src/HOL/Library/Convex_Euclidean_Space.thy
author paulson
Tue Oct 27 12:59:57 2009 +0000 (2009-10-27)
changeset 33269 3b7e2dbbd684
parent 32960 69916a850301
permissions -rw-r--r--
New theory SupInf of the supremum and infimum operators for sets of reals.
     1 (*  Title:      HOL/Library/Convex_Euclidean_Space.thy
     2     Author:     Robert Himmelmann, TU Muenchen
     3 *)
     4 
     5 header {* Convex sets, functions and related things. *}
     6 
     7 theory Convex_Euclidean_Space
     8 imports Topology_Euclidean_Space
     9 begin
    10 
    11 
    12 (* ------------------------------------------------------------------------- *)
    13 (* To be moved elsewhere                                                     *)
    14 (* ------------------------------------------------------------------------- *)
    15 
    16 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
    17 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
    18 declare dot_ladd[simp] dot_radd[simp] dot_lsub[simp] dot_rsub[simp]
    19 declare dot_lmult[simp] dot_rmult[simp] dot_lneg[simp] dot_rneg[simp]
    20 declare UNIV_1[simp]
    21 
    22 term "(x::real^'n \<Rightarrow> real) 0"
    23 
    24 lemma dim1in[intro]:"Suc 0 \<in> {1::nat .. CARD(1)}" by auto
    25 
    26 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_less_eq_def Cart_lambda_beta dest_vec1_def basis_component vector_uminus_component
    27 
    28 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul continuous_const continuous_sub continuous_at_id continuous_within_id
    29 
    30 lemmas continuous_on_intros = continuous_on_add continuous_on_const continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg continuous_on_sub
    31   uniformly_continuous_on_add uniformly_continuous_on_const uniformly_continuous_on_id uniformly_continuous_on_compose uniformly_continuous_on_cmul uniformly_continuous_on_neg uniformly_continuous_on_sub
    32 
    33 lemma dest_vec1_simps[simp]: fixes a::"real^1"
    34   shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
    35   "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
    36   by(auto simp add:vector_component_simps all_1 Cart_eq)
    37 
    38 lemma nequals0I:"x\<in>A \<Longrightarrow> A \<noteq> {}" by auto
    39 
    40 lemma norm_not_0:"(x::real^'n::finite)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
    41 
    42 lemma setsum_delta_notmem: assumes "x\<notin>s"
    43   shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
    44         "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
    45         "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
    46         "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
    47   apply(rule_tac [!] setsum_cong2) using assms by auto
    48 
    49 lemma setsum_delta'':
    50   fixes s::"'a::real_vector set" assumes "finite s"
    51   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)"
    52 proof-
    53   have *:"\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" by auto
    54   show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
    55 qed
    56 
    57 lemma not_disjointI:"x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A \<inter> B \<noteq> {}" by blast
    58 
    59 lemma if_smult:"(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" by auto
    60 
    61 lemma mem_interval_1: fixes x :: "real^1" shows
    62  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
    63  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
    64 by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def all_1)
    65 
    66 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n::finite)) ` {a..b} =
    67   (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
    68   using image_affinity_interval[of m 0 a b] by auto
    69 
    70 lemma dest_vec1_inverval:
    71   "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
    72   "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
    73   "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
    74   "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
    75   apply(rule_tac [!] equalityI)
    76   unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
    77   apply(rule_tac [!] allI)apply(rule_tac [!] impI)
    78   apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
    79   apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
    80   by (auto simp add: vector_less_def vector_less_eq_def all_1 dest_vec1_def
    81     vec1_dest_vec1[unfolded dest_vec1_def One_nat_def])
    82 
    83 lemma dest_vec1_setsum: assumes "finite S"
    84   shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
    85   using dest_vec1_sum[OF assms] by auto
    86 
    87 lemma dist_triangle_eq:
    88   fixes x y z :: "real ^ _"
    89   shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
    90 proof- have *:"x - y + (y - z) = x - z" by auto
    91   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded smult_conv_scaleR *]
    92     by(auto simp add:norm_minus_commute) qed
    93 
    94 lemma norm_eqI:"x = y \<Longrightarrow> norm x = norm y" by auto 
    95 lemma norm_minus_eqI:"(x::real^'n::finite) = - y \<Longrightarrow> norm x = norm y" by auto
    96 
    97 lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
    98   unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
    99 
   100 lemma dimindex_ge_1:"CARD(_::finite) \<ge> 1"
   101   using one_le_card_finite by auto
   102 
   103 lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1" 
   104   by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff) 
   105 
   106 lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
   107 
   108 subsection {* Affine set and affine hull.*}
   109 
   110 definition
   111   affine :: "'a::real_vector set \<Rightarrow> bool" where
   112   "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)"
   113 
   114 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)"
   115 proof- have *:"\<And>u v ::real. u + v = 1 \<longleftrightarrow> v = 1 - u" by auto
   116   { fix x y assume "x\<in>s" "y\<in>s"
   117     hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)" apply auto 
   118       apply(erule_tac[!] x="1 - u" in allE) unfolding * by auto  }
   119   thus ?thesis unfolding affine_def by auto qed
   120 
   121 lemma affine_empty[intro]: "affine {}"
   122   unfolding affine_def by auto
   123 
   124 lemma affine_sing[intro]: "affine {x}"
   125   unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
   126 
   127 lemma affine_UNIV[intro]: "affine UNIV"
   128   unfolding affine_def by auto
   129 
   130 lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
   131   unfolding affine_def by auto 
   132 
   133 lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
   134   unfolding affine_def by auto
   135 
   136 lemma affine_affine_hull: "affine(affine hull s)"
   137   unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
   138   unfolding mem_def by auto
   139 
   140 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   141 proof-
   142   { fix f assume "f \<subseteq> affine"
   143     hence "affine (\<Inter>f)" using affine_Inter[of f] unfolding subset_eq mem_def by auto  }
   144   thus ?thesis using hull_eq[unfolded mem_def, of affine s] by auto
   145 qed
   146 
   147 lemma setsum_restrict_set'': assumes "finite A"
   148   shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
   149   unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
   150 
   151 subsection {* Some explicit formulations (from Lars Schewe). *}
   152 
   153 lemma affine: fixes V::"'a::real_vector set"
   154   shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
   155 unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
   156 defer apply(rule, rule, rule, rule, rule) proof-
   157   fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
   158     "\<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"
   159   thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
   160     using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]] and as(1-3) 
   161     by(auto simp add: scaleR_left_distrib[THEN sym])
   162 next
   163   fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
   164     "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
   165   def n \<equiv> "card s"
   166   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   167   thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
   168     assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
   169     then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
   170     thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
   171       by(auto simp add: setsum_clauses(2))
   172   next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
   173       case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
   174       assume IA:"\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
   175                s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
   176         as:"Suc n \<equiv> 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"
   177            "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
   178       have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
   179         assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
   180         thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
   181           less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
   182       then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
   183 
   184       have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
   185       have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
   186       have **:"setsum u (s - {x}) = 1 - u x"
   187         using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
   188       have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
   189       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
   190         case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
   191           assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
   192           thus False using True by auto qed auto
   193         thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
   194         unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
   195       next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
   196         then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
   197         thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
   198           using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
   199       thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
   200          apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
   201          using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *\<^sub>R (\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa)"], 
   202          THEN spec[where x="u x"], THEN spec[where x="1 - u x"]] and rev_subsetD[OF `x\<in>s` `s\<subseteq>V`] and `u x \<noteq> 1` by auto
   203     qed auto
   204   next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
   205     thus ?thesis using as(4,5) by simp
   206   qed(insert `s\<noteq>{}` `finite s`, auto)
   207 qed
   208 
   209 lemma affine_hull_explicit:
   210   "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
   211   apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
   212   apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
   213   fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   214     apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
   215 next
   216   fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
   217   thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
   218 next
   219   show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}" unfolding affine_def
   220     apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
   221     fix u v ::real assume uv:"u + v = 1"
   222     fix x assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   223     then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
   224     fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
   225     then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
   226     have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
   227     have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
   228     show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
   229       apply(rule_tac x="sx \<union> sy" in exI)
   230       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)
   231       unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
   232       unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
   233       unfolding x y using x(1-3) y(1-3) uv by simp qed qed
   234 
   235 lemma affine_hull_finite:
   236   assumes "finite s"
   237   shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
   238   unfolding affine_hull_explicit and expand_set_eq and mem_Collect_eq apply (rule,rule)
   239   apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
   240   fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   241   thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
   242     apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
   243 next
   244   fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
   245   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"
   246   thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   247     unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
   248 
   249 subsection {* Stepping theorems and hence small special cases. *}
   250 
   251 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   252   apply(rule hull_unique) unfolding mem_def by auto
   253 
   254 lemma affine_hull_finite_step:
   255   fixes y :: "'a::real_vector"
   256   shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
   257   "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
   258                 (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
   259 proof-
   260   show ?th1 by simp
   261   assume ?as 
   262   { assume ?lhs
   263     then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
   264     have ?rhs proof(cases "a\<in>s")
   265       case True hence *:"insert a s = s" by auto
   266       show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
   267     next
   268       case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
   269     qed  } moreover
   270   { assume ?rhs
   271     then obtain v u where vu:"setsum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
   272     have *:"\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" by auto
   273     have ?lhs proof(cases "a\<in>s")
   274       case True thus ?thesis
   275         apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
   276         unfolding setsum_clauses(2)[OF `?as`]  apply simp
   277         unfolding scaleR_left_distrib and setsum_addf 
   278         unfolding vu and * and scaleR_zero_left
   279         by (auto simp add: setsum_delta[OF `?as`])
   280     next
   281       case False 
   282       hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
   283                "\<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
   284       from False show ?thesis
   285         apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
   286         unfolding setsum_clauses(2)[OF `?as`] and * using vu
   287         using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
   288         using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
   289     qed }
   290   ultimately show "?lhs = ?rhs" by blast
   291 qed
   292 
   293 lemma affine_hull_2:
   294   fixes a b :: "'a::real_vector"
   295   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
   296 proof-
   297   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   298          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   299   have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
   300     using affine_hull_finite[of "{a,b}"] by auto
   301   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
   302     by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
   303   also have "\<dots> = ?rhs" unfolding * by auto
   304   finally show ?thesis by auto
   305 qed
   306 
   307 lemma affine_hull_3:
   308   fixes a b c :: "'a::real_vector"
   309   shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
   310 proof-
   311   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   312          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   313   show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
   314     unfolding * apply auto
   315     apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
   316     apply(rule_tac x=u in exI) by(auto intro!: exI)
   317 qed
   318 
   319 subsection {* Some relations between affine hull and subspaces. *}
   320 
   321 lemma affine_hull_insert_subset_span:
   322   fixes a :: "real ^ _"
   323   shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
   324   unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq smult_conv_scaleR
   325   apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
   326   fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
   327   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
   328   thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
   329     apply(rule_tac x="x - a" in exI)
   330     apply (rule conjI, simp)
   331     apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
   332     apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
   333     apply (rule conjI) using as(1) apply simp
   334     apply (erule conjI)
   335     using as(1)
   336     apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib setsum_subtractf scaleR_left.setsum[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
   337     unfolding as by simp qed
   338 
   339 lemma affine_hull_insert_span:
   340   fixes a :: "real ^ _"
   341   assumes "a \<notin> s"
   342   shows "affine hull (insert a s) =
   343             {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
   344   apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
   345   unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
   346   fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
   347   then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y" unfolding span_explicit smult_conv_scaleR by auto
   348   def f \<equiv> "(\<lambda>x. x + a) ` t"
   349   have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt 
   350     by(auto simp add: setsum_reindex[unfolded inj_on_def])
   351   have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
   352   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"
   353     apply(rule_tac x="insert a f" in exI)
   354     apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
   355     using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
   356     unfolding setsum_cases[OF f(1), of "{a}", unfolded singleton_iff] and *
   357     by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps) qed
   358 
   359 lemma affine_hull_span:
   360   fixes a :: "real ^ _"
   361   assumes "a \<in> s"
   362   shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
   363   using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
   364 
   365 subsection {* Convexity. *}
   366 
   367 definition
   368   convex :: "'a::real_vector set \<Rightarrow> bool" where
   369   "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
   370 
   371 lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
   372 proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
   373   show ?thesis unfolding convex_def apply auto
   374     apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
   375     by (auto simp add: *) qed
   376 
   377 lemma mem_convex:
   378   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
   379   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
   380   using assms unfolding convex_alt by auto
   381 
   382 lemma convex_empty[intro]: "convex {}"
   383   unfolding convex_def by simp
   384 
   385 lemma convex_singleton[intro]: "convex {a}"
   386   unfolding convex_def by (auto simp add:scaleR_left_distrib[THEN sym])
   387 
   388 lemma convex_UNIV[intro]: "convex UNIV"
   389   unfolding convex_def by auto
   390 
   391 lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
   392   unfolding convex_def by auto
   393 
   394 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
   395   unfolding convex_def by auto
   396 
   397 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
   398   unfolding convex_def apply auto
   399   unfolding inner_add inner_scaleR
   400   by (metis real_convex_bound_le)
   401 
   402 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
   403 proof- have *:"{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" by auto
   404   show ?thesis apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
   405 
   406 lemma convex_hyperplane: "convex {x. inner a x = b}"
   407 proof-
   408   have *:"{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
   409   show ?thesis unfolding * apply(rule convex_Int)
   410     using convex_halfspace_le convex_halfspace_ge by auto
   411 qed
   412 
   413 lemma convex_halfspace_lt: "convex {x. inner a x < b}"
   414   unfolding convex_def
   415   by(auto simp add: real_convex_bound_lt inner_add)
   416 
   417 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
   418    using convex_halfspace_lt[of "-a" "-b"] by auto
   419 
   420 lemma convex_positive_orthant: "convex {x::real^'n::finite. (\<forall>i. 0 \<le> x$i)}"
   421   unfolding convex_def apply auto apply(erule_tac x=i in allE)+
   422   apply(rule add_nonneg_nonneg) by(auto simp add: mult_nonneg_nonneg)
   423 
   424 subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
   425 
   426 lemma convex: "convex s \<longleftrightarrow>
   427   (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
   428            \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
   429   unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
   430   fix x y u v assume as:"\<forall>(k::nat) u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> s"
   431     "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   432   show "u *\<^sub>R x + v *\<^sub>R y \<in> s" using as(1)[THEN spec[where x=2], THEN spec[where x="\<lambda>n. if n=1 then u else v"], THEN spec[where x="\<lambda>n. if n=1 then x else y"]] and as(2-)
   433     by (auto simp add: setsum_head_Suc) 
   434 next
   435   fix k u x assume as:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" 
   436   show "(\<forall>i::nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
   437   case (Suc k) show ?case proof(cases "u (Suc k) = 1")
   438     case True hence "(\<Sum>i = Suc 0..k. u i *\<^sub>R x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
   439       fix i assume i:"i \<in> {Suc 0..k}" "u i *\<^sub>R x i \<noteq> 0"
   440       hence ui:"u i \<noteq> 0" by auto
   441       hence "setsum (\<lambda>k. if k=i then u i else 0) {1 .. k} \<le> setsum u {1 .. k}" apply(rule_tac setsum_mono) using Suc(2) by auto
   442       hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta) 
   443       hence "setsum u {1 .. k} > 0"  using ui apply(rule_tac less_le_trans[of _ "u i"]) using Suc(2)[THEN spec[where x=i]] and i(1) by auto
   444       thus False using Suc(3) unfolding setsum_cl_ivl_Suc and True by simp qed
   445     thus ?thesis unfolding setsum_cl_ivl_Suc using True and Suc(2) by auto
   446   next
   447     have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
   448     have **:"u (Suc k) \<le> 1" apply(rule ccontr) unfolding not_le using Suc(3) using setsum_nonneg[of "{1..k}" u] using Suc(2) by auto
   449     have ***:"\<And>i k. (u i / (1 - u (Suc k))) *\<^sub>R x i = (inverse (1 - u (Suc k))) *\<^sub>R (u i *\<^sub>R x i)" unfolding real_divide_def by (auto simp add: algebra_simps)
   450     case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
   451     have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
   452       apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
   453     hence "(1 - u (Suc k)) *\<^sub>R (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) + u (Suc k) *\<^sub>R x (Suc k) \<in> s"
   454       apply(rule as[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using Suc(2)[THEN spec[where x="Suc k"]] and ** by auto
   455     thus ?thesis unfolding setsum_cl_ivl_Suc and *** and scaleR_right.setsum [symmetric] using nn by auto qed qed auto qed
   456 
   457 
   458 lemma convex_explicit:
   459   fixes s :: "'a::real_vector set"
   460   shows "convex s \<longleftrightarrow>
   461   (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
   462   unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
   463   fix x y u v assume as:"\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   464   show "u *\<^sub>R x + v *\<^sub>R y \<in> s" proof(cases "x=y")
   465     case True show ?thesis unfolding True and scaleR_left_distrib[THEN sym] using as(3,6) by auto next
   466     case False thus ?thesis using as(1)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>z. if z=x then u else v"]] and as(2-) by auto qed
   467 next 
   468   fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" "finite (t::'a set)"
   469   (*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
   470   from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" apply(induct_tac t rule:finite_induct)
   471     prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
   472     fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s"
   473     assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
   474     show "(\<Sum>x\<in>insert x f. u x *\<^sub>R x) \<in> s" proof(cases "u x = 1")
   475       case True hence "setsum (\<lambda>x. u x *\<^sub>R x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
   476         fix y assume y:"y \<in> f" "u y *\<^sub>R y \<noteq> 0"
   477         hence uy:"u y \<noteq> 0" by auto
   478         hence "setsum (\<lambda>k. if k=y then u y else 0) f \<le> setsum u f" apply(rule_tac setsum_mono) using as(4) by auto
   479         hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta) 
   480         hence "setsum u f > 0" using uy apply(rule_tac less_le_trans[of _ "u y"]) using as(4) and y(1) by auto
   481         thus False using as(2,5) unfolding setsum_clauses(2)[OF as(1)] and True by auto qed
   482       thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2,3) unfolding True by auto
   483     next
   484       have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
   485       have **:"u x \<le> 1" apply(rule ccontr) unfolding not_le using as(5)[unfolded setsum_clauses(2)[OF as(1)]] and as(2)
   486         using setsum_nonneg[of f u] and as(4) by auto
   487       case False hence "inverse (1 - u x) *\<^sub>R (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s" unfolding scaleR_right.setsum and scaleR_scaleR
   488         apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
   489         unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
   490       hence "u x *\<^sub>R x + (1 - u x) *\<^sub>R ((inverse (1 - u x)) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) f) \<in>s" 
   491         apply(rule_tac asm(1)[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using as and ** False by auto 
   492       thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
   493   qed auto thus "t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" by auto
   494 qed
   495 
   496 lemma convex_finite: assumes "finite s"
   497   shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
   498                       \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
   499   unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
   500   fix t u assume as:"\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
   501   have *:"s \<inter> t = t" using as(3) by auto
   502   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
   503     unfolding if_smult and setsum_cases[OF assms] and * using as(2-) by auto
   504 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
   505 
   506 subsection {* Cones. *}
   507 
   508 definition
   509   cone :: "'a::real_vector set \<Rightarrow> bool" where
   510   "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   511 
   512 lemma cone_empty[intro, simp]: "cone {}"
   513   unfolding cone_def by auto
   514 
   515 lemma cone_univ[intro, simp]: "cone UNIV"
   516   unfolding cone_def by auto
   517 
   518 lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
   519   unfolding cone_def by auto
   520 
   521 subsection {* Conic hull. *}
   522 
   523 lemma cone_cone_hull: "cone (cone hull s)"
   524   unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
   525   by (auto simp add: mem_def)
   526 
   527 lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
   528   apply(rule hull_eq[unfolded mem_def])
   529   using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
   530 
   531 subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
   532 
   533 definition
   534   affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
   535   "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
   536 
   537 lemma affine_dependent_explicit:
   538   "affine_dependent p \<longleftrightarrow>
   539     (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
   540     (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   541   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
   542   apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
   543 proof-
   544   fix x s u assume as:"x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   545   have "x\<notin>s" using as(1,4) by auto
   546   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"
   547     apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
   548     unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
   549 next
   550   fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
   551   have "s \<noteq> {v}" using as(3,6) by auto
   552   thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
   553     apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
   554     unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
   555 qed
   556 
   557 lemma affine_dependent_explicit_finite:
   558   fixes s :: "'a::real_vector set" assumes "finite s"
   559   shows "affine_dependent s \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
   560   (is "?lhs = ?rhs")
   561 proof
   562   have *:"\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::'a))" by auto
   563   assume ?lhs
   564   then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
   565     unfolding affine_dependent_explicit by auto
   566   thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   567     apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
   568     unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
   569 next
   570   assume ?rhs
   571   then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
   572   thus ?lhs unfolding affine_dependent_explicit using assms by auto
   573 qed
   574 
   575 subsection {* A general lemma. *}
   576 
   577 lemma convex_connected:
   578   fixes s :: "'a::real_normed_vector set"
   579   assumes "convex s" shows "connected s"
   580 proof-
   581   { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
   582     assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
   583     then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
   584     hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
   585 
   586     { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
   587       { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
   588           by (simp add: algebra_simps)
   589         assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
   590         hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
   591           unfolding * and scaleR_right_diff_distrib[THEN sym]
   592           unfolding less_divide_eq using n by auto  }
   593       hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
   594         apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
   595         apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
   596 
   597     have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
   598       apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
   599       using * apply(simp add: dist_norm)
   600       using as(1,2)[unfolded open_dist] apply simp
   601       using as(1,2)[unfolded open_dist] apply simp
   602       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
   603       using as(3) by auto
   604     then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
   605     hence False using as(4) 
   606       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
   607       using x1(2) x2(2) by auto  }
   608   thus ?thesis unfolding connected_def by auto
   609 qed
   610 
   611 subsection {* One rather trivial consequence. *}
   612 
   613 lemma connected_UNIV: "connected (UNIV :: 'a::real_normed_vector set)"
   614   by(simp add: convex_connected convex_UNIV)
   615 
   616 subsection {* Convex functions into the reals. *}
   617 
   618 definition
   619   convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
   620   "convex_on s f \<longleftrightarrow>
   621   (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
   622 
   623 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
   624   unfolding convex_on_def by auto
   625 
   626 lemma convex_add:
   627   assumes "convex_on s f" "convex_on s g"
   628   shows "convex_on s (\<lambda>x. f x + g x)"
   629 proof-
   630   { fix x y assume "x\<in>s" "y\<in>s" moreover
   631     fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
   632     ultimately have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
   633       using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
   634       using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
   635       apply - apply(rule add_mono) by auto
   636     hence "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps)  }
   637   thus ?thesis unfolding convex_on_def by auto 
   638 qed
   639 
   640 lemma convex_cmul: 
   641   assumes "0 \<le> (c::real)" "convex_on s f"
   642   shows "convex_on s (\<lambda>x. c * f x)"
   643 proof-
   644   have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: ring_simps)
   645   show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
   646 qed
   647 
   648 lemma convex_lower:
   649   assumes "convex_on s f"  "x\<in>s"  "y \<in> s"  "0 \<le> u"  "0 \<le> v"  "u + v = 1"
   650   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
   651 proof-
   652   let ?m = "max (f x) (f y)"
   653   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" apply(rule add_mono) 
   654     using assms(4,5) by(auto simp add: mult_mono1)
   655   also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
   656   finally show ?thesis using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
   657     using assms(2-6) by auto 
   658 qed
   659 
   660 lemma convex_local_global_minimum:
   661   fixes s :: "'a::real_normed_vector set"
   662   assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
   663   shows "\<forall>y\<in>s. f x \<le> f y"
   664 proof(rule ccontr)
   665   have "x\<in>s" using assms(1,3) by auto
   666   assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
   667   then obtain y where "y\<in>s" and y:"f x > f y" by auto
   668   hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
   669 
   670   then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
   671     using real_lbound_gt_zero[of 1 "e / dist x y"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
   672   hence "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
   673     using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]] by auto
   674   moreover
   675   have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
   676   have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
   677     using u unfolding pos_less_divide_eq[OF xy] by auto
   678   hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
   679   ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
   680 qed
   681 
   682 lemma convex_distance:
   683   fixes s :: "'a::real_normed_vector set"
   684   shows "convex_on s (\<lambda>x. dist a x)"
   685 proof(auto simp add: convex_on_def dist_norm)
   686   fix x y assume "x\<in>s" "y\<in>s"
   687   fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
   688   have "a = u *\<^sub>R a + v *\<^sub>R a" unfolding scaleR_left_distrib[THEN sym] and `u+v=1` by simp
   689   hence *:"a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
   690     by (auto simp add: algebra_simps)
   691   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
   692     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
   693     using `0 \<le> u` `0 \<le> v` by auto
   694 qed
   695 
   696 subsection {* Arithmetic operations on sets preserve convexity. *}
   697 
   698 lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *\<^sub>R x) ` s)"
   699   unfolding convex_def and image_iff apply auto
   700   apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by (auto simp add: algebra_simps)
   701 
   702 lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
   703   unfolding convex_def and image_iff apply auto
   704   apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by auto
   705 
   706 lemma convex_sums:
   707   assumes "convex s" "convex t"
   708   shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
   709 proof(auto simp add: convex_def image_iff scaleR_right_distrib)
   710   fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
   711   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   712   show "\<exists>x y. u *\<^sub>R xa + u *\<^sub>R ya + (v *\<^sub>R xb + v *\<^sub>R yb) = x + y \<and> x \<in> s \<and> y \<in> t"
   713     apply(rule_tac x="u *\<^sub>R xa + v *\<^sub>R xb" in exI) apply(rule_tac x="u *\<^sub>R ya + v *\<^sub>R yb" in exI)
   714     using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
   715     using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
   716     using uv xy by auto
   717 qed
   718 
   719 lemma convex_differences: 
   720   assumes "convex s" "convex t"
   721   shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
   722 proof-
   723   have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}" unfolding image_iff apply auto
   724     apply(rule_tac x=xa in exI) apply(rule_tac x="-y" in exI) apply simp
   725     apply(rule_tac x=xa in exI) apply(rule_tac x=xb in exI) by simp
   726   thus ?thesis using convex_sums[OF assms(1)  convex_negations[OF assms(2)]] by auto
   727 qed
   728 
   729 lemma convex_translation: assumes "convex s" shows "convex ((\<lambda>x. a + x) ` s)"
   730 proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
   731   thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
   732 
   733 lemma convex_affinity: assumes "convex s" shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
   734 proof- have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s" by auto
   735   thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
   736 
   737 lemma convex_linear_image:
   738   assumes c:"convex s" and l:"bounded_linear f"
   739   shows "convex(f ` s)"
   740 proof(auto simp add: convex_def)
   741   interpret f: bounded_linear f by fact
   742   fix x y assume xy:"x \<in> s" "y \<in> s"
   743   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   744   show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff
   745     apply(rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in bexI)
   746     unfolding f.add f.scaleR
   747     using c[unfolded convex_def] xy uv by auto
   748 qed
   749 
   750 subsection {* Balls, being convex, are connected. *}
   751 
   752 lemma convex_ball:
   753   fixes x :: "'a::real_normed_vector"
   754   shows "convex (ball x e)" 
   755 proof(auto simp add: convex_def)
   756   fix y z assume yz:"dist x y < e" "dist x z < e"
   757   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   758   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
   759     using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
   760   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using real_convex_bound_lt[OF yz uv] by auto 
   761 qed
   762 
   763 lemma convex_cball:
   764   fixes x :: "'a::real_normed_vector"
   765   shows "convex(cball x e)"
   766 proof(auto simp add: convex_def Ball_def mem_cball)
   767   fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
   768   fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
   769   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
   770     using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
   771   thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using real_convex_bound_le[OF yz uv] by auto 
   772 qed
   773 
   774 lemma connected_ball:
   775   fixes x :: "'a::real_normed_vector"
   776   shows "connected (ball x e)"
   777   using convex_connected convex_ball by auto
   778 
   779 lemma connected_cball:
   780   fixes x :: "'a::real_normed_vector"
   781   shows "connected(cball x e)"
   782   using convex_connected convex_cball by auto
   783 
   784 subsection {* Convex hull. *}
   785 
   786 lemma convex_convex_hull: "convex(convex hull s)"
   787   unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
   788   unfolding mem_def by auto
   789 
   790 lemma convex_hull_eq: "(convex hull s = s) \<longleftrightarrow> convex s" apply(rule hull_eq[unfolded mem_def])
   791   using convex_Inter[unfolded Ball_def mem_def] by auto
   792 
   793 lemma bounded_convex_hull:
   794   fixes s :: "'a::real_normed_vector set"
   795   assumes "bounded s" shows "bounded(convex hull s)"
   796 proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
   797   show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
   798     unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
   799     unfolding subset_eq mem_cball dist_norm using B by auto qed
   800 
   801 lemma finite_imp_bounded_convex_hull:
   802   fixes s :: "'a::real_normed_vector set"
   803   shows "finite s \<Longrightarrow> bounded(convex hull s)"
   804   using bounded_convex_hull finite_imp_bounded by auto
   805 
   806 subsection {* Stepping theorems for convex hulls of finite sets. *}
   807 
   808 lemma convex_hull_empty[simp]: "convex hull {} = {}"
   809   apply(rule hull_unique) unfolding mem_def by auto
   810 
   811 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
   812   apply(rule hull_unique) unfolding mem_def by auto
   813 
   814 lemma convex_hull_insert:
   815   fixes s :: "'a::real_vector set"
   816   assumes "s \<noteq> {}"
   817   shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
   818                                     b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
   819  apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
   820  fix x assume x:"x = a \<or> x \<in> s"
   821  thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
   822    apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
   823 next
   824   fix x assume "x\<in>?hull"
   825   then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
   826   have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
   827     using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
   828   thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
   829     apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
   830 next
   831   show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
   832     fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
   833     from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
   834     from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
   835     have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
   836     have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
   837     proof(cases "u * v1 + v * v2 = 0")
   838       have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
   839       case True hence **:"u * v1 = 0" "v * v2 = 0" apply- apply(rule_tac [!] ccontr)
   840         using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by auto
   841       hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
   842       thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
   843     next
   844       have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
   845       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
   846       also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
   847       case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
   848         apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
   849         using as(1,2) obt1(1,2) obt2(1,2) by auto 
   850       thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
   851         apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
   852         apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
   853         unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
   854         by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
   855     qed note * = this
   856     have u1:"u1 \<le> 1" apply(rule ccontr) unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
   857     have u2:"u2 \<le> 1" apply(rule ccontr) unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
   858     have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
   859       apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
   860     also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
   861     finally 
   862     show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
   863       apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
   864       using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
   865   qed
   866 qed
   867 
   868 
   869 subsection {* Explicit expression for convex hull. *}
   870 
   871 lemma convex_hull_indexed:
   872   fixes s :: "'a::real_vector set"
   873   shows "convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
   874                             (setsum u {1..k} = 1) \<and>
   875                             (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
   876   apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
   877   apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
   878 proof-
   879   fix x assume "x\<in>s"
   880   thus "x \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
   881 next
   882   fix t assume as:"s \<subseteq> t" "convex t"
   883   show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
   884     fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
   885     show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
   886       using assm(1,2) as(1) by auto qed
   887 next
   888   fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and xy:"x\<in>?hull" "y\<in>?hull"
   889   from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" by auto
   890   from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" by auto
   891   have *:"\<And>P (x1::'a) x2 s1 s2 i.(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
   892     "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
   893     prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
   894   have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
   895   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
   896     apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
   897     apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
   898     unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def
   899     unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
   900     fix i assume i:"i \<in> {1..k1+k2}"
   901     show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and> (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
   902     proof(cases "i\<in>{1..k1}")
   903       case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
   904     next def j \<equiv> "i - k1"
   905       case False with i have "j \<in> {1..k2}" unfolding j_def by auto
   906       thus ?thesis unfolding j_def[symmetric] using False
   907         using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
   908   qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
   909 qed
   910 
   911 lemma convex_hull_finite:
   912   fixes s :: "'a::real_vector set"
   913   assumes "finite s"
   914   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
   915          setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
   916 proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
   917   fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x" 
   918     apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
   919     unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
   920 next
   921   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   922   fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
   923   fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
   924   { fix x assume "x\<in>s"
   925     hence "0 \<le> u * ux x + v * uy x" using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
   926       by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
   927   moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
   928     unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
   929   moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
   930     unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
   931   ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
   932     apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
   933 next
   934   fix t assume t:"s \<subseteq> t" "convex t" 
   935   fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
   936   thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
   937     using assms and t(1) by auto
   938 qed
   939 
   940 subsection {* Another formulation from Lars Schewe. *}
   941 
   942 lemma setsum_constant_scaleR:
   943   fixes y :: "'a::real_vector"
   944   shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
   945 apply (cases "finite A")
   946 apply (induct set: finite)
   947 apply (simp_all add: algebra_simps)
   948 done
   949 
   950 lemma convex_hull_explicit:
   951   fixes p :: "'a::real_vector set"
   952   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
   953              (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
   954 proof-
   955   { fix x assume "x\<in>?lhs"
   956     then obtain k u y where obt:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
   957       unfolding convex_hull_indexed by auto
   958 
   959     have fin:"finite {1..k}" by auto
   960     have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
   961     { fix j assume "j\<in>{1..k}"
   962       hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
   963         using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
   964         apply(rule setsum_nonneg) using obt(1) by auto } 
   965     moreover
   966     have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
   967       unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
   968     moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
   969       using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
   970       unfolding scaleR_left.setsum using obt(3) by auto
   971     ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
   972       apply(rule_tac x="y ` {1..k}" in exI)
   973       apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
   974     hence "x\<in>?rhs" by auto  }
   975   moreover
   976   { fix y assume "y\<in>?rhs"
   977     then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
   978 
   979     obtain f where f:"inj_on f {1..card s}" "f ` {1..card s} = s" using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
   980     
   981     { fix i::nat assume "i\<in>{1..card s}"
   982       hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
   983       hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
   984     moreover have *:"finite {1..card s}" by auto
   985     { fix y assume "y\<in>s"
   986       then obtain i where "i\<in>{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"] by auto
   987       hence "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}" apply auto using f(1)[unfolded inj_on_def] apply(erule_tac x=x in ballE) by auto
   988       hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
   989       hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
   990             "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
   991         by (auto simp add: setsum_constant_scaleR)   }
   992 
   993     hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
   994       unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
   995       unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
   996       using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
   997     
   998     ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
   999       apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastsimp
  1000     hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
  1001   ultimately show ?thesis unfolding expand_set_eq by blast
  1002 qed
  1003 
  1004 subsection {* A stepping theorem for that expansion. *}
  1005 
  1006 lemma convex_hull_finite_step:
  1007   fixes s :: "'a::real_vector set" assumes "finite s"
  1008   shows "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
  1009      \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?lhs = ?rhs")
  1010 proof(rule, case_tac[!] "a\<in>s")
  1011   assume "a\<in>s" hence *:"insert a s = s" by auto
  1012   assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
  1013 next
  1014   assume ?lhs then obtain u where u:"\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
  1015   assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
  1016     apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
  1017 next
  1018   assume "a\<in>s" hence *:"insert a s = s" by auto
  1019   have fin:"finite (insert a s)" using assms by auto
  1020   assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
  1021   show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
  1022     unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
  1023 next
  1024   assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
  1025   moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
  1026     apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
  1027   ultimately show ?lhs apply(rule_tac x="\<lambda>x. if a = x then v else u x" in exI)  unfolding setsum_clauses(2)[OF assms] by auto
  1028 qed
  1029 
  1030 subsection {* Hence some special cases. *}
  1031 
  1032 lemma convex_hull_2:
  1033   "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}"
  1034 proof- have *:"\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" by auto have **:"finite {b}" by auto
  1035 show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
  1036   apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
  1037   apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
  1038 
  1039 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
  1040   unfolding convex_hull_2 unfolding Collect_def 
  1041 proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
  1042   fix x show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
  1043     unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
  1044 
  1045 lemma convex_hull_3:
  1046   "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}"
  1047 proof-
  1048   have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
  1049   have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
  1050          "\<And>x y z ::real^'n. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: ring_simps)
  1051   show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
  1052     unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
  1053     apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
  1054     apply(rule_tac x="1 - v - w" in exI) apply simp apply(rule_tac x=v in exI) apply simp apply(rule_tac x="\<lambda>x. w" in exI) by simp qed
  1055 
  1056 lemma convex_hull_3_alt:
  1057   "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}"
  1058 proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
  1059   show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply (simp add: algebra_simps)
  1060     apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
  1061 
  1062 subsection {* Relations among closure notions and corresponding hulls. *}
  1063 
  1064 text {* TODO: Generalize linear algebra concepts defined in @{text
  1065 Euclidean_Space.thy} so that we can generalize these lemmas. *}
  1066 
  1067 lemma subspace_imp_affine:
  1068   fixes s :: "(real ^ _) set" shows "subspace s \<Longrightarrow> affine s"
  1069   unfolding subspace_def affine_def smult_conv_scaleR by auto
  1070 
  1071 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
  1072   unfolding affine_def convex_def by auto
  1073 
  1074 lemma subspace_imp_convex:
  1075   fixes s :: "(real ^ _) set" shows "subspace s \<Longrightarrow> convex s"
  1076   using subspace_imp_affine affine_imp_convex by auto
  1077 
  1078 lemma affine_hull_subset_span:
  1079   fixes s :: "(real ^ _) set" shows "(affine hull s) \<subseteq> (span s)"
  1080   unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
  1081   using subspace_imp_affine  by auto
  1082 
  1083 lemma convex_hull_subset_span:
  1084   fixes s :: "(real ^ _) set" shows "(convex hull s) \<subseteq> (span s)"
  1085   unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
  1086   using subspace_imp_convex by auto
  1087 
  1088 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
  1089   unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
  1090   using affine_imp_convex by auto
  1091 
  1092 lemma affine_dependent_imp_dependent:
  1093   fixes s :: "(real ^ _) set" shows "affine_dependent s \<Longrightarrow> dependent s"
  1094   unfolding affine_dependent_def dependent_def 
  1095   using affine_hull_subset_span by auto
  1096 
  1097 lemma dependent_imp_affine_dependent:
  1098   fixes s :: "(real ^ _) set"
  1099   assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
  1100   shows "affine_dependent (insert a s)"
  1101 proof-
  1102   from assms(1)[unfolded dependent_explicit smult_conv_scaleR] obtain S u v 
  1103     where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
  1104   def t \<equiv> "(\<lambda>x. x + a) ` S"
  1105 
  1106   have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
  1107   have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
  1108   have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
  1109 
  1110   hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
  1111   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
  1112     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1113   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
  1114     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
  1115   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"
  1116     apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
  1117   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)"
  1118     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1119   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
  1120     unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
  1121     using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
  1122   hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
  1123     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *  vector_smult_lneg) 
  1124   ultimately show ?thesis unfolding affine_dependent_explicit
  1125     apply(rule_tac x="insert a t" in exI) by auto 
  1126 qed
  1127 
  1128 lemma convex_cone:
  1129   "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)" (is "?lhs = ?rhs")
  1130 proof-
  1131   { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
  1132     hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
  1133     hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
  1134       apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
  1135       apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
  1136   thus ?thesis unfolding convex_def cone_def by auto
  1137 qed
  1138 
  1139 lemma affine_dependent_biggerset: fixes s::"(real^'n::finite) set"
  1140   assumes "finite s" "card s \<ge> CARD('n) + 2"
  1141   shows "affine_dependent s"
  1142 proof-
  1143   have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
  1144   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1145   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1146     apply(rule card_image) unfolding inj_on_def by auto
  1147   also have "\<dots> > CARD('n)" using assms(2)
  1148     unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1149   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1150     apply(rule dependent_imp_affine_dependent)
  1151     apply(rule dependent_biggerset) by auto qed
  1152 
  1153 lemma affine_dependent_biggerset_general:
  1154   assumes "finite (s::(real^'n::finite) set)" "card s \<ge> dim s + 2"
  1155   shows "affine_dependent s"
  1156 proof-
  1157   from assms(2) have "s \<noteq> {}" by auto
  1158   then obtain a where "a\<in>s" by auto
  1159   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1160   have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1161     apply(rule card_image) unfolding inj_on_def by auto
  1162   have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
  1163     apply(rule subset_le_dim) unfolding subset_eq
  1164     using `a\<in>s` by (auto simp add:span_superset span_sub)
  1165   also have "\<dots> < dim s + 1" by auto
  1166   also have "\<dots> \<le> card (s - {a})" using assms
  1167     using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1168   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1169     apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
  1170 
  1171 subsection {* Caratheodory's theorem. *}
  1172 
  1173 lemma convex_hull_caratheodory: fixes p::"(real^'n::finite) set"
  1174   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and>
  1175   (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
  1176   unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
  1177 proof(rule,rule)
  1178   fix y let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
  1179   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"
  1180   then obtain N where "?P N" by auto
  1181   hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
  1182   then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
  1183   then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
  1184 
  1185   have "card s \<le> CARD('n) + 1" proof(rule ccontr, simp only: not_le)
  1186     assume "CARD('n) + 1 < card s"
  1187     hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
  1188     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"
  1189       using affine_dependent_explicit_finite[OF obt(1)] by auto
  1190     def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
  1191     have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
  1192       assume as:"\<forall>x\<in>s. 0 \<le> w x"
  1193       hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
  1194       hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
  1195         using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
  1196       thus False using wv(1) by auto
  1197     qed hence "i\<noteq>{}" unfolding i_def by auto
  1198 
  1199     hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
  1200       using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
  1201     have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
  1202       fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
  1203       show"0 \<le> u v + t * w v" proof(cases "w v < 0")
  1204         case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
  1205           using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
  1206         case True hence "t \<le> u v / (- w v)" using `v\<in>s`
  1207           unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
  1208         thus ?thesis unfolding real_0_le_add_iff
  1209           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
  1210       qed qed
  1211 
  1212     obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
  1213       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
  1214     hence a:"a\<in>s" "u a + t * w a = 0" by auto
  1215     have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'a::ring)" unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
  1216     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
  1217       unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
  1218     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" 
  1219       unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
  1220       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]]
  1221       by (simp add: vector_smult_lneg)
  1222     ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
  1223       apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a by (auto simp add: * scaleR_left_distrib)
  1224     thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
  1225   thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1
  1226     \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" using obt by auto
  1227 qed auto
  1228 
  1229 lemma caratheodory:
  1230  "convex hull p = {x::real^'n::finite. \<exists>s. finite s \<and> s \<subseteq> p \<and>
  1231       card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s}"
  1232   unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
  1233   fix x assume "x \<in> convex hull p"
  1234   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1"
  1235      "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"unfolding convex_hull_caratheodory by auto
  1236   thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
  1237     apply(rule_tac x=s in exI) using hull_subset[of s convex]
  1238   using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
  1239 next
  1240   fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
  1241   then obtain s where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1" "x \<in> convex hull s" by auto
  1242   thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
  1243 qed
  1244 
  1245 subsection {* Openness and compactness are preserved by convex hull operation. *}
  1246 
  1247 lemma open_convex_hull:
  1248   fixes s :: "'a::real_normed_vector set"
  1249   assumes "open s"
  1250   shows "open(convex hull s)"
  1251   unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10) 
  1252 proof(rule, rule) fix a
  1253   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"
  1254   then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
  1255 
  1256   from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
  1257     using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
  1258   have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
  1259 
  1260   show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
  1261     apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
  1262   proof-
  1263     show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
  1264       using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
  1265   next  fix y assume "y \<in> cball a (Min i)"
  1266     hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
  1267     { fix x assume "x\<in>t"
  1268       hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
  1269       hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
  1270       moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
  1271       ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by auto }
  1272     moreover
  1273     have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
  1274     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
  1275       unfolding setsum_reindex[OF *] o_def using obt(4) by auto
  1276     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
  1277       unfolding setsum_reindex[OF *] o_def using obt(4,5)
  1278       by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
  1279     ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
  1280       apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
  1281       using obt(1, 3) by auto
  1282   qed
  1283 qed
  1284 
  1285 lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
  1286 unfolding open_vector_def all_1
  1287 by (auto simp add: dest_vec1_def)
  1288 
  1289 lemma tendsto_dest_vec1 [tendsto_intros]:
  1290   "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
  1291   unfolding tendsto_def
  1292   apply clarify
  1293   apply (drule_tac x="dest_vec1 -` S" in spec)
  1294   apply (simp add: open_dest_vec1_vimage)
  1295   done
  1296 
  1297 lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
  1298   unfolding continuous_def by (rule tendsto_dest_vec1)
  1299 
  1300 (* TODO: move *)
  1301 lemma compact_real_interval:
  1302   fixes a b :: real shows "compact {a..b}"
  1303 proof -
  1304   have "continuous_on {vec1 a .. vec1 b} dest_vec1"
  1305     unfolding continuous_on
  1306     by (simp add: tendsto_dest_vec1 Lim_at_within Lim_ident_at)
  1307   moreover have "compact {vec1 a .. vec1 b}" by (rule compact_interval)
  1308   ultimately have "compact (dest_vec1 ` {vec1 a .. vec1 b})"
  1309     by (rule compact_continuous_image)
  1310   also have "dest_vec1 ` {vec1 a .. vec1 b} = {a..b}"
  1311     by (auto simp add: image_def Bex_def exists_vec1)
  1312   finally show ?thesis .
  1313 qed
  1314 
  1315 lemma compact_convex_combinations:
  1316   fixes s t :: "'a::real_normed_vector set"
  1317   assumes "compact s" "compact t"
  1318   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}"
  1319 proof-
  1320   let ?X = "{0..1} \<times> s \<times> t"
  1321   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  1322   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"
  1323     apply(rule set_ext) unfolding image_iff mem_Collect_eq
  1324     apply rule apply auto
  1325     apply (rule_tac x=u in rev_bexI, simp)
  1326     apply (erule rev_bexI, erule rev_bexI, simp)
  1327     by auto
  1328   have "continuous_on ({0..1} \<times> s \<times> t)
  1329      (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
  1330     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  1331   thus ?thesis unfolding *
  1332     apply (rule compact_continuous_image)
  1333     apply (intro compact_Times compact_real_interval assms)
  1334     done
  1335 qed
  1336 
  1337 lemma compact_convex_hull: fixes s::"(real^'n::finite) set"
  1338   assumes "compact s"  shows "compact(convex hull s)"
  1339 proof(cases "s={}")
  1340   case True thus ?thesis using compact_empty by simp
  1341 next
  1342   case False then obtain w where "w\<in>s" by auto
  1343   show ?thesis unfolding caratheodory[of s]
  1344   proof(induct "CARD('n) + 1")
  1345     have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
  1346       using compact_empty by (auto simp add: convex_hull_empty)
  1347     case 0 thus ?case unfolding * by simp
  1348   next
  1349     case (Suc n)
  1350     show ?case proof(cases "n=0")
  1351       case True have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
  1352         unfolding expand_set_eq and mem_Collect_eq proof(rule, rule)
  1353         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1354         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  1355         show "x\<in>s" proof(cases "card t = 0")
  1356           case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by(simp add: convex_hull_empty)
  1357         next
  1358           case False hence "card t = Suc 0" using t(3) `n=0` by auto
  1359           then obtain a where "t = {a}" unfolding card_Suc_eq by auto
  1360           thus ?thesis using t(2,4) by (simp add: convex_hull_singleton)
  1361         qed
  1362       next
  1363         fix x assume "x\<in>s"
  1364         thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1365           apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
  1366       qed thus ?thesis using assms by simp
  1367     next
  1368       case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
  1369         { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
  1370         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}}"
  1371         unfolding expand_set_eq and mem_Collect_eq proof(rule,rule)
  1372         fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  1373           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)"
  1374         then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
  1375           "0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n"  "v \<in> convex hull t" by auto
  1376         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
  1377           apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
  1378           using obt(7) and hull_mono[of t "insert u t"] by auto
  1379         ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1380           apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
  1381       next
  1382         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1383         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  1384         let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
  1385           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)"
  1386         show ?P proof(cases "card t = Suc n")
  1387           case False hence "card t \<le> n" using t(3) by auto
  1388           thus ?P apply(rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) using `w\<in>s` and t
  1389             by(auto intro!: exI[where x=t])
  1390         next
  1391           case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
  1392           show ?P proof(cases "u={}")
  1393             case True hence "x=a" using t(4)[unfolded au] by auto
  1394             show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
  1395               using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
  1396           next
  1397             case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
  1398               using t(4)[unfolded au convex_hull_insert[OF False]] by auto
  1399             have *:"1 - vx = ux" using obt(3) by auto
  1400             show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
  1401               using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
  1402               by(auto intro!: exI[where x=u])
  1403           qed
  1404         qed
  1405       qed
  1406       thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
  1407     qed
  1408   qed 
  1409 qed
  1410 
  1411 lemma finite_imp_compact_convex_hull:
  1412   fixes s :: "(real ^ _) set"
  1413   shows "finite s \<Longrightarrow> compact(convex hull s)"
  1414   apply(drule finite_imp_compact, drule compact_convex_hull) by assumption
  1415 
  1416 subsection {* Extremal points of a simplex are some vertices. *}
  1417 
  1418 lemma dist_increases_online:
  1419   fixes a b d :: "'a::real_inner"
  1420   assumes "d \<noteq> 0"
  1421   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
  1422 proof(cases "inner a d - inner b d > 0")
  1423   case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
  1424     apply(rule_tac add_pos_pos) using assms by auto
  1425   thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  1426     by (simp add: algebra_simps inner_commute)
  1427 next
  1428   case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
  1429     apply(rule_tac add_pos_nonneg) using assms by auto
  1430   thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
  1431     by (simp add: algebra_simps inner_commute)
  1432 qed
  1433 
  1434 lemma norm_increases_online:
  1435   fixes d :: "'a::real_inner"
  1436   shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
  1437   using dist_increases_online[of d a 0] unfolding dist_norm by auto
  1438 
  1439 lemma simplex_furthest_lt:
  1440   fixes s::"'a::real_inner set" assumes "finite s"
  1441   shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
  1442 proof(induct_tac rule: finite_induct[of s])
  1443   fix x s assume as:"finite s" "x\<notin>s" "\<forall>x\<in>convex hull s. x \<notin> s \<longrightarrow> (\<exists>y\<in>convex hull s. norm (x - a) < norm (y - a))"
  1444   show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow> (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
  1445   proof(rule,rule,cases "s = {}")
  1446     case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
  1447     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"
  1448       using y(1)[unfolded convex_hull_insert[OF False]] by auto
  1449     show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
  1450     proof(cases "y\<in>convex hull s")
  1451       case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
  1452         using as(3)[THEN bspec[where x=y]] and y(2) by auto
  1453       thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
  1454     next
  1455       case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
  1456         assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
  1457         thus ?thesis using False and obt(4) by auto
  1458       next
  1459         assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
  1460         thus ?thesis using y(2) by auto
  1461       next
  1462         assume "u\<noteq>0" "v\<noteq>0"
  1463         then obtain w where w:"w>0" "w<u" "w<v" using real_lbound_gt_zero[of u v] and obt(1,2) by auto
  1464         have "x\<noteq>b" proof(rule ccontr) 
  1465           assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
  1466             using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
  1467           thus False using obt(4) and False by simp qed
  1468         hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
  1469         show ?thesis using dist_increases_online[OF *, of a y]
  1470         proof(erule_tac disjE)
  1471           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
  1472           hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
  1473             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  1474           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
  1475             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  1476             apply(rule_tac x="u + w" in exI) apply rule defer 
  1477             apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  1478           ultimately show ?thesis by auto
  1479         next
  1480           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
  1481           hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
  1482             unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
  1483           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
  1484             unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  1485             apply(rule_tac x="u - w" in exI) apply rule defer 
  1486             apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  1487           ultimately show ?thesis by auto
  1488         qed
  1489       qed auto
  1490     qed
  1491   qed auto
  1492 qed (auto simp add: assms)
  1493 
  1494 lemma simplex_furthest_le:
  1495   fixes s :: "(real ^ _) set"
  1496   assumes "finite s" "s \<noteq> {}"
  1497   shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
  1498 proof-
  1499   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  1500   then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
  1501     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
  1502     unfolding dist_commute[of a] unfolding dist_norm by auto
  1503   thus ?thesis proof(cases "x\<in>s")
  1504     case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
  1505       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
  1506     thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
  1507   qed auto
  1508 qed
  1509 
  1510 lemma simplex_furthest_le_exists:
  1511   fixes s :: "(real ^ _) set"
  1512   shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
  1513   using simplex_furthest_le[of s] by (cases "s={}")auto
  1514 
  1515 lemma simplex_extremal_le:
  1516   fixes s :: "(real ^ _) set"
  1517   assumes "finite s" "s \<noteq> {}"
  1518   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)"
  1519 proof-
  1520   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  1521   then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
  1522     "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
  1523     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
  1524   thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
  1525     assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
  1526       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
  1527     thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
  1528   next
  1529     assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
  1530       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
  1531     thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
  1532       by (auto simp add: norm_minus_commute)
  1533   qed auto
  1534 qed 
  1535 
  1536 lemma simplex_extremal_le_exists:
  1537   fixes s :: "(real ^ _) set"
  1538   shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
  1539   \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
  1540   using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
  1541 
  1542 subsection {* Closest point of a convex set is unique, with a continuous projection. *}
  1543 
  1544 definition
  1545   closest_point :: "(real ^ 'n::finite) set \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 'n" where
  1546  "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
  1547 
  1548 lemma closest_point_exists:
  1549   assumes "closed s" "s \<noteq> {}"
  1550   shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
  1551   unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
  1552   using distance_attains_inf[OF assms(1,2), of a] by auto
  1553 
  1554 lemma closest_point_in_set:
  1555   "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
  1556   by(meson closest_point_exists)
  1557 
  1558 lemma closest_point_le:
  1559   "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
  1560   using closest_point_exists[of s] by auto
  1561 
  1562 lemma closest_point_self:
  1563   assumes "x \<in> s"  shows "closest_point s x = x"
  1564   unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
  1565   using assms by auto
  1566 
  1567 lemma closest_point_refl:
  1568  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
  1569   using closest_point_in_set[of s x] closest_point_self[of x s] by auto
  1570 
  1571 (* TODO: move *)
  1572 lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
  1573   unfolding norm_eq_sqrt_inner by simp
  1574 
  1575 (* TODO: move *)
  1576 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
  1577   unfolding norm_eq_sqrt_inner by simp
  1578 
  1579 lemma closer_points_lemma: fixes y::"real^'n::finite"
  1580   assumes "inner y z > 0"
  1581   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
  1582 proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
  1583   thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
  1584     fix v assume "0<v" "v \<le> inner y z / inner z z"
  1585     thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
  1586       by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
  1587   qed(rule divide_pos_pos, auto) qed
  1588 
  1589 lemma closer_point_lemma:
  1590   fixes x y z :: "real ^ 'n::finite"
  1591   assumes "inner (y - x) (z - x) > 0"
  1592   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
  1593 proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
  1594     using closer_points_lemma[OF assms] by auto
  1595   show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
  1596     unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
  1597 
  1598 lemma any_closest_point_dot:
  1599   fixes s :: "(real ^ _) set"
  1600   assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  1601   shows "inner (a - x) (y - x) \<le> 0"
  1602 proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
  1603   then obtain u where u:"u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
  1604   let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
  1605   thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
  1606 
  1607 (* TODO: move *)
  1608 lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> inner x x \<le> a\<twosuperior>"
  1609 proof -
  1610   have "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> norm x \<le> a"
  1611     using norm_ge_zero [of x] by arith
  1612   also have "\<dots> \<longleftrightarrow> 0 \<le> a \<and> (norm x)\<twosuperior> \<le> a\<twosuperior>"
  1613     by (auto intro: power_mono dest: power2_le_imp_le)
  1614   also have "\<dots> \<longleftrightarrow> 0 \<le> a \<and> inner x x \<le> a\<twosuperior>"
  1615     unfolding power2_norm_eq_inner ..
  1616   finally show ?thesis .
  1617 qed
  1618 
  1619 lemma any_closest_point_unique:
  1620   fixes s :: "(real ^ _) set"
  1621   assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
  1622   "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
  1623   shows "x = y" using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
  1624   unfolding norm_pths(1) and norm_le_square
  1625   by (auto simp add: algebra_simps)
  1626 
  1627 lemma closest_point_unique:
  1628   assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  1629   shows "x = closest_point s a"
  1630   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
  1631   using closest_point_exists[OF assms(2)] and assms(3) by auto
  1632 
  1633 lemma closest_point_dot:
  1634   assumes "convex s" "closed s" "x \<in> s"
  1635   shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
  1636   apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
  1637   using closest_point_exists[OF assms(2)] and assms(3) by auto
  1638 
  1639 lemma closest_point_lt:
  1640   assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
  1641   shows "dist a (closest_point s a) < dist a x"
  1642   apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
  1643   apply(rule closest_point_unique[OF assms(1-3), of a])
  1644   using closest_point_le[OF assms(2), of _ a] by fastsimp
  1645 
  1646 lemma closest_point_lipschitz:
  1647   assumes "convex s" "closed s" "s \<noteq> {}"
  1648   shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
  1649 proof-
  1650   have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
  1651        "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
  1652     apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
  1653     using closest_point_exists[OF assms(2-3)] by auto
  1654   thus ?thesis unfolding dist_norm and norm_le
  1655     using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
  1656     by (simp add: inner_add inner_diff inner_commute) qed
  1657 
  1658 lemma continuous_at_closest_point:
  1659   assumes "convex s" "closed s" "s \<noteq> {}"
  1660   shows "continuous (at x) (closest_point s)"
  1661   unfolding continuous_at_eps_delta 
  1662   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
  1663 
  1664 lemma continuous_on_closest_point:
  1665   assumes "convex s" "closed s" "s \<noteq> {}"
  1666   shows "continuous_on t (closest_point s)"
  1667   apply(rule continuous_at_imp_continuous_on) using continuous_at_closest_point[OF assms] by auto
  1668 
  1669 subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
  1670 
  1671 lemma supporting_hyperplane_closed_point:
  1672   fixes s :: "(real ^ _) set"
  1673   assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
  1674   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)"
  1675 proof-
  1676   from distance_attains_inf[OF assms(2-3)] obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x" by auto
  1677   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
  1678     apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
  1679     show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
  1680       unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
  1681   next
  1682     fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
  1683       using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
  1684     assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
  1685       "v>0" "v\<le>1" "dist (y + v *\<^sub>R (x - y)) z < dist y z" using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
  1686     thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
  1687   qed auto
  1688 qed
  1689 
  1690 lemma separating_hyperplane_closed_point:
  1691   fixes s :: "(real ^ _) set"
  1692   assumes "convex s" "closed s" "z \<notin> s"
  1693   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
  1694 proof(cases "s={}")
  1695   case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
  1696     using less_le_trans[OF _ inner_ge_zero[of z]] by auto
  1697 next
  1698   case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
  1699     using distance_attains_inf[OF assms(2) False] by auto
  1700   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
  1701     apply rule defer apply rule proof-
  1702     fix x assume "x\<in>s"
  1703     have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
  1704       assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
  1705       then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
  1706       thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
  1707         using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
  1708         using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
  1709     moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
  1710     hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
  1711     ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
  1712       unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
  1713   qed(insert `y\<in>s` `z\<notin>s`, auto)
  1714 qed
  1715 
  1716 lemma separating_hyperplane_closed_0:
  1717   assumes "convex (s::(real^'n::finite) set)" "closed s" "0 \<notin> s"
  1718   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
  1719   proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
  1720   case True have "norm ((basis a)::real^'n::finite) = 1" 
  1721     using norm_basis and dimindex_ge_1 by auto
  1722   thus ?thesis apply(rule_tac x="basis a" in exI, rule_tac x=1 in exI) using True by auto
  1723 next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
  1724     apply - apply(erule exE)+ unfolding dot_rzero apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
  1725 
  1726 subsection {* Now set-to-set for closed/compact sets. *}
  1727 
  1728 lemma separating_hyperplane_closed_compact:
  1729   assumes "convex (s::(real^'n::finite) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
  1730   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  1731 proof(cases "s={}")
  1732   case True
  1733   obtain b where b:"b>0" "\<forall>x\<in>t. norm x \<le> b" using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
  1734   obtain z::"real^'n" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
  1735   hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
  1736   then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
  1737     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
  1738   thus ?thesis using True by auto
  1739 next
  1740   case False then obtain y where "y\<in>s" by auto
  1741   obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
  1742     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
  1743     using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
  1744   hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
  1745   def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
  1746   show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
  1747     apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
  1748     from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
  1749       apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
  1750     hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
  1751     fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
  1752   next
  1753     fix x assume "x\<in>s" 
  1754     hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
  1755       using ab[THEN bspec[where x=x]] by auto
  1756     thus "k + b / 2 < inner a x" using `0 < b` by auto
  1757   qed
  1758 qed
  1759 
  1760 lemma separating_hyperplane_compact_closed:
  1761   fixes s :: "(real ^ _) set"
  1762   assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
  1763   shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
  1764 proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
  1765     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
  1766   thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
  1767 
  1768 subsection {* General case without assuming closure and getting non-strict separation. *}
  1769 
  1770 lemma separating_hyperplane_set_0:
  1771   assumes "convex s" "(0::real^'n::finite) \<notin> s"
  1772   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
  1773 proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> inner c x}"
  1774   have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
  1775     apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
  1776     defer apply(rule,rule,erule conjE) proof-
  1777     fix f assume as:"f \<subseteq> ?k ` s" "finite f"
  1778     obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
  1779     then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
  1780       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
  1781       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
  1782       using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
  1783     hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)" apply(rule_tac x="inverse(norm a) *\<^sub>R a" in exI)
  1784        using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
  1785        apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
  1786        by(auto simp add: inner_commute elim!: ballE)
  1787     thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
  1788   qed(insert closed_halfspace_ge, auto)
  1789   then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
  1790   thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
  1791 
  1792 lemma separating_hyperplane_sets:
  1793   assumes "convex s" "convex (t::(real^'n::finite) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
  1794   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)"
  1795 proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
  1796   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" 
  1797     using assms(3-5) by auto 
  1798   hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
  1799     by (force simp add: inner_diff)
  1800   thus ?thesis
  1801     apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
  1802     apply auto
  1803     apply (rule Sup[THEN isLubD2]) 
  1804     prefer 4
  1805     apply (rule Sup_least) 
  1806      using assms(3-5) apply (auto simp add: setle_def)
  1807     apply (metis COMBC_def Collect_def Collect_mem_eq) 
  1808     done
  1809 qed
  1810 
  1811 subsection {* More convexity generalities. *}
  1812 
  1813 lemma convex_closure:
  1814   fixes s :: "'a::real_normed_vector set"
  1815   assumes "convex s" shows "convex(closure s)"
  1816   unfolding convex_def Ball_def closure_sequential
  1817   apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
  1818   apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
  1819   apply(rule assms[unfolded convex_def, rule_format]) prefer 6
  1820   apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
  1821 
  1822 lemma convex_interior:
  1823   fixes s :: "'a::real_normed_vector set"
  1824   assumes "convex s" shows "convex(interior s)"
  1825   unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
  1826   fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
  1827   fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
  1828   show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
  1829     apply rule unfolding subset_eq defer apply rule proof-
  1830     fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
  1831     hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
  1832       apply(rule_tac assms[unfolded convex_alt, rule_format])
  1833       using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
  1834     thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
  1835 
  1836 lemma convex_hull_eq_empty: "convex hull s = {} \<longleftrightarrow> s = {}"
  1837   using hull_subset[of s convex] convex_hull_empty by auto
  1838 
  1839 subsection {* Moving and scaling convex hulls. *}
  1840 
  1841 lemma convex_hull_translation_lemma:
  1842   "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
  1843   apply(rule hull_minimal, rule image_mono, rule hull_subset) unfolding mem_def
  1844   using convex_translation[OF convex_convex_hull, of a s] by assumption
  1845 
  1846 lemma convex_hull_bilemma: fixes neg
  1847   assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
  1848   shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
  1849   \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
  1850   using assms by(metis subset_antisym) 
  1851 
  1852 lemma convex_hull_translation:
  1853   "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
  1854   apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
  1855 
  1856 lemma convex_hull_scaling_lemma:
  1857  "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  1858   apply(rule hull_minimal, rule image_mono, rule hull_subset)
  1859   unfolding mem_def by(rule convex_scaling, rule convex_convex_hull)
  1860 
  1861 lemma convex_hull_scaling:
  1862   "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
  1863   apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
  1864   unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv convex_hull_eq_empty)
  1865 
  1866 lemma convex_hull_affinity:
  1867   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
  1868   unfolding image_image[THEN sym] convex_hull_scaling convex_hull_translation  ..
  1869 
  1870 subsection {* Convex set as intersection of halfspaces. *}
  1871 
  1872 lemma convex_halfspace_intersection:
  1873   fixes s :: "(real ^ _) set"
  1874   assumes "closed s" "convex s"
  1875   shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
  1876   apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
  1877   fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
  1878   hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
  1879   thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
  1880     apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
  1881 qed auto
  1882 
  1883 subsection {* Radon's theorem (from Lars Schewe). *}
  1884 
  1885 lemma radon_ex_lemma:
  1886   assumes "finite c" "affine_dependent c"
  1887   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"
  1888 proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
  1889   thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
  1890     and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
  1891 
  1892 lemma radon_s_lemma:
  1893   assumes "finite s" "setsum f s = (0::real)"
  1894   shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
  1895 proof- have *:"\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" by auto
  1896   show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  1897     using assms(2) by assumption qed
  1898 
  1899 lemma radon_v_lemma:
  1900   assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::real^'n)"
  1901   shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
  1902 proof-
  1903   have *:"\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" using assms(3) by auto 
  1904   show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  1905     using assms(2) by assumption qed
  1906 
  1907 lemma radon_partition:
  1908   assumes "finite c" "affine_dependent c"
  1909   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
  1910   obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
  1911   have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
  1912   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}"
  1913   have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
  1914     case False hence "u v < 0" by auto
  1915     thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
  1916       case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
  1917     next
  1918       case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
  1919       thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
  1920   qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
  1921 
  1922   hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding real_less_def apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
  1923   moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
  1924     "(\<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)"
  1925     using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
  1926   hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
  1927    "(\<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)" 
  1928     unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
  1929   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
  1930     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
  1931 
  1932   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
  1933     apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
  1934     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
  1935     by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
  1936   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
  1937     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
  1938   hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
  1939     apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
  1940     using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
  1941     by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
  1942   ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
  1943 qed
  1944 
  1945 lemma radon: assumes "affine_dependent c"
  1946   obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
  1947 proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
  1948   hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
  1949   from radon_partition[OF *] guess m .. then guess p ..
  1950   thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
  1951 
  1952 subsection {* Helly's theorem. *}
  1953 
  1954 lemma helly_induct: fixes f::"(real^'n::finite) set set"
  1955   assumes "f hassize n" "n \<ge> CARD('n) + 1"
  1956   "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  1957   shows "\<Inter> f \<noteq> {}"
  1958   using assms unfolding hassize_def apply(erule_tac conjE) proof(induct n arbitrary: f)
  1959 case (Suc n)
  1960 show "\<Inter> f \<noteq> {}" apply(cases "n = CARD('n)") apply(rule Suc(4)[rule_format])
  1961   unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6) proof-
  1962   assume ng:"n \<noteq> CARD('n)" hence "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" apply(rule_tac bchoice) unfolding ex_in_conv
  1963     apply(rule, rule Suc(1)[rule_format])  unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6)
  1964     defer apply(rule Suc(3)[rule_format]) defer apply(rule Suc(4)[rule_format]) using Suc(2,5) by auto
  1965   then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
  1966   show ?thesis proof(cases "inj_on X f")
  1967     case False then obtain s t where st:"s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t" unfolding inj_on_def by auto
  1968     hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
  1969     show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
  1970       apply(rule, rule X[rule_format]) using X st by auto
  1971   next case True then obtain m p where mp:"m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
  1972       using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
  1973       unfolding card_image[OF True] and Suc(6) using Suc(2,5) and ng by auto
  1974     have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
  1975     then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
  1976     hence "f \<union> (g \<union> h) = f" by auto
  1977     hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
  1978       unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
  1979     have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
  1980     have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
  1981       apply(rule_tac [!] hull_minimal) using Suc(3) gh(3-4)  unfolding mem_def unfolding subset_eq
  1982       apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
  1983       fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
  1984       thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
  1985       fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
  1986       thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
  1987     qed(auto)
  1988     thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
  1989 qed(insert dimindex_ge_1, auto) qed(auto)
  1990 
  1991 lemma helly: fixes f::"(real^'n::finite) set set"
  1992   assumes "finite f" "card f \<ge> CARD('n) + 1" "\<forall>s\<in>f. convex s"
  1993           "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  1994   shows "\<Inter> f \<noteq>{}"
  1995   apply(rule helly_induct) unfolding hassize_def using assms by auto
  1996 
  1997 subsection {* Convex hull is "preserved" by a linear function. *}
  1998 
  1999 lemma convex_hull_linear_image:
  2000   assumes "bounded_linear f"
  2001   shows "f ` (convex hull s) = convex hull (f ` s)"
  2002   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  2003   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  2004   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  2005 proof-
  2006   interpret f: bounded_linear f by fact
  2007   show "convex {x. f x \<in> convex hull f ` s}" 
  2008   unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
  2009   interpret f: bounded_linear f by fact
  2010   show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
  2011     unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
  2012 qed auto
  2013 
  2014 lemma in_convex_hull_linear_image:
  2015   assumes "bounded_linear f" "x \<in> convex hull s"
  2016   shows "(f x) \<in> convex hull (f ` s)"
  2017 using convex_hull_linear_image[OF assms(1)] assms(2) by auto
  2018 
  2019 subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
  2020 
  2021 lemma compact_frontier_line_lemma:
  2022   fixes s :: "(real ^ _) set"
  2023   assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
  2024   obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
  2025 proof-
  2026   obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
  2027   let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
  2028   have A:"?A = (\<lambda>u. dest_vec1 u *\<^sub>R x) ` {0 .. vec1 (b / norm x)}"
  2029     unfolding image_image[of "\<lambda>u. u *\<^sub>R x" "\<lambda>x. dest_vec1 x", THEN sym]
  2030     unfolding dest_vec1_inverval vec1_dest_vec1 by auto
  2031   have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
  2032     apply(rule, rule continuous_vmul)
  2033     apply (rule continuous_dest_vec1)
  2034     apply(rule continuous_at_id) by(rule compact_interval)
  2035   moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
  2036     unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
  2037   ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
  2038     "y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
  2039 
  2040   have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
  2041   { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
  2042     hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
  2043       using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
  2044     hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
  2045       apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
  2046       using as(1) `u\<ge>0` by(auto simp add:field_simps) 
  2047     hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
  2048   } note u_max = this
  2049 
  2050   have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[THEN sym]
  2051     prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
  2052     fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
  2053     hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
  2054     thus False using u_max[OF _ as] by auto
  2055   qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
  2056   thus ?thesis apply(rule_tac that[of u]) apply(rule obt(1), assumption)
  2057     apply(rule,rule,rule ccontr) apply(rule u_max) by auto qed
  2058 
  2059 lemma starlike_compact_projective:
  2060   assumes "compact s" "cball (0::real^'n::finite) 1 \<subseteq> s "
  2061   "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
  2062   shows "s homeomorphic (cball (0::real^'n::finite) 1)"
  2063 proof-
  2064   have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
  2065   def pi \<equiv> "\<lambda>x::real^'n. inverse (norm x) *\<^sub>R x"
  2066   have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
  2067     using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
  2068   have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
  2069 
  2070   have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
  2071     apply rule unfolding pi_def
  2072     apply (rule continuous_mul)
  2073     apply (rule continuous_at_inv[unfolded o_def])
  2074     apply (rule continuous_at_norm)
  2075     apply simp
  2076     apply (rule continuous_at_id)
  2077     done
  2078   def sphere \<equiv> "{x::real^'n. norm x = 1}"
  2079   have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x" unfolding pi_def sphere_def by auto
  2080 
  2081   have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
  2082   have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
  2083     fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
  2084     hence "x\<noteq>0" using `0\<notin>frontier s` by auto
  2085     obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
  2086       using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
  2087     have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
  2088       assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
  2089       assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
  2090         using v and x and fs unfolding inverse_less_1_iff by auto qed
  2091     show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
  2092       assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
  2093         using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
  2094 
  2095   have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
  2096     apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
  2097     apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_ext,rule) 
  2098     unfolding inj_on_def prefer 3 apply(rule,rule,rule)
  2099   proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
  2100     thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
  2101   next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
  2102     then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
  2103       using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
  2104     thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
  2105   next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
  2106     hence xys:"x\<in>s" "y\<in>s" using fs by auto
  2107     from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
  2108     from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
  2109     from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto 
  2110     have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
  2111       unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
  2112     hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
  2113       using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
  2114       using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
  2115       using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
  2116     thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
  2117   qed(insert `0 \<notin> frontier s`, auto)
  2118   then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
  2119     "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
  2120   
  2121   have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
  2122     apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
  2123 
  2124   { fix x assume as:"x \<in> cball (0::real^'n) 1"
  2125     have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
  2126       case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
  2127       thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
  2128         apply(rule_tac fs[unfolded subset_eq, rule_format])
  2129         unfolding surf(5)[THEN sym] by auto
  2130     next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
  2131         unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
  2132 
  2133   { fix x assume "x\<in>s"
  2134     hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
  2135       case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
  2136     next let ?a = "inverse (norm (surf (pi x)))"
  2137       case False hence invn:"inverse (norm x) \<noteq> 0" by auto
  2138       from False have pix:"pi x\<in>sphere" using pi(1) by auto
  2139       hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
  2140       hence **:"norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
  2141       hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
  2142         apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
  2143       have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
  2144       hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
  2145         unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
  2146       moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))" 
  2147         unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
  2148       moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
  2149       hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
  2150         using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
  2151         using False `x\<in>s` by(auto simp add:field_simps)
  2152       ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
  2153         apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
  2154         unfolding pi(2)[OF `?a > 0`] by auto
  2155     qed } note hom2 = this
  2156 
  2157   show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
  2158     apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
  2159     prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
  2160     fix x::"real^'n" assume as:"x \<in> cball 0 1"
  2161     thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
  2162       case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
  2163         using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
  2164     next guess a using UNIV_witness[where 'a = 'n] ..
  2165       obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
  2166       hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis a" in ballE) defer apply(erule_tac x="basis a" in ballE)
  2167         unfolding Ball_def mem_cball dist_norm by (auto simp add: norm_basis[unfolded One_nat_def])
  2168       case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
  2169         apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
  2170         unfolding norm_0 scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
  2171         fix e and x::"real^'n" assume as:"norm x < e / B" "0 < norm x" "0<e"
  2172         hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
  2173         hence "norm (surf (pi x)) \<le> B" using B fs by auto
  2174         hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
  2175         also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
  2176         also have "\<dots> = e" using `B>0` by auto
  2177         finally show "norm x * norm (surf (pi x)) < e" by assumption
  2178       qed(insert `B>0`, auto) qed
  2179   next { fix x assume as:"surf (pi x) = 0"
  2180       have "x = 0" proof(rule ccontr)
  2181         assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
  2182         hence "surf (pi x) \<in> frontier s" using surf(5) by auto
  2183         thus False using `0\<notin>frontier s` unfolding as by simp qed
  2184     } note surf_0 = this
  2185     show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
  2186       fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
  2187       thus "x=y" proof(cases "x=0 \<or> y=0") 
  2188         case True thus ?thesis using as by(auto elim: surf_0) next
  2189         case False
  2190         hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
  2191           using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
  2192         moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
  2193         ultimately have *:"pi x = pi y" using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]] by auto 
  2194         moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
  2195         ultimately show ?thesis using injpi by auto qed qed
  2196   qed auto qed
  2197 
  2198 lemma homeomorphic_convex_compact_lemma: fixes s::"(real^'n::finite) set"
  2199   assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
  2200   shows "s homeomorphic (cball (0::real^'n) 1)"
  2201   apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
  2202   fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
  2203   hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
  2204     apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
  2205     unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
  2206     fix y assume "dist (u *\<^sub>R x) y < 1 - u"
  2207     hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
  2208       using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
  2209       unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
  2210       apply (rule mult_left_le_imp_le[of "1 - u"])
  2211       unfolding class_semiring.mul_a using `u<1` by auto
  2212     thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
  2213       using as unfolding scaleR_scaleR by auto qed auto
  2214   thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
  2215 
  2216 lemma homeomorphic_convex_compact_cball: fixes e::real and s::"(real^'n::finite) set"
  2217   assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
  2218   shows "s homeomorphic (cball (b::real^'n::finite) e)"
  2219 proof- obtain a where "a\<in>interior s" using assms(3) by auto
  2220   then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
  2221   let ?d = "inverse d" and ?n = "0::real^'n"
  2222   have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
  2223     apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
  2224     apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
  2225     by(auto simp add: mult_right_le_one_le)
  2226   hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
  2227     using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
  2228     using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
  2229   thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
  2230     apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
  2231     using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
  2232 
  2233 lemma homeomorphic_convex_compact: fixes s::"(real^'n::finite) set" and t::"(real^'n) set"
  2234   assumes "convex s" "compact s" "interior s \<noteq> {}"
  2235           "convex t" "compact t" "interior t \<noteq> {}"
  2236   shows "s homeomorphic t"
  2237   using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
  2238 
  2239 subsection {* Epigraphs of convex functions. *}
  2240 
  2241 definition "epigraph s (f::real^'n \<Rightarrow> real) = {xy. fstcart xy \<in> s \<and> f(fstcart xy) \<le> dest_vec1 (sndcart xy)}"
  2242 
  2243 lemma mem_epigraph: "(pastecart x (vec1 y)) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
  2244 
  2245 lemma convex_epigraph: 
  2246   "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
  2247   unfolding convex_def convex_on_def unfolding Ball_def forall_pastecart epigraph_def
  2248   unfolding mem_Collect_eq fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul [where 'a=real, unfolded smult_conv_scaleR] fstcart_add fstcart_cmul [where 'a=real, unfolded smult_conv_scaleR]
  2249   unfolding Ball_def[symmetric] unfolding dest_vec1_add dest_vec1_cmul [where 'a=real, unfolded smult_conv_scaleR]
  2250   apply(subst forall_dest_vec1[THEN sym])+ by(meson real_le_refl real_le_trans add_mono mult_left_mono) 
  2251 
  2252 lemma convex_epigraphI: assumes "convex_on s f" "convex s"
  2253   shows "convex(epigraph s f)" using assms unfolding convex_epigraph by auto
  2254 
  2255 lemma convex_epigraph_convex: "convex s \<Longrightarrow> (convex_on s f \<longleftrightarrow> convex(epigraph s f))"
  2256   using convex_epigraph by auto
  2257 
  2258 subsection {* Use this to derive general bound property of convex function. *}
  2259 
  2260 lemma forall_of_pastecart:
  2261   "(\<forall>p. P (\<lambda>x. fstcart (p x)) (\<lambda>x. sndcart (p x))) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
  2262   apply(erule_tac x="\<lambda>a. pastecart (x a) (y a)" in allE) unfolding o_def by auto
  2263 
  2264 lemma forall_of_pastecart':
  2265   "(\<forall>p. P (fstcart p) (sndcart p)) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
  2266   apply(erule_tac x="pastecart x y" in allE) unfolding o_def by auto
  2267 
  2268 lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
  2269   apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto 
  2270 
  2271 lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
  2272   apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule 
  2273   apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto
  2274 
  2275 lemma convex_on:
  2276   fixes s :: "(real ^ _) set"
  2277   assumes "convex s"
  2278   shows "convex_on s f \<longleftrightarrow> (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
  2279    f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
  2280   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
  2281   unfolding sndcart_setsum[OF finite_atLeastAtMost] fstcart_setsum[OF finite_atLeastAtMost] dest_vec1_setsum[OF finite_atLeastAtMost]
  2282   unfolding fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul [where 'a=real, unfolded smult_conv_scaleR] fstcart_add fstcart_cmul [where 'a=real, unfolded smult_conv_scaleR]
  2283   unfolding dest_vec1_add dest_vec1_cmul [where 'a=real, unfolded smult_conv_scaleR] apply(subst forall_of_pastecart)+ apply(subst forall_of_dest_vec1)+ apply rule
  2284   using assms[unfolded convex] apply simp apply(rule,rule,rule)
  2285   apply(erule_tac x=k in allE, erule_tac x=u in allE, erule_tac x=x in allE) apply rule apply rule apply rule defer
  2286   apply(rule_tac j="\<Sum>i = 1..k. u i * f (x i)" in real_le_trans)
  2287   defer apply(rule setsum_mono) apply(erule conjE)+ apply(erule_tac x=i in allE)apply(rule mult_left_mono)
  2288   using assms[unfolded convex] by auto
  2289 
  2290 subsection {* Convexity of general and special intervals. *}
  2291 
  2292 lemma is_interval_convex:
  2293   fixes s :: "(real ^ _) set"
  2294   assumes "is_interval s" shows "convex s"
  2295   unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
  2296   fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
  2297   hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
  2298   { fix a b assume "\<not> b \<le> u * a + v * b"
  2299     hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
  2300     hence "a < b" unfolding * using as(4) *(2) apply(rule_tac mult_left_less_imp_less[of "1 - v"]) by(auto simp add: field_simps)
  2301     hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
  2302   } moreover
  2303   { fix a b assume "\<not> u * a + v * b \<le> a"
  2304     hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
  2305     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: ring_simps)
  2306     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
  2307   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
  2308     using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
  2309 
  2310 lemma is_interval_connected:
  2311   fixes s :: "(real ^ _) set"
  2312   shows "is_interval s \<Longrightarrow> connected s"
  2313   using is_interval_convex convex_connected by auto
  2314 
  2315 lemma convex_interval: "convex {a .. b}" "convex {a<..<b::real^'n::finite}"
  2316   apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
  2317 
  2318 subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
  2319 
  2320 lemma is_interval_1:
  2321   "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b \<longrightarrow> x \<in> s)"
  2322   unfolding is_interval_def dest_vec1_def forall_1 by auto
  2323 
  2324 lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
  2325   apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
  2326   apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
  2327   fix a b x assume as:"connected s" "a \<in> s" "b \<in> s" "dest_vec1 a \<le> dest_vec1 x" "dest_vec1 x \<le> dest_vec1 b" "x\<notin>s"
  2328   hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
  2329   let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
  2330   { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
  2331     using as(6) `y\<in>s` by (auto simp add: inner_vector_def dest_vec1_eq [unfolded dest_vec1_def] dest_vec1_def) }
  2332   moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def dest_vec1_def)
  2333   hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
  2334   ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
  2335     apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
  2336     apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt) apply(rule, rule, rule ccontr)
  2337     by(auto simp add: basis_component field_simps) qed 
  2338 
  2339 lemma is_interval_convex_1:
  2340   "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
  2341   using is_interval_convex convex_connected is_interval_connected_1 by auto
  2342 
  2343 lemma convex_connected_1:
  2344   "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
  2345   using is_interval_convex convex_connected is_interval_connected_1 by auto
  2346 
  2347 subsection {* Another intermediate value theorem formulation. *}
  2348 
  2349 lemma ivt_increasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2350   assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f a)$k \<le> y" "y \<le> (f b)$k"
  2351   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2352 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
  2353     using assms(1) by(auto simp add: vector_less_eq_def dest_vec1_def)
  2354   thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
  2355     using connected_continuous_image[OF assms(2) convex_connected[OF convex_interval(1)]]
  2356     using assms by(auto intro!: imageI) qed
  2357 
  2358 lemma ivt_increasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2359   assumes "dest_vec1 a \<le> dest_vec1 b"
  2360   "\<forall>x\<in>{a .. b}. continuous (at x) f" "f a$k \<le> y" "y \<le> f b$k"
  2361   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2362   apply(rule ivt_increasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
  2363 
  2364 lemma ivt_decreasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2365   assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f b)$k \<le> y" "y \<le> (f a)$k"
  2366   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2367   apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
  2368   apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
  2369   by(auto simp add:vector_uminus_component)
  2370 
  2371 lemma ivt_decreasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2372   assumes "dest_vec1 a \<le> dest_vec1 b" "\<forall>x\<in>{a .. b}. continuous (at x) f" "f b$k \<le> y" "y \<le> f a$k"
  2373   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2374   apply(rule ivt_decreasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
  2375 
  2376 subsection {* A bound within a convex hull, and so an interval. *}
  2377 
  2378 lemma convex_on_convex_hull_bound:
  2379   fixes s :: "(real ^ _) set"
  2380   assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
  2381   shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
  2382   fix x assume "x\<in>convex hull s"
  2383   then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
  2384     unfolding convex_hull_indexed mem_Collect_eq by auto
  2385   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
  2386     unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
  2387     using assms(2) obt(1) by auto
  2388   thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
  2389     unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
  2390 
  2391 lemma unit_interval_convex_hull:
  2392   "{0::real^'n::finite .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
  2393 proof- have 01:"{0,1} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
  2394   { fix n x assume "x\<in>{0::real^'n .. 1}" "n \<le> CARD('n)" "card {i. x$i \<noteq> 0} \<le> n" 
  2395   hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
  2396     case 0 hence "x = 0" apply(subst Cart_eq) apply rule by auto
  2397     thus "x\<in>convex hull ?points" using 01 by auto
  2398   next
  2399     case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. x$i \<noteq> 0} = {}")
  2400       case True hence "x = 0" unfolding Cart_eq by auto
  2401       thus "x\<in>convex hull ?points" using 01 by auto
  2402     next
  2403       case False def xi \<equiv> "Min ((\<lambda>i. x$i) ` {i. x$i \<noteq> 0})"
  2404       have "xi \<in> (\<lambda>i. x$i) ` {i. x$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
  2405       then obtain i where i':"x$i = xi" "x$i \<noteq> 0" by auto
  2406       have i:"\<And>j. x$j > 0 \<Longrightarrow> x$i \<le> x$j"
  2407         unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
  2408         defer apply(rule_tac x=j in bexI) using i' by auto
  2409       have i01:"x$i \<le> 1" "x$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i] using i'(2) `x$i \<noteq> 0`
  2410         by(auto simp add: Cart_lambda_beta) 
  2411       show ?thesis proof(cases "x$i=1")
  2412         case True have "\<forall>j\<in>{i. x$i \<noteq> 0}. x$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq proof-
  2413           fix j assume "x $ j \<noteq> 0" "x $ j \<noteq> 1"
  2414           hence j:"x$j \<in> {0<..<1}" using Suc(2) by(auto simp add: vector_less_eq_def elim!:allE[where x=j])
  2415           hence "x$j \<in> op $ x ` {i. x $ i \<noteq> 0}" by auto 
  2416           hence "x$j \<ge> x$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
  2417           thus False using True Suc(2) j by(auto simp add: vector_less_eq_def elim!:ballE[where x=j]) qed
  2418         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
  2419           by(auto simp add: Cart_lambda_beta)
  2420       next let ?y = "\<lambda>j. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
  2421         case False hence *:"x = x$i *\<^sub>R (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *\<^sub>R (\<chi> j. ?y j)" unfolding Cart_eq
  2422           by(auto simp add: Cart_lambda_beta vector_add_component vector_smult_component vector_minus_component field_simps)
  2423         { fix j have "x$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $ j - x $ i) / (1 - x $ i)" "(x $ j - x $ i) / (1 - x $ i) \<le> 1"
  2424             apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
  2425             using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps Cart_lambda_beta) 
  2426           hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
  2427         moreover have "i\<in>{j. x$j \<noteq> 0} - {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" using i01 by(auto simp add: Cart_lambda_beta)
  2428         hence "{j. x$j \<noteq> 0} \<noteq> {j. ((\<chi> j. ?y j)::real^'n::finite) $ j \<noteq> 0}" by auto
  2429         hence **:"{j. ((\<chi> j. ?y j)::real^'n::finite) $ j \<noteq> 0} \<subset> {j. x$j \<noteq> 0}" apply - apply rule by(auto simp add: Cart_lambda_beta)  
  2430         have "card {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<le> n" using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
  2431         ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
  2432           apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
  2433           unfolding mem_interval using i01 Suc(3) by (auto simp add: Cart_lambda_beta)
  2434       qed qed qed } note * = this
  2435   show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
  2436     apply(rule_tac n2="CARD('n)" in *) prefer 3 apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
  2437     unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
  2438     by(auto simp add: vector_less_eq_def mem_def[of _ convex]) qed
  2439 
  2440 subsection {* And this is a finite set of vertices. *}
  2441 
  2442 lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. 1::real^'n::finite} = convex hull s"
  2443   apply(rule that[of "{x::real^'n::finite. \<forall>i. x$i=0 \<or> x$i=1}"])
  2444   apply(rule finite_subset[of _ "(\<lambda>s. (\<chi> i. if i\<in>s then 1::real else 0)::real^'n::finite) ` UNIV"])
  2445   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
  2446   fix x::"real^'n" assume as:"\<forall>i. x $ i = 0 \<or> x $ i = 1"
  2447   show "x \<in> (\<lambda>s. \<chi> i. if i \<in> s then 1 else 0) ` UNIV" apply(rule image_eqI[where x="{i. x$i = 1}"])
  2448     unfolding Cart_eq using as by(auto simp add:Cart_lambda_beta) qed auto
  2449 
  2450 subsection {* Hence any cube (could do any nonempty interval). *}
  2451 
  2452 lemma cube_convex_hull:
  2453   assumes "0 < d" obtains s::"(real^'n::finite) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" proof-
  2454   let ?d = "(\<chi> i. d)::real^'n"
  2455   have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. 1}" apply(rule set_ext, rule)
  2456     unfolding image_iff defer apply(erule bexE) proof-
  2457     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
  2458     { fix i::'n have "x $ i \<le> d + y $ i" "y $ i \<le> d + x $ i" using as[unfolded mem_interval, THEN spec[where x=i]]
  2459         by(auto simp add: vector_component)
  2460       hence "1 \<ge> inverse d * (x $ i - y $ i)" "1 \<ge> inverse d * (y $ i - x $ i)"
  2461         apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
  2462         using assms by(auto simp add: field_simps right_inverse) 
  2463       hence "inverse d * (x $ i * 2) \<le> 2 + inverse d * (y $ i * 2)"
  2464             "inverse d * (y $ i * 2) \<le> 2 + inverse d * (x $ i * 2)" by(auto simp add:field_simps) }
  2465     hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
  2466       by(auto simp add: Cart_eq vector_component_simps field_simps)
  2467     thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI) 
  2468       using assms by(auto simp add: Cart_eq vector_less_eq_def Cart_lambda_beta)
  2469   next
  2470     fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *\<^sub>R z" 
  2471     have "\<And>i. 0 \<le> d * z $ i \<and> d * z $ i \<le> d" using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
  2472       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
  2473       using assms by(auto simp add: vector_component_simps Cart_eq)
  2474     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
  2475       apply(erule_tac x=i in allE) using assms by(auto simp add:  vector_component_simps Cart_eq) qed
  2476   obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
  2477   thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
  2478 
  2479 subsection {* Bounded convex function on open set is continuous. *}
  2480 
  2481 lemma convex_on_bounded_continuous:
  2482   fixes s :: "(real ^ _) set"
  2483   assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
  2484   shows "continuous_on s f"
  2485   apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
  2486   fix x e assume "x\<in>s" "(0::real) < e"
  2487   def B \<equiv> "abs b + 1"
  2488   have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
  2489     unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
  2490   obtain k where "k>0"and k:"cball x k \<subseteq> s" using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]] using `x\<in>s` by auto
  2491   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
  2492     apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
  2493     fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)" 
  2494     show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
  2495       case False def t \<equiv> "k / norm (y - x)"
  2496       have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
  2497       have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
  2498         apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute) 
  2499       { def w \<equiv> "x + t *\<^sub>R (y - x)"
  2500         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  2501           unfolding t_def using `k>0` by auto
  2502         have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x" by (auto simp add: algebra_simps)
  2503         also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps)
  2504         finally have w:"(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
  2505         have  "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps) 
  2506         hence "(f w - f x) / t < e"
  2507           using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps) 
  2508         hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
  2509           using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
  2510           using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
  2511       moreover 
  2512       { def w \<equiv> "x - t *\<^sub>R (y - x)"
  2513         have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  2514           unfolding t_def using `k>0` by auto
  2515         have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x" by (auto simp add: algebra_simps)
  2516         also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
  2517         finally have w:"(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
  2518         have  "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps) 
  2519         hence *:"(f w - f y) / t < e" using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`] using `t>0` by(auto simp add:field_simps) 
  2520         have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y" 
  2521           using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
  2522           using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
  2523         also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding real_divide_def by (auto simp add:field_simps)
  2524         also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
  2525         finally have "f x - f y < e" by auto }
  2526       ultimately show ?thesis by auto 
  2527     qed(insert `0<e`, auto) 
  2528   qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
  2529 
  2530 subsection {* Upper bound on a ball implies upper and lower bounds. *}
  2531 
  2532 lemma convex_bounds_lemma:
  2533   fixes x :: "real ^ _"
  2534   assumes "convex_on (cball x e) f"  "\<forall>y \<in> cball x e. f y \<le> b"
  2535   shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
  2536   apply(rule) proof(cases "0 \<le> e") case True
  2537   fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *\<^sub>R x - y"
  2538   have *:"x - (2 *\<^sub>R x - y) = y - x" by vector
  2539   have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
  2540   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
  2541   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
  2542     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
  2543 next case False fix y assume "y\<in>cball x e" 
  2544   hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
  2545   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
  2546 
  2547 subsection {* Hence a convex function on an open set is continuous. *}
  2548 
  2549 lemma convex_on_continuous:
  2550   assumes "open (s::(real^'n::finite) set)" "convex_on s f" 
  2551   shows "continuous_on s f"
  2552   unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
  2553   note dimge1 = dimindex_ge_1[where 'a='n]
  2554   fix x assume "x\<in>s"
  2555   then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
  2556   def d \<equiv> "e / real CARD('n)"
  2557   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
  2558   let ?d = "(\<chi> i. d)::real^'n"
  2559   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
  2560   have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:vector_component_simps)
  2561   hence "c\<noteq>{}" apply(rule_tac ccontr) using c by(auto simp add:convex_hull_empty)
  2562   def k \<equiv> "Max (f ` c)"
  2563   have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
  2564     apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
  2565     fix z assume z:"z\<in>{x - ?d..x + ?d}"
  2566     have e:"e = setsum (\<lambda>i. d) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
  2567       by (metis card_enum field_simps d_def not_one_le_zero of_nat_le_iff real_eq_of_nat real_of_nat_1)
  2568     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
  2569       using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:field_simps vector_component_simps) qed
  2570   hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
  2571     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
  2572   have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 using real_dimindex_ge_1 by auto
  2573   hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
  2574   have conv:"convex_on (cball x d) f" apply(rule convex_on_subset, rule convex_on_subset[OF assms(2)]) apply(rule e(1)) using dsube by auto
  2575   hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
  2576     fix y assume y:"y\<in>cball x d"
  2577     { fix i::'n have "x $ i - d \<le> y $ i"  "y $ i \<le> x $ i + d" 
  2578         using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add: vector_component)  }
  2579     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
  2580       by(auto simp add: vector_component_simps) qed
  2581   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
  2582     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
  2583     apply force
  2584     done
  2585   thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
  2586     using `d>0` by auto 
  2587 qed
  2588 
  2589 subsection {* Line segments, Starlike Sets, etc.*}
  2590 
  2591 (* Use the same overloading tricks as for intervals, so that 
  2592    segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
  2593 
  2594 definition
  2595   midpoint :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 'n" where
  2596   "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
  2597 
  2598 definition
  2599   open_segment :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> (real ^ 'n) set" where
  2600   "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real.  0 < u \<and> u < 1}"
  2601 
  2602 definition
  2603   closed_segment :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> (real ^ 'n) set" where
  2604   "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
  2605 
  2606 definition "between = (\<lambda> (a,b). closed_segment a b)"
  2607 
  2608 lemmas segment = open_segment_def closed_segment_def
  2609 
  2610 definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
  2611 
  2612 lemma midpoint_refl: "midpoint x x = x"
  2613   unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto
  2614 
  2615 lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
  2616 
  2617 lemma dist_midpoint:
  2618   "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
  2619   "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
  2620   "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
  2621   "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
  2622 proof-
  2623   have *: "\<And>x y::real^'n::finite. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
  2624   have **:"\<And>x y::real^'n::finite. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2" by auto
  2625   note scaleR_right_distrib [simp]
  2626   show ?t1 unfolding midpoint_def dist_norm apply (rule **) by(auto,vector)
  2627   show ?t2 unfolding midpoint_def dist_norm apply (rule *)  by(auto,vector)
  2628   show ?t3 unfolding midpoint_def dist_norm apply (rule *)  by(auto,vector)
  2629   show ?t4 unfolding midpoint_def dist_norm apply (rule **) by(auto,vector) qed
  2630 
  2631 lemma midpoint_eq_endpoint:
  2632   "midpoint a b = a \<longleftrightarrow> a = (b::real^'n::finite)"
  2633   "midpoint a b = b \<longleftrightarrow> a = b"
  2634   unfolding dist_eq_0_iff[where 'a="real^'n", THEN sym] dist_midpoint by auto
  2635 
  2636 lemma convex_contains_segment:
  2637   "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. closed_segment a b \<subseteq> s)"
  2638   unfolding convex_alt closed_segment_def by auto
  2639 
  2640 lemma convex_imp_starlike:
  2641   "convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
  2642   unfolding convex_contains_segment starlike_def by auto
  2643 
  2644 lemma segment_convex_hull:
  2645  "closed_segment a b = convex hull {a,b}" proof-
  2646   have *:"\<And>x. {x} \<noteq> {}" by auto
  2647   have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
  2648   show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_ext)
  2649     unfolding mem_Collect_eq apply(rule,erule exE) 
  2650     apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
  2651     apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
  2652 
  2653 lemma convex_segment: "convex (closed_segment a b)"
  2654   unfolding segment_convex_hull by(rule convex_convex_hull)
  2655 
  2656 lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
  2657   unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto
  2658 
  2659 lemma segment_furthest_le:
  2660   assumes "x \<in> closed_segment a b" shows "norm(y - x) \<le> norm(y - a) \<or>  norm(y - x) \<le> norm(y - b)" proof-
  2661   obtain z where "z\<in>{a, b}" "norm (x - y) \<le> norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
  2662     using assms[unfolded segment_convex_hull] by auto
  2663   thus ?thesis by(auto simp add:norm_minus_commute) qed
  2664 
  2665 lemma segment_bound:
  2666   assumes "x \<in> closed_segment a b"
  2667   shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
  2668   using segment_furthest_le[OF assms, of a]
  2669   using segment_furthest_le[OF assms, of b]
  2670   by (auto simp add:norm_minus_commute) 
  2671 
  2672 lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
  2673 
  2674 lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
  2675   unfolding between_def mem_def by auto
  2676 
  2677 lemma between:"between (a,b) (x::real^'n::finite) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
  2678 proof(cases "a = b")
  2679   case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
  2680     by(auto simp add:segment_refl dist_commute) next
  2681   case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto 
  2682   have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
  2683   show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
  2684     apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
  2685       fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" 
  2686       hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
  2687         unfolding as(1) by(auto simp add:algebra_simps)
  2688       show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
  2689         unfolding norm_minus_commute[of x a] * Cart_eq using as(2,3)
  2690         by(auto simp add: vector_component_simps field_simps)
  2691     next assume as:"dist a b = dist a x + dist x b"
  2692       have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2] unfolding as[unfolded dist_norm] norm_ge_zero by auto 
  2693       thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
  2694         unfolding dist_norm Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
  2695           fix i::'n have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i =
  2696             ((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"
  2697             using Fal by(auto simp add:vector_component_simps field_simps)
  2698           also have "\<dots> = x$i" apply(rule divide_eq_imp[OF Fal])
  2699             unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
  2700             by(auto simp add:field_simps vector_component_simps)
  2701           finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i" by auto
  2702         qed(insert Fal2, auto) qed qed
  2703 
  2704 lemma between_midpoint: fixes a::"real^'n::finite" shows
  2705   "between (a,b) (midpoint a b)" (is ?t1) 
  2706   "between (b,a) (midpoint a b)" (is ?t2)
  2707 proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
  2708   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
  2709     by(auto simp add:field_simps Cart_eq vector_component_simps) qed
  2710 
  2711 lemma between_mem_convex_hull:
  2712   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
  2713   unfolding between_mem_segment segment_convex_hull ..
  2714 
  2715 subsection {* Shrinking towards the interior of a convex set. *}
  2716 
  2717 lemma mem_interior_convex_shrink:
  2718   fixes s :: "(real ^ _) set"
  2719   assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
  2720   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  2721 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  2722   show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
  2723     apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
  2724     fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
  2725     have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
  2726     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)"
  2727       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule norm_eqI) using `e>0`
  2728       by(auto simp add:vector_component_simps Cart_eq field_simps) 
  2729     also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:norm_eqI simp add: algebra_simps)
  2730     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
  2731       by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
  2732     finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
  2733       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
  2734   qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
  2735 
  2736 lemma mem_interior_closure_convex_shrink:
  2737   fixes s :: "(real ^ _) set"
  2738   assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
  2739   shows "x - e *\<^sub>R (x - c) \<in> interior s"
  2740 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  2741   have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
  2742     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  2743     case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
  2744     show ?thesis proof(cases "e=1")
  2745       case True obtain y where "y\<in>s" "y \<noteq> x" "dist y x < 1"
  2746         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  2747       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  2748       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  2749         using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  2750       then obtain y where "y\<in>s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
  2751         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  2752       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  2753   then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
  2754   def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
  2755   have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
  2756   have "z\<in>interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
  2757     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
  2758     by(auto simp add:field_simps norm_minus_commute)
  2759   thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink) 
  2760     using assms(1,4-5) `y\<in>s` by auto qed
  2761 
  2762 subsection {* Some obvious but surprisingly hard simplex lemmas. *}
  2763 
  2764 lemma simplex:
  2765   assumes "finite s" "0 \<notin> s"
  2766   shows "convex hull (insert 0 s) =  { y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y)}"
  2767   unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_ext, rule) unfolding mem_Collect_eq
  2768   apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
  2769   apply(rule_tac x=u in exI) defer apply(rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI) using assms(2)
  2770   unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
  2771 
  2772 lemma std_simplex:
  2773   "convex hull (insert 0 { basis i | i. i\<in>UNIV}) =
  2774         {x::real^'n::finite . (\<forall>i. 0 \<le> x$i) \<and> setsum (\<lambda>i. x$i) UNIV \<le> 1 }" (is "convex hull (insert 0 ?p) = ?s")
  2775 proof- let ?D = "UNIV::'n set"
  2776   have "0\<notin>?p" by(auto simp add: basis_nonzero)
  2777   have "{(basis i)::real^'n |i. i \<in> ?D} = basis ` ?D" by auto
  2778   note sumbas = this  setsum_reindex[OF basis_inj, unfolded o_def]
  2779   show ?thesis unfolding simplex[OF finite_stdbasis `0\<notin>?p`] apply(rule set_ext) unfolding mem_Collect_eq apply rule
  2780     apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
  2781     fix x::"real^'n" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x" "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
  2782     have *:"\<forall>i. u (basis i) = x$i" using as(3) unfolding sumbas and basis_expansion_unique [where 'a=real, unfolded smult_conv_scaleR] by auto
  2783     hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $ x) ?D" unfolding sumbas by(rule_tac setsum_cong, auto)
  2784     show " (\<forall>i. 0 \<le> x $ i) \<and> setsum (op $ x) ?D \<le> 1" apply - proof(rule,rule)
  2785       fix i::'n show "0 \<le> x$i" unfolding *[rule_format,of i,THEN sym] apply(rule_tac as(1)[rule_format]) by auto
  2786     qed(insert as(2)[unfolded **], auto)
  2787   next fix x::"real^'n" assume as:"\<forall>i. 0 \<le> x $ i" "setsum (op $ x) ?D \<le> 1"
  2788     show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and> setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
  2789       apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE) using as(1) apply(erule_tac x=i in allE) 
  2790       unfolding sumbas using as(2) and basis_expansion_unique [where 'a=real, unfolded smult_conv_scaleR] by(auto simp add:inner_basis) qed qed 
  2791 
  2792 lemma interior_std_simplex:
  2793   "interior (convex hull (insert 0 { basis i| i. i\<in>UNIV})) =
  2794   {x::real^'n::finite. (\<forall>i. 0 < x$i) \<and> setsum (\<lambda>i. x$i) UNIV < 1 }"
  2795   apply(rule set_ext) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
  2796   unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
  2797   fix x::"real^'n" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x. 0 \<le> xa $ x) \<and> setsum (op $ xa) UNIV \<le> 1"
  2798   show "(\<forall>xa. 0 < x $ xa) \<and> setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
  2799     fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
  2800       unfolding dist_norm by(auto simp add: norm_basis vector_component_simps basis_component elim:allE[where x=i])
  2801   next guess a using UNIV_witness[where 'a='n] ..
  2802     have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e" using  `e>0` and norm_basis[of a]
  2803       unfolding dist_norm by(auto simp add: vector_component_simps basis_component intro!: mult_strict_left_mono_comm)
  2804     have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
  2805     hence *:"setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto) 
  2806     have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV" unfolding * setsum_addf
  2807       using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
  2808     also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
  2809     finally show "setsum (op $ x) UNIV < 1" by auto qed
  2810 next
  2811   fix x::"real^'n::finite" assume as:"\<forall>i. 0 < x $ i" "setsum (op $ x) UNIV < 1"
  2812   guess a using UNIV_witness[where 'a='b] ..
  2813   let ?d = "(1 - setsum (op $ x) UNIV) / real (CARD('n))"
  2814   have "Min ((op $ x) ` UNIV) > 0" apply(rule Min_grI) using as(1) dimindex_ge_1 by auto
  2815   moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) using dimindex_ge_1 by(auto simp add: Suc_le_eq)
  2816   ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1"
  2817     apply(rule_tac x="min (Min ((op $ x) ` UNIV)) ?D" in exI) apply rule defer apply(rule,rule) proof-
  2818     fix y assume y:"dist x y < min (Min (op $ x ` UNIV)) ?d"
  2819     have "setsum (op $ y) UNIV \<le> setsum (\<lambda>i. x$i + ?d) UNIV" proof(rule setsum_mono)
  2820       fix i::'n have "abs (y$i - x$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
  2821         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add:vector_component_simps norm_minus_commute)
  2822       thus "y $ i \<le> x $ i + ?d" by auto qed
  2823     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat using dimindex_ge_1 by(auto simp add: Suc_le_eq)
  2824     finally show "(\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1" apply- proof(rule,rule)
  2825       fix i::'n have "norm (x - y) < x$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
  2826         using Min_gr_iff[of "op $ x ` dimset x"] dimindex_ge_1 by auto
  2827       thus "0 \<le> y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by(auto simp add: vector_component_simps)
  2828     qed auto qed auto qed
  2829 
  2830 lemma interior_std_simplex_nonempty: obtains a::"real^'n::finite" where
  2831   "a \<in> interior(convex hull (insert 0 {basis i | i . i \<in> UNIV}))" proof-
  2832   let ?D = "UNIV::'n set" let ?a = "setsum (\<lambda>b::real^'n. inverse (2 * real CARD('n)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
  2833   have *:"{basis i :: real ^ 'n | i. i \<in> ?D} = basis ` ?D" by auto
  2834   { fix i have "?a $ i = inverse (2 * real CARD('n))"
  2835     unfolding setsum_component vector_smult_component and * and setsum_reindex[OF basis_inj] and o_def
  2836     apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real CARD('n)) else 0) ?D"]) apply(rule setsum_cong2)
  2837       unfolding setsum_delta'[OF finite_UNIV[where 'a='n]] and real_dimindex_ge_1[where 'n='n] by(auto simp add: basis_component[of i]) }
  2838   note ** = this
  2839   show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof(rule,rule)
  2840     fix i::'n show "0 < ?a $ i" unfolding ** using dimindex_ge_1 by(auto simp add: Suc_le_eq) next
  2841     have "setsum (op $ ?a) ?D = setsum (\<lambda>i. inverse (2 * real CARD('n))) ?D" by(rule setsum_cong2, rule **) 
  2842     also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat real_divide_def[THEN sym] by (auto simp add:field_simps)
  2843     finally show "setsum (op $ ?a) ?D < 1" by auto qed qed
  2844 
  2845 subsection {* Paths. *}
  2846 
  2847 definition "path (g::real^1 \<Rightarrow> real^'n::finite) \<longleftrightarrow> continuous_on {0 .. 1} g"
  2848 
  2849 definition "pathstart (g::real^1 \<Rightarrow> real^'n) = g 0"
  2850 
  2851 definition "pathfinish (g::real^1 \<Rightarrow> real^'n) = g 1"
  2852 
  2853 definition "path_image (g::real^1 \<Rightarrow> real^'n) = g ` {0 .. 1}"
  2854 
  2855 definition "reversepath (g::real^1 \<Rightarrow> real^'n) = (\<lambda>x. g(1 - x))"
  2856 
  2857 definition joinpaths:: "(real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n)" (infixr "+++" 75)
  2858   where "joinpaths g1 g2 = (\<lambda>x. if dest_vec1 x \<le> ((1 / 2)::real) then g1 (2 *\<^sub>R x) else g2(2 *\<^sub>R x - 1))"
  2859 definition "simple_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
  2860   (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
  2861 
  2862 definition "injective_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
  2863   (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)"
  2864 
  2865 subsection {* Some lemmas about these concepts. *}
  2866 
  2867 lemma injective_imp_simple_path:
  2868   "injective_path g \<Longrightarrow> simple_path g"
  2869   unfolding injective_path_def simple_path_def by auto
  2870 
  2871 lemma path_image_nonempty: "path_image g \<noteq> {}"
  2872   unfolding path_image_def image_is_empty interval_eq_empty by auto 
  2873 
  2874 lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g"
  2875   unfolding pathstart_def path_image_def apply(rule imageI)
  2876   unfolding mem_interval_1 vec_1[THEN sym] dest_vec1_vec by auto
  2877 
  2878 lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g"
  2879   unfolding pathfinish_def path_image_def apply(rule imageI)
  2880   unfolding mem_interval_1 vec_1[THEN sym] dest_vec1_vec by auto
  2881 
  2882 lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"
  2883   unfolding path_def path_image_def apply(rule connected_continuous_image, assumption)
  2884   by(rule convex_connected, rule convex_interval)
  2885 
  2886 lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"
  2887   unfolding path_def path_image_def apply(rule compact_continuous_image, assumption)
  2888   by(rule compact_interval)
  2889 
  2890 lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g"
  2891   unfolding reversepath_def by auto
  2892 
  2893 lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g"
  2894   unfolding pathstart_def reversepath_def pathfinish_def by auto
  2895 
  2896 lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"
  2897   unfolding pathstart_def reversepath_def pathfinish_def by auto
  2898 
  2899 lemma pathstart_join[simp]: "pathstart(g1 +++ g2) = pathstart g1"
  2900   unfolding pathstart_def joinpaths_def pathfinish_def by auto
  2901 
  2902 lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2" proof-
  2903   have "2 *\<^sub>R 1 - 1 = (1::real^1)" unfolding Cart_eq by(auto simp add:vector_component_simps)
  2904   thus ?thesis unfolding pathstart_def joinpaths_def pathfinish_def
  2905     unfolding vec_1[THEN sym] dest_vec1_vec by auto qed
  2906 
  2907 lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g" proof-
  2908   have *:"\<And>g. path_image(reversepath g) \<subseteq> path_image g"
  2909     unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply(rule,rule,erule bexE)  
  2910     apply(rule_tac x="1 - xa" in bexI) by(auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
  2911   show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
  2912 
  2913 lemma path_reversepath[simp]: "path(reversepath g) \<longleftrightarrow> path g" proof-
  2914   have *:"\<And>g. path g \<Longrightarrow> path(reversepath g)" unfolding path_def reversepath_def
  2915     apply(rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
  2916     apply(rule continuous_on_sub, rule continuous_on_const, rule continuous_on_id)
  2917     apply(rule continuous_on_subset[of "{0..1}"], assumption)
  2918     by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
  2919   show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
  2920 
  2921 lemmas reversepath_simps = path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
  2922 
  2923 lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow>  path g1 \<and> path g2"
  2924   unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof-
  2925   assume as:"continuous_on {0..1} (g1 +++ g2)"
  2926   have *:"g1 = (\<lambda>x. g1 (2 *\<^sub>R x)) \<circ> (\<lambda>x. (1/2) *\<^sub>R x)" 
  2927          "g2 = (\<lambda>x. g2 (2 *\<^sub>R x - 1)) \<circ> (\<lambda>x. (1/2) *\<^sub>R (x + 1))" unfolding o_def by auto
  2928   have "op *\<^sub>R (1 / 2) ` {0::real^1..1} \<subseteq> {0..1}"  "(\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {(0::real^1)..1} \<subseteq> {0..1}"
  2929     unfolding image_smult_interval by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
  2930   thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule
  2931     apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose)
  2932     apply (rule continuous_on_cmul, rule continuous_on_add, rule continuous_on_id, rule continuous_on_const) defer
  2933     apply (rule continuous_on_cmul, rule continuous_on_id) apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3
  2934     apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption)
  2935     apply(rule) defer apply rule proof-
  2936     fix x assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real^1..1}"
  2937     hence "dest_vec1 x \<le> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
  2938     thus "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto next
  2939     fix x assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real^1..1}"
  2940     hence "dest_vec1 x \<ge> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
  2941     thus "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)" proof(cases "dest_vec1 x = 1 / 2")
  2942       case True hence "x = (1/2) *\<^sub>R 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
  2943       thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by auto
  2944     qed (auto simp add:le_less joinpaths_def) qed
  2945 next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"
  2946   have *:"{0 .. 1::real^1} = {0.. (1/2)*\<^sub>R 1} \<union> {(1/2) *\<^sub>R 1 .. 1}" by(auto simp add: vector_component_simps) 
  2947   have **:"op *\<^sub>R 2 ` {0..(1 / 2) *\<^sub>R 1} = {0..1::real^1}" apply(rule set_ext, rule) unfolding image_iff 
  2948     defer apply(rule_tac x="(1/2)*\<^sub>R x" in bexI) by(auto simp add: vector_component_simps)
  2949   have ***:"(\<lambda>x. 2 *\<^sub>R x - 1) ` {(1 / 2) *\<^sub>R 1..1} = {0..1::real^1}"
  2950     unfolding image_affinity_interval[of _ "- 1", unfolded diff_def[symmetric]] and interval_eq_empty_1
  2951     by(auto simp add: vector_component_simps)
  2952   have ****:"\<And>x::real^1. x $ 1 * 2 = 1 \<longleftrightarrow> x = (1/2) *\<^sub>R 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
  2953   show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply(rule closed_interval)+ proof-
  2954     show "continuous_on {0..(1 / 2) *\<^sub>R 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *\<^sub>R x)"]) defer
  2955       unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply(rule continuous_on_cmul, rule continuous_on_id)
  2956       unfolding ** apply(rule as(1)) unfolding joinpaths_def by(auto simp add: vector_component_simps) next
  2957     show "continuous_on {(1/2)*\<^sub>R1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *\<^sub>R x - 1)"]) defer
  2958       apply(rule continuous_on_compose) apply(rule continuous_on_sub, rule continuous_on_cmul, rule continuous_on_id, rule continuous_on_const)
  2959       unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def]
  2960       by(auto simp add: vector_component_simps ****) qed qed
  2961 
  2962 lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof
  2963   fix x assume "x \<in> path_image (g1 +++ g2)"
  2964   then obtain y where y:"y\<in>{0..1}" "x = (if dest_vec1 y \<le> 1 / 2 then g1 (2 *\<^sub>R y) else g2 (2 *\<^sub>R y - 1))"
  2965     unfolding path_image_def image_iff joinpaths_def by auto
  2966   thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "dest_vec1 y \<le> 1/2")
  2967     apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1)
  2968     by(auto intro!: imageI simp add: vector_component_simps) qed
  2969 
  2970 lemma subset_path_image_join:
  2971   assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s" shows "path_image(g1 +++ g2) \<subseteq> s"
  2972   using path_image_join_subset[of g1 g2] and assms by auto
  2973 
  2974 lemma path_image_join:
  2975   assumes "path g1" "path g2" "pathfinish g1 = pathstart g2"
  2976   shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"
  2977 apply(rule, rule path_image_join_subset, rule) unfolding Un_iff proof(erule disjE)
  2978   fix x assume "x \<in> path_image g1"
  2979   then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto
  2980   thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
  2981     apply(rule_tac x="(1/2) *\<^sub>R y" in bexI) by(auto simp add: vector_component_simps) next
  2982   fix x assume "x \<in> path_image g2"
  2983   then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto
  2984   moreover have *:"y $ 1 = 0 \<Longrightarrow> y = 0" unfolding Cart_eq by auto
  2985   ultimately show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
  2986     apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]
  2987     by(auto simp add: vector_component_simps) qed 
  2988 
  2989 lemma not_in_path_image_join:
  2990   assumes "x \<notin> path_image g1" "x \<notin> path_image g2" shows "x \<notin> path_image(g1 +++ g2)"
  2991   using assms and path_image_join_subset[of g1 g2] by auto
  2992 
  2993 lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)"
  2994   using assms unfolding simple_path_def reversepath_def apply- apply(rule ballI)+
  2995   apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
  2996   unfolding mem_interval_1 by(auto simp add:vector_component_simps)
  2997 
  2998 lemma dest_vec1_scaleR [simp]:
  2999   "dest_vec1 (scaleR a x) = scaleR a (dest_vec1 x)"
  3000 unfolding dest_vec1_def by simp
  3001 
  3002 lemma simple_path_join_loop:
  3003   assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
  3004   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
  3005   shows "simple_path(g1 +++ g2)"
  3006 unfolding simple_path_def proof((rule ballI)+, rule impI) let ?g = "g1 +++ g2"
  3007   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
  3008   fix x y::"real^1" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
  3009   show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" proof(case_tac "x$1 \<le> 1/2",case_tac[!] "y$1 \<le> 1/2", unfold not_le)
  3010     assume as:"x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2"
  3011     hence "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
  3012     moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as
  3013       unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
  3014     ultimately show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
  3015   next assume as:"x $ 1 > 1 / 2" "y $ 1 > 1 / 2"
  3016     hence "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
  3017     moreover have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}" using xy(1,2) as
  3018       unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
  3019     ultimately show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
  3020   next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2"
  3021     hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
  3022       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  3023     moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
  3024       using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)[unfolded mem_interval_1]
  3025       apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
  3026     ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
  3027     hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)[unfolded mem_interval_1]
  3028       using inj(1)[of "2 *\<^sub>R x" 0] by(auto simp add:vector_component_simps)
  3029     moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
  3030       unfolding joinpaths_def pathfinish_def using as(2) and xy(2)[unfolded mem_interval_1]
  3031       using inj(2)[of "2 *\<^sub>R y - 1" 1] by (auto simp add:vector_component_simps Cart_eq)
  3032     ultimately show ?thesis by auto 
  3033   next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2"
  3034     hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
  3035       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  3036     moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
  3037       using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)[unfolded mem_interval_1]
  3038       apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
  3039     ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
  3040     hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)[unfolded mem_interval_1]
  3041       using inj(1)[of "2 *\<^sub>R y" 0] by(auto simp add:vector_component_simps)
  3042     moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
  3043       unfolding joinpaths_def pathfinish_def using as(1) and xy(1)[unfolded mem_interval_1]
  3044       using inj(2)[of "2 *\<^sub>R x - 1" 1] by(auto simp add:vector_component_simps Cart_eq)
  3045     ultimately show ?thesis by auto qed qed
  3046 
  3047 lemma injective_path_join:
  3048   assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
  3049   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
  3050   shows "injective_path(g1 +++ g2)"
  3051   unfolding injective_path_def proof(rule,rule,rule) let ?g = "g1 +++ g2"
  3052   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
  3053   fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
  3054   show "x = y" proof(cases "x$1 \<le> 1/2", case_tac[!] "y$1 \<le> 1/2", unfold not_le)
  3055     assume "x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2" thus ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy
  3056       unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
  3057   next assume "x $ 1 > 1 / 2" "y $ 1 > 1 / 2" thus ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy
  3058       unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
  3059   next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2" 
  3060     hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
  3061       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  3062     hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto
  3063     thus ?thesis using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2)
  3064       unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
  3065       by(auto simp add:vector_component_simps Cart_eq forall_1)
  3066   next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2" 
  3067     hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
  3068       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  3069     hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto
  3070     thus ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2)
  3071       unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
  3072       by(auto simp add:vector_component_simps forall_1 Cart_eq) qed qed
  3073 
  3074 lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
  3075  
  3076 subsection {* Reparametrizing a closed curve to start at some chosen point. *}
  3077 
  3078 definition "shiftpath a (f::real^1 \<Rightarrow> real^'n) =
  3079   (\<lambda>x. if dest_vec1 (a + x) \<le> 1 then f(a + x) else f(a + x - 1))"
  3080 
  3081 lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"
  3082   unfolding pathstart_def shiftpath_def by auto
  3083 
  3084 (** move this **)
  3085 declare forall_1[simp] ex_1[simp]
  3086 
  3087 lemma pathfinish_shiftpath: assumes "0 \<le> a" "pathfinish g = pathstart g"
  3088   shows "pathfinish(shiftpath a g) = g a"
  3089   using assms unfolding pathstart_def pathfinish_def shiftpath_def
  3090   by(auto simp add: vector_component_simps)
  3091 
  3092 lemma endpoints_shiftpath:
  3093   assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}" 
  3094   shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"
  3095   using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)
  3096 
  3097 lemma closed_shiftpath:
  3098   assumes "pathfinish g = pathstart g" "a \<in> {0..1}"
  3099   shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)"
  3100   using endpoints_shiftpath[OF assms] by auto
  3101 
  3102 lemma path_shiftpath:
  3103   assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
  3104   shows "path(shiftpath a g)" proof-
  3105   have *:"{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by(auto simp add: vector_component_simps)
  3106   have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
  3107     using assms(2)[unfolded pathfinish_def pathstart_def] by auto
  3108   show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union)
  3109     apply(rule closed_interval)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
  3110     apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
  3111     apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+
  3112     apply(rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
  3113     using assms(3) and ** by(auto simp add:vector_component_simps field_simps Cart_eq) qed
  3114 
  3115 lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" 
  3116   shows "shiftpath (1 - a) (shiftpath a g) x = g x"
  3117   using assms unfolding pathfinish_def pathstart_def shiftpath_def 
  3118   by(auto simp add: vector_component_simps)
  3119 
  3120 lemma path_image_shiftpath:
  3121   assumes "a \<in> {0..1}" "pathfinish g = pathstart g"
  3122   shows "path_image(shiftpath a g) = path_image g" proof-
  3123   { fix x assume as:"g 1 = g 0" "x \<in> {0..1::real^1}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a $ 1 + x $ 1 \<le> 1}. g x \<noteq> g (a + y - 1)" 
  3124     hence "\<exists>y\<in>{0..1} \<inter> {x. a $ 1 + x $ 1 \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x")
  3125       case False thus ?thesis apply(rule_tac x="1 + x - a" in bexI)
  3126         using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
  3127         by(auto simp add:vector_component_simps field_simps atomize_not) next
  3128       case True thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
  3129         by(auto simp add:vector_component_simps field_simps) qed }
  3130   thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def 
  3131     by(auto simp add:vector_component_simps image_iff) qed
  3132 
  3133 subsection {* Special case of straight-line paths. *}
  3134 
  3135 definition
  3136   linepath :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 1 \<Rightarrow> real ^ 'n" where
  3137   "linepath a b = (\<lambda>x. (1 - dest_vec1 x) *\<^sub>R a + dest_vec1 x *\<^sub>R b)"
  3138 
  3139 lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
  3140   unfolding pathstart_def linepath_def by auto
  3141 
  3142 lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b"
  3143   unfolding pathfinish_def linepath_def by auto
  3144 
  3145 lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
  3146   unfolding linepath_def
  3147   by (intro continuous_intros continuous_dest_vec1)
  3148 
  3149 lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
  3150   using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
  3151 
  3152 lemma path_linepath[intro]: "path(linepath a b)"
  3153   unfolding path_def by(rule continuous_on_linepath)
  3154 
  3155 lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
  3156   unfolding path_image_def segment linepath_def apply (rule set_ext, rule) defer
  3157   unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *\<^sub>R 1" in bexI)
  3158   by(auto simp add:vector_component_simps)
  3159 
  3160 lemma reversepath_linepath[simp]:  "reversepath(linepath a b) = linepath b a"
  3161   unfolding reversepath_def linepath_def by(rule ext, auto simp add:vector_component_simps)
  3162 
  3163 lemma injective_path_linepath: assumes "a \<noteq> b" shows "injective_path(linepath a b)" proof- 
  3164   { obtain i where i:"a$i \<noteq> b$i" using assms[unfolded Cart_eq] by auto
  3165     fix x y::"real^1" assume "x $ 1 *\<^sub>R b + y $ 1 *\<^sub>R a = x $ 1 *\<^sub>R a + y $ 1 *\<^sub>R b"
  3166     hence "x$1 * (b$i - a$i) = y$1 * (b$i - a$i)" unfolding Cart_eq by(auto simp add:field_simps vector_component_simps)
  3167     hence "x = y" unfolding mult_cancel_right Cart_eq using i(1) by(auto simp add:field_simps) }
  3168   thus ?thesis unfolding injective_path_def linepath_def by(auto simp add:vector_component_simps algebra_simps) qed
  3169 
  3170 lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath)
  3171 
  3172 subsection {* Bounding a point away from a path. *}
  3173 
  3174 lemma not_on_path_ball: assumes "path g" "z \<notin> path_image g"
  3175   shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof-
  3176   obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y"
  3177     using distance_attains_inf[OF _ path_image_nonempty, of g z]
  3178     using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
  3179   thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed
  3180 
  3181 lemma not_on_path_cball: assumes "path g" "z \<notin> path_image g"
  3182   shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof-
  3183   obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto
  3184   moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto
  3185   ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed
  3186 
  3187 subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
  3188 
  3189 definition "path_component s x y \<longleftrightarrow> (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
  3190 
  3191 lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def 
  3192 
  3193 lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s"
  3194   using assms unfolding path_defs by auto
  3195 
  3196 lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x"
  3197   unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms 
  3198   by(auto intro!:continuous_on_intros)    
  3199 
  3200 lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
  3201   by(auto intro!: path_component_mem path_component_refl) 
  3202 
  3203 lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
  3204   using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI) 
  3205   by(auto simp add: reversepath_simps)
  3206 
  3207 lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z"
  3208   using assms unfolding path_component_def apply- apply(erule exE)+ apply(rule_tac x="g +++ ga" in exI) by(auto simp add: path_image_join)
  3209 
  3210 lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow>  path_component s x y \<Longrightarrow> path_component t x y"
  3211   unfolding path_component_def by auto
  3212 
  3213 subsection {* Can also consider it as a set, as the name suggests. *}
  3214 
  3215 lemma path_component_set: "path_component s x = { y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y )}"
  3216   apply(rule set_ext) unfolding mem_Collect_eq unfolding mem_def path_component_def by auto
  3217 
  3218 lemma mem_path_component_set:"x \<in> path_component s y \<longleftrightarrow> path_component s y x" unfolding mem_def by auto
  3219 
  3220 lemma path_component_subset: "(path_component s x) \<subseteq> s"
  3221   apply(rule, rule path_component_mem(2)) by(auto simp add:mem_def)
  3222 
  3223 lemma path_component_eq_empty: "path_component s x = {} \<longleftrightarrow> x \<notin> s"
  3224   apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_path_component_set
  3225   apply(drule path_component_mem(1)) using path_component_refl by auto
  3226 
  3227 subsection {* Path connectedness of a space. *}
  3228 
  3229 definition "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> (path_image g) \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
  3230 
  3231 lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
  3232   unfolding path_connected_def path_component_def by auto
  3233 
  3234 lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component s x = s)" 
  3235   unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset) 
  3236   unfolding subset_eq mem_path_component_set Ball_def mem_def by auto
  3237 
  3238 subsection {* Some useful lemmas about path-connectedness. *}
  3239 
  3240 lemma convex_imp_path_connected: assumes "convex s" shows "path_connected s"
  3241   unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI)
  3242   unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto
  3243 
  3244 lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s"
  3245   unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof-
  3246   fix e1 e2 assume as:"open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
  3247   then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto
  3248   then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
  3249     using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
  3250   have *:"connected {0..1::real^1}" by(auto intro!: convex_connected convex_interval)
  3251   have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" using as(3) g(2)[unfolded path_defs] by blast
  3252   moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto 
  3253   moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" using g(3,4)[unfolded path_defs] using obt by(auto intro!: exI)
  3254   ultimately show False using *[unfolded connected_local not_ex,rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
  3255     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]
  3256     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed
  3257 
  3258 lemma open_path_component: assumes "open s" shows "open(path_component s x)"
  3259   unfolding open_contains_ball proof
  3260   fix y assume as:"y \<in> path_component s x"
  3261   hence "y\<in>s" apply- apply(rule path_component_mem(2)) unfolding mem_def by auto
  3262   then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
  3263   show "\<exists>e>0. ball y e \<subseteq> path_component s x" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule) unfolding mem_ball mem_path_component_set proof-
  3264     fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer 
  3265       apply(rule path_component_of_subset[OF e(2)]) apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) using `e>0`
  3266       using as[unfolded mem_def] by auto qed qed
  3267 
  3268 lemma open_non_path_component: assumes "open s" shows "open(s - path_component s x)" unfolding open_contains_ball proof
  3269   fix y assume as:"y\<in>s - path_component s x" 
  3270   then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
  3271   show "\<exists>e>0. ball y e \<subseteq> s - path_component s x" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule,rule) defer proof(rule ccontr)
  3272     fix z assume "z\<in>ball y e" "\<not> z \<notin> path_component s x" 
  3273     hence "y \<in> path_component s x" unfolding not_not mem_path_component_set using `e>0` 
  3274       apply- apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)])
  3275       apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto
  3276     thus False using as by auto qed(insert e(2), auto) qed
  3277 
  3278 lemma connected_open_path_connected: assumes "open s" "connected s" shows "path_connected s"
  3279   unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule)
  3280   fix x y assume "x \<in> s" "y \<in> s" show "y \<in> path_component s x" proof(rule ccontr)
  3281     assume "y \<notin> path_component s x" moreover
  3282     have "path_component s x \<inter> s \<noteq> {}" using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto
  3283     ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
  3284     using assms(2)[unfolded connected_def not_ex, rule_format, of"path_component s x" "s - path_component s x"] by auto
  3285 qed qed
  3286 
  3287 lemma path_connected_continuous_image:
  3288   assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)"
  3289   unfolding path_connected_def proof(rule,rule)
  3290   fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s"
  3291   then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
  3292   guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] ..
  3293   thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"
  3294     unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs
  3295     using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed
  3296 
  3297 lemma homeomorphic_path_connectedness:
  3298   "s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"
  3299   unfolding homeomorphic_def homeomorphism_def apply(erule exE|erule conjE)+ apply rule
  3300   apply(drule_tac f=f in path_connected_continuous_image) prefer 3
  3301   apply(drule_tac f=g in path_connected_continuous_image) by auto
  3302 
  3303 lemma path_connected_empty: "path_connected {}"
  3304   unfolding path_connected_def by auto
  3305 
  3306 lemma path_connected_singleton: "path_connected {a}"
  3307   unfolding path_connected_def apply(rule,rule)
  3308   apply(rule_tac x="linepath a a" in exI) by(auto simp add:segment scaleR_left_diff_distrib)
  3309 
  3310 lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
  3311   shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule)
  3312   fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t" 
  3313   from assms(3) obtain z where "z \<in> s \<inter> t" by auto
  3314   thus "path_component (s \<union> t) x y" using as using assms(1-2)[unfolded path_connected_component] apply- 
  3315     apply(erule_tac[!] UnE)+ apply(rule_tac[2-3] path_component_trans[of _ _ z])
  3316     by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed
  3317 
  3318 subsection {* sphere is path-connected. *}
  3319 
  3320 lemma path_connected_punctured_universe:
  3321  assumes "2 \<le> CARD('n::finite)" shows "path_connected((UNIV::(real^'n::finite) set) - {a})" proof-
  3322   obtain \<psi> where \<psi>:"bij_betw \<psi> {1..CARD('n)} (UNIV::'n set)" using ex_bij_betw_nat_finite_1[OF finite_UNIV] by auto
  3323   let ?U = "UNIV::(real^'n) set" let ?u = "?U - {0}"
  3324   let ?basis = "\<lambda>k. basis (\<psi> k)"
  3325   let ?A = "\<lambda>k. {x::real^'n. \<exists>i\<in>{1..k}. inner (basis (\<psi> i)) x \<noteq> 0}"
  3326   have "\<forall>k\<in>{2..CARD('n)}. path_connected (?A k)" proof
  3327     have *:"\<And>k. ?A (Suc k) = {x. inner (?basis (Suc k)) x < 0} \<union> {x. inner (?basis (Suc k)) x > 0} \<union> ?A k" apply(rule set_ext,rule) defer
  3328       apply(erule UnE)+  unfolding mem_Collect_eq apply(rule_tac[1-2] x="Suc k" in bexI)
  3329       by(auto elim!: ballE simp add: not_less le_Suc_eq)
  3330     fix k assume "k \<in> {2..CARD('n)}" thus "path_connected (?A k)" proof(induct k)
  3331       case (Suc k) show ?case proof(cases "k = 1")
  3332         case False from Suc have d:"k \<in> {1..CARD('n)}" "Suc k \<in> {1..CARD('n)}" by auto
  3333         hence "\<psi> k \<noteq> \<psi> (Suc k)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=k]] by auto
  3334         hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < inner (?basis (Suc k)) x} \<inter> (?A k)" 
  3335           "?basis k - ?basis (Suc k) \<in> {x. 0 > inner (?basis (Suc k)) x} \<inter> ({x. 0 < inner (?basis (Suc k)) x} \<union> (?A k))" using d
  3336           by(auto simp add: inner_basis vector_component_simps intro!:bexI[where x=k])
  3337         show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un) 
  3338           prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt)
  3339           apply(rule Suc(1)) apply(rule_tac[2-3] ccontr) using d ** False by auto
  3340       next case True hence d:"1\<in>{1..CARD('n)}" "2\<in>{1..CARD('n)}" using Suc(2) by auto
  3341         have ***:"Suc 1 = 2" by auto
  3342         have **:"\<And>s t P Q. s \<union> t \<union> {x. P x \<or> Q x} = (s \<union> {x. P x}) \<union> (t \<union> {x. Q x})" by auto
  3343         have "\<psi> 2 \<noteq> \<psi> (Suc 0)" apply(rule ccontr) using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=2]] using assms by auto
  3344         thus ?thesis unfolding * True unfolding ** neq_iff bex_disj_distrib apply -
  3345           apply(rule path_connected_Un, rule_tac[1-2] path_connected_Un) defer 3 apply(rule_tac[1-4] convex_imp_path_connected) 
  3346           apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I)
  3347           apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I)
  3348           apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I)
  3349           using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps inner_basis)
  3350   qed qed auto qed note lem = this
  3351 
  3352   have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. inner (basis (\<psi> i)) x \<noteq> 0) \<longleftrightarrow> (\<exists>i. inner (basis i) x \<noteq> 0)"
  3353     apply rule apply(erule bexE) apply(rule_tac x="\<psi> i" in exI) defer apply(erule exE) proof- 
  3354     fix x::"real^'n" and i assume as:"inner (basis i) x \<noteq> 0"
  3355     have "i\<in>\<psi> ` {1..CARD('n)}" using \<psi>[unfolded bij_betw_def, THEN conjunct2] by auto
  3356     then obtain j where "j\<in>{1..CARD('n)}" "\<psi> j = i" by auto
  3357     thus "\<exists>i\<in>{1..CARD('n)}. inner (basis (\<psi> i)) x \<noteq> 0" apply(rule_tac x=j in bexI) using as by auto qed auto
  3358   have *:"?U - {a} = (\<lambda>x. x + a) ` {x. x \<noteq> 0}" apply(rule set_ext) unfolding image_iff 
  3359     apply rule apply(rule_tac x="x - a" in bexI) by auto
  3360   have **:"\<And>x::real^'n. x\<noteq>0 \<longleftrightarrow> (\<exists>i. inner (basis i) x \<noteq> 0)" unfolding Cart_eq by(auto simp add: inner_basis)
  3361   show ?thesis unfolding * apply(rule path_connected_continuous_image) apply(rule continuous_on_intros)+ 
  3362     unfolding ** apply(rule lem[THEN bspec[where x="CARD('n)"], unfolded ***]) using assms by auto qed
  3363 
  3364 lemma path_connected_sphere: assumes "2 \<le> CARD('n::finite)" shows "path_connected {x::real^'n::finite. norm(x - a) = r}" proof(cases "r\<le>0")
  3365   case True thus ?thesis proof(cases "r=0") 
  3366     case False hence "{x::real^'n. norm(x - a) = r} = {}" using True by auto
  3367     thus ?thesis using path_connected_empty by auto
  3368   qed(auto intro!:path_connected_singleton) next
  3369   case False hence *:"{x::real^'n. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" unfolding not_le apply -apply(rule set_ext,rule)
  3370     unfolding image_iff apply(rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) unfolding mem_Collect_eq norm_scaleR by (auto simp add: scaleR_right_diff_distrib)
  3371   have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" apply(rule set_ext,rule)
  3372     unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq by(auto split:split_if_asm)
  3373   have "continuous_on (UNIV - {0}) (\<lambda>x::real^'n. 1 / norm x)" unfolding o_def continuous_on_eq_continuous_within
  3374     apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
  3375     apply(rule continuous_at_norm[unfolded o_def]) by auto
  3376   thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
  3377     by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
  3378 
  3379 lemma connected_sphere: "2 \<le> CARD('n) \<Longrightarrow> connected {x::real^'n::finite. norm(x - a) = r}"
  3380   using path_connected_sphere path_connected_imp_connected by auto
  3381  
  3382 (** In continuous_at_vec1_norm : Use \<And> instead of \<forall>. **)
  3383 
  3384 end