding span_explicit by auto
   798   def f \<equiv> "(\<lambda>x. x + a) ` t"
   799   have f: "finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a"
   800     unfolding f_def using obt by (auto simp add: setsum.reindex[unfolded inj_on_def])
   801   have *: "f \<inter> {a} = {}" "f \<inter> - {a} = f"
   802     using f(2) assms by auto
   803   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"
   804     apply (rule_tac x = "insert a f" in exI)
   805     apply (rule_tac x = "\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
   806     using assms and f
   807     unfolding setsum_clauses(2)[OF f(1)] and if_smult
   808     unfolding setsum.If_cases[OF f(1), of "\<lambda>x. x = a"]
   809     apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)
   810     done
   811 qed
   812 
   813 lemma affine_hull_span:
   814   assumes "a \<in> s"
   815   shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
   816   using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
   817 
   818 
   819 subsubsection {* Parallel affine sets *}
   820 
   821 definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
   822   where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
   823 
   824 lemma affine_parallel_expl_aux:
   825   fixes S T :: "'a::real_vector set"
   826   assumes "\<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
   827   shows "T = (\<lambda>x. a + x) ` S"
   828 proof -
   829   {
   830     fix x
   831     assume "x \<in> T"
   832     then have "( - a) + x \<in> S"
   833       using assms by auto
   834     then have "x \<in> ((\<lambda>x. a + x) ` S)"
   835       using imageI[of "-a+x" S "(\<lambda>x. a+x)"] by auto
   836   }
   837   moreover have "T \<ge> (\<lambda>x. a + x) ` S"
   838     using assms by auto
   839   ultimately show ?thesis by auto
   840 qed
   841 
   842 lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
   843   unfolding affine_parallel_def
   844   using affine_parallel_expl_aux[of S _ T] by auto
   845 
   846 lemma affine_parallel_reflex: "affine_parallel S S"
   847   unfolding affine_parallel_def
   848   apply (rule exI[of _ "0"])
   849   apply auto
   850   done
   851 
   852 lemma affine_parallel_commut:
   853   assumes "affine_parallel A B"
   854   shows "affine_parallel B A"
   855 proof -
   856   from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
   857     unfolding affine_parallel_def by auto
   858   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
   859   from B show ?thesis
   860     using translation_galois [of B a A]
   861     unfolding affine_parallel_def by auto
   862 qed
   863 
   864 lemma affine_parallel_assoc:
   865   assumes "affine_parallel A B"
   866     and "affine_parallel B C"
   867   shows "affine_parallel A C"
   868 proof -
   869   from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
   870     unfolding affine_parallel_def by auto
   871   moreover
   872   from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
   873     unfolding affine_parallel_def by auto
   874   ultimately show ?thesis
   875     using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
   876 qed
   877 
   878 lemma affine_translation_aux:
   879   fixes a :: "'a::real_vector"
   880   assumes "affine ((\<lambda>x. a + x) ` S)"
   881   shows "affine S"
   882 proof -
   883   {
   884     fix x y u v
   885     assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
   886     then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
   887       by auto
   888     then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
   889       using xy assms unfolding affine_def by auto
   890     have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
   891       by (simp add: algebra_simps)
   892     also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
   893       using `u + v = 1` by auto
   894     ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
   895       using h1 by auto
   896     then have "u *\<^sub>R x + v *\<^sub>R y : S" by auto
   897   }
   898   then show ?thesis unfolding affine_def by auto
   899 qed
   900 
   901 lemma affine_translation:
   902   fixes a :: "'a::real_vector"
   903   shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
   904 proof -
   905   have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
   906     using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
   907     using translation_assoc[of "-a" a S] by auto
   908   then show ?thesis using affine_translation_aux by auto
   909 qed
   910 
   911 lemma parallel_is_affine:
   912   fixes S T :: "'a::real_vector set"
   913   assumes "affine S" "affine_parallel S T"
   914   shows "affine T"
   915 proof -
   916   from assms obtain a where "T = (\<lambda>x. a + x) ` S"
   917     unfolding affine_parallel_def by auto
   918   then show ?thesis
   919     using affine_translation assms by auto
   920 qed
   921 
   922 lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
   923   unfolding subspace_def affine_def by auto
   924 
   925 
   926 subsubsection {* Subspace parallel to an affine set *}
   927 
   928 lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
   929 proof -
   930   have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
   931     using subspace_imp_affine[of S] subspace_0 by auto
   932   {
   933     assume assm: "affine S \<and> 0 \<in> S"
   934     {
   935       fix c :: real
   936       fix x
   937       assume x: "x \<in> S"
   938       have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
   939       moreover
   940       have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
   941         using affine_alt[of S] assm x by auto
   942       ultimately have "c *\<^sub>R x \<in> S" by auto
   943     }
   944     then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
   945 
   946     {
   947       fix x y
   948       assume xy: "x \<in> S" "y \<in> S"
   949       def u == "(1 :: real)/2"
   950       have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
   951         by auto
   952       moreover
   953       have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
   954         by (simp add: algebra_simps)
   955       moreover
   956       have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
   957         using affine_alt[of S] assm xy by auto
   958       ultimately
   959       have "(1/2) *\<^sub>R (x+y) \<in> S"
   960         using u_def by auto
   961       moreover
   962       have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
   963         by auto
   964       ultimately
   965       have "x + y \<in> S"
   966         using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
   967     }
   968     then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
   969       by auto
   970     then have "subspace S"
   971       using h1 assm unfolding subspace_def by auto
   972   }
   973   then show ?thesis using h0 by metis
   974 qed
   975 
   976 lemma affine_diffs_subspace:
   977   assumes "affine S" "a \<in> S"
   978   shows "subspace ((\<lambda>x. (-a)+x) ` S)"
   979 proof -
   980   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
   981   have "affine ((\<lambda>x. (-a)+x) ` S)"
   982     using  affine_translation assms by auto
   983   moreover have "0 : ((\<lambda>x. (-a)+x) ` S)"
   984     using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
   985   ultimately show ?thesis using subspace_affine by auto
   986 qed
   987 
   988 lemma parallel_subspace_explicit:
   989   assumes "affine S"
   990     and "a \<in> S"
   991   assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
   992   shows "subspace L \<and> affine_parallel S L"
   993 proof -
   994   from assms have "L = plus (- a) ` S" by auto
   995   then have par: "affine_parallel S L"
   996     unfolding affine_parallel_def ..
   997   then have "affine L" using assms parallel_is_affine by auto
   998   moreover have "0 \<in> L"
   999     using assms by auto
  1000   ultimately show ?thesis
  1001     using subspace_affine par by auto
  1002 qed
  1003 
  1004 lemma parallel_subspace_aux:
  1005   assumes "subspace A"
  1006     and "subspace B"
  1007     and "affine_parallel A B"
  1008   shows "A \<supseteq> B"
  1009 proof -
  1010   from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
  1011     using affine_parallel_expl[of A B] by auto
  1012   then have "-a \<in> A"
  1013     using assms subspace_0[of B] by auto
  1014   then have "a \<in> A"
  1015     using assms subspace_neg[of A "-a"] by auto
  1016   then show ?thesis
  1017     using assms a unfolding subspace_def by auto
  1018 qed
  1019 
  1020 lemma parallel_subspace:
  1021   assumes "subspace A"
  1022     and "subspace B"
  1023     and "affine_parallel A B"
  1024   shows "A = B"
  1025 proof
  1026   show "A \<supseteq> B"
  1027     using assms parallel_subspace_aux by auto
  1028   show "A \<subseteq> B"
  1029     using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
  1030 qed
  1031 
  1032 lemma affine_parallel_subspace:
  1033   assumes "affine S" "S \<noteq> {}"
  1034   shows "\<exists>!L. subspace L \<and> affine_parallel S L"
  1035 proof -
  1036   have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
  1037     using assms parallel_subspace_explicit by auto
  1038   {
  1039     fix L1 L2
  1040     assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
  1041     then have "affine_parallel L1 L2"
  1042       using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
  1043     then have "L1 = L2"
  1044       using ass parallel_subspace by auto
  1045   }
  1046   then show ?thesis using ex by auto
  1047 qed
  1048 
  1049 
  1050 subsection {* Cones *}
  1051 
  1052 definition cone :: "'a::real_vector set \<Rightarrow> bool"
  1053   where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
  1054 
  1055 lemma cone_empty[intro, simp]: "cone {}"
  1056   unfolding cone_def by auto
  1057 
  1058 lemma cone_univ[intro, simp]: "cone UNIV"
  1059   unfolding cone_def by auto
  1060 
  1061 lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
  1062   unfolding cone_def by auto
  1063 
  1064 
  1065 subsubsection {* Conic hull *}
  1066 
  1067 lemma cone_cone_hull: "cone (cone hull s)"
  1068   unfolding hull_def by auto
  1069 
  1070 lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
  1071   apply (rule hull_eq)
  1072   using cone_Inter
  1073   unfolding subset_eq
  1074   apply auto
  1075   done
  1076 
  1077 lemma mem_cone:
  1078   assumes "cone S" "x \<in> S" "c \<ge> 0"
  1079   shows "c *\<^sub>R x : S"
  1080   using assms cone_def[of S] by auto
  1081 
  1082 lemma cone_contains_0:
  1083   assumes "cone S"
  1084   shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
  1085 proof -
  1086   {
  1087     assume "S \<noteq> {}"
  1088     then obtain a where "a \<in> S" by auto
  1089     then have "0 \<in> S"
  1090       using assms mem_cone[of S a 0] by auto
  1091   }
  1092   then show ?thesis by auto
  1093 qed
  1094 
  1095 lemma cone_0: "cone {0}"
  1096   unfolding cone_def by auto
  1097 
  1098 lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (Union f)"
  1099   unfolding cone_def by blast
  1100 
  1101 lemma cone_iff:
  1102   assumes "S \<noteq> {}"
  1103   shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
  1104 proof -
  1105   {
  1106     assume "cone S"
  1107     {
  1108       fix c :: real
  1109       assume "c > 0"
  1110       {
  1111         fix x
  1112         assume "x \<in> S"
  1113         then have "x \<in> (op *\<^sub>R c) ` S"
  1114           unfolding image_def
  1115           using `cone S` `c>0` mem_cone[of S x "1/c"]
  1116             exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
  1117           by auto
  1118       }
  1119       moreover
  1120       {
  1121         fix x
  1122         assume "x \<in> (op *\<^sub>R c) ` S"
  1123         then have "x \<in> S"
  1124           using `cone S` `c > 0`
  1125           unfolding cone_def image_def `c > 0` by auto
  1126       }
  1127       ultimately have "(op *\<^sub>R c) ` S = S" by auto
  1128     }
  1129     then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
  1130       using `cone S` cone_contains_0[of S] assms by auto
  1131   }
  1132   moreover
  1133   {
  1134     assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
  1135     {
  1136       fix x
  1137       assume "x \<in> S"
  1138       fix c1 :: real
  1139       assume "c1 \<ge> 0"
  1140       then have "c1 = 0 \<or> c1 > 0" by auto
  1141       then have "c1 *\<^sub>R x \<in> S" using a `x \<in> S` by auto
  1142     }
  1143     then have "cone S" unfolding cone_def by auto
  1144   }
  1145   ultimately show ?thesis by blast
  1146 qed
  1147 
  1148 lemma cone_hull_empty: "cone hull {} = {}"
  1149   by (metis cone_empty cone_hull_eq)
  1150 
  1151 lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
  1152   by (metis bot_least cone_hull_empty hull_subset xtrans(5))
  1153 
  1154 lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
  1155   using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
  1156   by auto
  1157 
  1158 lemma mem_cone_hull:
  1159   assumes "x \<in> S" "c \<ge> 0"
  1160   shows "c *\<^sub>R x \<in> cone hull S"
  1161   by (metis assms cone_cone_hull hull_inc mem_cone)
  1162 
  1163 lemma cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
  1164   (is "?lhs = ?rhs")
  1165 proof -
  1166   {
  1167     fix x
  1168     assume "x \<in> ?rhs"
  1169     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
  1170       by auto
  1171     fix c :: real
  1172     assume c: "c \<ge> 0"
  1173     then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
  1174       using x by (simp add: algebra_simps)
  1175     moreover
  1176     have "c * cx \<ge> 0" using c x by auto
  1177     ultimately
  1178     have "c *\<^sub>R x \<in> ?rhs" using x by auto
  1179   }
  1180   then have "cone ?rhs"
  1181     unfolding cone_def by auto
  1182   then have "?rhs \<in> Collect cone"
  1183     unfolding mem_Collect_eq by auto
  1184   {
  1185     fix x
  1186     assume "x \<in> S"
  1187     then have "1 *\<^sub>R x \<in> ?rhs"
  1188       apply auto
  1189       apply (rule_tac x = 1 in exI)
  1190       apply auto
  1191       done
  1192     then have "x \<in> ?rhs" by auto
  1193   }
  1194   then have "S \<subseteq> ?rhs" by auto
  1195   then have "?lhs \<subseteq> ?rhs"
  1196     using `?rhs \<in> Collect cone` hull_minimal[of S "?rhs" "cone"] by auto
  1197   moreover
  1198   {
  1199     fix x
  1200     assume "x \<in> ?rhs"
  1201     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
  1202       by auto
  1203     then have "xx \<in> cone hull S"
  1204       using hull_subset[of S] by auto
  1205     then have "x \<in> ?lhs"
  1206       using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
  1207   }
  1208   ultimately show ?thesis by auto
  1209 qed
  1210 
  1211 lemma cone_closure:
  1212   fixes S :: "'a::real_normed_vector set"
  1213   assumes "cone S"
  1214   shows "cone (closure S)"
  1215 proof (cases "S = {}")
  1216   case True
  1217   then show ?thesis by auto
  1218 next
  1219   case False
  1220   then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
  1221     using cone_iff[of S] assms by auto
  1222   then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` closure S = closure S)"
  1223     using closure_subset by (auto simp add: closure_scaleR)
  1224   then show ?thesis
  1225     using cone_iff[of "closure S"] by auto
  1226 qed
  1227 
  1228 
  1229 subsection {* Affine dependence and consequential theorems (from Lars Schewe) *}
  1230 
  1231 definition affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
  1232   where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
  1233 
  1234 lemma affine_dependent_explicit:
  1235   "affine_dependent p \<longleftrightarrow>
  1236     (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
  1237       (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
  1238   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq
  1239   apply rule
  1240   apply (erule bexE, erule exE, erule exE)
  1241   apply (erule conjE)+
  1242   defer
  1243   apply (erule exE, erule exE)
  1244   apply (erule conjE)+
  1245   apply (erule bexE)
  1246 proof -
  1247   fix x s u
  1248   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"
  1249   have "x \<notin> s" using as(1,4) by auto
  1250   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"
  1251     apply (rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
  1252     unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as
  1253     using as
  1254     apply auto
  1255     done
  1256 next
  1257   fix s u v
  1258   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"
  1259   have "s \<noteq> {v}"
  1260     using as(3,6) by auto
  1261   then show "\<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"
  1262     apply (rule_tac x=v in bexI)
  1263     apply (rule_tac x="s - {v}" in exI)
  1264     apply (rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
  1265     unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
  1266     unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]
  1267     using as
  1268     apply auto
  1269     done
  1270 qed
  1271 
  1272 lemma affine_dependent_explicit_finite:
  1273   fixes s :: "'a::real_vector set"
  1274   assumes "finite s"
  1275   shows "affine_dependent s \<longleftrightarrow>
  1276     (\<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)"
  1277   (is "?lhs = ?rhs")
  1278 proof
  1279   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)"
  1280     by auto
  1281   assume ?lhs
  1282   then obtain t u v where
  1283     "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"
  1284     unfolding affine_dependent_explicit by auto
  1285   then show ?rhs
  1286     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
  1287     apply auto unfolding * and setsum.inter_restrict[OF assms, symmetric]
  1288     unfolding Int_absorb1[OF `t\<subseteq>s`]
  1289     apply auto
  1290     done
  1291 next
  1292   assume ?rhs
  1293   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"
  1294     by auto
  1295   then show ?lhs unfolding affine_dependent_explicit
  1296     using assms by auto
  1297 qed
  1298 
  1299 
  1300 subsection {* Connectedness of convex sets *}
  1301 
  1302 lemma connectedD:
  1303   "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
  1304   by (metis connected_def)
  1305 
  1306 lemma convex_connected:
  1307   fixes s :: "'a::real_normed_vector set"
  1308   assumes "convex s"
  1309   shows "connected s"
  1310 proof (rule connectedI)
  1311   fix A B
  1312   assume "open A" "open B" "A \<inter> B \<inter> s = {}" "s \<subseteq> A \<union> B"
  1313   moreover
  1314   assume "A \<inter> s \<noteq> {}" "B \<inter> s \<noteq> {}"
  1315   then obtain a b where a: "a \<in> A" "a \<in> s" and b: "b \<in> B" "b \<in> s" by auto
  1316   def f \<equiv> "\<lambda>u. u *\<^sub>R a + (1 - u) *\<^sub>R b"
  1317   then have "continuous_on {0 .. 1} f"
  1318     by (auto intro!: continuous_intros)
  1319   then have "connected (f ` {0 .. 1})"
  1320     by (auto intro!: connected_continuous_image)
  1321   note connectedD[OF this, of A B]
  1322   moreover have "a \<in> A \<inter> f ` {0 .. 1}"
  1323     using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
  1324   moreover have "b \<in> B \<inter> f ` {0 .. 1}"
  1325     using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
  1326   moreover have "f ` {0 .. 1} \<subseteq> s"
  1327     using `convex s` a b unfolding convex_def f_def by auto
  1328   ultimately show False by auto
  1329 qed
  1330 
  1331 text {* One rather trivial consequence. *}
  1332 
  1333 lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
  1334   by(simp add: convex_connected convex_UNIV)
  1335 
  1336 text {* Balls, being convex, are connected. *}
  1337 
  1338 lemma convex_prod:
  1339   assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
  1340   shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
  1341   using assms unfolding convex_def
  1342   by (auto simp: inner_add_left)
  1343 
  1344 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
  1345   by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
  1346 
  1347 lemma convex_local_global_minimum:
  1348   fixes s :: "'a::real_normed_vector set"
  1349   assumes "e > 0"
  1350     and "convex_on s f"
  1351     and "ball x e \<subseteq> s"
  1352     and "\<forall>y\<in>ball x e. f x \<le> f y"
  1353   shows "\<forall>y\<in>s. f x \<le> f y"
  1354 proof (rule ccontr)
  1355   have "x \<in> s" using assms(1,3) by auto
  1356   assume "\<not> ?thesis"
  1357   then obtain y where "y\<in>s" and y: "f x > f y" by auto
  1358   then have xy: "0 < dist x y"
  1359     by (auto simp add: dist_nz[symmetric])
  1360 
  1361   then obtain u where "0 < u" "u \<le> 1" and u: "u < e / dist x y"
  1362     using real_lbound_gt_zero[of 1 "e / dist x y"] xy `e>0` by auto
  1363   then have "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y"
  1364     using `x\<in>s` `y\<in>s`
  1365     using assms(2)[unfolded convex_on_def,
  1366       THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
  1367     by auto
  1368   moreover
  1369   have *: "x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)"
  1370     by (simp add: algebra_simps)
  1371   have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e"
  1372     unfolding mem_ball dist_norm
  1373     unfolding * and norm_scaleR and abs_of_pos[OF `0<u`]
  1374     unfolding dist_norm[symmetric]
  1375     using u
  1376     unfolding pos_less_divide_eq[OF xy]
  1377     by auto
  1378   then have "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)"
  1379     using assms(4) by auto
  1380   ultimately show False
  1381     using mult_strict_left_mono[OF y `u>0`]
  1382     unfolding left_diff_distrib
  1383     by auto
  1384 qed
  1385 
  1386 lemma convex_ball:
  1387   fixes x :: "'a::real_normed_vector"
  1388   shows "convex (ball x e)"
  1389 proof (auto simp add: convex_def)
  1390   fix y z
  1391   assume yz: "dist x y < e" "dist x z < e"
  1392   fix u v :: real
  1393   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
  1394   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
  1395     using uv yz
  1396     using convex_on_dist [of "ball x e" x, unfolded convex_on_def,
  1397       THEN bspec[where x=y], THEN bspec[where x=z]]
  1398     by auto
  1399   then show "dist x (u *\<^sub>R y + v *\<^sub>R z) < e"
  1400     using convex_bound_lt[OF yz uv] by auto
  1401 qed
  1402 
  1403 lemma convex_cball:
  1404   fixes x :: "'a::real_normed_vector"
  1405   shows "convex (cball x e)"
  1406 proof -
  1407   {
  1408     fix y z
  1409     assume yz: "dist x y \<le> e" "dist x z \<le> e"
  1410     fix u v :: real
  1411     assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
  1412     have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
  1413       using uv yz
  1414       using convex_on_dist [of "cball x e" x, unfolded convex_on_def,
  1415         THEN bspec[where x=y], THEN bspec[where x=z]]
  1416       by auto
  1417     then have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e"
  1418       using convex_bound_le[OF yz uv] by auto
  1419   }
  1420   then show ?thesis by (auto simp add: convex_def Ball_def)
  1421 qed
  1422 
  1423 lemma connected_ball:
  1424   fixes x :: "'a::real_normed_vector"
  1425   shows "connected (ball x e)"
  1426   using convex_connected convex_ball by auto
  1427 
  1428 lemma connected_cball:
  1429   fixes x :: "'a::real_normed_vector"
  1430   shows "connected (cball x e)"
  1431   using convex_connected convex_cball by auto
  1432 
  1433 
  1434 subsection {* Convex hull *}
  1435 
  1436 lemma convex_convex_hull: "convex (convex hull s)"
  1437   unfolding hull_def
  1438   using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
  1439   by auto
  1440 
  1441 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
  1442   by (metis convex_convex_hull hull_same)
  1443 
  1444 lemma bounded_convex_hull:
  1445   fixes s :: "'a::real_normed_vector set"
  1446   assumes "bounded s"
  1447   shows "bounded (convex hull s)"
  1448 proof -
  1449   from assms obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
  1450     unfolding bounded_iff by auto
  1451   show ?thesis
  1452     apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
  1453     unfolding subset_hull[of convex, OF convex_cball]
  1454     unfolding subset_eq mem_cball dist_norm using B
  1455     apply auto
  1456     done
  1457 qed
  1458 
  1459 lemma finite_imp_bounded_convex_hull:
  1460   fixes s :: "'a::real_normed_vector set"
  1461   shows "finite s \<Longrightarrow> bounded (convex hull s)"
  1462   using bounded_convex_hull finite_imp_bounded
  1463   by auto
  1464 
  1465 
  1466 subsubsection {* Convex hull is "preserved" by a linear function *}
  1467 
  1468 lemma convex_hull_linear_image:
  1469   assumes f: "linear f"
  1470   shows "f ` (convex hull s) = convex hull (f ` s)"
  1471 proof
  1472   show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
  1473     by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
  1474   show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
  1475   proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
  1476     show "s \<subseteq> f -` (convex hull (f ` s))"
  1477       by (fast intro: hull_inc)
  1478     show "convex (f -` (convex hull (f ` s)))"
  1479       by (intro convex_linear_vimage [OF f] convex_convex_hull)
  1480   qed
  1481 qed
  1482 
  1483 lemma in_convex_hull_linear_image:
  1484   assumes "linear f"
  1485     and "x \<in> convex hull s"
  1486   shows "f x \<in> convex hull (f ` s)"
  1487   using convex_hull_linear_image[OF assms(1)] assms(2) by auto
  1488 
  1489 lemma convex_hull_Times:
  1490   "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
  1491 proof
  1492   show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
  1493     by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
  1494   have "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)"
  1495   proof (intro hull_induct)
  1496     fix x y assume "x \<in> s" and "y \<in> t"
  1497     then show "(x, y) \<in> convex hull (s \<times> t)"
  1498       by (simp add: hull_inc)
  1499   next
  1500     fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
  1501     have "convex ?S"
  1502       by (intro convex_linear_vimage convex_translation convex_convex_hull,
  1503         simp add: linear_iff)
  1504     also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
  1505       by (auto simp add: image_def Bex_def)
  1506     finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
  1507   next
  1508     show "convex {x. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)}"
  1509     proof (unfold Collect_ball_eq, rule convex_INT [rule_format])
  1510       fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
  1511       have "convex ?S"
  1512       by (intro convex_linear_vimage convex_translation convex_convex_hull,
  1513         simp add: linear_iff)
  1514       also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
  1515         by (auto simp add: image_def Bex_def)
  1516       finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
  1517     qed
  1518   qed
  1519   then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
  1520     unfolding subset_eq split_paired_Ball_Sigma .
  1521 qed
  1522 
  1523 
  1524 subsubsection {* Stepping theorems for convex hulls of finite sets *}
  1525 
  1526 lemma convex_hull_empty[simp]: "convex hull {} = {}"
  1527   by (rule hull_unique) auto
  1528 
  1529 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
  1530   by (rule hull_unique) auto
  1531 
  1532 lemma convex_hull_insert:
  1533   fixes s :: "'a::real_vector set"
  1534   assumes "s \<noteq> {}"
  1535   shows "convex hull (insert a s) =
  1536     {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
  1537   (is "_ = ?hull")
  1538   apply (rule, rule hull_minimal, rule)
  1539   unfolding insert_iff
  1540   prefer 3
  1541   apply rule
  1542 proof -
  1543   fix x
  1544   assume x: "x = a \<or> x \<in> s"
  1545   then show "x \<in> ?hull"
  1546     apply rule
  1547     unfolding mem_Collect_eq
  1548     apply (rule_tac x=1 in exI)
  1549     defer
  1550     apply (rule_tac x=0 in exI)
  1551     using assms hull_subset[of s convex]
  1552     apply auto
  1553     done
  1554 next
  1555   fix x
  1556   assume "x \<in> ?hull"
  1557   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"
  1558     by auto
  1559   have "a \<in> convex hull insert a s" "b \<in> convex hull insert a s"
  1560     using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4)
  1561     by auto
  1562   then show "x \<in> convex hull insert a s"
  1563     unfolding obt(5) using obt(1-3)
  1564     by (rule convexD [OF convex_convex_hull])
  1565 next
  1566   show "convex ?hull"
  1567   proof (rule convexI)
  1568     fix x y u v
  1569     assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
  1570     from as(4) obtain u1 v1 b1 where
  1571       obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
  1572       by auto
  1573     from as(5) obtain u2 v2 b2 where
  1574       obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
  1575       by auto
  1576     have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
  1577       by (auto simp add: algebra_simps)
  1578     have **: "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y =
  1579       (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
  1580     proof (cases "u * v1 + v * v2 = 0")
  1581       case True
  1582       have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
  1583         by (auto simp add: algebra_simps)
  1584       from True have ***: "u * v1 = 0" "v * v2 = 0"
  1585         using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`]
  1586         by arith+
  1587       then have "u * u1 + v * u2 = 1"
  1588         using as(3) obt1(3) obt2(3) by auto
  1589       then show ?thesis
  1590         unfolding obt1(5) obt2(5) *
  1591         using assms hull_subset[of s convex]
  1592         by (auto simp add: *** scaleR_right_distrib)
  1593     next
  1594       case False
  1595       have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
  1596         using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
  1597       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
  1598         using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
  1599       also have "\<dots> = u * v1 + v * v2"
  1600         by simp
  1601       finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
  1602       have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
  1603         using as(1,2) obt1(1,2) obt2(1,2) by auto
  1604       then show ?thesis
  1605         unfolding obt1(5) obt2(5)
  1606         unfolding * and **
  1607         using False
  1608         apply (rule_tac
  1609           x = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI)
  1610         defer
  1611         apply (rule convexD [OF convex_convex_hull])
  1612         using obt1(4) obt2(4)
  1613         unfolding add_divide_distrib[symmetric] and zero_le_divide_iff
  1614         apply (auto simp add: scaleR_left_distrib scaleR_right_distrib)
  1615         done
  1616     qed
  1617     have u1: "u1 \<le> 1"
  1618       unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
  1619     have u2: "u2 \<le> 1"
  1620       unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
  1621     have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
  1622       apply (rule add_mono)
  1623       apply (rule_tac [!] mult_right_mono)
  1624       using as(1,2) obt1(1,2) obt2(1,2)
  1625       apply auto
  1626       done
  1627     also have "\<dots> \<le> 1"
  1628       unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
  1629     finally show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
  1630       unfolding mem_Collect_eq
  1631       apply (rule_tac x="u * u1 + v * u2" in exI)
  1632       apply (rule conjI)
  1633       defer
  1634       apply (rule_tac x="1 - u * u1 - v * u2" in exI)
  1635       unfolding Bex_def
  1636       using as(1,2) obt1(1,2) obt2(1,2) **
  1637       apply (auto simp add: algebra_simps)
  1638       done
  1639   qed
  1640 qed
  1641 
  1642 
  1643 subsubsection {* Explicit expression for convex hull *}
  1644 
  1645 lemma convex_hull_indexed:
  1646   fixes s :: "'a::real_vector set"
  1647   shows "convex hull s =
  1648     {y. \<exists>k u x.
  1649       (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
  1650       (setsum u {1..k} = 1) \<and> (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}"
  1651   (is "?xyz = ?hull")
  1652   apply (rule hull_unique)
  1653   apply rule
  1654   defer
  1655   apply (rule convexI)
  1656 proof -
  1657   fix x
  1658   assume "x\<in>s"
  1659   then show "x \<in> ?hull"
  1660     unfolding mem_Collect_eq
  1661     apply (rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI)
  1662     apply auto
  1663     done
  1664 next
  1665   fix t
  1666   assume as: "s \<subseteq> t" "convex t"
  1667   show "?hull \<subseteq> t"
  1668     apply rule
  1669     unfolding mem_Collect_eq
  1670     apply (elim exE conjE)
  1671   proof -
  1672     fix x k u y
  1673     assume assm:
  1674       "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s"
  1675       "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
  1676     show "x\<in>t"
  1677       unfolding assm(3) [symmetric]
  1678       apply (rule as(2)[unfolded convex, rule_format])
  1679       using assm(1,2) as(1) apply auto
  1680       done
  1681   qed
  1682 next
  1683   fix x y u v
  1684   assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
  1685   assume xy: "x \<in> ?hull" "y \<in> ?hull"
  1686   from xy obtain k1 u1 x1 where
  1687     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"
  1688     by auto
  1689   from xy obtain k2 u2 x2 where
  1690     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"
  1691     by auto
  1692   have *: "\<And>P (x1::'a) x2 s1 s2 i.
  1693     (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)"
  1694     "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
  1695     prefer 3
  1696     apply (rule, rule)
  1697     unfolding image_iff
  1698     apply (rule_tac x = "x - k1" in bexI)
  1699     apply (auto simp add: not_le)
  1700     done
  1701   have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
  1702     unfolding inj_on_def by auto
  1703   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
  1704     apply rule
  1705     apply (rule_tac x="k1 + k2" in exI)
  1706     apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
  1707     apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI)
  1708     apply (rule, rule)
  1709     defer
  1710     apply rule
  1711     unfolding * and setsum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
  1712       setsum.reindex[OF inj] and o_def Collect_mem_eq
  1713     unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]
  1714   proof -
  1715     fix i
  1716     assume i: "i \<in> {1..k1+k2}"
  1717     show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and>
  1718       (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
  1719     proof (cases "i\<in>{1..k1}")
  1720       case True
  1721       then show ?thesis
  1722         using uv(1) x(1)[THEN bspec[where x=i]] by auto
  1723     next
  1724       case False
  1725       def j \<equiv> "i - k1"
  1726       from i False have "j \<in> {1..k2}"
  1727         unfolding j_def by auto
  1728       then show ?thesis
  1729         using False uv(2) y(1)[THEN bspec[where x=j]]
  1730         by (auto simp: j_def[symmetric])
  1731     qed
  1732   qed (auto simp add: not_le x(2,3) y(2,3) uv(3))
  1733 qed
  1734 
  1735 lemma convex_hull_finite:
  1736   fixes s :: "'a::real_vector set"
  1737   assumes "finite s"
  1738   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
  1739     setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
  1740   (is "?HULL = ?set")
  1741 proof (rule hull_unique, auto simp add: convex_def[of ?set])
  1742   fix x
  1743   assume "x \<in> s"
  1744   then show "\<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"
  1745     apply (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI)
  1746     apply auto
  1747     unfolding setsum.delta'[OF assms] and setsum_delta''[OF assms]
  1748     apply auto
  1749     done
  1750 next
  1751   fix u v :: real
  1752   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
  1753   fix ux assume ux: "\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
  1754   fix uy assume uy: "\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
  1755   {
  1756     fix x
  1757     assume "x\<in>s"
  1758     then have "0 \<le> u * ux x + v * uy x"
  1759       using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
  1760       by auto
  1761   }
  1762   moreover
  1763   have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
  1764     unfolding setsum.distrib and setsum_right_distrib[symmetric] and ux(2) uy(2)
  1765     using uv(3) by auto
  1766   moreover
  1767   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)"
  1768     unfolding scaleR_left_distrib and setsum.distrib and scaleR_scaleR[symmetric]
  1769       and scaleR_right.setsum [symmetric]
  1770     by auto
  1771   ultimately
  1772   show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and>
  1773       (\<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)"
  1774     apply (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI)
  1775     apply auto
  1776     done
  1777 next
  1778   fix t
  1779   assume t: "s \<subseteq> t" "convex t"
  1780   fix u
  1781   assume u: "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
  1782   then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t"
  1783     using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
  1784     using assms and t(1) by auto
  1785 qed
  1786 
  1787 
  1788 subsubsection {* Another formulation from Lars Schewe *}
  1789 
  1790 lemma setsum_constant_scaleR:
  1791   fixes y :: "'a::real_vector"
  1792   shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
  1793   apply (cases "finite A")
  1794   apply (induct set: finite)
  1795   apply (simp_all add: algebra_simps)
  1796   done
  1797 
  1798 lemma convex_hull_explicit:
  1799   fixes p :: "'a::real_vector set"
  1800   shows "convex hull p =
  1801     {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  1802   (is "?lhs = ?rhs")
  1803 proof -
  1804   {
  1805     fix x
  1806     assume "x\<in>?lhs"
  1807     then obtain k u y where
  1808         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"
  1809       unfolding convex_hull_indexed by auto
  1810 
  1811     have fin: "finite {1..k}" by auto
  1812     have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
  1813     {
  1814       fix j
  1815       assume "j\<in>{1..k}"
  1816       then have "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
  1817         using obt(1)[THEN bspec[where x=j]] and obt(2)
  1818         apply simp
  1819         apply (rule setsum_nonneg)
  1820         using obt(1)
  1821         apply auto
  1822         done
  1823     }
  1824     moreover
  1825     have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
  1826       unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
  1827     moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
  1828       using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
  1829       unfolding scaleR_left.setsum using obt(3) by auto
  1830     ultimately
  1831     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"
  1832       apply (rule_tac x="y ` {1..k}" in exI)
  1833       apply (rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI)
  1834       apply auto
  1835       done
  1836     then have "x\<in>?rhs" by auto
  1837   }
  1838   moreover
  1839   {
  1840     fix y
  1841     assume "y\<in>?rhs"
  1842     then obtain s u where
  1843       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"
  1844       by auto
  1845 
  1846     obtain f where f: "inj_on f {1..card s}" "f ` {1..card s} = s"
  1847       using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
  1848 
  1849     {
  1850       fix i :: nat
  1851       assume "i\<in>{1..card s}"
  1852       then have "f i \<in> s"
  1853         apply (subst f(2)[symmetric])
  1854         apply auto
  1855         done
  1856       then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
  1857     }
  1858     moreover have *: "finite {1..card s}" by auto
  1859     {
  1860       fix y
  1861       assume "y\<in>s"
  1862       then obtain i where "i\<in>{1..card s}" "f i = y"
  1863         using f using image_iff[of y f "{1..card s}"]
  1864         by auto
  1865       then have "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}"
  1866         apply auto
  1867         using f(1)[unfolded inj_on_def]
  1868         apply(erule_tac x=x in ballE)
  1869         apply auto
  1870         done
  1871       then have "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
  1872       then have "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
  1873           "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
  1874         by (auto simp add: setsum_constant_scaleR)
  1875     }
  1876     then have "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
  1877       unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
  1878         and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
  1879       unfolding f
  1880       using setsum.cong [of s 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"]
  1881       using setsum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u]
  1882       unfolding obt(4,5)
  1883       by auto
  1884     ultimately
  1885     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>
  1886         (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
  1887       apply (rule_tac x="card s" in exI)
  1888       apply (rule_tac x="u \<circ> f" in exI)
  1889       apply (rule_tac x=f in exI)
  1890       apply fastforce
  1891       done
  1892     then have "y \<in> ?lhs"
  1893       unfolding convex_hull_indexed by auto
  1894   }
  1895   ultimately show ?thesis
  1896     unfolding set_eq_iff by blast
  1897 qed
  1898 
  1899 
  1900 subsubsection {* A stepping theorem for that expansion *}
  1901 
  1902 lemma convex_hull_finite_step:
  1903   fixes s :: "'a::real_vector set"
  1904   assumes "finite s"
  1905   shows
  1906     "(\<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)
  1907       \<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)"
  1908   (is "?lhs = ?rhs")
  1909 proof (rule, case_tac[!] "a\<in>s")
  1910   assume "a \<in> s"
  1911   then have *: "insert a s = s" by auto
  1912   assume ?lhs
  1913   then show ?rhs
  1914     unfolding *
  1915     apply (rule_tac x=0 in exI)
  1916     apply auto
  1917     done
  1918 next
  1919   assume ?lhs
  1920   then obtain u where
  1921       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"
  1922     by auto
  1923   assume "a \<notin> s"
  1924   then show ?rhs
  1925     apply (rule_tac x="u a" in exI)
  1926     using u(1)[THEN bspec[where x=a]]
  1927     apply simp
  1928     apply (rule_tac x=u in exI)
  1929     using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s`
  1930     apply auto
  1931     done
  1932 next
  1933   assume "a \<in> s"
  1934   then have *: "insert a s = s" by auto
  1935   have fin: "finite (insert a s)" using assms by auto
  1936   assume ?rhs
  1937   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"
  1938     by auto
  1939   show ?lhs
  1940     apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
  1941     unfolding scaleR_left_distrib and setsum.distrib and setsum_delta''[OF fin] and setsum.delta'[OF fin]
  1942     unfolding setsum_clauses(2)[OF assms]
  1943     using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s`
  1944     apply auto
  1945     done
  1946 next
  1947   assume ?rhs
  1948   then obtain v u where
  1949     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"
  1950     by auto
  1951   moreover
  1952   assume "a \<notin> s"
  1953   moreover
  1954   have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s"
  1955     and "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
  1956     apply (rule_tac setsum.cong) apply rule
  1957     defer
  1958     apply (rule_tac setsum.cong) apply rule
  1959     using `a \<notin> s`
  1960     apply auto
  1961     done
  1962   ultimately show ?lhs
  1963     apply (rule_tac x="\<lambda>x. if a = x then v else u x" in exI)
  1964     unfolding setsum_clauses(2)[OF assms]
  1965     apply auto
  1966     done
  1967 qed
  1968 
  1969 
  1970 subsubsection {* Hence some special cases *}
  1971 
  1972 lemma convex_hull_2:
  1973   "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}"
  1974 proof -
  1975   have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
  1976     by auto
  1977   have **: "finite {b}" by auto
  1978   show ?thesis
  1979     apply (simp add: convex_hull_finite)
  1980     unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
  1981     apply auto
  1982     apply (rule_tac x=v in exI)
  1983     apply (rule_tac x="1 - v" in exI)
  1984     apply simp
  1985     apply (rule_tac x=u in exI)
  1986     apply simp
  1987     apply (rule_tac x="\<lambda>x. v" in exI)
  1988     apply simp
  1989     done
  1990 qed
  1991 
  1992 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
  1993   unfolding convex_hull_2
  1994 proof (rule Collect_cong)
  1995   have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
  1996     by auto
  1997   fix x
  1998   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) \<longleftrightarrow>
  1999     (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
  2000     unfolding *
  2001     apply auto
  2002     apply (rule_tac[!] x=u in exI)
  2003     apply (auto simp add: algebra_simps)
  2004     done
  2005 qed
  2006 
  2007 lemma convex_hull_3:
  2008   "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}"
  2009 proof -
  2010   have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
  2011     by auto
  2012   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
  2013     by (auto simp add: field_simps)
  2014   show ?thesis
  2015     unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
  2016     unfolding convex_hull_finite_step[OF fin(3)]
  2017     apply (rule Collect_cong)
  2018     apply simp
  2019     apply auto
  2020     apply (rule_tac x=va in exI)
  2021     apply (rule_tac x="u c" in exI)
  2022     apply simp
  2023     apply (rule_tac x="1 - v - w" in exI)
  2024     apply simp
  2025     apply (rule_tac x=v in exI)
  2026     apply simp
  2027     apply (rule_tac x="\<lambda>x. w" in exI)
  2028     apply simp
  2029     done
  2030 qed
  2031 
  2032 lemma convex_hull_3_alt:
  2033   "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}"
  2034 proof -
  2035   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
  2036     by auto
  2037   show ?thesis
  2038     unfolding convex_hull_3
  2039     apply (auto simp add: *)
  2040     apply (rule_tac x=v in exI)
  2041     apply (rule_tac x=w in exI)
  2042     apply (simp add: algebra_simps)
  2043     apply (rule_tac x=u in exI)
  2044     apply (rule_tac x=v in exI)
  2045     apply (simp add: algebra_simps)
  2046     done
  2047 qed
  2048 
  2049 
  2050 subsection {* Relations among closure notions and corresponding hulls *}
  2051 
  2052 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
  2053   unfolding affine_def convex_def by auto
  2054 
  2055 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
  2056   using subspace_imp_affine affine_imp_convex by auto
  2057 
  2058 lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
  2059   by (metis hull_minimal span_inc subspace_imp_affine subspace_span)
  2060 
  2061 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
  2062   by (metis hull_minimal span_inc subspace_imp_convex subspace_span)
  2063 
  2064 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
  2065   by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
  2066 
  2067 
  2068 lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
  2069   unfolding affine_dependent_def dependent_def
  2070   using affine_hull_subset_span by auto
  2071 
  2072 lemma dependent_imp_affine_dependent:
  2073   assumes "dependent {x - a| x . x \<in> s}"
  2074     and "a \<notin> s"
  2075   shows "affine_dependent (insert a s)"
  2076 proof -
  2077   from assms(1)[unfolded dependent_explicit] obtain S u v
  2078     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"
  2079     by auto
  2080   def t \<equiv> "(\<lambda>x. x + a) ` S"
  2081 
  2082   have inj: "inj_on (\<lambda>x. x + a) S"
  2083     unfolding inj_on_def by auto
  2084   have "0 \<notin> S"
  2085     using obt(2) assms(2) unfolding subset_eq by auto
  2086   have fin: "finite t" and "t \<subseteq> s"
  2087     unfolding t_def using obt(1,2) by auto
  2088   then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
  2089     by auto
  2090   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
  2091     apply (rule setsum.cong)
  2092     using `a\<notin>s` `t\<subseteq>s`
  2093     apply auto
  2094     done
  2095   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
  2096     unfolding setsum_clauses(2)[OF fin]
  2097     using `a\<notin>s` `t\<subseteq>s`
  2098     apply auto
  2099     unfolding *
  2100     apply auto
  2101     done
  2102   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"
  2103     apply (rule_tac x="v + a" in bexI)
  2104     using obt(3,4) and `0\<notin>S`
  2105     unfolding t_def
  2106     apply auto
  2107     done
  2108   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)"
  2109     apply (rule setsum.cong)
  2110     using `a\<notin>s` `t\<subseteq>s`
  2111     apply auto
  2112     done
  2113   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
  2114     unfolding scaleR_left.setsum
  2115     unfolding t_def and setsum.reindex[OF inj] and o_def
  2116     using obt(5)
  2117     by (auto simp add: setsum.distrib scaleR_right_distrib)
  2118   then have "(\<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"
  2119     unfolding setsum_clauses(2)[OF fin]
  2120     using `a\<notin>s` `t\<subseteq>s`
  2121     by (auto simp add: *)
  2122   ultimately show ?thesis
  2123     unfolding affine_dependent_explicit
  2124     apply (rule_tac x="insert a t" in exI)
  2125     apply auto
  2126     done
  2127 qed
  2128 
  2129 lemma convex_cone:
  2130   "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)"
  2131   (is "?lhs = ?rhs")
  2132 proof -
  2133   {
  2134     fix x y
  2135     assume "x\<in>s" "y\<in>s" and ?lhs
  2136     then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
  2137       unfolding cone_def by auto
  2138     then have "x + y \<in> s"
  2139       using `?lhs`[unfolded convex_def, THEN conjunct1]
  2140       apply (erule_tac x="2*\<^sub>R x" in ballE)
  2141       apply (erule_tac x="2*\<^sub>R y" in ballE)
  2142       apply (erule_tac x="1/2" in allE)
  2143       apply simp
  2144       apply (erule_tac x="1/2" in allE)
  2145       apply auto
  2146       done
  2147   }
  2148   then show ?thesis
  2149     unfolding convex_def cone_def by blast
  2150 qed
  2151 
  2152 lemma affine_dependent_biggerset:
  2153   fixes s :: "'a::euclidean_space set"
  2154   assumes "finite s" "card s \<ge> DIM('a) + 2"
  2155   shows "affine_dependent s"
  2156 proof -
  2157   have "s \<noteq> {}" using assms by auto
  2158   then obtain a where "a\<in>s" by auto
  2159   have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
  2160     by auto
  2161   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
  2162     unfolding *
  2163     apply (rule card_image)
  2164     unfolding inj_on_def
  2165     apply auto
  2166     done
  2167   also have "\<dots> > DIM('a)" using assms(2)
  2168     unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  2169   finally show ?thesis
  2170     apply (subst insert_Diff[OF `a\<in>s`, symmetric])
  2171     apply (rule dependent_imp_affine_dependent)
  2172     apply (rule dependent_biggerset)
  2173     apply auto
  2174     done
  2175 qed
  2176 
  2177 lemma affine_dependent_biggerset_general:
  2178   assumes "finite (s :: 'a::euclidean_space set)"
  2179     and "card s \<ge> dim s + 2"
  2180   shows "affine_dependent s"
  2181 proof -
  2182   from assms(2) have "s \<noteq> {}" by auto
  2183   then obtain a where "a\<in>s" by auto
  2184   have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
  2185     by auto
  2186   have **: "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
  2187     unfolding *
  2188     apply (rule card_image)
  2189     unfolding inj_on_def
  2190     apply auto
  2191     done
  2192   have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
  2193     apply (rule subset_le_dim)
  2194     unfolding subset_eq
  2195     using `a\<in>s`
  2196     apply (auto simp add:span_superset span_sub)
  2197     done
  2198   also have "\<dots> < dim s + 1" by auto
  2199   also have "\<dots> \<le> card (s - {a})"
  2200     using assms
  2201     using card_Diff_singleton[OF assms(1) `a\<in>s`]
  2202     by auto
  2203   finally show ?thesis
  2204     apply (subst insert_Diff[OF `a\<in>s`, symmetric])
  2205     apply (rule dependent_imp_affine_dependent)
  2206     apply (rule dependent_biggerset_general)
  2207     unfolding **
  2208     apply auto
  2209     done
  2210 qed
  2211 
  2212 
  2213 subsection {* Caratheodory's theorem. *}
  2214 
  2215 lemma convex_hull_caratheodory:
  2216   fixes p :: "('a::euclidean_space) set"
  2217   shows "convex hull p =
  2218     {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
  2219       (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  2220   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
  2221 proof (rule, rule)
  2222   fix y
  2223   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>
  2224     setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
  2225   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"
  2226   then obtain N where "?P N" by auto
  2227   then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
  2228     apply (rule_tac ex_least_nat_le)
  2229     apply auto
  2230     done
  2231   then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
  2232     by blast
  2233   then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
  2234     "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
  2235 
  2236   have "card s \<le> DIM('a) + 1"
  2237   proof (rule ccontr, simp only: not_le)
  2238     assume "DIM('a) + 1 < card s"
  2239     then have "affine_dependent s"
  2240       using affine_dependent_biggerset[OF obt(1)] by auto
  2241     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"
  2242       using affine_dependent_explicit_finite[OF obt(1)] by auto
  2243     def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
  2244     def t \<equiv> "Min i"
  2245     have "\<exists>x\<in>s. w x < 0"
  2246     proof (rule ccontr, simp add: not_less)
  2247       assume as:"\<forall>x\<in>s. 0 \<le> w x"
  2248       then have "setsum w (s - {v}) \<ge> 0"
  2249         apply (rule_tac setsum_nonneg)
  2250         apply auto
  2251         done
  2252       then have "setsum w s > 0"
  2253         unfolding setsum.remove[OF obt(1) `v\<in>s`]
  2254         using as[THEN bspec[where x=v]] and `v\<in>s`
  2255         using `w v \<noteq> 0`
  2256         by auto
  2257       then show False using wv(1) by auto
  2258     qed
  2259     then have "i \<noteq> {}" unfolding i_def by auto
  2260 
  2261     then have "t \<ge> 0"
  2262       using Min_ge_iff[of i 0 ] and obt(1)
  2263       unfolding t_def i_def
  2264       using obt(4)[unfolded le_less]
  2265       by (auto simp: divide_le_0_iff)
  2266     have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
  2267     proof
  2268       fix v
  2269       assume "v \<in> s"
  2270       then have v: "0 \<le> u v"
  2271         using obt(4)[THEN bspec[where x=v]] by auto
  2272       show "0 \<le> u v + t * w v"
  2273       proof (cases "w v < 0")
  2274         case False
  2275         thus ?thesis using v `t\<ge>0` by auto
  2276       next
  2277         case True
  2278         then have "t \<le> u v / (- w v)"
  2279           using `v\<in>s`
  2280           unfolding t_def i_def
  2281           apply (rule_tac Min_le)
  2282           using obt(1)
  2283           apply auto
  2284           done
  2285         then show ?thesis
  2286           unfolding real_0_le_add_iff
  2287           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
  2288           by auto
  2289       qed
  2290     qed
  2291 
  2292     obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
  2293       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
  2294     then have a: "a \<in> s" "u a + t * w a = 0" by auto
  2295     have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
  2296       unfolding setsum.remove[OF obt(1) `a\<in>s`] by auto
  2297     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
  2298       unfolding setsum.distrib wv(1) setsum_right_distrib[symmetric] obt(5) by auto
  2299     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"
  2300       unfolding setsum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
  2301       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
  2302     ultimately have "?P (n - 1)"
  2303       apply (rule_tac x="(s - {a})" in exI)
  2304       apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
  2305       using obt(1-3) and t and a
  2306       apply (auto simp add: * scaleR_left_distrib)
  2307       done
  2308     then show False
  2309       using smallest[THEN spec[where x="n - 1"]] by auto
  2310   qed
  2311   then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
  2312       (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
  2313     using obt by auto
  2314 qed auto
  2315 
  2316 lemma caratheodory:
  2317   "convex hull p =
  2318     {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
  2319       card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
  2320   unfolding set_eq_iff
  2321   apply rule
  2322   apply rule
  2323   unfolding mem_Collect_eq
  2324 proof -
  2325   fix x
  2326   assume "x \<in> convex hull p"
  2327   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
  2328     "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
  2329     unfolding convex_hull_caratheodory by auto
  2330   then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  2331     apply (rule_tac x=s in exI)
  2332     using hull_subset[of s convex]
  2333     using convex_convex_hull[unfolded convex_explicit, of s,
  2334       THEN spec[where x=s], THEN spec[where x=u]]
  2335     apply auto
  2336     done
  2337 next
  2338   fix x
  2339   assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
  2340   then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
  2341     by auto
  2342   then show "x \<in> convex hull p"
  2343     using hull_mono[OF `s\<subseteq>p`] by auto
  2344 qed
  2345 
  2346 
  2347 subsection {* Some Properties of Affine Dependent Sets *}
  2348 
  2349 lemma affine_independent_empty: "\<not> affine_dependent {}"
  2350   by (simp add: affine_dependent_def)
  2351 
  2352 lemma affine_independent_sing: "\<not> affine_dependent {a}"
  2353   by (simp add: affine_dependent_def)
  2354 
  2355 lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
  2356 proof -
  2357   have "affine ((\<lambda>x. a + x) ` (affine hull S))"
  2358     using affine_translation affine_affine_hull by auto
  2359   moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
  2360     using hull_subset[of S] by auto
  2361   ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
  2362     by (metis hull_minimal)
  2363   have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
  2364     using affine_translation affine_affine_hull by (auto simp del: uminus_add_conv_diff)
  2365   moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
  2366     using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
  2367   moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
  2368     using translation_assoc[of "-a" a] by auto
  2369   ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
  2370     by (metis hull_minimal)
  2371   then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
  2372     by auto
  2373   then show ?thesis using h1 by auto
  2374 qed
  2375 
  2376 lemma affine_dependent_translation:
  2377   assumes "affine_dependent S"
  2378   shows "affine_dependent ((\<lambda>x. a + x) ` S)"
  2379 proof -
  2380   obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
  2381     using assms affine_dependent_def by auto
  2382   have "op + a ` (S - {x}) = op + a ` S - {a + x}"
  2383     by auto
  2384   then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
  2385     using affine_hull_translation[of a "S - {x}"] x by auto
  2386   moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
  2387     using x by auto
  2388   ultimately show ?thesis
  2389     unfolding affine_dependent_def by auto
  2390 qed
  2391 
  2392 lemma affine_dependent_translation_eq:
  2393   "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
  2394 proof -
  2395   {
  2396     assume "affine_dependent ((\<lambda>x. a + x) ` S)"
  2397     then have "affine_dependent S"
  2398       using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
  2399       by auto
  2400   }
  2401   then show ?thesis
  2402     using affine_dependent_translation by auto
  2403 qed
  2404 
  2405 lemma affine_hull_0_dependent:
  2406   assumes "0 \<in> affine hull S"
  2407   shows "dependent S"
  2408 proof -
  2409   obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
  2410     using assms affine_hull_explicit[of S] by auto
  2411   then have "\<exists>v\<in>s. u v \<noteq> 0"
  2412     using setsum_not_0[of "u" "s"] by auto
  2413   then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
  2414     using s_u by auto
  2415   then show ?thesis
  2416     unfolding dependent_explicit[of S] by auto
  2417 qed
  2418 
  2419 lemma affine_dependent_imp_dependent2:
  2420   assumes "affine_dependent (insert 0 S)"
  2421   shows "dependent S"
  2422 proof -
  2423   obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
  2424     using affine_dependent_def[of "(insert 0 S)"] assms by blast
  2425   then have "x \<in> span (insert 0 S - {x})"
  2426     using affine_hull_subset_span by auto
  2427   moreover have "span (insert 0 S - {x}) = span (S - {x})"
  2428     using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
  2429   ultimately have "x \<in> span (S - {x})" by auto
  2430   then have "x \<noteq> 0 \<Longrightarrow> dependent S"
  2431     using x dependent_def by auto
  2432   moreover
  2433   {
  2434     assume "x = 0"
  2435     then have "0 \<in> affine hull S"
  2436       using x hull_mono[of "S - {0}" S] by auto
  2437     then have "dependent S"
  2438       using affine_hull_0_dependent by auto
  2439   }
  2440   ultimately show ?thesis by auto
  2441 qed
  2442 
  2443 lemma affine_dependent_iff_dependent:
  2444   assumes "a \<notin> S"
  2445   shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
  2446 proof -
  2447   have "(op + (- a) ` S) = {x - a| x . x : S}" by auto
  2448   then show ?thesis
  2449     using affine_dependent_translation_eq[of "(insert a S)" "-a"]
  2450       affine_dependent_imp_dependent2 assms
  2451       dependent_imp_affine_dependent[of a S]
  2452     by (auto simp del: uminus_add_conv_diff)
  2453 qed
  2454 
  2455 lemma affine_dependent_iff_dependent2:
  2456   assumes "a \<in> S"
  2457   shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
  2458 proof -
  2459   have "insert a (S - {a}) = S"
  2460     using assms by auto
  2461   then show ?thesis
  2462     using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
  2463 qed
  2464 
  2465 lemma affine_hull_insert_span_gen:
  2466   "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
  2467 proof -
  2468   have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
  2469     by auto
  2470   {
  2471     assume "a \<notin> s"
  2472     then have ?thesis
  2473       using affine_hull_insert_span[of a s] h1 by auto
  2474   }
  2475   moreover
  2476   {
  2477     assume a1: "a \<in> s"
  2478     have "\<exists>x. x \<in> s \<and> -a+x=0"
  2479       apply (rule exI[of _ a])
  2480       using a1
  2481       apply auto
  2482       done
  2483     then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
  2484       by auto
  2485     then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
  2486       using span_insert_0[of "op + (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
  2487     moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
  2488       by auto
  2489     moreover have "insert a (s - {a}) = insert a s"
  2490       using assms by auto
  2491     ultimately have ?thesis
  2492       using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
  2493   }
  2494   ultimately show ?thesis by auto
  2495 qed
  2496 
  2497 lemma affine_hull_span2:
  2498   assumes "a \<in> s"
  2499   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
  2500   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
  2501   by auto
  2502 
  2503 lemma affine_hull_span_gen:
  2504   assumes "a \<in> affine hull s"
  2505   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
  2506 proof -
  2507   have "affine hull (insert a s) = affine hull s"
  2508     using hull_redundant[of a affine s] assms by auto
  2509   then show ?thesis
  2510     using affine_hull_insert_span_gen[of a "s"] by auto
  2511 qed
  2512 
  2513 lemma affine_hull_span_0:
  2514   assumes "0 \<in> affine hull S"
  2515   shows "affine hull S = span S"
  2516   using affine_hull_span_gen[of "0" S] assms by auto
  2517 
  2518 
  2519 lemma extend_to_affine_basis:
  2520   fixes S V :: "'n::euclidean_space set"
  2521   assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
  2522   shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
  2523 proof -
  2524   obtain a where a: "a \<in> S"
  2525     using assms by auto
  2526   then have h0: "independent  ((\<lambda>x. -a + x) ` (S-{a}))"
  2527     using affine_dependent_iff_dependent2 assms by auto
  2528   then obtain B where B:
  2529     "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
  2530      using maximal_independent_subset_extend[of "(\<lambda>x. -a+x) ` (S-{a})" "(\<lambda>x. -a + x) ` V"] assms
  2531      by blast
  2532   def T \<equiv> "(\<lambda>x. a+x) ` insert 0 B"
  2533   then have "T = insert a ((\<lambda>x. a+x) ` B)"
  2534     by auto
  2535   then have "affine hull T = (\<lambda>x. a+x) ` span B"
  2536     using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
  2537     by auto
  2538   then have "V \<subseteq> affine hull T"
  2539     using B assms translation_inverse_subset[of a V "span B"]
  2540     by auto
  2541   moreover have "T \<subseteq> V"
  2542     using T_def B a assms by auto
  2543   ultimately have "affine hull T = affine hull V"
  2544     by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
  2545   moreover have "S \<subseteq> T"
  2546     using T_def B translation_inverse_subset[of a "S-{a}" B]
  2547     by auto
  2548   moreover have "\<not> affine_dependent T"
  2549     using T_def affine_dependent_translation_eq[of "insert 0 B"]
  2550       affine_dependent_imp_dependent2 B
  2551     by auto
  2552   ultimately show ?thesis using `T \<subseteq> V` by auto
  2553 qed
  2554 
  2555 lemma affine_basis_exists:
  2556   fixes V :: "'n::euclidean_space set"
  2557   shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
  2558 proof (cases "V = {}")
  2559   case True
  2560   then show ?thesis
  2561     using affine_independent_empty by auto
  2562 next
  2563   case False
  2564   then obtain x where "x \<in> V" by auto
  2565   then show ?thesis
  2566     using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}" V]
  2567     by auto
  2568 qed
  2569 
  2570 
  2571 subsection {* Affine Dimension of a Set *}
  2572 
  2573 definition "aff_dim V =
  2574   (SOME d :: int.
  2575     \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
  2576 
  2577 lemma aff_dim_basis_exists:
  2578   fixes V :: "('n::euclidean_space) set"
  2579   shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
  2580 proof -
  2581   obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
  2582     using affine_basis_exists[of V] by auto
  2583   then show ?thesis
  2584     unfolding aff_dim_def
  2585       some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
  2586     apply auto
  2587     apply (rule exI[of _ "int (card B) - (1 :: int)"])
  2588     apply (rule exI[of _ "B"])
  2589     apply auto
  2590     done
  2591 qed
  2592 
  2593 lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
  2594 proof -
  2595   have "S = {} \<Longrightarrow> affine hull S = {}"
  2596     using affine_hull_empty by auto
  2597   moreover have "affine hull S = {} \<Longrightarrow> S = {}"
  2598     unfolding hull_def by auto
  2599   ultimately show ?thesis by blast
  2600 qed
  2601 
  2602 lemma aff_dim_parallel_subspace_aux:
  2603   fixes B :: "'n::euclidean_space set"
  2604   assumes "\<not> affine_dependent B" "a \<in> B"
  2605   shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
  2606 proof -
  2607   have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
  2608     using affine_dependent_iff_dependent2 assms by auto
  2609   then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
  2610     "finite ((\<lambda>x. -a + x) ` (B - {a}))"
  2611     using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
  2612   show ?thesis
  2613   proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
  2614     case True
  2615     have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
  2616       using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
  2617     then have "B = {a}" using True by auto
  2618     then show ?thesis using assms fin by auto
  2619   next
  2620     case False
  2621     then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
  2622       using fin by auto
  2623     moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
  2624        apply (rule card_image)
  2625        using translate_inj_on
  2626        apply (auto simp del: uminus_add_conv_diff)
  2627        done
  2628     ultimately have "card (B-{a}) > 0" by auto
  2629     then have *: "finite (B - {a})"
  2630       using card_gt_0_iff[of "(B - {a})"] by auto
  2631     then have "card (B - {a}) = card B - 1"
  2632       using card_Diff_singleton assms by auto
  2633     with * show ?thesis using fin h1 by auto
  2634   qed
  2635 qed
  2636 
  2637 lemma aff_dim_parallel_subspace:
  2638   fixes V L :: "'n::euclidean_space set"
  2639   assumes "V \<noteq> {}"
  2640     and "subspace L"
  2641     and "affine_parallel (affine hull V) L"
  2642   shows "aff_dim V = int (dim L)"
  2643 proof -
  2644   obtain B where
  2645     B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
  2646     using aff_dim_basis_exists by auto
  2647   then have "B \<noteq> {}"
  2648     using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
  2649     by auto
  2650   then obtain a where a: "a \<in> B" by auto
  2651   def Lb \<equiv> "span ((\<lambda>x. -a+x) ` (B-{a}))"
  2652   moreover have "affine_parallel (affine hull B) Lb"
  2653     using Lb_def B assms affine_hull_span2[of a B] a
  2654       affine_parallel_commut[of "Lb" "(affine hull B)"]
  2655     unfolding affine_parallel_def
  2656     by auto
  2657   moreover have "subspace Lb"
  2658     using Lb_def subspace_span by auto
  2659   moreover have "affine hull B \<noteq> {}"
  2660     using assms B affine_hull_nonempty[of V] by auto
  2661   ultimately have "L = Lb"
  2662     using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
  2663     by auto
  2664   then have "dim L = dim Lb"
  2665     by auto
  2666   moreover have "card B - 1 = dim Lb" and "finite B"
  2667     using Lb_def aff_dim_parallel_subspace_aux a B by auto
  2668   ultimately show ?thesis
  2669     using B `B \<noteq> {}` card_gt_0_iff[of B] by auto
  2670 qed
  2671 
  2672 lemma aff_independent_finite:
  2673   fixes B :: "'n::euclidean_space set"
  2674   assumes "\<not> affine_dependent B"
  2675   shows "finite B"
  2676 proof -
  2677   {
  2678     assume "B \<noteq> {}"
  2679     then obtain a where "a \<in> B" by auto
  2680     then have ?thesis
  2681       using aff_dim_parallel_subspace_aux assms by auto
  2682   }
  2683   then show ?thesis by auto
  2684 qed
  2685 
  2686 lemma independent_finite:
  2687   fixes B :: "'n::euclidean_space set"
  2688   assumes "independent B"
  2689   shows "finite B"
  2690   using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms
  2691   by auto
  2692 
  2693 lemma subspace_dim_equal:
  2694   assumes "subspace (S :: ('n::euclidean_space) set)"
  2695     and "subspace T"
  2696     and "S \<subseteq> T"
  2697     and "dim S \<ge> dim T"
  2698   shows "S = T"
  2699 proof -
  2700   obtain B where B: "B \<le> S" "independent B \<and> S \<subseteq> span B" "card B = dim S"
  2701     using basis_exists[of S] by auto
  2702   then have "span B \<subseteq> S"
  2703     using span_mono[of B S] span_eq[of S] assms by metis
  2704   then have "span B = S"
  2705     using B by auto
  2706   have "dim S = dim T"
  2707     using assms dim_subset[of S T] by auto
  2708   then have "T \<subseteq> span B"
  2709     using card_eq_dim[of B T] B independent_finite assms by auto
  2710   then show ?thesis
  2711     using assms `span B = S` by auto
  2712 qed
  2713 
  2714 lemma span_substd_basis:
  2715   assumes d: "d \<subseteq> Basis"
  2716   shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
  2717   (is "_ = ?B")
  2718 proof -
  2719   have "d \<subseteq> ?B"
  2720     using d by (auto simp: inner_Basis)
  2721   moreover have s: "subspace ?B"
  2722     using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
  2723   ultimately have "span d \<subseteq> ?B"
  2724     using span_mono[of d "?B"] span_eq[of "?B"] by blast
  2725   moreover have *: "card d \<le> dim (span d)"
  2726     using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms] span_inc[of d]
  2727     by auto
  2728   moreover from * have "dim ?B \<le> dim (span d)"
  2729     using dim_substandard[OF assms] by auto
  2730   ultimately show ?thesis
  2731     using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
  2732 qed
  2733 
  2734 lemma basis_to_substdbasis_subspace_isomorphism:
  2735   fixes B :: "'a::euclidean_space set"
  2736   assumes "independent B"
  2737   shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
  2738     f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
  2739 proof -
  2740   have B: "card B = dim B"
  2741     using dim_unique[of B B "card B"] assms span_inc[of B] by auto
  2742   have "dim B \<le> card (Basis :: 'a set)"
  2743     using dim_subset_UNIV[of B] by simp
  2744   from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
  2745     by auto
  2746   let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
  2747   have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
  2748     apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "d"])
  2749     apply (rule subspace_span)
  2750     apply (rule subspace_substandard)
  2751     defer
  2752     apply (rule span_inc)
  2753     apply (rule assms)
  2754     defer
  2755     unfolding dim_span[of B]
  2756     apply(rule B)
  2757     unfolding span_substd_basis[OF d, symmetric]
  2758     apply (rule span_inc)
  2759     apply (rule independent_substdbasis[OF d])
  2760     apply rule
  2761     apply assumption
  2762     unfolding t[symmetric] span_substd_basis[OF d] dim_substandard[OF d]
  2763     apply auto
  2764     done
  2765   with t `card B = dim B` d show ?thesis by auto
  2766 qed
  2767 
  2768 lemma aff_dim_empty:
  2769   fixes S :: "'n::euclidean_space set"
  2770   shows "S = {} \<longleftrightarrow> aff_dim S = -1"
  2771 proof -
  2772   obtain B where *: "affine hull B = affine hull S"
  2773     and "\<not> affine_dependent B"
  2774     and "int (card B) = aff_dim S + 1"
  2775     using aff_dim_basis_exists by auto
  2776   moreover
  2777   from * have "S = {} \<longleftrightarrow> B = {}"
  2778     using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
  2779   ultimately show ?thesis
  2780     using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
  2781 qed
  2782 
  2783 lemma aff_dim_affine_hull: "aff_dim (affine hull S) = aff_dim S"
  2784   unfolding aff_dim_def using hull_hull[of _ S] by auto
  2785 
  2786 lemma aff_dim_affine_hull2:
  2787   assumes "affine hull S = affine hull T"
  2788   shows "aff_dim S = aff_dim T"
  2789   unfolding aff_dim_def using assms by auto
  2790 
  2791 lemma aff_dim_unique:
  2792   fixes B V :: "'n::euclidean_space set"
  2793   assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
  2794   shows "of_nat (card B) = aff_dim V + 1"
  2795 proof (cases "B = {}")
  2796   case True
  2797   then have "V = {}"
  2798     using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
  2799     by auto
  2800   then have "aff_dim V = (-1::int)"
  2801     using aff_dim_empty by auto
  2802   then show ?thesis
  2803     using `B = {}` by auto
  2804 next
  2805   case False
  2806   then obtain a where a: "a \<in> B" by auto
  2807   def Lb \<equiv> "span ((\<lambda>x. -a+x) ` (B-{a}))"
  2808   have "affine_parallel (affine hull B) Lb"
  2809     using Lb_def affine_hull_span2[of a B] a
  2810       affine_parallel_commut[of "Lb" "(affine hull B)"]
  2811     unfolding affine_parallel_def by auto
  2812   moreover have "subspace Lb"
  2813     using Lb_def subspace_span by auto
  2814   ultimately have "aff_dim B = int(dim Lb)"
  2815     using aff_dim_parallel_subspace[of B Lb] `B \<noteq> {}` by auto
  2816   moreover have "(card B) - 1 = dim Lb" "finite B"
  2817     using Lb_def aff_dim_parallel_subspace_aux a assms by auto
  2818   ultimately have "of_nat (card B) = aff_dim B + 1"
  2819     using `B \<noteq> {}` card_gt_0_iff[of B] by auto
  2820   then show ?thesis
  2821     using aff_dim_affine_hull2 assms by auto
  2822 qed
  2823 
  2824 lemma aff_dim_affine_independent:
  2825   fixes B :: "'n::euclidean_space set"
  2826   assumes "\<not> affine_dependent B"
  2827   shows "of_nat (card B) = aff_dim B + 1"
  2828   using aff_dim_unique[of B B] assms by auto
  2829 
  2830 lemma aff_dim_sing:
  2831   fixes a :: "'n::euclidean_space"
  2832   shows "aff_dim {a} = 0"
  2833   using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
  2834 
  2835 lemma aff_dim_inner_basis_exists:
  2836   fixes V :: "('n::euclidean_space) set"
  2837   shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
  2838     \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
  2839 proof -
  2840   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
  2841     using affine_basis_exists[of V] by auto
  2842   then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
  2843   with B show ?thesis by auto
  2844 qed
  2845 
  2846 lemma aff_dim_le_card:
  2847   fixes V :: "'n::euclidean_space set"
  2848   assumes "finite V"
  2849   shows "aff_dim V \<le> of_nat (card V) - 1"
  2850 proof -
  2851   obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
  2852     using aff_dim_inner_basis_exists[of V] by auto
  2853   then have "card B \<le> card V"
  2854     using assms card_mono by auto
  2855   with B show ?thesis by auto
  2856 qed
  2857 
  2858 lemma aff_dim_parallel_eq:
  2859   fixes S T :: "'n::euclidean_space set"
  2860   assumes "affine_parallel (affine hull S) (affine hull T)"
  2861   shows "aff_dim S = aff_dim T"
  2862 proof -
  2863   {
  2864     assume "T \<noteq> {}" "S \<noteq> {}"
  2865     then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
  2866       using affine_parallel_subspace[of "affine hull T"]
  2867         affine_affine_hull[of T] affine_hull_nonempty
  2868       by auto
  2869     then have "aff_dim T = int (dim L)"
  2870       using aff_dim_parallel_subspace `T \<noteq> {}` by auto
  2871     moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
  2872        using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
  2873     moreover from * have "aff_dim S = int (dim L)"
  2874       using aff_dim_parallel_subspace `S \<noteq> {}` by auto
  2875     ultimately have ?thesis by auto
  2876   }
  2877   moreover
  2878   {
  2879     assume "S = {}"
  2880     then have "S = {}" and "T = {}"
  2881       using assms affine_hull_nonempty
  2882       unfolding affine_parallel_def
  2883       by auto
  2884     then have ?thesis using aff_dim_empty by auto
  2885   }
  2886   moreover
  2887   {
  2888     assume "T = {}"
  2889     then have "S = {}" and "T = {}"
  2890       using assms affine_hull_nonempty
  2891       unfolding affine_parallel_def
  2892       by auto
  2893     then have ?thesis
  2894       using aff_dim_empty by auto
  2895   }
  2896   ultimately show ?thesis by blast
  2897 qed
  2898 
  2899 lemma aff_dim_translation_eq:
  2900   fixes a :: "'n::euclidean_space"
  2901   shows "aff_dim ((\<lambda>x. a + x) ` S) = aff_dim S"
  2902 proof -
  2903   have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
  2904     unfolding affine_parallel_def
  2905     apply (rule exI[of _ "a"])
  2906     using affine_hull_translation[of a S]
  2907     apply auto
  2908     done
  2909   then show ?thesis
  2910     using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
  2911 qed
  2912 
  2913 lemma aff_dim_affine:
  2914   fixes S L :: "'n::euclidean_space set"
  2915   assumes "S \<noteq> {}"
  2916     and "affine S"
  2917     and "subspace L"
  2918     and "affine_parallel S L"
  2919   shows "aff_dim S = int (dim L)"
  2920 proof -
  2921   have *: "affine hull S = S"
  2922     using assms affine_hull_eq[of S] by auto
  2923   then have "affine_parallel (affine hull S) L"
  2924     using assms by (simp add: *)
  2925   then show ?thesis
  2926     using assms aff_dim_parallel_subspace[of S L] by blast
  2927 qed
  2928 
  2929 lemma dim_affine_hull:
  2930   fixes S :: "'n::euclidean_space set"
  2931   shows "dim (affine hull S) = dim S"
  2932 proof -
  2933   have "dim (affine hull S) \<ge> dim S"
  2934     using dim_subset by auto
  2935   moreover have "dim (span S) \<ge> dim (affine hull S)"
  2936     using dim_subset affine_hull_subset_span by auto
  2937   moreover have "dim (span S) = dim S"
  2938     using dim_span by auto
  2939   ultimately show ?thesis by auto
  2940 qed
  2941 
  2942 lemma aff_dim_subspace:
  2943   fixes S :: "'n::euclidean_space set"
  2944   assumes "S \<noteq> {}"
  2945     and "subspace S"
  2946   shows "aff_dim S = int (dim S)"
  2947   using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S]
  2948   by auto
  2949 
  2950 lemma aff_dim_zero:
  2951   fixes S :: "'n::euclidean_space set"
  2952   assumes "0 \<in> affine hull S"
  2953   shows "aff_dim S = int (dim S)"
  2954 proof -
  2955   have "subspace (affine hull S)"
  2956     using subspace_affine[of "affine hull S"] affine_affine_hull assms
  2957     by auto
  2958   then have "aff_dim (affine hull S) = int (dim (affine hull S))"
  2959     using assms aff_dim_subspace[of "affine hull S"] by auto
  2960   then show ?thesis
  2961     using aff_dim_affine_hull[of S] dim_affine_hull[of S]
  2962     by auto
  2963 qed
  2964 
  2965 lemma aff_dim_univ: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
  2966   using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
  2967     dim_UNIV[where 'a="'n::euclidean_space"]
  2968   by auto
  2969 
  2970 lemma aff_dim_geq:
  2971   fixes V :: "'n::euclidean_space set"
  2972   shows "aff_dim V \<ge> -1"
  2973 proof -
  2974   obtain B where "affine hull B = affine hull V"
  2975     and "\<not> affine_dependent B"
  2976     and "int (card B) = aff_dim V + 1"
  2977     using aff_dim_basis_exists by auto
  2978   then show ?thesis by auto
  2979 qed
  2980 
  2981 lemma independent_card_le_aff_dim:
  2982   fixes B :: "'n::euclidean_space set"
  2983   assumes "B \<subseteq> V"
  2984   assumes "\<not> affine_dependent B"
  2985   shows "int (card B) \<le> aff_dim V + 1"
  2986 proof (cases "B = {}")
  2987   case True
  2988   then have "-1 \<le> aff_dim V"
  2989     using aff_dim_geq by auto
  2990   with True show ?thesis by auto
  2991 next
  2992   case False
  2993   then obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
  2994     using assms extend_to_affine_basis[of B V] by auto
  2995   then have "of_nat (card T) = aff_dim V + 1"
  2996     using aff_dim_unique by auto
  2997   then show ?thesis
  2998     using T card_mono[of T B] aff_independent_finite[of T] by auto
  2999 qed
  3000 
  3001 lemma aff_dim_subset:
  3002   fixes S T :: "'n::euclidean_space set"
  3003   assumes "S \<subseteq> T"
  3004   shows "aff_dim S \<le> aff_dim T"
  3005 proof -
  3006   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
  3007     "of_nat (card B) = aff_dim S + 1"
  3008     using aff_dim_inner_basis_exists[of S] by auto
  3009   then have "int (card B) \<le> aff_dim T + 1"
  3010     using assms independent_card_le_aff_dim[of B T] by auto
  3011   with B show ?thesis by auto
  3012 qed
  3013 
  3014 lemma aff_dim_subset_univ:
  3015   fixes S :: "'n::euclidean_space set"
  3016   shows "aff_dim S \<le> int (DIM('n))"
  3017 proof -
  3018   have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
  3019     using aff_dim_univ by auto
  3020   then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
  3021     using assms aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
  3022 qed
  3023 
  3024 lemma affine_dim_equal:
  3025   fixes S :: "'n::euclidean_space set"
  3026   assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
  3027   shows "S = T"
  3028 proof -
  3029   obtain a where "a \<in> S" using assms by auto
  3030   then have "a \<in> T" using assms by auto
  3031   def LS \<equiv> "{y. \<exists>x \<in> S. (-a) + x = y}"
  3032   then have ls: "subspace LS" "affine_parallel S LS"
  3033     using assms parallel_subspace_explicit[of S a LS] `a \<in> S` by auto
  3034   then have h1: "int(dim LS) = aff_dim S"
  3035     using assms aff_dim_affine[of S LS] by auto
  3036   have "T \<noteq> {}" using assms by auto
  3037   def LT \<equiv> "{y. \<exists>x \<in> T. (-a) + x = y}"
  3038   then have lt: "subspace LT \<and> affine_parallel T LT"
  3039     using assms parallel_subspace_explicit[of T a LT] `a \<in> T` by auto
  3040   then have "int(dim LT) = aff_dim T"
  3041     using assms aff_dim_affine[of T LT] `T \<noteq> {}` by auto
  3042   then have "dim LS = dim LT"
  3043     using h1 assms by auto
  3044   moreover have "LS \<le> LT"
  3045     using LS_def LT_def assms by auto
  3046   ultimately have "LS = LT"
  3047     using subspace_dim_equal[of LS LT] ls lt by auto
  3048   moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
  3049     using LS_def by auto
  3050   moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
  3051     using LT_def by auto
  3052   ultimately show ?thesis by auto
  3053 qed
  3054 
  3055 lemma affine_hull_univ:
  3056   fixes S :: "'n::euclidean_space set"
  3057   assumes "aff_dim S = int(DIM('n))"
  3058   shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
  3059 proof -
  3060   have "S \<noteq> {}"
  3061     using assms aff_dim_empty[of S] by auto
  3062   have h0: "S \<subseteq> affine hull S"
  3063     using hull_subset[of S _] by auto
  3064   have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
  3065     using aff_dim_univ assms by auto
  3066   then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
  3067     using aff_dim_subset_univ[of "affine hull S"] assms h0 by auto
  3068   have h3: "aff_dim S \<le> aff_dim (affine hull S)"
  3069     using h0 aff_dim_subset[of S "affine hull S"] assms by auto
  3070   then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
  3071     using h0 h1 h2 by auto
  3072   then show ?thesis
  3073     using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
  3074       affine_affine_hull[of S] affine_UNIV assms h4 h0 `S \<noteq> {}`
  3075     by auto
  3076 qed
  3077 
  3078 lemma aff_dim_convex_hull:
  3079   fixes S :: "'n::euclidean_space set"
  3080   shows "aff_dim (convex hull S) = aff_dim S"
  3081   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
  3082     hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
  3083     aff_dim_subset[of "convex hull S" "affine hull S"]
  3084   by auto
  3085 
  3086 lemma aff_dim_cball:
  3087   fixes a :: "'n::euclidean_space"
  3088   assumes "e > 0"
  3089   shows "aff_dim (cball a e) = int (DIM('n))"
  3090 proof -
  3091   have "(\<lambda>x. a + x) ` (cball 0 e) \<subseteq> cball a e"
  3092     unfolding cball_def dist_norm by auto
  3093   then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \<le> aff_dim (cball a e)"
  3094     using aff_dim_translation_eq[of a "cball 0 e"]
  3095           aff_dim_subset[of "op + a ` cball 0 e" "cball a e"]
  3096     by auto
  3097   moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
  3098     using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
  3099       centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
  3100     by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
  3101   ultimately show ?thesis
  3102     using aff_dim_subset_univ[of "cball a e"] by auto
  3103 qed
  3104 
  3105 lemma aff_dim_open:
  3106   fixes S :: "'n::euclidean_space set"
  3107   assumes "open S"
  3108     and "S \<noteq> {}"
  3109   shows "aff_dim S = int (DIM('n))"
  3110 proof -
  3111   obtain x where "x \<in> S"
  3112     using assms by auto
  3113   then obtain e where e: "e > 0" "cball x e \<subseteq> S"
  3114     using open_contains_cball[of S] assms by auto
  3115   then have "aff_dim (cball x e) \<le> aff_dim S"
  3116     using aff_dim_subset by auto
  3117   with e show ?thesis
  3118     using aff_dim_cball[of e x] aff_dim_subset_univ[of S] by auto
  3119 qed
  3120 
  3121 lemma low_dim_interior:
  3122   fixes S :: "'n::euclidean_space set"
  3123   assumes "\<not> aff_dim S = int (DIM('n))"
  3124   shows "interior S = {}"
  3125 proof -
  3126   have "aff_dim(interior S) \<le> aff_dim S"
  3127     using interior_subset aff_dim_subset[of "interior S" S] by auto
  3128   then show ?thesis
  3129     using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto
  3130 qed
  3131 
  3132 subsection {* Relative interior of a set *}
  3133 
  3134 definition "rel_interior S =
  3135   {x. \<exists>T. openin (subtopology euclidean (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
  3136 
  3137 lemma rel_interior:
  3138   "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
  3139   unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
  3140   apply auto
  3141 proof -
  3142   fix x T
  3143   assume *: "x \<in> S" "open T" "x \<in> T" "T \<inter> affine hull S \<subseteq> S"
  3144   then have **: "x \<in> T \<inter> affine hull S"
  3145     using hull_inc by auto
  3146   show "\<exists>Tb. (\<exists>Ta. open Ta \<and> Tb = affine hull S \<inter> Ta) \<and> x \<in> Tb \<and> Tb \<subseteq> S"
  3147     apply (rule_tac x = "T \<inter> (affine hull S)" in exI)
  3148     using * **
  3149     apply auto
  3150     done
  3151 qed
  3152 
  3153 lemma mem_rel_interior: "x \<in> rel_interior S \<longleftrightarrow> (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
  3154   by (auto simp add: rel_interior)
  3155 
  3156 lemma mem_rel_interior_ball:
  3157   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S)"
  3158   apply (simp add: rel_interior, safe)
  3159   apply (force simp add: open_contains_ball)
  3160   apply (rule_tac x = "ball x e" in exI)
  3161   apply simp
  3162   done
  3163 
  3164 lemma rel_interior_ball:
  3165   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S}"
  3166   using mem_rel_interior_ball [of _ S] by auto
  3167 
  3168 lemma mem_rel_interior_cball:
  3169   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S)"
  3170   apply (simp add: rel_interior, safe)
  3171   apply (force simp add: open_contains_cball)
  3172   apply (rule_tac x = "ball x e" in exI)
  3173   apply (simp add: subset_trans [OF ball_subset_cball])
  3174   apply auto
  3175   done
  3176 
  3177 lemma rel_interior_cball:
  3178   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S}"
  3179   using mem_rel_interior_cball [of _ S] by auto
  3180 
  3181 lemma rel_interior_empty: "rel_interior {} = {}"
  3182    by (auto simp add: rel_interior_def)
  3183 
  3184 lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"
  3185   by (metis affine_hull_eq affine_sing)
  3186 
  3187 lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"
  3188   unfolding rel_interior_ball affine_hull_sing
  3189   apply auto
  3190   apply (rule_tac x = "1 :: real" in exI)
  3191   apply simp
  3192   done
  3193 
  3194 lemma subset_rel_interior:
  3195   fixes S T :: "'n::euclidean_space set"
  3196   assumes "S \<subseteq> T"
  3197     and "affine hull S = affine hull T"
  3198   shows "rel_interior S \<subseteq> rel_interior T"
  3199   using assms by (auto simp add: rel_interior_def)
  3200 
  3201 lemma rel_interior_subset: "rel_interior S \<subseteq> S"
  3202   by (auto simp add: rel_interior_def)
  3203 
  3204 lemma rel_interior_subset_closure: "rel_interior S \<subseteq> closure S"
  3205   using rel_interior_subset by (auto simp add: closure_def)
  3206 
  3207 lemma interior_subset_rel_interior: "interior S \<subseteq> rel_interior S"
  3208   by (auto simp add: rel_interior interior_def)
  3209 
  3210 lemma interior_rel_interior:
  3211   fixes S :: "'n::euclidean_space set"
  3212   assumes "aff_dim S = int(DIM('n))"
  3213   shows "rel_interior S = interior S"
  3214 proof -
  3215   have "affine hull S = UNIV"
  3216     using assms affine_hull_univ[of S] by auto
  3217   then show ?thesis
  3218     unfolding rel_interior interior_def by auto
  3219 qed
  3220 
  3221 lemma rel_interior_open:
  3222   fixes S :: "'n::euclidean_space set"
  3223   assumes "open S"
  3224   shows "rel_interior S = S"
  3225   by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
  3226 
  3227 lemma interior_rel_interior_gen:
  3228   fixes S :: "'n::euclidean_space set"
  3229   shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
  3230   by (metis interior_rel_interior low_dim_interior)
  3231 
  3232 lemma rel_interior_univ:
  3233   fixes S :: "'n::euclidean_space set"
  3234   shows "rel_interior (affine hull S) = affine hull S"
  3235 proof -
  3236   have *: "rel_interior (affine hull S) \<subseteq> affine hull S"
  3237     using rel_interior_subset by auto
  3238   {
  3239     fix x
  3240     assume x: "x \<in> affine hull S"
  3241     def e \<equiv> "1::real"
  3242     then have "e > 0" "ball x e \<inter> affine hull (affine hull S) \<subseteq> affine hull S"
  3243       using hull_hull[of _ S] by auto
  3244     then have "x \<in> rel_interior (affine hull S)"
  3245       using x rel_interior_ball[of "affine hull S"] by auto
  3246   }
  3247   then show ?thesis using * by auto
  3248 qed
  3249 
  3250 lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
  3251   by (metis open_UNIV rel_interior_open)
  3252 
  3253 lemma rel_interior_convex_shrink:
  3254   fixes S :: "'a::euclidean_space set"
  3255   assumes "convex S"
  3256     and "c \<in> rel_interior S"
  3257     and "x \<in> S"
  3258     and "0 < e"
  3259     and "e \<le> 1"
  3260   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
  3261 proof -
  3262   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
  3263     using assms(2) unfolding  mem_rel_interior_ball by auto
  3264   {
  3265     fix y
  3266     assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d" "y \<in> affine hull S"
  3267     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
  3268       using `e > 0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
  3269     have "x \<in> affine hull S"
  3270       using assms hull_subset[of S] by auto
  3271     moreover have "1 / e + - ((1 - e) / e) = 1"
  3272       using `e > 0` left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
  3273     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \<in> affine hull S"
  3274       using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
  3275       by (simp add: algebra_simps)
  3276     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)"
  3277       unfolding dist_norm norm_scaleR[symmetric]
  3278       apply (rule arg_cong[where f=norm])
  3279       using `e > 0`
  3280       apply (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
  3281       done
  3282     also have "\<dots> = abs (1/e) * norm (x - e *\<^sub>R (x - c) - y)"
  3283       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
  3284     also have "\<dots> < d"
  3285       using as[unfolded dist_norm] and `e > 0`
  3286       by (auto simp add:pos_divide_less_eq[OF `e > 0`] mult.commute)
  3287     finally have "y \<in> S"
  3288       apply (subst *)
  3289       apply (rule assms(1)[unfolded convex_alt,rule_format])
  3290       apply (rule d[unfolded subset_eq,rule_format])
  3291       unfolding mem_ball
  3292       using assms(3-5) **
  3293       apply auto
  3294       done
  3295   }
  3296   then have "ball (x - e *\<^sub>R (x - c)) (e*d) \<inter> affine hull S \<subseteq> S"
  3297     by auto
  3298   moreover have "e * d > 0"
  3299     using `e > 0` `d > 0` by simp
  3300   moreover have c: "c \<in> S"
  3301     using assms rel_interior_subset by auto
  3302   moreover from c have "x - e *\<^sub>R (x - c) \<in> S"
  3303     using mem_convex[of S x c e]
  3304     apply (simp add: algebra_simps)
  3305     using assms
  3306     apply auto
  3307     done
  3308   ultimately show ?thesis
  3309     using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e > 0` by auto
  3310 qed
  3311 
  3312 lemma interior_real_semiline:
  3313   fixes a :: real
  3314   shows "interior {a..} = {a<..}"
  3315 proof -
  3316   {
  3317     fix y
  3318     assume "a < y"
  3319     then have "y \<in> interior {a..}"
  3320       apply (simp add: mem_interior)
  3321       apply (rule_tac x="(y-a)" in exI)
  3322       apply (auto simp add: dist_norm)
  3323       done
  3324   }
  3325   moreover
  3326   {
  3327     fix y
  3328     assume "y \<in> interior {a..}"
  3329     then obtain e where e: "e > 0" "cball y e \<subseteq> {a..}"
  3330       using mem_interior_cball[of y "{a..}"] by auto
  3331     moreover from e have "y - e \<in> cball y e"
  3332       by (auto simp add: cball_def dist_norm)
  3333     ultimately have "a \<le> y - e" by auto
  3334     then have "a < y" using e by auto
  3335   }
  3336   ultimately show ?thesis by auto
  3337 qed
  3338 
  3339 lemma rel_interior_real_box:
  3340   fixes a b :: real
  3341   assumes "a < b"
  3342   shows "rel_interior {a .. b} = {a <..< b}"
  3343 proof -
  3344   have "box a b \<noteq> {}"
  3345     using assms
  3346     unfolding set_eq_iff
  3347     by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
  3348   then show ?thesis
  3349     using interior_rel_interior_gen[of "cbox a b", symmetric]
  3350     by (simp split: split_if_asm del: box_real add: box_real[symmetric] interior_cbox)
  3351 qed
  3352 
  3353 lemma rel_interior_real_semiline:
  3354   fixes a :: real
  3355   shows "rel_interior {a..} = {a<..}"
  3356 proof -
  3357   have *: "{a<..} \<noteq> {}"
  3358     unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
  3359   then show ?thesis using interior_real_semiline interior_rel_interior_gen[of "{a..}"]
  3360     by (auto split: split_if_asm)
  3361 qed
  3362 
  3363 subsubsection {* Relative open sets *}
  3364 
  3365 definition "rel_open S \<longleftrightarrow> rel_interior S = S"
  3366 
  3367 lemma rel_open: "rel_open S \<longleftrightarrow> openin (subtopology euclidean (affine hull S)) S"
  3368   unfolding rel_open_def rel_interior_def
  3369   apply auto
  3370   using openin_subopen[of "subtopology euclidean (affine hull S)" S]
  3371   apply auto
  3372   done
  3373 
  3374 lemma opein_rel_interior: "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
  3375   apply (simp add: rel_interior_def)
  3376   apply (subst openin_subopen)
  3377   apply blast
  3378   done
  3379 
  3380 lemma affine_rel_open:
  3381   fixes S :: "'n::euclidean_space set"
  3382   assumes "affine S"
  3383   shows "rel_open S"
  3384   unfolding rel_open_def
  3385   using assms rel_interior_univ[of S] affine_hull_eq[of S]
  3386   by metis
  3387 
  3388 lemma affine_closed:
  3389   fixes S :: "'n::euclidean_space set"
  3390   assumes "affine S"
  3391   shows "closed S"
  3392 proof -
  3393   {
  3394     assume "S \<noteq> {}"
  3395     then obtain L where L: "subspace L" "affine_parallel S L"
  3396       using assms affine_parallel_subspace[of S] by auto
  3397     then obtain a where a: "S = (op + a ` L)"
  3398       using affine_parallel_def[of L S] affine_parallel_commut by auto
  3399     from L have "closed L" using closed_subspace by auto
  3400     then have "closed S"
  3401       using closed_translation a by auto
  3402   }
  3403   then show ?thesis by auto
  3404 qed
  3405 
  3406 lemma closure_affine_hull:
  3407   fixes S :: "'n::euclidean_space set"
  3408   shows "closure S \<subseteq> affine hull S"
  3409   by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
  3410 
  3411 lemma closure_same_affine_hull:
  3412   fixes S :: "'n::euclidean_space set"
  3413   shows "affine hull (closure S) = affine hull S"
  3414 proof -
  3415   have "affine hull (closure S) \<subseteq> affine hull S"
  3416     using hull_mono[of "closure S" "affine hull S" "affine"]
  3417       closure_affine_hull[of S] hull_hull[of "affine" S]
  3418     by auto
  3419   moreover have "affine hull (closure S) \<supseteq> affine hull S"
  3420     using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
  3421   ultimately show ?thesis by auto
  3422 qed
  3423 
  3424 lemma closure_aff_dim:
  3425   fixes S :: "'n::euclidean_space set"
  3426   shows "aff_dim (closure S) = aff_dim S"
  3427 proof -
  3428   have "aff_dim S \<le> aff_dim (closure S)"
  3429     using aff_dim_subset closure_subset by auto
  3430   moreover have "aff_dim (closure S) \<le> aff_dim (affine hull S)"
  3431     using aff_dim_subset closure_affine_hull by auto
  3432   moreover have "aff_dim (affine hull S) = aff_dim S"
  3433     using aff_dim_affine_hull by auto
  3434   ultimately show ?thesis by auto
  3435 qed
  3436 
  3437 lemma rel_interior_closure_convex_shrink:
  3438   fixes S :: "_::euclidean_space set"
  3439   assumes "convex S"
  3440     and "c \<in> rel_interior S"
  3441     and "x \<in> closure S"
  3442     and "e > 0"
  3443     and "e \<le> 1"
  3444   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
  3445 proof -
  3446   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
  3447     using assms(2) unfolding mem_rel_interior_ball by auto
  3448   have "\<exists>y \<in> S. norm (y - x) * (1 - e) < e * d"
  3449   proof (cases "x \<in> S")
  3450     case True
  3451     then show ?thesis using `e > 0` `d > 0`
  3452       apply (rule_tac bexI[where x=x])
  3453       apply (auto)
  3454       done
  3455   next
  3456     case False
  3457     then have x: "x islimpt S"
  3458       using assms(3)[unfolded closure_def] by auto
  3459     show ?thesis
  3460     proof (cases "e = 1")
  3461       case True
  3462       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
  3463         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  3464       then show ?thesis
  3465         apply (rule_tac x=y in bexI)
  3466         unfolding True
  3467         using `d > 0`
  3468         apply auto
  3469         done
  3470     next
  3471       case False
  3472       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
  3473         using `e \<le> 1` `e > 0` `d > 0` by (auto)
  3474       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
  3475         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  3476       then show ?thesis
  3477         apply (rule_tac x=y in bexI)
  3478         unfolding dist_norm
  3479         using pos_less_divide_eq[OF *]
  3480         apply auto
  3481         done
  3482     qed
  3483   qed
  3484   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
  3485     by auto
  3486   def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
  3487   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
  3488     unfolding z_def using `e > 0`
  3489     by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
  3490   have zball: "z \<in> ball c d"
  3491     using mem_ball z_def dist_norm[of c]
  3492     using y and assms(4,5)
  3493     by (auto simp add:field_simps norm_minus_commute)
  3494   have "x \<in> affine hull S"
  3495     using closure_affine_hull assms by auto
  3496   moreover have "y \<in> affine hull S"
  3497     using `y \<in> S` hull_subset[of S] by auto
  3498   moreover have "c \<in> affine hull S"
  3499     using assms rel_interior_subset hull_subset[of S] by auto
  3500   ultimately have "z \<in> affine hull S"
  3501     using z_def affine_affine_hull[of S]
  3502       mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
  3503       assms
  3504     by (auto simp add: field_simps)
  3505   then have "z \<in> S" using d zball by auto
  3506   obtain d1 where "d1 > 0" and d1: "ball z d1 \<le> ball c d"
  3507     using zball open_ball[of c d] openE[of "ball c d" z] by auto
  3508   then have "ball z d1 \<inter> affine hull S \<subseteq> ball c d \<inter> affine hull S"
  3509     by auto
  3510   then have "ball z d1 \<inter> affine hull S \<subseteq> S"
  3511     using d by auto
  3512   then have "z \<in> rel_interior S"
  3513     using mem_rel_interior_ball using `d1 > 0` `z \<in> S` by auto
  3514   then have "y - e *\<^sub>R (y - z) \<in> rel_interior S"
  3515     using rel_interior_convex_shrink[of S z y e] assms `y \<in> S` by auto
  3516   then show ?thesis using * by auto
  3517 qed
  3518 
  3519 
  3520 subsubsection{* Relative interior preserves under linear transformations *}
  3521 
  3522 lemma rel_interior_translation_aux:
  3523   fixes a :: "'n::euclidean_space"
  3524   shows "((\<lambda>x. a + x) ` rel_interior S) \<subseteq> rel_interior ((\<lambda>x. a + x) ` S)"
  3525 proof -
  3526   {
  3527     fix x
  3528     assume x: "x \<in> rel_interior S"
  3529     then obtain T where "open T" "x \<in> T \<inter> S" "T \<inter> affine hull S \<subseteq> S"
  3530       using mem_rel_interior[of x S] by auto
  3531     then have "open ((\<lambda>x. a + x) ` T)"
  3532       and "a + x \<in> ((\<lambda>x. a + x) ` T) \<inter> ((\<lambda>x. a + x) ` S)"
  3533       and "((\<lambda>x. a + x) ` T) \<inter> affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` S"
  3534       using affine_hull_translation[of a S] open_translation[of T a] x by auto
  3535     then have "a + x \<in> rel_interior ((\<lambda>x. a + x) ` S)"
  3536       using mem_rel_interior[of "a+x" "((\<lambda>x. a + x) ` S)"] by auto
  3537   }
  3538   then show ?thesis by auto
  3539 qed
  3540 
  3541 lemma rel_interior_translation:
  3542   fixes a :: "'n::euclidean_space"
  3543   shows "rel_interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` rel_interior S"
  3544 proof -
  3545   have "(\<lambda>x. (-a) + x) ` rel_interior ((\<lambda>x. a + x) ` S) \<subseteq> rel_interior S"
  3546     using rel_interior_translation_aux[of "-a" "(\<lambda>x. a + x) ` S"]
  3547       translation_assoc[of "-a" "a"]
  3548     by auto
  3549   then have "((\<lambda>x. a + x) ` rel_interior S) \<supseteq> rel_interior ((\<lambda>x. a + x) ` S)"
  3550     using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"]
  3551     by auto
  3552   then show ?thesis
  3553     using rel_interior_translation_aux[of a S] by auto
  3554 qed
  3555 
  3556 
  3557 lemma affine_hull_linear_image:
  3558   assumes "bounded_linear f"
  3559   shows "f ` (affine hull s) = affine hull f ` s"
  3560   apply rule
  3561   unfolding subset_eq ball_simps
  3562   apply (rule_tac[!] hull_induct, rule hull_inc)
  3563   prefer 3
  3564   apply (erule imageE)
  3565   apply (rule_tac x=xa in image_eqI)
  3566   apply assumption
  3567   apply (rule hull_subset[unfolded subset_eq, rule_format])
  3568   apply assumption
  3569 proof -
  3570   interpret f: bounded_linear f by fact
  3571   show "affine {x. f x \<in> affine hull f ` s}"
  3572     unfolding affine_def
  3573     by (auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
  3574   show "affine {x. x \<in> f ` (affine hull s)}"
  3575     using affine_affine_hull[unfolded affine_def, of s]
  3576     unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  3577 qed auto
  3578 
  3579 
  3580 lemma rel_interior_injective_on_span_linear_image:
  3581   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
  3582     and S :: "'m::euclidean_space set"
  3583   assumes "bounded_linear f"
  3584     and "inj_on f (span S)"
  3585   shows "rel_interior (f ` S) = f ` (rel_interior S)"
  3586 proof -
  3587   {
  3588     fix z
  3589     assume z: "z \<in> rel_interior (f ` S)"
  3590     then have "z \<in> f ` S"
  3591       using rel_interior_subset[of "f ` S"] by auto
  3592     then obtain x where x: "x \<in> S" "f x = z" by auto
  3593     obtain e2 where e2: "e2 > 0" "cball z e2 \<inter> affine hull (f ` S) \<subseteq> (f ` S)"
  3594       using z rel_interior_cball[of "f ` S"] by auto
  3595     obtain K where K: "K > 0" "\<And>x. norm (f x) \<le> norm x * K"
  3596      using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
  3597     def e1 \<equiv> "1 / K"
  3598     then have e1: "e1 > 0" "\<And>x. e1 * norm (f x) \<le> norm x"
  3599       using K pos_le_divide_eq[of e1] by auto
  3600     def e \<equiv> "e1 * e2"
  3601     then have "e > 0" using e1 e2 by auto
  3602     {
  3603       fix y
  3604       assume y: "y \<in> cball x e \<inter> affine hull S"
  3605       then have h1: "f y \<in> affine hull (f ` S)"
  3606         using affine_hull_linear_image[of f S] assms by auto
  3607       from y have "norm (x-y) \<le> e1 * e2"
  3608         using cball_def[of x e] dist_norm[of x y] e_def by auto
  3609       moreover have "f x - f y = f (x - y)"
  3610         using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
  3611       moreover have "e1 * norm (f (x-y)) \<le> norm (x - y)"
  3612         using e1 by auto
  3613       ultimately have "e1 * norm ((f x)-(f y)) \<le> e1 * e2"
  3614         by auto
  3615       then have "f y \<in> cball z e2"
  3616         using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
  3617       then have "f y \<in> f ` S"
  3618         using y e2 h1 by auto
  3619       then have "y \<in> S"
  3620         using assms y hull_subset[of S] affine_hull_subset_span
  3621           inj_on_image_mem_iff[of f "span S" S y]
  3622         by auto
  3623     }
  3624     then have "z \<in> f ` (rel_interior S)"
  3625       using mem_rel_interior_cball[of x S] `e > 0` x by auto
  3626   }
  3627   moreover
  3628   {
  3629     fix x
  3630     assume x: "x \<in> rel_interior S"
  3631     then obtain e2 where e2: "e2 > 0" "cball x e2 \<inter> affine hull S \<subseteq> S"
  3632       using rel_interior_cball[of S] by auto
  3633     have "x \<in> S" using x rel_interior_subset by auto
  3634     then have *: "f x \<in> f ` S" by auto
  3635     have "\<forall>x\<in>span S. f x = 0 \<longrightarrow> x = 0"
  3636       using assms subspace_span linear_conv_bounded_linear[of f]
  3637         linear_injective_on_subspace_0[of f "span S"]
  3638       by auto
  3639     then obtain e1 where e1: "e1 > 0" "\<forall>x \<in> span S. e1 * norm x \<le> norm (f x)"
  3640       using assms injective_imp_isometric[of "span S" f]
  3641         subspace_span[of S] closed_subspace[of "span S"]
  3642       by auto
  3643     def e \<equiv> "e1 * e2"
  3644     hence "e > 0" using e1 e2 by auto
  3645     {
  3646       fix y
  3647       assume y: "y \<in> cball (f x) e \<inter> affine hull (f ` S)"
  3648       then have "y \<in> f ` (affine hull S)"
  3649         using affine_hull_linear_image[of f S] assms by auto
  3650       then obtain xy where xy: "xy \<in> affine hull S" "f xy = y" by auto
  3651       with y have "norm (f x - f xy) \<le> e1 * e2"
  3652         using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
  3653       moreover have "f x - f xy = f (x - xy)"
  3654         using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
  3655       moreover have *: "x - xy \<in> span S"
  3656         using subspace_sub[of "span S" x xy] subspace_span `x \<in> S` xy
  3657           affine_hull_subset_span[of S] span_inc
  3658         by auto
  3659       moreover from * have "e1 * norm (x - xy) \<le> norm (f (x - xy))"
  3660         using e1 by auto
  3661       ultimately have "e1 * norm (x - xy) \<le> e1 * e2"
  3662         by auto
  3663       then have "xy \<in> cball x e2"
  3664         using cball_def[of x e2] dist_norm[of x xy] e1 by auto
  3665       then have "y \<in> f ` S"
  3666         using xy e2 by auto
  3667     }
  3668     then have "f x \<in> rel_interior (f ` S)"
  3669       using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e > 0` by auto
  3670   }
  3671   ultimately show ?thesis by auto
  3672 qed
  3673 
  3674 lemma rel_interior_injective_linear_image:
  3675   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
  3676   assumes "bounded_linear f"
  3677     and "inj f"
  3678   shows "rel_interior (f ` S) = f ` (rel_interior S)"
  3679   using assms rel_interior_injective_on_span_linear_image[of f S]
  3680     subset_inj_on[of f "UNIV" "span S"]
  3681   by auto
  3682 
  3683 
  3684 subsection{* Some Properties of subset of standard basis *}
  3685 
  3686 lemma affine_hull_substd_basis:
  3687   assumes "d \<subseteq> Basis"
  3688   shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
  3689   (is "affine hull (insert 0 ?A) = ?B")
  3690 proof -
  3691   have *: "\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A"
  3692     by auto
  3693   show ?thesis
  3694     unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
  3695 qed
  3696 
  3697 lemma affine_hull_convex_hull: "affine hull (convex hull S) = affine hull S"
  3698   by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
  3699 
  3700 
  3701 subsection {* Openness and compactness are preserved by convex hull operation. *}
  3702 
  3703 lemma open_convex_hull[intro]:
  3704   fixes s :: "'a::real_normed_vector set"
  3705   assumes "open s"
  3706   shows "open (convex hull s)"
  3707   unfolding open_contains_cball convex_hull_explicit
  3708   unfolding mem_Collect_eq ball_simps(8)
  3709 proof (rule, rule)
  3710   fix a
  3711   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"
  3712   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"
  3713     by auto
  3714 
  3715   from assms[unfolded open_contains_cball] obtain b
  3716     where b: "\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
  3717     using bchoice[of s "\<lambda>x e. e > 0 \<and> cball x e \<subseteq> s"] by auto
  3718   have "b ` t \<noteq> {}"
  3719     using obt by auto
  3720   def i \<equiv> "b ` t"
  3721 
  3722   show "\<exists>e > 0.
  3723     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}"
  3724     apply (rule_tac x = "Min i" in exI)
  3725     unfolding subset_eq
  3726     apply rule
  3727     defer
  3728     apply rule
  3729     unfolding mem_Collect_eq
  3730   proof -
  3731     show "0 < Min i"
  3732       unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
  3733       using b
  3734       apply simp
  3735       apply rule
  3736       apply (erule_tac x=x in ballE)
  3737       using `t\<subseteq>s`
  3738       apply auto
  3739       done
  3740   next
  3741     fix y
  3742     assume "y \<in> cball a (Min i)"
  3743     then have y: "norm (a - y) \<le> Min i"
  3744       unfolding dist_norm[symmetric] by auto
  3745     {
  3746       fix x
  3747       assume "x \<in> t"
  3748       then have "Min i \<le> b x"
  3749         unfolding i_def
  3750         apply (rule_tac Min_le)
  3751         using obt(1)
  3752         apply auto
  3753         done
  3754       then have "x + (y - a) \<in> cball x (b x)"
  3755         using y unfolding mem_cball dist_norm by auto
  3756       moreover from `x\<in>t` have "x \<in> s"
  3757         using obt(2) by auto
  3758       ultimately have "x + (y - a) \<in> s"
  3759         using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast
  3760     }
  3761     moreover
  3762     have *: "inj_on (\<lambda>v. v + (y - a)) t"
  3763       unfolding inj_on_def by auto
  3764     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
  3765       unfolding setsum.reindex[OF *] o_def using obt(4) by auto
  3766     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
  3767       unfolding setsum.reindex[OF *] o_def using obt(4,5)
  3768       by (simp add: setsum.distrib setsum_subtractf scaleR_left.setsum[symmetric] scaleR_right_distrib)
  3769     ultimately
  3770     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"
  3771       apply (rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI)
  3772       apply (rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
  3773       using obt(1, 3)
  3774       apply auto
  3775       done
  3776   qed
  3777 qed
  3778 
  3779 lemma compact_convex_combinations:
  3780   fixes s t :: "'a::real_normed_vector set"
  3781   assumes "compact s" "compact t"
  3782   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}"
  3783 proof -
  3784   let ?X = "{0..1} \<times> s \<times> t"
  3785   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  3786   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"
  3787     apply (rule set_eqI)
  3788     unfolding image_iff mem_Collect_eq
  3789     apply rule
  3790     apply auto
  3791     apply (rule_tac x=u in rev_bexI)
  3792     apply simp
  3793     apply (erule rev_bexI)
  3794     apply (erule rev_bexI)
  3795     apply simp
  3796     apply auto
  3797     done
  3798   have "continuous_on ?X (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  3799     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  3800   then show ?thesis
  3801     unfolding *
  3802     apply (rule compact_continuous_image)
  3803     apply (intro compact_Times compact_Icc assms)
  3804     done
  3805 qed
  3806 
  3807 lemma finite_imp_compact_convex_hull:
  3808   fixes s :: "'a::real_normed_vector set"
  3809   assumes "finite s"
  3810   shows "compact (convex hull s)"
  3811 proof (cases "s = {}")
  3812   case True
  3813   then show ?thesis by simp
  3814 next
  3815   case False
  3816   with assms show ?thesis
  3817   proof (induct rule: finite_ne_induct)
  3818     case (singleton x)
  3819     show ?case by simp
  3820   next
  3821     case (insert x A)
  3822     let ?f = "\<lambda>(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
  3823     let ?T = "{0..1::real} \<times> (convex hull A)"
  3824     have "continuous_on ?T ?f"
  3825       unfolding split_def continuous_on by (intro ballI tendsto_intros)
  3826     moreover have "compact ?T"
  3827       by (intro compact_Times compact_Icc insert)
  3828     ultimately have "compact (?f ` ?T)"
  3829       by (rule compact_continuous_image)
  3830     also have "?f ` ?T = convex hull (insert x A)"
  3831       unfolding convex_hull_insert [OF `A \<noteq> {}`]
  3832       apply safe
  3833       apply (rule_tac x=a in exI, simp)
  3834       apply (rule_tac x="1 - a" in exI, simp)
  3835       apply fast
  3836       apply (rule_tac x="(u, b)" in image_eqI, simp_all)
  3837       done
  3838     finally show "compact (convex hull (insert x A))" .
  3839   qed
  3840 qed
  3841 
  3842 lemma compact_convex_hull:
  3843   fixes s :: "'a::euclidean_space set"
  3844   assumes "compact s"
  3845   shows "compact (convex hull s)"
  3846 proof (cases "s = {}")
  3847   case True
  3848   then show ?thesis using compact_empty by simp
  3849 next
  3850   case False
  3851   then obtain w where "w \<in> s" by auto
  3852   show ?thesis
  3853     unfolding caratheodory[of s]
  3854   proof (induct ("DIM('a) + 1"))
  3855     case 0
  3856     have *: "{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
  3857       using compact_empty by auto
  3858     from 0 show ?case unfolding * by simp
  3859   next
  3860     case (Suc n)
  3861     show ?case
  3862     proof (cases "n = 0")
  3863       case True
  3864       have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
  3865         unfolding set_eq_iff and mem_Collect_eq
  3866       proof (rule, rule)
  3867         fix x
  3868         assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  3869         then obtain t where t: "finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t"
  3870           by auto
  3871         show "x \<in> s"
  3872         proof (cases "card t = 0")
  3873           case True
  3874           then show ?thesis
  3875             using t(4) unfolding card_0_eq[OF t(1)] by simp
  3876         next
  3877           case False
  3878           then have "card t = Suc 0" using t(3) `n=0` by auto
  3879           then obtain a where "t = {a}" unfolding card_Suc_eq by auto
  3880           then show ?thesis using t(2,4) by simp
  3881         qed
  3882       next
  3883         fix x assume "x\<in>s"
  3884         then show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  3885           apply (rule_tac x="{x}" in exI)
  3886           unfolding convex_hull_singleton
  3887           apply auto
  3888           done
  3889       qed
  3890       then show ?thesis using assms by simp
  3891     next
  3892       case False
  3893       have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
  3894         {(1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
  3895           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}}"
  3896         unfolding set_eq_iff and mem_Collect_eq
  3897       proof (rule, rule)
  3898         fix x
  3899         assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  3900           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)"
  3901         then obtain u v c t where obt: "x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
  3902           "0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n"  "v \<in> convex hull t"
  3903           by auto
  3904         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
  3905           apply (rule mem_convex)
  3906           using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
  3907           using obt(7) and hull_mono[of t "insert u t"]
  3908           apply auto
  3909           done
  3910         ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  3911           apply (rule_tac x="insert u t" in exI)
  3912           apply (auto simp add: card_insert_if)
  3913           done
  3914       next
  3915         fix x
  3916         assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  3917         then obtain t where t: "finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t"
  3918           by auto
  3919         show "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  3920           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)"
  3921         proof (cases "card t = Suc n")
  3922           case False
  3923           then have "card t \<le> n" using t(3) by auto
  3924           then show ?thesis
  3925             apply (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI)
  3926             using `w\<in>s` and t
  3927             apply (auto intro!: exI[where x=t])
  3928             done
  3929         next
  3930           case True
  3931           then obtain a u where au: "t = insert a u" "a\<notin>u"
  3932             apply (drule_tac card_eq_SucD)
  3933             apply auto
  3934             done
  3935           show ?thesis
  3936           proof (cases "u = {}")
  3937             case True
  3938             then have "x = a" using t(4)[unfolded au] by auto
  3939             show ?thesis unfolding `x = a`
  3940               apply (rule_tac x=a in exI)
  3941               apply (rule_tac x=a in exI)
  3942               apply (rule_tac x=1 in exI)
  3943               using t and `n \<noteq> 0`
  3944               unfolding au
  3945               apply (auto intro!: exI[where x="{a}"])
  3946               done
  3947           next
  3948             case False
  3949             obtain ux vx b where obt: "ux\<ge>0" "vx\<ge>0" "ux + vx = 1"
  3950               "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
  3951               using t(4)[unfolded au convex_hull_insert[OF False]]
  3952               by auto
  3953             have *: "1 - vx = ux" using obt(3) by auto
  3954             show ?thesis
  3955               apply (rule_tac x=a in exI)
  3956               apply (rule_tac x=b in exI)
  3957               apply (rule_tac x=vx in exI)
  3958               using obt and t(1-3)
  3959               unfolding au and * using card_insert_disjoint[OF _ au(2)]
  3960               apply (auto intro!: exI[where x=u])
  3961               done
  3962           qed
  3963         qed
  3964       qed
  3965       then show ?thesis
  3966         using compact_convex_combinations[OF assms Suc] by simp
  3967     qed
  3968   qed
  3969 qed
  3970 
  3971 
  3972 subsection {* Extremal points of a simplex are some vertices. *}
  3973 
  3974 lemma dist_increases_online:
  3975   fixes a b d :: "'a::real_inner"
  3976   assumes "d \<noteq> 0"
  3977   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
  3978 proof (cases "inner a d - inner b d > 0")
  3979   case True
  3980   then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
  3981     apply (rule_tac add_pos_pos)
  3982     using assms
  3983     apply auto
  3984     done
  3985   then show ?thesis
  3986     apply (rule_tac disjI2)
  3987     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  3988     apply  (simp add: algebra_simps inner_commute)
  3989     done
  3990 next
  3991   case False
  3992   then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
  3993     apply (rule_tac add_pos_nonneg)
  3994     using assms
  3995     apply auto
  3996     done
  3997   then show ?thesis
  3998     apply (rule_tac disjI1)
  3999     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  4000     apply (simp add: algebra_simps inner_commute)
  4001     done
  4002 qed
  4003 
  4004 lemma norm_increases_online:
  4005   fixes d :: "'a::real_inner"
  4006   shows "d \<noteq> 0 \<Longrightarrow> norm (a + d) > norm a \<or> norm(a - d) > norm a"
  4007   using dist_increases_online[of d a 0] unfolding dist_norm by auto
  4008 
  4009 lemma simplex_furthest_lt:
  4010   fixes s :: "'a::real_inner set"
  4011   assumes "finite s"
  4012   shows "\<forall>x \<in> convex hull s.  x \<notin> s \<longrightarrow> (\<exists>y \<in> convex hull s. norm (x - a) < norm(y - a))"
  4013   using assms
  4014 proof induct
  4015   fix x s
  4016   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))"
  4017   show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow>
  4018     (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
  4019   proof (rule, rule, cases "s = {}")
  4020     case False
  4021     fix y
  4022     assume y: "y \<in> convex hull insert x s" "y \<notin> insert x s"
  4023     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"
  4024       using y(1)[unfolded convex_hull_insert[OF False]] by auto
  4025     show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
  4026     proof (cases "y \<in> convex hull s")
  4027       case True
  4028       then obtain z where "z \<in> convex hull s" "norm (y - a) < norm (z - a)"
  4029         using as(3)[THEN bspec[where x=y]] and y(2) by auto
  4030       then show ?thesis
  4031         apply (rule_tac x=z in bexI)
  4032         unfolding convex_hull_insert[OF False]
  4033         apply auto
  4034         done
  4035     next
  4036       case False
  4037       show ?thesis
  4038         using obt(3)
  4039       proof (cases "u = 0", case_tac[!] "v = 0")
  4040         assume "u = 0" "v \<noteq> 0"
  4041         then have "y = b" using obt by auto
  4042         then show ?thesis using False and obt(4) by auto
  4043       next
  4044         assume "u \<noteq> 0" "v = 0"
  4045         then have "y = x" using obt by auto
  4046         then show ?thesis using y(2) by auto
  4047       next
  4048         assume "u \<noteq> 0" "v \<noteq> 0"
  4049         then obtain w where w: "w>0" "w<u" "w<v"
  4050           using real_lbound_gt_zero[of u v] and obt(1,2) by auto
  4051         have "x \<noteq> b"
  4052         proof
  4053           assume "x = b"
  4054           then have "y = b" unfolding obt(5)
  4055             using obt(3) by (auto simp add: scaleR_left_distrib[symmetric])
  4056           then show False using obt(4) and False by simp
  4057         qed
  4058         then have *: "w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
  4059         show ?thesis
  4060           using dist_increases_online[OF *, of a y]
  4061         proof (elim disjE)
  4062           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
  4063           then have "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
  4064             unfolding dist_commute[of a]
  4065             unfolding dist_norm obt(5)
  4066             by (simp add: algebra_simps)
  4067           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
  4068             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  4069             apply (rule_tac x="u + w" in exI)
  4070             apply rule
  4071             defer
  4072             apply (rule_tac x="v - w" in exI)
  4073             using `u \<ge> 0` and w and obt(3,4)
  4074             apply auto
  4075             done
  4076           ultimately show ?thesis by auto
  4077         next
  4078           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
  4079           then have "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
  4080             unfolding dist_commute[of a]
  4081             unfolding dist_norm obt(5)
  4082             by (simp add: algebra_simps)
  4083           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
  4084             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  4085             apply (rule_tac x="u - w" in exI)
  4086             apply rule
  4087             defer
  4088             apply (rule_tac x="v + w" in exI)
  4089             using `u \<ge> 0` and w and obt(3,4)
  4090             apply auto
  4091             done
  4092           ultimately show ?thesis by auto
  4093         qed
  4094       qed auto
  4095     qed
  4096   qed auto
  4097 qed (auto simp add: assms)
  4098 
  4099 lemma simplex_furthest_le:
  4100   fixes s :: "'a::real_inner set"
  4101   assumes "finite s"
  4102     and "s \<noteq> {}"
  4103   shows "\<exists>y\<in>s. \<forall>x\<in> convex hull s. norm (x - a) \<le> norm (y - a)"
  4104 proof -
  4105   have "convex hull s \<noteq> {}"
  4106     using hull_subset[of s convex] and assms(2) by auto
  4107   then obtain x where x: "x \<in> convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
  4108     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
  4109     unfolding dist_commute[of a]
  4110     unfolding dist_norm
  4111     by auto
  4112   show ?thesis
  4113   proof (cases "x \<in> s")
  4114     case False
  4115     then obtain y where "y \<in> convex hull s" "norm (x - a) < norm (y - a)"
  4116       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
  4117       by auto
  4118     then show ?thesis
  4119       using x(2)[THEN bspec[where x=y]] by auto
  4120   next
  4121     case True
  4122     with x show ?thesis by auto
  4123   qed
  4124 qed
  4125 
  4126 lemma simplex_furthest_le_exists:
  4127   fixes s :: "('a::real_inner) set"
  4128   shows "finite s \<Longrightarrow> \<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm (x - a) \<le> norm (y - a)"
  4129   using simplex_furthest_le[of s] by (cases "s = {}") auto
  4130 
  4131 lemma simplex_extremal_le:
  4132   fixes s :: "'a::real_inner set"
  4133   assumes "finite s"
  4134     and "s \<noteq> {}"
  4135   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)"
  4136 proof -
  4137   have "convex hull s \<noteq> {}"
  4138     using hull_subset[of s convex] and assms(2) by auto
  4139   then obtain u v where obt: "u \<in> convex hull s" "v \<in> convex hull s"
  4140     "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
  4141     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
  4142     by (auto simp: dist_norm)
  4143   then show ?thesis
  4144   proof (cases "u\<notin>s \<or> v\<notin>s", elim disjE)
  4145     assume "u \<notin> s"
  4146     then obtain y where "y \<in> convex hull s" "norm (u - v) < norm (y - v)"
  4147       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
  4148       by auto
  4149     then show ?thesis
  4150       using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
  4151       by auto
  4152   next
  4153     assume "v \<notin> s"
  4154     then obtain y where "y \<in> convex hull s" "norm (v - u) < norm (y - u)"
  4155       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
  4156       by auto
  4157     then show ?thesis
  4158       using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
  4159       by (auto simp add: norm_minus_commute)
  4160   qed auto
  4161 qed
  4162 
  4163 lemma simplex_extremal_le_exists:
  4164   fixes s :: "'a::real_inner set"
  4165   shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s \<Longrightarrow>
  4166     \<exists>u\<in>s. \<exists>v\<in>s. norm (x - y) \<le> norm (u - v)"
  4167   using convex_hull_empty simplex_extremal_le[of s]
  4168   by(cases "s = {}") auto
  4169 
  4170 
  4171 subsection {* Closest point of a convex set is unique, with a continuous projection. *}
  4172 
  4173 definition closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a"
  4174   where "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
  4175 
  4176 lemma closest_point_exists:
  4177   assumes "closed s"
  4178     and "s \<noteq> {}"
  4179   shows "closest_point s a \<in> s"
  4180     and "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
  4181   unfolding closest_point_def
  4182   apply(rule_tac[!] someI2_ex)
  4183   using distance_attains_inf[OF assms(1,2), of a]
  4184   apply auto
  4185   done
  4186 
  4187 lemma closest_point_in_set: "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> closest_point s a \<in> s"
  4188   by (meson closest_point_exists)
  4189 
  4190 lemma closest_point_le: "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
  4191   using closest_point_exists[of s] by auto
  4192 
  4193 lemma closest_point_self:
  4194   assumes "x \<in> s"
  4195   shows "closest_point s x = x"
  4196   unfolding closest_point_def
  4197   apply (rule some1_equality, rule ex1I[of _ x])
  4198   using assms
  4199   apply auto
  4200   done
  4201 
  4202 lemma closest_point_refl: "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> closest_point s x = x \<longleftrightarrow> x \<in> s"
  4203   using closest_point_in_set[of s x] closest_point_self[of x s]
  4204   by auto
  4205 
  4206 lemma closer_points_lemma:
  4207   assumes "inner y z > 0"
  4208   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
  4209 proof -
  4210   have z: "inner z z > 0"
  4211     unfolding inner_gt_zero_iff using assms by auto
  4212   then show ?thesis
  4213     using assms
  4214     apply (rule_tac x = "inner y z / inner z z" in exI)
  4215     apply rule
  4216     defer
  4217   proof rule+
  4218     fix v
  4219     assume "0 < v" and "v \<le> inner y z / inner z z"
  4220     then show "norm (v *\<^sub>R z - y) < norm y"
  4221       unfolding norm_lt using z and assms
  4222       by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
  4223   qed auto
  4224 qed
  4225 
  4226 lemma closer_point_lemma:
  4227   assumes "inner (y - x) (z - x) > 0"
  4228   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
  4229 proof -
  4230   obtain u where "u > 0"
  4231     and u: "\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
  4232     using closer_points_lemma[OF assms] by auto
  4233   show ?thesis
  4234     apply (rule_tac x="min u 1" in exI)
  4235     using u[THEN spec[where x="min u 1"]] and `u > 0`
  4236     unfolding dist_norm by (auto simp add: norm_minus_commute field_simps)
  4237 qed
  4238 
  4239 lemma any_closest_point_dot:
  4240   assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  4241   shows "inner (a - x) (y - x) \<le> 0"
  4242 proof (rule ccontr)
  4243   assume "\<not> ?thesis"
  4244   then obtain u where u: "u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a"
  4245     using closer_point_lemma[of a x y] by auto
  4246   let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y"
  4247   have "?z \<in> s"
  4248     using mem_convex[OF assms(1,3,4), of u] using u by auto
  4249   then show False
  4250     using assms(5)[THEN bspec[where x="?z"]] and u(3)
  4251     by (auto simp add: dist_commute algebra_simps)
  4252 qed
  4253 
  4254 lemma any_closest_point_unique:
  4255   fixes x :: "'a::real_inner"
  4256   assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
  4257     "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
  4258   shows "x = y"
  4259   using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
  4260   unfolding norm_pths(1) and norm_le_square
  4261   by (auto simp add: algebra_simps)
  4262 
  4263 lemma closest_point_unique:
  4264   assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  4265   shows "x = closest_point s a"
  4266   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"]
  4267   using closest_point_exists[OF assms(2)] and assms(3) by auto
  4268 
  4269 lemma closest_point_dot:
  4270   assumes "convex s" "closed s" "x \<in> s"
  4271   shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
  4272   apply (rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
  4273   using closest_point_exists[OF assms(2)] and assms(3)
  4274   apply auto
  4275   done
  4276 
  4277 lemma closest_point_lt:
  4278   assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
  4279   shows "dist a (closest_point s a) < dist a x"
  4280   apply (rule ccontr)
  4281   apply (rule_tac notE[OF assms(4)])
  4282   apply (rule closest_point_unique[OF assms(1-3), of a])
  4283   using closest_point_le[OF assms(2), of _ a]
  4284   apply fastforce
  4285   done
  4286 
  4287 lemma closest_point_lipschitz:
  4288   assumes "convex s"
  4289     and "closed s" "s \<noteq> {}"
  4290   shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
  4291 proof -
  4292   have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
  4293     and "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
  4294     apply (rule_tac[!] any_closest_point_dot[OF assms(1-2)])
  4295     using closest_point_exists[OF assms(2-3)]
  4296     apply auto
  4297     done
  4298   then show ?thesis unfolding dist_norm and norm_le
  4299     using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
  4300     by (simp add: inner_add inner_diff inner_commute)
  4301 qed
  4302 
  4303 lemma continuous_at_closest_point:
  4304   assumes "convex s"
  4305     and "closed s"
  4306     and "s \<noteq> {}"
  4307   shows "continuous (at x) (closest_point s)"
  4308   unfolding continuous_at_eps_delta
  4309   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
  4310 
  4311 lemma continuous_on_closest_point:
  4312   assumes "convex s"
  4313     and "closed s"
  4314     and "s \<noteq> {}"
  4315   shows "continuous_on t (closest_point s)"
  4316   by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
  4317 
  4318 
  4319 subsubsection {* Various point-to-set separating/supporting hyperplane theorems. *}
  4320 
  4321 lemma supporting_hyperplane_closed_point:
  4322   fixes z :: "'a::{real_inner,heine_borel}"
  4323   assumes "convex s"
  4324     and "closed s"
  4325     and "s \<noteq> {}"
  4326     and "z \<notin> s"
  4327   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)"
  4328 proof -
  4329   from distance_attains_inf[OF assms(2-3)]
  4330   obtain y where "y \<in> s" and y: "\<forall>x\<in>s. dist z y \<le> dist z x"
  4331     by auto
  4332   show ?thesis
  4333     apply (rule_tac x="y - z" in exI)
  4334     apply (rule_tac x="inner (y - z) y" in exI)
  4335     apply (rule_tac x=y in bexI)
  4336     apply rule
  4337     defer
  4338     apply rule
  4339     defer
  4340     apply rule
  4341     apply (rule ccontr)
  4342     using `y \<in> s`
  4343   proof -
  4344     show "inner (y - z) z < inner (y - z) y"
  4345       apply (subst diff_less_iff(1)[symmetric])
  4346       unfolding inner_diff_right[symmetric] and inner_gt_zero_iff
  4347       using `y\<in>s` `z\<notin>s`
  4348       apply auto
  4349       done
  4350   next
  4351     fix x
  4352     assume "x \<in> s"
  4353     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)"
  4354       using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
  4355     assume "\<not> inner (y - z) y \<le> inner (y - z) x"
  4356     then obtain v where "v > 0" "v \<le> 1" "dist (y + v *\<^sub>R (x - y)) z < dist y z"
  4357       using closer_point_lemma[of z y x] by (auto simp add: inner_diff)
  4358     then show False
  4359       using *[THEN spec[where x=v]] by (auto simp add: dist_commute algebra_simps)
  4360   qed auto
  4361 qed
  4362 
  4363 lemma separating_hyperplane_closed_point:
  4364   fixes z :: "'a::{real_inner,heine_borel}"
  4365   assumes "convex s"
  4366     and "closed s"
  4367     and "z \<notin> s"
  4368   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
  4369 proof (cases "s = {}")
  4370   case True
  4371   then show ?thesis
  4372     apply (rule_tac x="-z" in exI)
  4373     apply (rule_tac x=1 in exI)
  4374     using less_le_trans[OF _ inner_ge_zero[of z]]
  4375     apply auto
  4376     done
  4377 next
  4378   case False
  4379   obtain y where "y \<in> s" and y: "\<forall>x\<in>s. dist z y \<le> dist z x"
  4380     using distance_attains_inf[OF assms(2) False] by auto
  4381   show ?thesis
  4382     apply (rule_tac x="y - z" in exI)
  4383     apply (rule_tac x="inner (y - z) z + (norm (y - z))\<^sup>2 / 2" in exI)
  4384     apply rule
  4385     defer
  4386     apply rule
  4387   proof -
  4388     fix x
  4389     assume "x \<in> s"
  4390     have "\<not> 0 < inner (z - y) (x - y)"
  4391       apply (rule notI)
  4392       apply (drule closer_point_lemma)
  4393     proof -
  4394       assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
  4395       then obtain u where "u > 0" "u \<le> 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
  4396         by auto
  4397       then show False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
  4398         using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
  4399         using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps)
  4400     qed
  4401     moreover have "0 < (norm (y - z))\<^sup>2"
  4402       using `y\<in>s` `z\<notin>s` by auto
  4403     then have "0 < inner (y - z) (y - z)"
  4404       unfolding power2_norm_eq_inner by simp
  4405     ultimately show "inner (y - z) z + (norm (y - z))\<^sup>2 / 2 < inner (y - z) x"
  4406       unfolding power2_norm_eq_inner and not_less
  4407       by (auto simp add: field_simps inner_commute inner_diff)
  4408   qed (insert `y\<in>s` `z\<notin>s`, auto)
  4409 qed
  4410 
  4411 lemma separating_hyperplane_closed_0:
  4412   assumes "convex (s::('a::euclidean_space) set)"
  4413     and "closed s"
  4414     and "0 \<notin> s"
  4415   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
  4416 proof (cases "s = {}")
  4417   case True
  4418   have "norm ((SOME i. i\<in>Basis)::'a) = 1" "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
  4419     defer
  4420     apply (subst norm_le_zero_iff[symmetric])
  4421     apply (auto simp: SOME_Basis)
  4422     done
  4423   then show ?thesis
  4424     apply (rule_tac x="SOME i. i\<in>Basis" in exI)
  4425     apply (rule_tac x=1 in exI)
  4426     using True using DIM_positive[where 'a='a]
  4427     apply auto
  4428     done
  4429 next
  4430   case False
  4431   then show ?thesis
  4432     using False using separating_hyperplane_closed_point[OF assms]
  4433     apply (elim exE)
  4434     unfolding inner_zero_right
  4435     apply (rule_tac x=a in exI)
  4436     apply (rule_tac x=b in exI)
  4437     apply auto
  4438     done
  4439 qed
  4440 
  4441 
  4442 subsubsection {* Now set-to-set for closed/compact sets *}
  4443 
  4444 lemma separating_hyperplane_closed_compact:
  4445   fixes s :: "'a::euclidean_space set"
  4446   assumes "convex s"
  4447     and "closed s"
  4448     and "convex t"
  4449     and "compact t"
  4450     and "t \<noteq> {}"
  4451     and "s \<inter> t = {}"
  4452   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  4453 proof (cases "s = {}")
  4454   case True
  4455   obtain b where b: "b > 0" "\<forall>x\<in>t. norm x \<le> b"
  4456     using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
  4457   obtain z :: 'a where z: "norm z = b + 1"
  4458     using vector_choose_size[of "b + 1"] and b(1) by auto
  4459   then have "z \<notin> t" using b(2)[THEN bspec[where x=z]] by auto
  4460   then obtain a b where ab: "inner a z < b" "\<forall>x\<in>t. b < inner a x"
  4461     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
  4462     by auto
  4463   then show ?thesis
  4464     using True by auto
  4465 next
  4466   case False
  4467   then obtain y where "y \<in> s" by auto
  4468   obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
  4469     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
  4470     using closed_compact_differences[OF assms(2,4)]
  4471     using assms(6) by auto blast
  4472   then have ab: "\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x"
  4473     apply -
  4474     apply rule
  4475     apply rule
  4476     apply (erule_tac x="x - y" in ballE)
  4477     apply (auto simp add: inner_diff)
  4478     done
  4479   def k \<equiv> "SUP x:t. a \<bullet> x"
  4480   show ?thesis
  4481     apply (rule_tac x="-a" in exI)
  4482     apply (rule_tac x="-(k + b / 2)" in exI)
  4483     apply (intro conjI ballI)
  4484     unfolding inner_minus_left and neg_less_iff_less
  4485   proof -
  4486     fix x assume "x \<in> t"
  4487     then have "inner a x - b / 2 < k"
  4488       unfolding k_def
  4489     proof (subst less_cSUP_iff)
  4490       show "t \<noteq> {}" by fact
  4491       show "bdd_above (op \<bullet> a ` t)"
  4492         using ab[rule_format, of y] `y \<in> s`
  4493         by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
  4494     qed (auto intro!: bexI[of _ x] `0<b`)
  4495     then show "inner a x < k + b / 2"
  4496       by auto
  4497   next
  4498     fix x
  4499     assume "x \<in> s"
  4500     then have "k \<le> inner a x - b"
  4501       unfolding k_def
  4502       apply (rule_tac cSUP_least)
  4503       using assms(5)
  4504       using ab[THEN bspec[where x=x]]
  4505       apply auto
  4506       done
  4507     then show "k + b / 2 < inner a x"
  4508       using `0 < b` by auto
  4509   qed
  4510 qed
  4511 
  4512 lemma separating_hyperplane_compact_closed:
  4513   fixes s :: "'a::euclidean_space set"
  4514   assumes "convex s"
  4515     and "compact s"
  4516     and "s \<noteq> {}"
  4517     and "convex t"
  4518     and "closed t"
  4519     and "s \<inter> t = {}"
  4520   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  4521 proof -
  4522   obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
  4523     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6)
  4524     by auto
  4525   then show ?thesis
  4526     apply (rule_tac x="-a" in exI)
  4527     apply (rule_tac x="-b" in exI)
  4528     apply auto
  4529     done
  4530 qed
  4531 
  4532 
  4533 subsubsection {* General case without assuming closure and getting non-strict separation *}
  4534 
  4535 lemma separating_hyperplane_set_0:
  4536   assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
  4537   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
  4538 proof -
  4539   let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
  4540   have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
  4541     apply (rule compact_imp_fip)
  4542     apply (rule compact_frontier[OF compact_cball])
  4543     defer
  4544     apply rule
  4545     apply rule
  4546     apply (erule conjE)
  4547   proof -
  4548     fix f
  4549     assume as: "f \<subseteq> ?k ` s" "finite f"
  4550     obtain c where c: "f = ?k ` c" "c \<subseteq> s" "finite c"
  4551       using finite_subset_image[OF as(2,1)] by auto
  4552     then obtain a b where ab: "a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < inner a x"
  4553       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
  4554       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
  4555       using subset_hull[of convex, OF assms(1), symmetric, of c]
  4556       by auto
  4557     then have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
  4558       apply (rule_tac x = "inverse(norm a) *\<^sub>R a" in exI)
  4559       using hull_subset[of c convex]
  4560       unfolding subset_eq and inner_scaleR
  4561       by (auto simp add: inner_commute del: ballE elim!: ballE)
  4562     then show "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}"
  4563       unfolding c(1) frontier_cball dist_norm by auto
  4564   qed (insert closed_halfspace_ge, auto)
  4565   then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y"
  4566     unfolding frontier_cball dist_norm by auto
  4567   then show ?thesis
  4568     apply (rule_tac x=x in exI)
  4569     apply (auto simp add: inner_commute)
  4570     done
  4571 qed
  4572 
  4573 lemma separating_hyperplane_sets:
  4574   fixes s t :: "'a::euclidean_space set"
  4575   assumes "convex s"
  4576     and "convex t"
  4577     and "s \<noteq> {}"
  4578     and "t \<noteq> {}"
  4579     and "s \<inter> t = {}"
  4580   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)"
  4581 proof -
  4582   from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
  4583   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"
  4584     using assms(3-5) by auto
  4585   then have *: "\<And>x y. x \<in> t \<Longrightarrow> y \<in> s \<Longrightarrow> inner a y \<le> inner a x"
  4586     by (force simp add: inner_diff)
  4587   then have bdd: "bdd_above ((op \<bullet> a)`s)"
  4588     using `t \<noteq> {}` by (auto intro: bdd_aboveI2[OF *])
  4589   show ?thesis
  4590     using `a\<noteq>0`
  4591     by (intro exI[of _ a] exI[of _ "SUP x:s. a \<bullet> x"])
  4592        (auto intro!: cSUP_upper bdd cSUP_least `a \<noteq> 0` `s \<noteq> {}` *)
  4593 qed
  4594 
  4595 
  4596 subsection {* More convexity generalities *}
  4597 
  4598 lemma convex_closure:
  4599   fixes s :: "'a::real_normed_vector set"
  4600   assumes "convex s"
  4601   shows "convex (closure s)"
  4602   apply (rule convexI)
  4603   apply (unfold closure_sequential, elim exE)
  4604   apply (rule_tac x="\<lambda>n. u *\<^sub>R xa n + v *\<^sub>R xb n" in exI)
  4605   apply (rule,rule)
  4606   apply (rule convexD [OF assms])
  4607   apply (auto del: tendsto_const intro!: tendsto_intros)
  4608   done
  4609 
  4610 lemma convex_interior:
  4611   fixes s :: "'a::real_normed_vector set"
  4612   assumes "convex s"
  4613   shows "convex (interior s)"
  4614   unfolding convex_alt Ball_def mem_interior
  4615   apply (rule,rule,rule,rule,rule,rule)
  4616   apply (elim exE conjE)
  4617 proof -
  4618   fix x y u
  4619   assume u: "0 \<le> u" "u \<le> (1::real)"
  4620   fix e d
  4621   assume ed: "ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e"
  4622   show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s"
  4623     apply (rule_tac x="min d e" in exI)
  4624     apply rule
  4625     unfolding subset_eq
  4626     defer
  4627     apply rule
  4628   proof -
  4629     fix z
  4630     assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
  4631     then have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
  4632       apply (rule_tac assms[unfolded convex_alt, rule_format])
  4633       using ed(1,2) and u
  4634       unfolding subset_eq mem_ball Ball_def dist_norm
  4635       apply (auto simp add: algebra_simps)
  4636       done
  4637     then show "z \<in> s"
  4638       using u by (auto simp add: algebra_simps)
  4639   qed(insert u ed(3-4), auto)
  4640 qed
  4641 
  4642 lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
  4643   using hull_subset[of s convex] convex_hull_empty by auto
  4644 
  4645 
  4646 subsection {* Moving and scaling convex hulls. *}
  4647 
  4648 lemma convex_hull_set_plus:
  4649   "convex hull (s + t) = convex hull s + convex hull t"
  4650   unfolding set_plus_image
  4651   apply (subst convex_hull_linear_image [symmetric])
  4652   apply (simp add: linear_iff scaleR_right_distrib)
  4653   apply (simp add: convex_hull_Times)
  4654   done
  4655 
  4656 lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` t = {a} + t"
  4657   unfolding set_plus_def by auto
  4658 
  4659 lemma convex_hull_translation:
  4660   "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
  4661   unfolding translation_eq_singleton_plus
  4662   by (simp only: convex_hull_set_plus convex_hull_singleton)
  4663 
  4664 lemma convex_hull_scaling:
  4665   "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  4666   using linear_scaleR by (rule convex_hull_linear_image [symmetric])
  4667 
  4668 lemma convex_hull_affinity:
  4669   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
  4670   by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
  4671 
  4672 
  4673 subsection {* Convexity of cone hulls *}
  4674 
  4675 lemma convex_cone_hull:
  4676   assumes "convex S"
  4677   shows "convex (cone hull S)"
  4678 proof (rule convexI)
  4679   fix x y
  4680   assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
  4681   then have "S \<noteq> {}"
  4682     using cone_hull_empty_iff[of S] by auto
  4683   fix u v :: real
  4684   assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
  4685   then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
  4686     using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
  4687   from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
  4688     using cone_hull_expl[of S] by auto
  4689   from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
  4690     using cone_hull_expl[of S] by auto
  4691   {
  4692     assume "cx + cy \<le> 0"
  4693     then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
  4694       using x y by auto
  4695     then have "u *\<^sub>R x + v *\<^sub>R y = 0"
  4696       by auto
  4697     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
  4698       using cone_hull_contains_0[of S] `S \<noteq> {}` by auto
  4699   }
  4700   moreover
  4701   {
  4702     assume "cx + cy > 0"
  4703     then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
  4704       using assms mem_convex_alt[of S xx yy cx cy] x y by auto
  4705     then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
  4706       using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] `cx+cy>0`
  4707       by (auto simp add: scaleR_right_distrib)
  4708     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
  4709       using x y by auto
  4710   }
  4711   moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
  4712   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
  4713 qed
  4714 
  4715 lemma cone_convex_hull:
  4716   assumes "cone S"
  4717   shows "cone (convex hull S)"
  4718 proof (cases "S = {}")
  4719   case True
  4720   then show ?thesis by auto
  4721 next
  4722   case False
  4723   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
  4724     using cone_iff[of S] assms by auto
  4725   {
  4726     fix c :: real
  4727     assume "c > 0"
  4728     then have "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
  4729       using convex_hull_scaling[of _ S] by auto
  4730     also have "\<dots> = convex hull S"
  4731       using * `c > 0` by auto
  4732     finally have "op *\<^sub>R c ` (convex hull S) = convex hull S"
  4733       by auto
  4734   }
  4735   then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> (op *\<^sub>R c ` (convex hull S)) = (convex hull S)"
  4736     using * hull_subset[of S convex] by auto
  4737   then show ?thesis
  4738     using `S \<noteq> {}` cone_iff[of "convex hull S"] by auto
  4739 qed
  4740 
  4741 subsection {* Convex set as intersection of halfspaces *}
  4742 
  4743 lemma convex_halfspace_intersection:
  4744   fixes s :: "('a::euclidean_space) set"
  4745   assumes "closed s" "convex s"
  4746   shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
  4747   apply (rule set_eqI)
  4748   apply rule
  4749   unfolding Inter_iff Ball_def mem_Collect_eq
  4750   apply (rule,rule,erule conjE)
  4751 proof -
  4752   fix x
  4753   assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
  4754   then have "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}"
  4755     by blast
  4756   then show "x \<in> s"
  4757     apply (rule_tac ccontr)
  4758     apply (drule separating_hyperplane_closed_point[OF assms(2,1)])
  4759     apply (erule exE)+
  4760     apply (erule_tac x="-a" in allE)
  4761     apply (erule_tac x="-b" in allE)
  4762     apply auto
  4763     done
  4764 qed auto
  4765 
  4766 
  4767 subsection {* Radon's theorem (from Lars Schewe) *}
  4768 
  4769 lemma radon_ex_lemma:
  4770   assumes "finite c" "affine_dependent c"
  4771   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"
  4772 proof -
  4773   from assms(2)[unfolded affine_dependent_explicit]
  4774   obtain s u where
  4775       "finite s" "s \<subseteq> c" "setsum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
  4776     by blast
  4777   then show ?thesis
  4778     apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
  4779     unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms(1), symmetric]
  4780     apply (auto simp add: Int_absorb1)
  4781     done
  4782 qed
  4783 
  4784 lemma radon_s_lemma:
  4785   assumes "finite s"
  4786     and "setsum f s = (0::real)"
  4787   shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
  4788 proof -
  4789   have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
  4790     by auto
  4791   show ?thesis
  4792     unfolding real_add_eq_0_iff[symmetric] and setsum.inter_filter[OF assms(1)]
  4793       and setsum.distrib[symmetric] and *
  4794     using assms(2)
  4795     apply assumption
  4796     done
  4797 qed
  4798 
  4799 lemma radon_v_lemma:
  4800   assumes "finite s"
  4801     and "setsum f s = 0"
  4802     and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
  4803   shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
  4804 proof -
  4805   have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
  4806     using assms(3) by auto
  4807   show ?thesis
  4808     unfolding eq_neg_iff_add_eq_0 and setsum.inter_filter[OF assms(1)]
  4809       and setsum.distrib[symmetric] and *
  4810     using assms(2)
  4811     apply assumption
  4812     done
  4813 qed
  4814 
  4815 lemma radon_partition:
  4816   assumes "finite c" "affine_dependent c"
  4817   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
  4818 proof -
  4819   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"
  4820     using radon_ex_lemma[OF assms] by auto
  4821   have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
  4822     using assms(1) by auto
  4823   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}"
  4824   have "setsum u {x \<in> c. 0 < u x} \<noteq> 0"
  4825   proof (cases "u v \<ge> 0")
  4826     case False
  4827     then have "u v < 0" by auto
  4828     then show ?thesis
  4829     proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
  4830       case True
  4831       then show ?thesis
  4832         using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
  4833     next
  4834       case False
  4835       then have "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c"
  4836         apply (rule_tac setsum_mono)
  4837         apply auto
  4838         done
  4839       then show ?thesis
  4840         unfolding setsum.delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto
  4841     qed
  4842   qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
  4843 
  4844   then have *: "setsum u {x\<in>c. u x > 0} > 0"
  4845     unfolding less_le
  4846     apply (rule_tac conjI)
  4847     apply (rule_tac setsum_nonneg)
  4848     apply auto
  4849     done
  4850   moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
  4851     "(\<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)"
  4852     using assms(1)
  4853     apply (rule_tac[!] setsum.mono_neutral_left)
  4854     apply auto
  4855     done
  4856   then have "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
  4857     "(\<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)"
  4858     unfolding eq_neg_iff_add_eq_0
  4859     using uv(1,4)
  4860     by (auto simp add: setsum.union_inter_neutral[OF fin, symmetric])
  4861   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x"
  4862     apply rule
  4863     apply (rule mult_nonneg_nonneg)
  4864     using *
  4865     apply auto
  4866     done
  4867   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
  4868     unfolding convex_hull_explicit mem_Collect_eq
  4869     apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
  4870     apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
  4871     using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
  4872     apply (auto simp add: setsum_negf setsum_right_distrib[symmetric])
  4873     done
  4874   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x"
  4875     apply rule
  4876     apply (rule mult_nonneg_nonneg)
  4877     using *
  4878     apply auto
  4879     done
  4880   then have "z \<in> convex hull {v \<in> c. u v > 0}"
  4881     unfolding convex_hull_explicit mem_Collect_eq
  4882     apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
  4883     apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
  4884     using assms(1)
  4885     unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
  4886     using *
  4887     apply (auto simp add: setsum_negf setsum_right_distrib[symmetric])
  4888     done
  4889   ultimately show ?thesis
  4890     apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
  4891     apply (rule_tac x="{v\<in>c. u v > 0}" in exI)
  4892     apply auto
  4893     done
  4894 qed
  4895 
  4896 lemma radon:
  4897   assumes "affine_dependent c"
  4898   obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
  4899 proof -
  4900   from assms[unfolded affine_dependent_explicit]
  4901   obtain s u where
  4902       "finite s" "s \<subseteq> c" "setsum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
  4903     by blast
  4904   then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
  4905     unfolding affine_dependent_explicit by auto
  4906   from radon_partition[OF *]
  4907   obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
  4908     by blast
  4909   then show ?thesis
  4910     apply (rule_tac that[of p m])
  4911     using s
  4912     apply auto
  4913     done
  4914 qed
  4915 
  4916 
  4917 subsection {* Helly's theorem *}
  4918 
  4919 lemma helly_induct:
  4920   fixes f :: "'a::euclidean_space set set"
  4921   assumes "card f = n"
  4922     and "n \<ge> DIM('a) + 1"
  4923     and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  4924   shows "\<Inter>f \<noteq> {}"
  4925   using assms
  4926 proof (induct n arbitrary: f)
  4927   case 0
  4928   then show ?case by auto
  4929 next
  4930   case (Suc n)
  4931   have "finite f"
  4932     using `card f = Suc n` by (auto intro: card_ge_0_finite)
  4933   show "\<Inter>f \<noteq> {}"
  4934     apply (cases "n = DIM('a)")
  4935     apply (rule Suc(5)[rule_format])
  4936     unfolding `card f = Suc n`
  4937   proof -
  4938     assume ng: "n \<noteq> DIM('a)"
  4939     then have "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})"
  4940       apply (rule_tac bchoice)
  4941       unfolding ex_in_conv
  4942       apply (rule, rule Suc(1)[rule_format])
  4943       unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
  4944       defer
  4945       defer
  4946       apply (rule Suc(4)[rule_format])
  4947       defer
  4948       apply (rule Suc(5)[rule_format])
  4949       using Suc(3) `finite f`
  4950       apply auto
  4951       done
  4952     then obtain X where X: "\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
  4953     show ?thesis
  4954     proof (cases "inj_on X f")
  4955       case False
  4956       then obtain s t where st: "s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t"
  4957         unfolding inj_on_def by auto
  4958       then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
  4959       show ?thesis
  4960         unfolding *
  4961         unfolding ex_in_conv[symmetric]
  4962         apply (rule_tac x="X s" in exI)
  4963         apply rule
  4964         apply (rule X[rule_format])
  4965         using X st
  4966         apply auto
  4967         done
  4968     next
  4969       case True
  4970       then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
  4971         using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
  4972         unfolding card_image[OF True] and `card f = Suc n`
  4973         using Suc(3) `finite f` and ng
  4974         by auto
  4975       have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
  4976         using mp(2) by auto
  4977       then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
  4978         unfolding subset_image_iff by auto
  4979       then have "f \<union> (g \<union> h) = f" by auto
  4980       then have f: "f = g \<union> h"
  4981         using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
  4982         unfolding mp(2)[unfolded image_Un[symmetric] gh]
  4983         by auto
  4984       have *: "g \<inter> h = {}"
  4985         using mp(1)
  4986         unfolding gh
  4987         using inj_on_image_Int[OF True gh(3,4)]
  4988         by auto
  4989       have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
  4990         apply (rule_tac [!] hull_minimal)
  4991         using Suc gh(3-4)
  4992         unfolding subset_eq
  4993         apply (rule_tac [2] convex_Inter, rule_tac [4] convex_Inter)
  4994         apply rule
  4995         prefer 3
  4996         apply rule
  4997       proof -
  4998         fix x
  4999         assume "x \<in> X ` g"
  5000         then obtain y where "y \<in> g" "x = X y"
  5001           unfolding image_iff ..
  5002         then show "x \<in> \<Inter>h"
  5003           using X[THEN bspec[where x=y]] using * f by auto
  5004       next
  5005         fix x
  5006         assume "x \<in> X ` h"
  5007         then obtain y where "y \<in> h" "x = X y"
  5008           unfolding image_iff ..
  5009         then show "x \<in> \<Inter>g"
  5010           using X[THEN bspec[where x=y]] using * f by auto
  5011       qed auto
  5012       then show ?thesis
  5013         unfolding f using mp(3)[unfolded gh] by blast
  5014     qed
  5015   qed auto
  5016 qed
  5017 
  5018 lemma helly:
  5019   fixes f :: "'a::euclidean_space set set"
  5020   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
  5021     and "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  5022   shows "\<Inter>f \<noteq> {}"
  5023   apply (rule helly_induct)
  5024   using assms
  5025   apply auto
  5026   done
  5027 
  5028 
  5029 subsection {* Homeomorphism of all convex compact sets with nonempty interior *}
  5030 
  5031 lemma compact_frontier_line_lemma:
  5032   fixes s :: "'a::euclidean_space set"
  5033   assumes "compact s"
  5034     and "0 \<in> s"
  5035     and "x \<noteq> 0"
  5036   obtains u where "0 \<le> u" and "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
  5037 proof -
  5038   obtain b where b: "b > 0" "\<forall>x\<in>s. norm x \<le> b"
  5039     using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
  5040   let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
  5041   have A: "?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
  5042     by auto
  5043   have *: "\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
  5044   have "compact ?A"
  5045     unfolding A
  5046     apply (rule compact_continuous_image)
  5047     apply (rule continuous_at_imp_continuous_on)
  5048     apply rule
  5049     apply (intro continuous_intros)
  5050     apply (rule compact_Icc)
  5051     done
  5052   moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}"
  5053     apply(rule *[OF _ assms(2)])
  5054     unfolding mem_Collect_eq
  5055     using `b > 0` assms(3)
  5056     apply auto
  5057     done
  5058   ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
  5059     "y \<in> ?A" "y \<in> s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y"
  5060     using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0]
  5061     by auto
  5062 
  5063   have "norm x > 0"
  5064     using assms(3)[unfolded zero_less_norm_iff[symmetric]] by auto
  5065   {
  5066     fix v
  5067     assume as: "v > u" "v *\<^sub>R x \<in> s"
  5068     then have "v \<le> b / norm x"
  5069       using b(2)[rule_format, OF as(2)]
  5070       using `u\<ge>0`
  5071       unfolding pos_le_divide_eq[OF `norm x > 0`]
  5072       by auto
  5073     then have "norm (v *\<^sub>R x) \<le> norm y"
  5074       apply (rule_tac obt(6)[rule_format, unfolded dist_0_norm])
  5075       apply (rule IntI)
  5076       defer
  5077       apply (rule as(2))
  5078       unfolding mem_Collect_eq
  5079       apply (rule_tac x=v in exI)
  5080       using as(1) `u\<ge>0`
  5081       apply (auto simp add: field_simps)
  5082       done
  5083     then have False
  5084       unfolding obt(3) using `u\<ge>0` `norm x > 0` `v > u`
  5085       by (auto simp add:field_simps)
  5086   } note u_max = this
  5087 
  5088   have "u *\<^sub>R x \<in> frontier s"
  5089     unfolding frontier_straddle
  5090     apply (rule,rule,rule)
  5091     apply (rule_tac x="u *\<^sub>R x" in bexI)
  5092     unfolding obt(3)[symmetric]
  5093     prefer 3
  5094     apply (rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI)
  5095     apply (rule, rule)
  5096   proof -
  5097     fix e
  5098     assume "e > 0" and as: "(u + e / 2 / norm x) *\<^sub>R x \<in> s"
  5099     then have "u + e / 2 / norm x > u"
  5100       using `norm x > 0` by (auto simp del:zero_less_norm_iff)
  5101     then show False using u_max[OF _ as] by auto
  5102   qed (insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
  5103   then show ?thesis by(metis that[of u] u_max obt(1))
  5104 qed
  5105 
  5106 lemma starlike_compact_projective:
  5107   assumes "compact s"
  5108     and "cball (0::'a::euclidean_space) 1 \<subseteq> s "
  5109     and "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> u *\<^sub>R x \<in> s - frontier s"
  5110   shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
  5111 proof -
  5112   have fs: "frontier s \<subseteq> s"
  5113     apply (rule frontier_subset_closed)
  5114     using compact_imp_closed[OF assms(1)]
  5115     apply simp
  5116     done
  5117   def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
  5118   have "0 \<notin> frontier s"
  5119     unfolding frontier_straddle
  5120     apply (rule notI)
  5121     apply (erule_tac x=1 in allE)
  5122     using assms(2)[unfolded subset_eq Ball_def mem_cball]
  5123     apply auto
  5124     done
  5125   have injpi: "\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y"
  5126     unfolding pi_def by auto
  5127 
  5128   have contpi: "continuous_on (UNIV - {0}) pi"
  5129     apply (rule continuous_at_imp_continuous_on)
  5130     apply rule unfolding pi_def
  5131     apply (intro continuous_intros)
  5132     apply simp
  5133     done
  5134   def sphere \<equiv> "{x::'a. norm x = 1}"
  5135   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"
  5136     unfolding pi_def sphere_def by auto
  5137 
  5138   have "0 \<in> s"
  5139     using assms(2) and centre_in_cball[of 0 1] by auto
  5140   have front_smul: "\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1"
  5141   proof (rule,rule,rule)
  5142     fix x and u :: real
  5143     assume x: "x \<in> frontier s" and "0 \<le> u"
  5144     then have "x \<noteq> 0"
  5145       using `0 \<notin> frontier s` by auto
  5146     obtain v where v: "0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
  5147       using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
  5148     have "v = 1"
  5149       apply (rule ccontr)
  5150       unfolding neq_iff
  5151       apply (erule disjE)
  5152     proof -
  5153       assume "v < 1"
  5154       then show False
  5155         using v(3)[THEN spec[where x=1]] using x and fs by auto
  5156     next
  5157       assume "v > 1"
  5158       then show False
  5159         using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
  5160         using v and x and fs
  5161         unfolding inverse_less_1_iff by auto
  5162     qed
  5163     show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1"
  5164       apply rule
  5165       using v(3)[unfolded `v=1`, THEN spec[where x=u]]
  5166     proof -
  5167       assume "u \<le> 1"
  5168       then show "u *\<^sub>R x \<in> s"
  5169       apply (cases "u = 1")
  5170         using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]]
  5171         using `0\<le>u` and x and fs
  5172         apply auto
  5173         done
  5174     qed auto
  5175   qed
  5176 
  5177   have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
  5178     apply (rule homeomorphism_compact)
  5179     apply (rule compact_frontier[OF assms(1)])
  5180     apply (rule continuous_on_subset[OF contpi])
  5181     defer
  5182     apply (rule set_eqI)
  5183     apply rule
  5184     unfolding inj_on_def
  5185     prefer 3
  5186     apply(rule,rule,rule)
  5187   proof -
  5188     fix x
  5189     assume "x \<in> pi ` frontier s"
  5190     then obtain y where "y \<in> frontier s" "x = pi y" by auto
  5191     then show "x \<in> sphere"
  5192       using pi(1)[of y] and `0 \<notin> frontier s` by auto
  5193   next
  5194     fix x
  5195     assume "x \<in> sphere"
  5196     then have "norm x = 1" "x \<noteq> 0"
  5197       unfolding sphere_def by auto
  5198     then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
  5199       using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
  5200     then show "x \<in> pi ` frontier s"
  5201       unfolding image_iff le_less pi_def
  5202       apply (rule_tac x="u *\<^sub>R x" in bexI)
  5203       using `norm x = 1` `0 \<notin> frontier s`
  5204       apply auto
  5205       done
  5206   next
  5207     fix x y
  5208     assume as: "x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
  5209     then have xys: "x \<in> s" "y \<in> s"
  5210       using fs by auto
  5211     from as(1,2) have nor: "norm x \<noteq> 0" "norm y \<noteq> 0"
  5212       using `0\<notin>frontier s` by auto
  5213     from nor have x: "x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)"
  5214       unfolding as(3)[unfolded pi_def, symmetric] by auto
  5215     from nor have y: "y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)"
  5216       unfolding as(3)[unfolded pi_def] by auto
  5217     have "0 \<le> norm y * inverse (norm x)" and "0 \<le> norm x * inverse (norm y)"
  5218       using nor
  5219       apply auto
  5220       done
  5221     then have "norm x = norm y"
  5222       apply -
  5223       apply (rule ccontr)
  5224       unfolding neq_iff
  5225       using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
  5226       using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
  5227       using xys nor
  5228       apply (auto simp add: field_simps)
  5229       done
  5230     then show "x = y"
  5231       apply (subst injpi[symmetric])
  5232       using as(3)
  5233       apply auto
  5234       done
  5235   qed (insert `0 \<notin> frontier s`, auto)
  5236   then obtain surf where
  5237     surf: "\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
  5238     "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf"
  5239     unfolding homeomorphism_def by auto
  5240 
  5241   have cont_surfpi: "continuous_on (UNIV -  {0}) (surf \<circ> pi)"
  5242     apply (rule continuous_on_compose)
  5243     apply (rule contpi)
  5244     apply (rule continuous_on_subset[of sphere])
  5245     apply (rule surf(6))
  5246     using pi(1)
  5247     apply auto
  5248     done
  5249 
  5250   {
  5251     fix x
  5252     assume as: "x \<in> cball (0::'a) 1"
  5253     have "norm x *\<^sub>R surf (pi x) \<in> s"
  5254     proof (cases "x=0 \<or> norm x = 1")
  5255       case False
  5256       then have "pi x \<in> sphere" "norm x < 1"
  5257         using pi(1)[of x] as by(auto simp add: dist_norm)
  5258       then show ?thesis
  5259         apply (rule_tac assms(3)[rule_format, THEN DiffD1])
  5260         apply (rule_tac fs[unfolded subset_eq, rule_format])
  5261         unfolding surf(5)[symmetric]
  5262         apply auto
  5263         done
  5264     next
  5265       case True
  5266       then show ?thesis
  5267         apply rule
  5268         defer
  5269         unfolding pi_def
  5270         apply (rule fs[unfolded subset_eq, rule_format])
  5271         unfolding surf(5)[unfolded sphere_def, symmetric]
  5272         using `0\<in>s`
  5273         apply auto
  5274         done
  5275     qed
  5276   } note hom = this
  5277 
  5278   {
  5279     fix x
  5280     assume "x \<in> s"
  5281     then have "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1"
  5282     proof (cases "x = 0")
  5283       case True
  5284       show ?thesis
  5285         unfolding image_iff True
  5286         apply (rule_tac x=0 in bexI)
  5287         apply auto
  5288         done
  5289     next
  5290       let ?a = "inverse (norm (surf (pi x)))"
  5291       case False
  5292       then have invn: "inverse (norm x) \<noteq> 0" by auto
  5293       from False have pix: "pi x\<in>sphere" using pi(1) by auto
  5294       then have "pi (surf (pi x)) = pi x"
  5295         apply (rule_tac surf(4)[rule_format])
  5296         apply assumption
  5297         done
  5298       then have **: "norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x"
  5299         apply (rule_tac scaleR_left_imp_eq[OF invn])
  5300         unfolding pi_def
  5301         using invn
  5302         apply auto
  5303         done
  5304       then have *: "?a * norm x > 0" and "?a > 0" "?a \<noteq> 0"
  5305         using surf(5) `0\<notin>frontier s`
  5306         apply -
  5307         apply (rule mult_pos_pos)
  5308         using False[unfolded zero_less_norm_iff[symmetric]]
  5309         apply auto
  5310         done
  5311       have "norm (surf (pi x)) \<noteq> 0"
  5312         using ** False by auto
  5313       then have "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
  5314         unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
  5315       moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))"
  5316         unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
  5317       moreover have "surf (pi x) \<in> frontier s"
  5318         using surf(5) pix by auto
  5319       then have "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1"
  5320         unfolding dist_norm
  5321         using ** and *
  5322         using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
  5323         using False `x\<in>s`
  5324         by (auto simp add: field_simps)
  5325       ultimately show ?thesis
  5326         unfolding image_iff
  5327         apply (rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
  5328         apply (subst injpi[symmetric])
  5329         unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
  5330         unfolding pi(2)[OF `?a > 0`]
  5331         apply auto
  5332         done
  5333     qed
  5334   } note hom2 = this
  5335 
  5336   show ?thesis
  5337     apply (subst homeomorphic_sym)
  5338     apply (rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
  5339     apply (rule compact_cball)
  5340     defer
  5341     apply (rule set_eqI)
  5342     apply rule
  5343     apply (erule imageE)
  5344     apply (drule hom)
  5345     prefer 4
  5346     apply (rule continuous_at_imp_continuous_on)
  5347     apply rule
  5348     apply (rule_tac [3] hom2)
  5349   proof -
  5350     fix x :: 'a
  5351     assume as: "x \<in> cball 0 1"
  5352     then show "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))"
  5353     proof (cases "x = 0")
  5354       case False
  5355       then show ?thesis
  5356         apply (intro continuous_intros)
  5357         using cont_surfpi
  5358         unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def
  5359         apply auto
  5360         done
  5361     next
  5362       case True
  5363       obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
  5364         using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
  5365       then have "B > 0"
  5366         using assms(2)
  5367         unfolding subset_eq
  5368         apply (erule_tac x="SOME i. i\<in>Basis" in ballE)
  5369         defer
  5370         apply (erule_tac x="SOME i. i\<in>Basis" in ballE)
  5371         unfolding Ball_def mem_cball dist_norm
  5372         using DIM_positive[where 'a='a]
  5373         apply (auto simp: SOME_Basis)
  5374         done
  5375       show ?thesis
  5376         unfolding True continuous_at Lim_at
  5377         apply(rule,rule)
  5378         apply(rule_tac x="e / B" in exI)
  5379         apply rule
  5380         apply (rule divide_pos_pos)
  5381         prefer 3
  5382         apply(rule,rule,erule conjE)
  5383         unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel
  5384       proof -
  5385         fix e and x :: 'a
  5386         assume as: "norm x < e / B" "0 < norm x" "e > 0"
  5387         then have "surf (pi x) \<in> frontier s"
  5388           using pi(1)[of x] unfolding surf(5)[symmetric] by auto
  5389         then have "norm (surf (pi x)) \<le> B"
  5390           using B fs by auto
  5391         then have "norm x * norm (surf (pi x)) \<le> norm x * B"
  5392           using as(2) by auto
  5393         also have "\<dots> < e / B * B"
  5394           apply (rule mult_strict_right_mono)
  5395           using as(1) `B>0`
  5396           apply auto
  5397           done
  5398         also have "\<dots> = e" using `B > 0` by auto
  5399         finally show "norm x * norm (surf (pi x)) < e" .
  5400       qed (insert `B>0`, auto)
  5401     qed
  5402   next
  5403     {
  5404       fix x
  5405       assume as: "surf (pi x) = 0"
  5406       have "x = 0"
  5407       proof (rule ccontr)
  5408         assume "x \<noteq> 0"
  5409         then have "pi x \<in> sphere"
  5410           using pi(1) by auto
  5411         then have "surf (pi x) \<in> frontier s"
  5412           using surf(5) by auto
  5413         then show False
  5414           using `0\<notin>frontier s` unfolding as by simp
  5415       qed
  5416     } note surf_0 = this
  5417     show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)"
  5418       unfolding inj_on_def
  5419     proof (rule,rule,rule)
  5420       fix x y
  5421       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)"
  5422       then show "x = y"
  5423       proof (cases "x=0 \<or> y=0")
  5424         case True
  5425         then show ?thesis
  5426           using as by (auto elim: surf_0)
  5427       next
  5428         case False
  5429         then have "pi (surf (pi x)) = pi (surf (pi y))"
  5430           using as(3)
  5431           using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"]
  5432           by auto
  5433         moreover have "pi x \<in> sphere" "pi y \<in> sphere"
  5434           using pi(1) False by auto
  5435         ultimately have *: "pi x = pi y"
  5436           using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]]
  5437           by auto
  5438         moreover have "norm x = norm y"
  5439           using as(3)[unfolded *] using False
  5440           by (auto dest:surf_0)
  5441         ultimately show ?thesis
  5442           using injpi by auto
  5443       qed
  5444     qed
  5445   qed auto
  5446 qed
  5447 
  5448 lemma homeomorphic_convex_compact_lemma:
  5449   fixes s :: "'a::euclidean_space set"
  5450   assumes "convex s"
  5451     and "compact s"
  5452     and "cball 0 1 \<subseteq> s"
  5453   shows "s homeomorphic (cball (0::'a) 1)"
  5454 proof (rule starlike_compact_projective[OF assms(2-3)], clarify)
  5455   fix x u
  5456   assume "x \<in> s" and "0 \<le> u" and "u < (1::real)"
  5457   have "open (ball (u *\<^sub>R x) (1 - u))"
  5458     by (rule open_ball)
  5459   moreover have "u *\<^sub>R x \<in> ball (u *\<^sub>R x) (1 - u)"
  5460     unfolding centre_in_ball using `u < 1` by simp
  5461   moreover have "ball (u *\<^sub>R x) (1 - u) \<subseteq> s"
  5462   proof
  5463     fix y
  5464     assume "y \<in> ball (u *\<^sub>R x) (1 - u)"
  5465     then have "dist (u *\<^sub>R x) y < 1 - u"
  5466       unfolding mem_ball .
  5467     with `u < 1` have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> cball 0 1"
  5468       by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
  5469     with assms(3) have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s" ..
  5470     with assms(1) have "(1 - u) *\<^sub>R ((y - u *\<^sub>R x) /\<^sub>R (1 - u)) + u *\<^sub>R x \<in> s"
  5471       using `x \<in> s` `0 \<le> u` `u < 1` [THEN less_imp_le] by (rule mem_convex)
  5472     then show "y \<in> s" using `u < 1`
  5473       by simp
  5474   qed
  5475   ultimately have "u *\<^sub>R x \<in> interior s" ..
  5476   then show "u *\<^sub>R x \<in> s - frontier s"
  5477     using frontier_def and interior_subset by auto
  5478 qed
  5479 
  5480 lemma homeomorphic_convex_compact_cball:
  5481   fixes e :: real
  5482     and s :: "'a::euclidean_space set"
  5483   assumes "convex s"
  5484     and "compact s"
  5485     and "interior s \<noteq> {}"
  5486     and "e > 0"
  5487   shows "s homeomorphic (cball (b::'a) e)"
  5488 proof -
  5489   obtain a where "a \<in> interior s"
  5490     using assms(3) by auto
  5491   then obtain d where "d > 0" and d: "cball a d \<subseteq> s"
  5492     unfolding mem_interior_cball by auto
  5493   let ?d = "inverse d" and ?n = "0::'a"
  5494   have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
  5495     apply rule
  5496     apply (rule_tac x="d *\<^sub>R x + a" in image_eqI)
  5497     defer
  5498     apply (rule d[unfolded subset_eq, rule_format])
  5499     using `d > 0`
  5500     unfolding mem_cball dist_norm
  5501     apply (auto simp add: mult_right_le_one_le)
  5502     done
  5503   then have "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
  5504     using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s",
  5505       OF convex_affinity compact_affinity]
  5506     using assms(1,2)
  5507     by (auto simp add: scaleR_right_diff_distrib)
  5508   then show ?thesis
  5509     apply (rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
  5510     apply (rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
  5511     using `d>0` `e>0`
  5512     apply (auto simp add: scaleR_right_diff_distrib)
  5513     done
  5514 qed
  5515 
  5516 lemma homeomorphic_convex_compact:
  5517   fixes s :: "'a::euclidean_space set"
  5518     and t :: "'a set"
  5519   assumes "convex s" "compact s" "interior s \<noteq> {}"
  5520     and "convex t" "compact t" "interior t \<noteq> {}"
  5521   shows "s homeomorphic t"
  5522   using assms
  5523   by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
  5524 
  5525 
  5526 subsection {* Epigraphs of convex functions *}
  5527 
  5528 definition "epigraph s (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
  5529 
  5530 lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y"
  5531   unfolding epigraph_def by auto
  5532 
  5533 lemma convex_epigraph: "convex (epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
  5534   unfolding convex_def convex_on_def
  5535   unfolding Ball_def split_paired_All epigraph_def
  5536   unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
  5537   apply safe
  5538   defer
  5539   apply (erule_tac x=x in allE)
  5540   apply (erule_tac x="f x" in allE)
  5541   apply safe
  5542   apply (erule_tac x=xa in allE)
  5543   apply (erule_tac x="f xa" in allE)
  5544   prefer 3
  5545   apply (rule_tac y="u * f a + v * f aa" in order_trans)
  5546   defer
  5547   apply (auto intro!:mult_left_mono add_mono)
  5548   done
  5549 
  5550 lemma convex_epigraphI: "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex (epigraph s f)"
  5551   unfolding convex_epigraph by auto
  5552 
  5553 lemma convex_epigraph_convex: "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
  5554   by (simp add: convex_epigraph)
  5555 
  5556 
  5557 subsubsection {* Use this to derive general bound property of convex function *}
  5558 
  5559 lemma convex_on:
  5560   assumes "convex s"
  5561   shows "convex_on s f \<longleftrightarrow>
  5562     (\<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>
  5563       f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k})"
  5564   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
  5565   unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
  5566   apply safe
  5567   apply (drule_tac x=k in spec)
  5568   apply (drule_tac x=u in spec)
  5569   apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
  5570   apply simp
  5571   using assms[unfolded convex]
  5572   apply simp
  5573   apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
  5574   defer
  5575   apply (rule setsum_mono)
  5576   apply (erule_tac x=i in allE)
  5577   unfolding real_scaleR_def
  5578   apply (rule mult_left_mono)
  5579   using assms[unfolded convex]
  5580   apply auto
  5581   done
  5582 
  5583 
  5584 subsection {* Convexity of general and special intervals *}
  5585 
  5586 lemma is_interval_convex:
  5587   fixes s :: "'a::euclidean_space set"
  5588   assumes "is_interval s"
  5589   shows "convex s"
  5590 proof (rule convexI)
  5591   fix x y and u v :: real
  5592   assume as: "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
  5593   then have *: "u = 1 - v" "1 - v \<ge> 0" and **: "v = 1 - u" "1 - u \<ge> 0"
  5594     by auto
  5595   {
  5596     fix a b
  5597     assume "\<not> b \<le> u * a + v * b"
  5598     then have "u * a < (1 - v) * b"
  5599       unfolding not_le using as(4) by (auto simp add: field_simps)
  5600     then have "a < b"
  5601       unfolding * using as(4) *(2)
  5602       apply (rule_tac mult_left_less_imp_less[of "1 - v"])
  5603       apply (auto simp add: field_simps)
  5604       done
  5605     then have "a \<le> u * a + v * b"
  5606       unfolding * using as(4)
  5607       by (auto simp add: field_simps intro!:mult_right_mono)
  5608   }
  5609   moreover
  5610   {
  5611     fix a b
  5612     assume "\<not> u * a + v * b \<le> a"
  5613     then have "v * b > (1 - u) * a"
  5614       unfolding not_le using as(4) by (auto simp add: field_simps)
  5615     then have "a < b"
  5616       unfolding * using as(4)
  5617       apply (rule_tac mult_left_less_imp_less)
  5618       apply (auto simp add: field_simps)
  5619       done
  5620     then have "u * a + v * b \<le> b"
  5621       unfolding **
  5622       using **(2) as(3)
  5623       by (auto simp add: field_simps intro!:mult_right_mono)
  5624   }
  5625   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s"
  5626     apply -
  5627     apply (rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
  5628     using as(3-) DIM_positive[where 'a='a]
  5629     apply (auto simp: inner_simps)
  5630     done
  5631 qed
  5632 
  5633 lemma is_interval_connected:
  5634   fixes s :: "'a::euclidean_space set"
  5635   shows "is_interval s \<Longrightarrow> connected s"
  5636   using is_interval_convex convex_connected by auto
  5637 
  5638 lemma convex_box: "convex (cbox a b)" "convex (box a (b::'a::euclidean_space))"
  5639   apply (rule_tac[!] is_interval_convex)+
  5640   using is_interval_box is_interval_cbox
  5641   apply auto
  5642   done
  5643 
  5644 subsection {* On @{text "real"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
  5645 
  5646 lemma is_interval_1:
  5647   "is_interval (s::real set) \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> s)"
  5648   unfolding is_interval_def by auto
  5649 
  5650 lemma is_interval_connected_1:
  5651   fixes s :: "real set"
  5652   shows "is_interval s \<longleftrightarrow> connected s"
  5653   apply rule
  5654   apply (rule is_interval_connected, assumption)
  5655   unfolding is_interval_1
  5656   apply rule
  5657   apply rule
  5658   apply rule
  5659   apply rule
  5660   apply (erule conjE)
  5661   apply (rule ccontr)
  5662 proof -
  5663   fix a b x
  5664   assume as: "connected s" "a \<in> s" "b \<in> s" "a \<le> x" "x \<le> b" "x \<notin> s"
  5665   then have *: "a < x" "x < b"
  5666     unfolding not_le [symmetric] by auto
  5667   let ?halfl = "{..<x} "
  5668   let ?halfr = "{x<..}"
  5669   {
  5670     fix y
  5671     assume "y \<in> s"
  5672     with `x \<notin> s` have "x \<noteq> y" by auto
  5673     then have "y \<in> ?halfr \<union> ?halfl" by auto
  5674   }
  5675   moreover have "a \<in> ?halfl" "b \<in> ?halfr" using * by auto
  5676   then have "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"
  5677     using as(2-3) by auto
  5678   ultimately show False
  5679     apply (rule_tac notE[OF as(1)[unfolded connected_def]])
  5680     apply (rule_tac x = ?halfl in exI)
  5681     apply (rule_tac x = ?halfr in exI)
  5682     apply rule
  5683     apply (rule open_lessThan)
  5684     apply rule
  5685     apply (rule open_greaterThan)
  5686     apply auto
  5687     done
  5688 qed
  5689 
  5690 lemma is_interval_convex_1:
  5691   fixes s :: "real set"
  5692   shows "is_interval s \<longleftrightarrow> convex s"
  5693   by (metis is_interval_convex convex_connected is_interval_connected_1)
  5694 
  5695 lemma convex_connected_1:
  5696   fixes s :: "real set"
  5697   shows "connected s \<longleftrightarrow> convex s"
  5698   by (metis is_interval_convex convex_connected is_interval_connected_1)
  5699 
  5700 
  5701 subsection {* Another intermediate value theorem formulation *}
  5702 
  5703 lemma ivt_increasing_component_on_1:
  5704   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
  5705   assumes "a \<le> b"
  5706     and "continuous_on (cbox a b) f"
  5707     and "(f a)\<bullet>k \<le> y" "y \<le> (f b)\<bullet>k"
  5708   shows "\<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
  5709 proof -
  5710   have "f a \<in> f ` cbox a b" "f b \<in> f ` cbox a b"
  5711     apply (rule_tac[!] imageI)
  5712     using assms(1)
  5713     apply auto
  5714     done
  5715   then show ?thesis
  5716     using connected_ivt_component[of "f ` cbox a b" "f a" "f b" k y]
  5717     using connected_continuous_image[OF assms(2) convex_connected[OF convex_box(1)]]
  5718     using assms
  5719     by auto
  5720 qed
  5721 
  5722 lemma ivt_increasing_component_1:
  5723   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
  5724   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>cbox a b. continuous (at x) f \<Longrightarrow>
  5725     f a\<bullet>k \<le> y \<Longrightarrow> y \<le> f b\<bullet>k \<Longrightarrow> \<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
  5726   by (rule ivt_increasing_component_on_1) (auto simp add: continuous_at_imp_continuous_on)
  5727 
  5728 lemma ivt_decreasing_component_on_1:
  5729   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
  5730   assumes "a \<le> b"
  5731     and "continuous_on (cbox a b) f"
  5732     and "(f b)\<bullet>k \<le> y"
  5733     and "y \<le> (f a)\<bullet>k"
  5734   shows "\<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
  5735   apply (subst neg_equal_iff_equal[symmetric])
  5736   using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"]
  5737   using assms using continuous_on_minus
  5738   apply auto
  5739   done
  5740 
  5741 lemma ivt_decreasing_component_1:
  5742   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
  5743   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>cbox a b. continuous (at x) f \<Longrightarrow>
  5744     f b\<bullet>k \<le> y \<Longrightarrow> y \<le> f a\<bullet>k \<Longrightarrow> \<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
  5745   by (rule ivt_decreasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
  5746 
  5747 
  5748 subsection {* A bound within a convex hull, and so an interval *}
  5749 
  5750 lemma convex_on_convex_hull_bound:
  5751   assumes "convex_on (convex hull s) f"
  5752     and "\<forall>x\<in>s. f x \<le> b"
  5753   shows "\<forall>x\<in> convex hull s. f x \<le> b"
  5754 proof
  5755   fix x
  5756   assume "x \<in> convex hull s"
  5757   then obtain k u v where
  5758     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"
  5759     unfolding convex_hull_indexed mem_Collect_eq by auto
  5760   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
  5761     using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
  5762     unfolding setsum_left_distrib[symmetric] obt(2) mult_1
  5763     apply (drule_tac meta_mp)
  5764     apply (rule mult_left_mono)
  5765     using assms(2) obt(1)
  5766     apply auto
  5767     done
  5768   then show "f x \<le> b"
  5769     using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
  5770     unfolding obt(2-3)
  5771     using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
  5772     by auto
  5773 qed
  5774 
  5775 lemma inner_setsum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
  5776   by (simp add: inner_setsum_left setsum.If_cases inner_Basis)
  5777 
  5778 lemma convex_set_plus:
  5779   assumes "convex s" and "convex t" shows "convex (s + t)"
  5780 proof -
  5781   have "convex {x + y |x y. x \<in> s \<and> y \<in> t}"
  5782     using assms by (rule convex_sums)
  5783   moreover have "{x + y |x y. x \<in> s \<and> y \<in> t} = s + t"
  5784     unfolding set_plus_def by auto
  5785   finally show "convex (s + t)" .
  5786 qed
  5787 
  5788 lemma convex_set_setsum:
  5789   assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
  5790   shows "convex (\<Sum>i\<in>A. B i)"
  5791 proof (cases "finite A")
  5792   case True then show ?thesis using assms
  5793     by induct (auto simp: convex_set_plus)
  5794 qed auto
  5795 
  5796 lemma finite_set_setsum:
  5797   assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
  5798   using assms by (induct set: finite, simp, simp add: finite_set_plus)
  5799 
  5800 lemma set_setsum_eq:
  5801   "finite A \<Longrightarrow> (\<Sum>i\<in>A. B i) = {\<Sum>i\<in>A. f i |f. \<forall>i\<in>A. f i \<in> B i}"
  5802   apply (induct set: finite)
  5803   apply simp
  5804   apply simp
  5805   apply (safe elim!: set_plus_elim)
  5806   apply (rule_tac x="fun_upd f x a" in exI)
  5807   apply simp
  5808   apply (rule_tac f="\<lambda>x. a + x" in arg_cong)
  5809   apply (rule setsum.cong [OF refl])
  5810   apply clarsimp
  5811   apply fast
  5812   done
  5813 
  5814 lemma box_eq_set_setsum_Basis:
  5815   shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
  5816   apply (subst set_setsum_eq [OF finite_Basis])
  5817   apply safe
  5818   apply (fast intro: euclidean_representation [symmetric])
  5819   apply (subst inner_setsum_left)
  5820   apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
  5821   apply (drule (1) bspec)
  5822   apply clarsimp
  5823   apply (frule setsum.remove [OF finite_Basis])
  5824   apply (erule trans)
  5825   apply simp
  5826   apply (rule setsum.neutral)
  5827   apply clarsimp
  5828   apply (frule_tac x=i in bspec, assumption)
  5829   apply (drule_tac x=x in bspec, assumption)
  5830   apply clarsimp
  5831   apply (cut_tac u=x and v=i in inner_Basis, assumption+)
  5832   apply (rule ccontr)
  5833   apply simp
  5834   done
  5835 
  5836 lemma convex_hull_set_setsum:
  5837   "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
  5838 proof (cases "finite A")
  5839   assume "finite A" then show ?thesis
  5840     by (induct set: finite, simp, simp add: convex_hull_set_plus)
  5841 qed simp
  5842 
  5843 lemma convex_hull_eq_real_cbox:
  5844   fixes x y :: real assumes "x \<le> y"
  5845   shows "convex hull {x, y} = cbox x y"
  5846 proof (rule hull_unique)
  5847   show "{x, y} \<subseteq> cbox x y" using `x \<le> y` by auto
  5848   show "convex (cbox x y)"
  5849     by (rule convex_box)
  5850 next
  5851   fix s assume "{x, y} \<subseteq> s" and "convex s"
  5852   then show "cbox x y \<subseteq> s"
  5853     unfolding is_interval_convex_1 [symmetric] is_interval_def Basis_real_def
  5854     by - (clarify, simp (no_asm_use), fast)
  5855 qed
  5856 
  5857 lemma unit_interval_convex_hull:
  5858   "cbox (0::'a::euclidean_space) One = convex hull {x. \<forall>i\<in>Basis. (x\<bullet>i = 0) \<or> (x\<bullet>i = 1)}"
  5859   (is "?int = convex hull ?points")
  5860 proof -
  5861   have One[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> One \<bullet> i = 1"
  5862     by (simp add: One_def inner_setsum_left setsum.If_cases inner_Basis)
  5863   have "?int = {x. \<forall>i\<in>Basis. x \<bullet> i \<in> cbox 0 1}"
  5864     by (auto simp: cbox_def)
  5865   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` cbox 0 1)"
  5866     by (simp only: box_eq_set_setsum_Basis)
  5867   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (convex hull {0, 1}))"
  5868     by (simp only: convex_hull_eq_real_cbox zero_le_one)
  5869   also have "\<dots> = (\<Sum>i\<in>Basis. convex hull ((\<lambda>x. x *\<^sub>R i) ` {0, 1}))"
  5870     by (simp only: convex_hull_linear_image linear_scaleR_left)
  5871   also have "\<dots> = convex hull (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` {0, 1})"
  5872     by (simp only: convex_hull_set_setsum)
  5873   also have "\<dots> = convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}}"
  5874     by (simp only: box_eq_set_setsum_Basis)
  5875   also have "convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}} = convex hull ?points"
  5876     by simp
  5877   finally show ?thesis .
  5878 qed
  5879 
  5880 text {* And this is a finite set of vertices. *}
  5881 
  5882 lemma unit_cube_convex_hull:
  5883   obtains s :: "'a::euclidean_space set"
  5884     where "finite s" and "cbox 0 (\<Sum>Basis) = convex hull s"
  5885   apply (rule that[of "{x::'a. \<forall>i\<in>Basis. x\<bullet>i=0 \<or> x\<bullet>i=1}"])
  5886   apply (rule finite_subset[of _ "(\<lambda>s. (\<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i)::'a) ` Pow Basis"])
  5887   prefer 3
  5888   apply (rule unit_interval_convex_hull)
  5889   apply rule
  5890   unfolding mem_Collect_eq
  5891 proof -
  5892   fix x :: 'a
  5893   assume as: "\<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1"
  5894   show "x \<in> (\<lambda>s. \<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i) ` Pow Basis"
  5895     apply (rule image_eqI[where x="{i. i\<in>Basis \<and> x\<bullet>i = 1}"])
  5896     using as
  5897     apply (subst euclidean_eq_iff)
  5898     apply auto
  5899     done
  5900 qed auto
  5901 
  5902 text {* Hence any cube (could do any nonempty interval). *}
  5903 
  5904 lemma cube_convex_hull:
  5905   assumes "d > 0"
  5906   obtains s :: "'a::euclidean_space set" where
  5907     "finite s" and "cbox (x - (\<Sum>i\<in>Basis. d*\<^sub>Ri)) (x + (\<Sum>i\<in>Basis. d*\<^sub>Ri)) = convex hull s"
  5908 proof -
  5909   let ?d = "(\<Sum>i\<in>Basis. d*\<^sub>Ri)::'a"
  5910   have *: "cbox (x - ?d) (x + ?d) = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` cbox 0 (\<Sum>Basis)"
  5911     apply (rule set_eqI, rule)
  5912     unfolding image_iff
  5913     defer
  5914     apply (erule bexE)
  5915   proof -
  5916     fix y
  5917     assume as: "y\<in>cbox (x - ?d) (x + ?d)"
  5918     then have "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> cbox 0 (\<Sum>Basis)"
  5919       using assms by (simp add: mem_box field_simps inner_simps)
  5920     with `0 < d` show "\<exists>z\<in>cbox 0 (\<Sum>Basis). y = x - ?d + (2 * d) *\<^sub>R z"
  5921       by (intro bexI[of _ "inverse (2 * d) *\<^sub>R (y - (x - ?d))"]) auto
  5922   next
  5923     fix y z
  5924     assume as: "z\<in>cbox 0 (\<Sum>Basis)" "y = x - ?d + (2*d) *\<^sub>R z"
  5925     have "\<And>i. i\<in>Basis \<Longrightarrow> 0 \<le> d * (z \<bullet> i) \<and> d * (z \<bullet> i) \<le> d"
  5926       using assms as(1)[unfolded mem_box]
  5927       apply (erule_tac x=i in ballE)
  5928       apply rule
  5929       prefer 2
  5930       apply (rule mult_right_le_one_le)
  5931       using assms
  5932       apply auto
  5933       done
  5934     then show "y \<in> cbox (x - ?d) (x + ?d)"
  5935       unfolding as(2) mem_box
  5936       apply -
  5937       apply rule
  5938       using as(1)[unfolded mem_box]
  5939       apply (erule_tac x=i in ballE)
  5940       using assms
  5941       apply (auto simp: inner_simps)
  5942       done
  5943   qed
  5944   obtain s where "finite s" "cbox 0 (\<Sum>Basis::'a) = convex hull s"
  5945     using unit_cube_convex_hull by auto
  5946   then show ?thesis
  5947     apply (rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"])
  5948     unfolding * and convex_hull_affinity
  5949     apply auto
  5950     done
  5951 qed
  5952 
  5953 
  5954 subsection {* Bounded convex function on open set is continuous *}
  5955 
  5956 lemma convex_on_bounded_continuous:
  5957   fixes s :: "('a::real_normed_vector) set"
  5958   assumes "open s"
  5959     and "convex_on s f"
  5960     and "\<forall>x\<in>s. abs(f x) \<le> b"
  5961   shows "continuous_on s f"
  5962   apply (rule continuous_at_imp_continuous_on)
  5963   unfolding continuous_at_real_range
  5964 proof (rule,rule,rule)
  5965   fix x and e :: real
  5966   assume "x \<in> s" "e > 0"
  5967   def B \<equiv> "abs b + 1"
  5968   have B: "0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
  5969     unfolding B_def
  5970     defer
  5971     apply (drule assms(3)[rule_format])
  5972     apply auto
  5973     done
  5974   obtain k where "k > 0" and k: "cball x k \<subseteq> s"
  5975     using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]]
  5976     using `x\<in>s` by auto
  5977   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
  5978     apply (rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI)
  5979     apply rule
  5980     defer
  5981   proof (rule, rule)
  5982     fix y
  5983     assume as: "norm (y - x) < min (k / 2) (e / (2 * B) * k)"
  5984     show "\<bar>f y - f x\<bar> < e"
  5985     proof (cases "y = x")
  5986       case False
  5987       def t \<equiv> "k / norm (y - x)"
  5988       have "2 < t" "0<t"
  5989         unfolding t_def using as False and `k>0`
  5990         by (auto simp add:field_simps)
  5991       have "y \<in> s"
  5992         apply (rule k[unfolded subset_eq,rule_format])
  5993         unfolding mem_cball dist_norm
  5994         apply (rule order_trans[of _ "2 * norm (x - y)"])
  5995         using as
  5996         by (auto simp add: field_simps norm_minus_commute)
  5997       {
  5998         def w \<equiv> "x + t *\<^sub>R (y - x)"
  5999         have "w \<in> s"
  6000           unfolding w_def
  6001           apply (rule k[unfolded subset_eq,rule_format])
  6002           unfolding mem_cball dist_norm
  6003           unfolding t_def
  6004           using `k>0`
  6005           apply auto
  6006           done
  6007         have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x"
  6008           by (auto simp add: algebra_simps)
  6009         also have "\<dots> = 0"
  6010           using `t > 0` by (auto simp add:field_simps)
  6011         finally have w: "(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y"
  6012           unfolding w_def using False and `t > 0`
  6013           by (auto simp add: algebra_simps)
  6014         have  "2 * B < e * t"
  6015           unfolding t_def using `0 < e` `0 < k` `B > 0` and as and False
  6016           by (auto simp add:field_simps)
  6017         then have "(f w - f x) / t < e"
  6018           using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`]
  6019           using `t > 0` by (auto simp add:field_simps)
  6020         then have th1: "f y - f x < e"
  6021           apply -
  6022           apply (rule le_less_trans)
  6023           defer
  6024           apply assumption
  6025           using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
  6026           using `0 < t` `2 < t` and `x \<in> s` `w \<in> s`
  6027           by (auto simp add:field_simps)
  6028       }
  6029       moreover
  6030       {
  6031         def w \<equiv> "x - t *\<^sub>R (y - x)"
  6032         have "w \<in> s"
  6033           unfolding w_def
  6034           apply (rule k[unfolded subset_eq,rule_format])
  6035           unfolding mem_cball dist_norm
  6036           unfolding t_def
  6037           using `k > 0`
  6038           apply auto
  6039           done
  6040         have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x"
  6041           by (auto simp add: algebra_simps)
  6042         also have "\<dots> = x"
  6043           using `t > 0` by (auto simp add:field_simps)
  6044         finally have w: "(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x"
  6045           unfolding w_def using False and `t > 0`
  6046           by (auto simp add: algebra_simps)
  6047         have "2 * B < e * t"
  6048           unfolding t_def
  6049           using `0 < e` `0 < k` `B > 0` and as and False
  6050           by (auto simp add:field_simps)
  6051         then have *: "(f w - f y) / t < e"
  6052           using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`]
  6053           using `t > 0`
  6054           by (auto simp add:field_simps)
  6055         have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
  6056           using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
  6057           using `0 < t` `2 < t` and `y \<in> s` `w \<in> s`
  6058           by (auto simp add:field_simps)
  6059         also have "\<dots> = (f w + t * f y) / (1 + t)"
  6060           using `t > 0` by (auto simp add: divide_simps)
  6061         also have "\<dots> < e + f y"
  6062           using `t > 0` * `e > 0` by (auto simp add: field_simps)
  6063         finally have "f x - f y < e" by auto
  6064       }
  6065       ultimately show ?thesis by auto
  6066     qed (insert `0<e`, auto)
  6067   qed (insert `0<e` `0<k` `0<B`, auto simp: field_simps)
  6068 qed
  6069 
  6070 
  6071 subsection {* Upper bound on a ball implies upper and lower bounds *}
  6072 
  6073 lemma convex_bounds_lemma:
  6074   fixes x :: "'a::real_normed_vector"
  6075   assumes "convex_on (cball x e) f"
  6076     and "\<forall>y \<in> cball x e. f y \<le> b"
  6077   shows "\<forall>y \<in> cball x e. abs (f y) \<le> b + 2 * abs (f x)"
  6078   apply rule
  6079 proof (cases "0 \<le> e")
  6080   case True
  6081   fix y
  6082   assume y: "y \<in> cball x e"
  6083   def z \<equiv> "2 *\<^sub>R x - y"
  6084   have *: "x - (2 *\<^sub>R x - y) = y - x"
  6085     by (simp add: scaleR_2)
  6086   have z: "z \<in> cball x e"
  6087     using y unfolding z_def mem_cball dist_norm * by (auto simp add: norm_minus_commute)
  6088   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x"
  6089     unfolding z_def by (auto simp add: algebra_simps)
  6090   then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
  6091     using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
  6092     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z]
  6093     by (auto simp add:field_simps)
  6094 next
  6095   case False
  6096   fix y
  6097   assume "y \<in> cball x e"
  6098   then have "dist x y < 0"
  6099     using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
  6100   then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
  6101     using zero_le_dist[of x y] by auto
  6102 qed
  6103 
  6104 
  6105 subsubsection {* Hence a convex function on an open set is continuous *}
  6106 
  6107 lemma real_of_nat_ge_one_iff: "1 \<le> real (n::nat) \<longleftrightarrow> 1 \<le> n"
  6108   by auto
  6109 
  6110 lemma convex_on_continuous:
  6111   assumes "open (s::('a::euclidean_space) set)" "convex_on s f"
  6112   shows "continuous_on s f"
  6113   unfolding continuous_on_eq_continuous_at[OF assms(1)]
  6114 proof
  6115   note dimge1 = DIM_positive[where 'a='a]
  6116   fix x
  6117   assume "x \<in> s"
  6118   then obtain e where e: "cball x e \<subseteq> s" "e > 0"
  6119     using assms(1) unfolding open_contains_cball by auto
  6120   def d \<equiv> "e / real DIM('a)"
  6121   have "0 < d"
  6122     unfolding d_def using `e > 0` dimge1 by auto
  6123   let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
  6124   obtain c
  6125     where c: "finite c"
  6126     and c1: "convex hull c \<subseteq> cball x e"
  6127     and c2: "cball x d \<subseteq> convex hull c"
  6128   proof
  6129     def c \<equiv> "\<Sum>i\<in>Basis. (\<lambda>a. a *\<^sub>R i) ` {x\<bullet>i - d, x\<bullet>i + d}"
  6130     show "finite c"
  6131       unfolding c_def by (simp add: finite_set_setsum)
  6132     have 1: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cbox (x \<bullet> i - d) (x \<bullet> i + d)}"
  6133       unfolding box_eq_set_setsum_Basis
  6134       unfolding c_def convex_hull_set_setsum
  6135       apply (subst convex_hull_linear_image [symmetric])
  6136       apply (simp add: linear_iff scaleR_add_left)
  6137       apply (rule setsum.cong [OF refl])
  6138       apply (rule im