src/HOL/Library/Convex_Euclidean_Space.thy
author huffman
Thu Jun 11 09:03:24 2009 -0700 (2009-06-11)
changeset 31563 ded2364d14d4
parent 31561 a5e168fd2bb9
child 31565 da5a5589418e
permissions -rw-r--r--
cleaned up some proofs
     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 vector_unminus_smult[simp]: "(-1::real) *s x = -x"
    43   unfolding vector_sneg_minus1 by simp
    44   (* TODO: move to Euclidean_Space.thy *)
    45 
    46 lemma setsum_delta_notmem: assumes "x\<notin>s"
    47   shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
    48         "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
    49         "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
    50         "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
    51   apply(rule_tac [!] setsum_cong2) using assms by auto
    52 
    53 lemma setsum_delta'': fixes s::"(real^'n) set" assumes "finite s"
    54   shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *s x) = (if y\<in>s then (f y) *s y else 0)"
    55 proof-
    56   have *:"\<And>x y. (if y = x then f x else (0::real)) *s x = (if x=y then (f x) *s x else 0)" by auto
    57   show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *s x"] by auto
    58 qed
    59 
    60 lemma not_disjointI:"x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A \<inter> B \<noteq> {}" by blast
    61 
    62 lemma if_smult:"(if P then x else (y::real)) *s v = (if P then x *s v else y *s v)" by auto
    63 
    64 lemma mem_interval_1: fixes x :: "real^1" shows
    65  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
    66  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
    67 by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def all_1)
    68 
    69 lemma image_smult_interval:"(\<lambda>x. m *s (x::real^'n::finite)) ` {a..b} =
    70   (if {a..b} = {} then {} else if 0 \<le> m then {m *s a..m *s b} else {m *s b..m *s a})"
    71   using image_affinity_interval[of m 0 a b] by auto
    72 
    73 lemma dest_vec1_inverval:
    74   "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
    75   "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
    76   "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
    77   "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
    78   apply(rule_tac [!] equalityI)
    79   unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
    80   apply(rule_tac [!] allI)apply(rule_tac [!] impI)
    81   apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
    82   apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
    83   by (auto simp add: vector_less_def vector_less_eq_def all_1 dest_vec1_def
    84     vec1_dest_vec1[unfolded dest_vec1_def One_nat_def])
    85 
    86 lemma dest_vec1_setsum: assumes "finite S"
    87   shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
    88   using dest_vec1_sum[OF assms] by auto
    89 
    90 lemma dist_triangle_eq:"dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *s (y - z) = norm (y - z) *s (x - y)"
    91 proof- have *:"x - y + (y - z) = x - z" by auto
    92   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *] 
    93     by(auto simp add:norm_minus_commute) qed
    94 
    95 lemma norm_eqI:"x = y \<Longrightarrow> norm x = norm y" by auto 
    96 lemma norm_minus_eqI:"(x::real^'n::finite) = - y \<Longrightarrow> norm x = norm y" by auto
    97 
    98 lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
    99   unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
   100 
   101 lemma dimindex_ge_1:"CARD(_::finite) \<ge> 1"
   102   using one_le_card_finite by auto
   103 
   104 lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1" 
   105   by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff) 
   106 
   107 lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
   108 
   109 subsection {* Affine set and affine hull.*}
   110 
   111 definition "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v::real. u + v = 1 \<longrightarrow> (u *s x + v *s y) \<in> s)"
   112 
   113 lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *s x + u *s y \<in> s)"
   114 proof- have *:"\<And>u v ::real. u + v = 1 \<longleftrightarrow> v = 1 - u" by auto
   115   { fix x y assume "x\<in>s" "y\<in>s"
   116     hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *s x + u *s y \<in> s)" apply auto 
   117       apply(erule_tac[!] x="1 - u" in allE) unfolding * by auto  }
   118   thus ?thesis unfolding affine_def by auto qed
   119 
   120 lemma affine_empty[intro]: "affine {}"
   121   unfolding affine_def by auto
   122 
   123 lemma affine_sing[intro]: "affine {x}"
   124   unfolding affine_alt by (auto simp add: vector_sadd_rdistrib[THEN sym]) 
   125 
   126 lemma affine_UNIV[intro]: "affine UNIV"
   127   unfolding affine_def by auto
   128 
   129 lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
   130   unfolding affine_def by auto 
   131 
   132 lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
   133   unfolding affine_def by auto
   134 
   135 lemma affine_affine_hull: "affine(affine hull s)"
   136   unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
   137   unfolding mem_def by auto
   138 
   139 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   140 proof-
   141   { fix f assume "f \<subseteq> affine"
   142     hence "affine (\<Inter>f)" using affine_Inter[of f] unfolding subset_eq mem_def by auto  }
   143   thus ?thesis using hull_eq[unfolded mem_def, of affine s] by auto
   144 qed
   145 
   146 lemma setsum_restrict_set'': assumes "finite A"
   147   shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
   148   unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
   149 
   150 subsection {* Some explicit formulations (from Lars Schewe). *}
   151 
   152 lemma affine: fixes V::"(real^'n) set"
   153   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) *s x)) s \<in> V)"
   154 unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
   155 defer apply(rule, rule, rule, rule, rule) proof-
   156   fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
   157     "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> V"
   158   thus "u *s x + v *s y \<in> V" apply(cases "x=y")
   159     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) 
   160     by(auto simp add: vector_sadd_rdistrib[THEN sym])
   161 next
   162   fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> V"
   163     "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
   164   def n \<equiv> "card s"
   165   have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
   166   thus "(\<Sum>x\<in>s. u x *s x) \<in> V" proof(auto simp only: disjE)
   167     assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
   168     then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
   169     thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
   170       by(auto simp add: setsum_clauses(2))
   171   next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
   172       case (Suc n) fix s::"(real^'n) set" and u::"real^'n\<Rightarrow> real"
   173       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 *s x + v *s y \<in> V; finite s;
   174                s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> V" and
   175 	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 *s x + v *s y \<in> V"
   176            "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
   177       have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
   178 	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
   179 	thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
   180 	  less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
   181       then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
   182 
   183       have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
   184       have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
   185       have **:"setsum u (s - {x}) = 1 - u x"
   186 	using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
   187       have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
   188       have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *s xa) \<in> V" proof(cases "card (s - {x}) > 2")
   189 	case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
   190 	  assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
   191 	  thus False using True by auto qed auto
   192 	thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
   193 	unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
   194       next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
   195 	then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
   196 	thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
   197 	  using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
   198       thus "(\<Sum>x\<in>s. u x *s x) \<in> V" unfolding vector_smult_assoc[THEN sym] and setsum_cmul
   199  	 apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
   200 	 using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *s (\<Sum>xa\<in>s - {x}. u xa *s xa)"], 
   201 	 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
   202     qed auto
   203   next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
   204     thus ?thesis using as(4,5) by simp
   205   qed(insert `s\<noteq>{}` `finite s`, auto)
   206 qed
   207 
   208 lemma affine_hull_explicit:
   209   "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) *s v) s = y}"
   210   apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
   211   apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
   212   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 *s v) = x"
   213     apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
   214 next
   215   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 *s v) = x" 
   216   thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
   217 next
   218   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 *s v) = y}" unfolding affine_def
   219     apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
   220     fix u v ::real assume uv:"u + v = 1"
   221     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 *s v) = x"
   222     then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *s v) = x" by auto
   223     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 *s v) = y"
   224     then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *s v) = y" by auto
   225     have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
   226     have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
   227     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 *s v) = u *s x + v *s y"
   228       apply(rule_tac x="sx \<union> sy" in exI)
   229       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)
   230       unfolding vector_sadd_rdistrib setsum_addf if_smult vector_smult_lzero  ** setsum_restrict_set[OF xy, THEN sym]
   231       unfolding vector_smult_assoc[THEN sym] setsum_cmul and setsum_right_distrib[THEN sym]
   232       unfolding x y using x(1-3) y(1-3) uv by simp qed qed
   233 
   234 lemma affine_hull_finite:
   235   assumes "finite s"
   236   shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
   237   unfolding affine_hull_explicit and expand_set_eq and mem_Collect_eq apply (rule,rule)
   238   apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
   239   fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"
   240   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 *s v) = x"
   241     apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
   242 next
   243   fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
   244   assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *s v) = x"
   245   thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   246     unfolding if_smult vector_smult_lzero and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
   247 
   248 subsection {* Stepping theorems and hence small special cases. *}
   249 
   250 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   251   apply(rule hull_unique) unfolding mem_def by auto
   252 
   253 lemma affine_hull_finite_step:
   254   shows "(\<exists>u::real^'n=>real. setsum u {} = w \<and> setsum (\<lambda>x. u x *s x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
   255   "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *s x) (insert a s) = y) \<longleftrightarrow>
   256                 (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *s x) s = y - v *s a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
   257 proof-
   258   show ?th1 by simp
   259   assume ?as 
   260   { assume ?lhs
   261     then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *s x) = y" by auto
   262     have ?rhs proof(cases "a\<in>s")
   263       case True hence *:"insert a s = s" by auto
   264       show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
   265     next
   266       case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
   267     qed  } moreover
   268   { assume ?rhs
   269     then obtain v u where vu:"setsum u s = w - v"  "(\<Sum>x\<in>s. u x *s x) = y - v *s a" by auto
   270     have *:"\<And>x M. (if x = a then v else M) *s x = (if x = a then v *s x else M *s x)" by auto
   271     have ?lhs proof(cases "a\<in>s")
   272       case True thus ?thesis
   273 	apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
   274 	unfolding setsum_clauses(2)[OF `?as`]  apply simp
   275 	unfolding vector_sadd_rdistrib and setsum_addf 
   276 	unfolding vu and * and vector_smult_lzero
   277 	by (auto simp add: setsum_delta[OF `?as`])
   278     next
   279       case False 
   280       hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
   281                "\<And>x. x \<in> s \<Longrightarrow> u x *s x = (if x = a then v *s x else u x *s x)" by auto
   282       from False show ?thesis
   283 	apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
   284 	unfolding setsum_clauses(2)[OF `?as`] and * using vu
   285 	using setsum_cong2[of s "\<lambda>x. u x *s x" "\<lambda>x. if x = a then v *s x else u x *s x", OF **(2)]
   286 	using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
   287     qed }
   288   ultimately show "?lhs = ?rhs" by blast
   289 qed
   290 
   291 lemma affine_hull_2: "affine hull {a,b::real^'n} = {u *s a + v *s b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
   292 proof-
   293   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   294          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real^'n)" by auto
   295   have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *s v) = y}"
   296     using affine_hull_finite[of "{a,b}"] by auto
   297   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *s b = y - v *s a}"
   298     by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
   299   also have "\<dots> = ?rhs" unfolding * by auto
   300   finally show ?thesis by auto
   301 qed
   302 
   303 lemma affine_hull_3: "affine hull {a,b,c::real^'n} = { u *s a + v *s b + w *s c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
   304 proof-
   305   have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
   306          "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real^'n)" by auto
   307   show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
   308     unfolding * apply auto
   309     apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
   310     apply(rule_tac x=u in exI) by(auto intro!: exI)
   311 qed
   312 
   313 subsection {* Some relations between affine hull and subspaces. *}
   314 
   315 lemma affine_hull_insert_subset_span:
   316   "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
   317   unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
   318   apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
   319   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 *s v) = x"
   320   have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
   321   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 *s v) = v)"
   322     apply(rule_tac x="x - a" in exI) apply rule defer apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
   323     apply(rule_tac x="\<lambda>x. u (x + a)" in exI) using as(1)
   324     apply(simp add: setsum_reindex[unfolded inj_on_def] setsum_subtractf setsum_diff1 setsum_vmul[THEN sym])
   325     unfolding as by simp_all qed
   326 
   327 lemma affine_hull_insert_span:
   328   assumes "a \<notin> s"
   329   shows "affine hull (insert a s) =
   330             {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
   331   apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
   332   unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
   333   fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
   334   then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *s v) = y" unfolding span_explicit by auto
   335   def f \<equiv> "(\<lambda>x. x + a) ` t"
   336   have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *s (v - a)) = y - a" unfolding f_def using obt 
   337     by(auto simp add: setsum_reindex[unfolded inj_on_def])
   338   have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
   339   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 *s v) = y"
   340     apply(rule_tac x="insert a f" in exI)
   341     apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
   342     using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
   343     unfolding setsum_cases[OF f(1), of "{a}", unfolded singleton_iff] and *
   344     by (auto simp add: setsum_subtractf setsum_vmul field_simps) qed
   345 
   346 lemma affine_hull_span:
   347   assumes "a \<in> s"
   348   shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
   349   using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
   350 
   351 subsection {* Convexity. *}
   352 
   353 definition "convex (s::(real^'n) set) \<longleftrightarrow>
   354         (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. (u + v = 1) \<longrightarrow> (u *s x + v *s y) \<in> s)"
   355 
   356 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) *s x + u *s y) \<in> s)"
   357 proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
   358   show ?thesis unfolding convex_def apply auto
   359     apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
   360     by (auto simp add: *) qed
   361 
   362 lemma mem_convex:
   363   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
   364   shows "((1 - u) *s a + u *s b) \<in> s"
   365   using assms unfolding convex_alt by auto
   366 
   367 lemma convex_empty[intro]: "convex {}"
   368   unfolding convex_def by simp
   369 
   370 lemma convex_singleton[intro]: "convex {a}"
   371   unfolding convex_def by (auto simp add:vector_sadd_rdistrib[THEN sym])
   372 
   373 lemma convex_UNIV[intro]: "convex UNIV"
   374   unfolding convex_def by auto
   375 
   376 lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
   377   unfolding convex_def by auto
   378 
   379 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
   380   unfolding convex_def by auto
   381 
   382 lemma convex_halfspace_le: "convex {x. a \<bullet> x \<le> b}"
   383   unfolding convex_def apply auto
   384   unfolding dot_radd dot_rmult by (metis real_convex_bound_le) 
   385 
   386 lemma convex_halfspace_ge: "convex {x. a \<bullet> x \<ge> b}"
   387 proof- have *:"{x. a \<bullet> x \<ge> b} = {x. -a \<bullet> x \<le> -b}" by auto
   388   show ?thesis apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
   389 
   390 lemma convex_hyperplane: "convex {x. a \<bullet> x = b}"
   391 proof-
   392   have *:"{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}" by auto
   393   show ?thesis unfolding * apply(rule convex_Int)
   394     using convex_halfspace_le convex_halfspace_ge by auto
   395 qed
   396 
   397 lemma convex_halfspace_lt: "convex {x. a \<bullet> x < b}"
   398   unfolding convex_def by(auto simp add: real_convex_bound_lt dot_radd dot_rmult)
   399 
   400 lemma convex_halfspace_gt: "convex {x. a \<bullet> x > b}"
   401    using convex_halfspace_lt[of "-a" "-b"] by(auto simp add: dot_lneg neg_less_iff_less)
   402 
   403 lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
   404   unfolding convex_def apply auto apply(erule_tac x=i in allE)+
   405   apply(rule add_nonneg_nonneg) by(auto simp add: mult_nonneg_nonneg)
   406 
   407 subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
   408 
   409 lemma convex: "convex s \<longleftrightarrow>
   410   (\<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)
   411            \<longrightarrow> setsum (\<lambda>i. u i *s x i) {1..k} \<in> s)"
   412   unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
   413   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 *s x i) \<in> s"
   414     "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   415   show "u *s x + v *s 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-)
   416     by (auto simp add: setsum_head_Suc) 
   417 next
   418   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 *s x + v *s y \<in> s" 
   419   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 *s x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
   420   case (Suc k) show ?case proof(cases "u (Suc k) = 1")
   421     case True hence "(\<Sum>i = Suc 0..k. u i *s x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
   422       fix i assume i:"i \<in> {Suc 0..k}" "u i *s x i \<noteq> 0"
   423       hence ui:"u i \<noteq> 0" by auto
   424       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
   425       hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta) 
   426       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
   427       thus False using Suc(3) unfolding setsum_cl_ivl_Suc and True by simp qed
   428     thus ?thesis unfolding setsum_cl_ivl_Suc using True and Suc(2) by auto
   429   next
   430     have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
   431     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
   432     have ***:"\<And>i k. (u i / (1 - u (Suc k))) *s x i = (inverse (1 - u (Suc k))) *s (u i *s x i)" unfolding real_divide_def by auto
   433     case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
   434     have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
   435       apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
   436     hence "(1 - u (Suc k)) *s (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) + u (Suc k) *s x (Suc k) \<in> s"
   437       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
   438     thus ?thesis unfolding setsum_cl_ivl_Suc and *** and setsum_cmul using nn by auto qed qed auto qed
   439 
   440 
   441 lemma convex_explicit: "convex (s::(real^'n) set) \<longleftrightarrow>
   442   (\<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 *s x) t \<in> s)"
   443   unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
   444   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 *s x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   445   show "u *s x + v *s y \<in> s" proof(cases "x=y")
   446     case True show ?thesis unfolding True and vector_sadd_rdistrib[THEN sym] using as(3,6) by auto next
   447     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
   448 next 
   449   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 *s x + v *s y \<in> s" "finite (t::(real^'n) set)"
   450   (*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
   451   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 *s x) \<in> s" apply(induct_tac t rule:finite_induct)
   452     prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
   453     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 *s x) \<in> s"
   454     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)"
   455     show "(\<Sum>x\<in>insert x f. u x *s x) \<in> s" proof(cases "u x = 1")
   456       case True hence "setsum (\<lambda>x. u x *s x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
   457 	fix y assume y:"y \<in> f" "u y *s y \<noteq> 0"
   458 	hence uy:"u y \<noteq> 0" by auto
   459 	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
   460 	hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta) 
   461 	hence "setsum u f > 0" using uy apply(rule_tac less_le_trans[of _ "u y"]) using as(4) and y(1) by auto
   462 	thus False using as(2,5) unfolding setsum_clauses(2)[OF as(1)] and True by auto qed
   463       thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2,3) unfolding True by auto
   464     next
   465       have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
   466       have **:"u x \<le> 1" apply(rule ccontr) unfolding not_le using as(5)[unfolded setsum_clauses(2)[OF as(1)]] and as(2)
   467 	using setsum_nonneg[of f u] and as(4) by auto
   468       case False hence "inverse (1 - u x) *s (\<Sum>x\<in>f. u x *s x) \<in> s" unfolding setsum_cmul[THEN sym] and vector_smult_assoc
   469 	apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
   470 	unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
   471       hence "u x *s x + (1 - u x) *s ((inverse (1 - u x)) *s setsum (\<lambda>x. u x *s x) f) \<in>s" 
   472 	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 
   473       thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
   474   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 *s x) \<in> s" by auto
   475 qed
   476 
   477 lemma convex_finite: assumes "finite s"
   478   shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
   479                       \<longrightarrow> setsum (\<lambda>x. u x *s x) s \<in> s)"
   480   unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
   481   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 *s x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
   482   have *:"s \<inter> t = t" using as(3) by auto
   483   show "(\<Sum>x\<in>t. u x *s x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
   484     unfolding if_smult and setsum_cases[OF assms] and * using as(2-) by auto
   485 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
   486 
   487 subsection {* Cones. *}
   488 
   489 definition "cone (s::(real^'n) set) \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *s x) \<in> s)"
   490 
   491 lemma cone_empty[intro, simp]: "cone {}"
   492   unfolding cone_def by auto
   493 
   494 lemma cone_univ[intro, simp]: "cone UNIV"
   495   unfolding cone_def by auto
   496 
   497 lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
   498   unfolding cone_def by auto
   499 
   500 subsection {* Conic hull. *}
   501 
   502 lemma cone_cone_hull: "cone (cone hull s)"
   503   unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
   504   by (auto simp add: mem_def)
   505 
   506 lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
   507   apply(rule hull_eq[unfolded mem_def])
   508   using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
   509 
   510 subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
   511 
   512 definition "affine_dependent (s::(real^'n) set) \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
   513 
   514 lemma affine_dependent_explicit:
   515   "affine_dependent p \<longleftrightarrow>
   516     (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
   517     (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) s = 0)"
   518   unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
   519   apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
   520 proof-
   521   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 *s v) = x"
   522   have "x\<notin>s" using as(1,4) by auto
   523   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 *s v) = 0"
   524     apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
   525     unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
   526 next
   527   fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *s v) = 0" "v \<in> s" "u v \<noteq> 0"
   528   have "s \<noteq> {v}" using as(3,6) by auto
   529   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 *s v) = x" 
   530     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)
   531     unfolding vector_smult_assoc[THEN sym] and setsum_cmul unfolding setsum_right_distrib[THEN sym] and setsum_diff1_ring[OF as(1,5)] using as by auto
   532 qed
   533 
   534 lemma affine_dependent_explicit_finite:
   535   assumes "finite (s::(real^'n) set)"
   536   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 *s v) s = 0)"
   537   (is "?lhs = ?rhs")
   538 proof
   539   have *:"\<And>vt u v. (if vt then u v else 0) *s v = (if vt then (u v) *s v else (0::real^'n))" by auto
   540   assume ?lhs
   541   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 *s v) = 0"
   542     unfolding affine_dependent_explicit by auto
   543   thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
   544     apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
   545     unfolding Int_absorb2[OF `t\<subseteq>s`, unfolded Int_commute] by auto
   546 next
   547   assume ?rhs
   548   then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *s v) = 0" by auto
   549   thus ?lhs unfolding affine_dependent_explicit using assms by auto
   550 qed
   551 
   552 subsection {* A general lemma. *}
   553 
   554 lemma convex_connected:
   555   assumes "convex s" shows "connected s"
   556 proof-
   557   { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
   558     assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
   559     then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
   560     hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
   561 
   562     { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
   563       { fix y have *:"(1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2) = (y - x) *s x1 - (y - x) *s x2"
   564 	  by(simp add: ring_simps vector_sadd_rdistrib vector_sub_rdistrib)
   565 	assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
   566 	hence "norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
   567 	  unfolding * and vector_ssub_ldistrib[THEN sym] and norm_mul 
   568 	  unfolding less_divide_eq using n by auto  }
   569       hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
   570 	apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
   571 	apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
   572 
   573     have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e2"
   574       apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
   575       using * apply(simp add: dist_norm)
   576       using as(1,2)[unfolded open_dist] apply simp
   577       using as(1,2)[unfolded open_dist] apply simp
   578       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
   579       using as(3) by auto
   580     then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *s x1 + x *s x2 \<notin> e1"  "(1 - x) *s x1 + x *s x2 \<notin> e2" by auto
   581     hence False using as(4) 
   582       using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
   583       using x1(2) x2(2) by auto  }
   584   thus ?thesis unfolding connected_def by auto
   585 qed
   586 
   587 subsection {* One rather trivial consequence. *}
   588 
   589 lemma connected_UNIV: "connected (UNIV :: (real ^ _) set)"
   590   by(simp add: convex_connected convex_UNIV)
   591 
   592 subsection {* Convex functions into the reals. *}
   593 
   594 definition "convex_on s (f::real^'n \<Rightarrow> real) = 
   595   (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *s x + v *s y) \<le> u * f x + v * f y)"
   596 
   597 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
   598   unfolding convex_on_def by auto
   599 
   600 lemma convex_add:
   601   assumes "convex_on s f" "convex_on s g"
   602   shows "convex_on s (\<lambda>x. f x + g x)"
   603 proof-
   604   { fix x y assume "x\<in>s" "y\<in>s" moreover
   605     fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
   606     ultimately have "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
   607       using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
   608       using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
   609       apply - apply(rule add_mono) by auto
   610     hence "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps)  }
   611   thus ?thesis unfolding convex_on_def by auto 
   612 qed
   613 
   614 lemma convex_cmul: 
   615   assumes "0 \<le> (c::real)" "convex_on s f"
   616   shows "convex_on s (\<lambda>x. c * f x)"
   617 proof-
   618   have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: ring_simps)
   619   show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
   620 qed
   621 
   622 lemma convex_lower:
   623   assumes "convex_on s f"  "x\<in>s"  "y \<in> s"  "0 \<le> u"  "0 \<le> v"  "u + v = 1"
   624   shows "f (u *s x + v *s y) \<le> max (f x) (f y)"
   625 proof-
   626   let ?m = "max (f x) (f y)"
   627   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" apply(rule add_mono) 
   628     using assms(4,5) by(auto simp add: mult_mono1)
   629   also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
   630   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]]
   631     using assms(2-6) by auto 
   632 qed
   633 
   634 lemma convex_local_global_minimum:
   635   assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
   636   shows "\<forall>y\<in>s. f x \<le> f y"
   637 proof(rule ccontr)
   638   have "x\<in>s" using assms(1,3) by auto
   639   assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
   640   then obtain y where "y\<in>s" and y:"f x > f y" by auto
   641   hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
   642 
   643   then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
   644     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
   645   hence "f ((1-u) *s x + u *s y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
   646     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
   647   moreover
   648   have *:"x - ((1 - u) *s x + u *s y) = u *s (x - y)" by (simp add: vector_ssub_ldistrib vector_sub_rdistrib)
   649   have "(1 - u) *s x + u *s y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_mul and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
   650     using u unfolding pos_less_divide_eq[OF xy] by auto
   651   hence "f x \<le> f ((1 - u) *s x + u *s y)" using assms(4) by auto
   652   ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
   653 qed
   654 
   655 lemma convex_distance: "convex_on s (\<lambda>x. dist a x)"
   656 proof(auto simp add: convex_on_def dist_norm)
   657   fix x y assume "x\<in>s" "y\<in>s"
   658   fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
   659   have "a = u *s a + v *s a" unfolding vector_sadd_rdistrib[THEN sym] and `u+v=1` by simp
   660   hence *:"a - (u *s x + v *s y) = (u *s (a - x)) + (v *s (a - y))" by auto
   661   show "norm (a - (u *s x + v *s y)) \<le> u * norm (a - x) + v * norm (a - y)"
   662     unfolding * using norm_triangle_ineq[of "u *s (a - x)" "v *s (a - y)"] unfolding norm_mul
   663     using `0 \<le> u` `0 \<le> v` by auto
   664 qed
   665 
   666 subsection {* Arithmetic operations on sets preserve convexity. *}
   667 
   668 lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *s x) ` s)"
   669   unfolding convex_def and image_iff apply auto
   670   apply (rule_tac x="u *s x+v *s y" in bexI) by (auto simp add: field_simps)
   671 
   672 lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
   673   unfolding convex_def and image_iff apply auto
   674   apply (rule_tac x="u *s x+v *s y" in bexI) by auto
   675 
   676 lemma convex_sums:
   677   assumes "convex s" "convex t"
   678   shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
   679 proof(auto simp add: convex_def image_iff)
   680   fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
   681   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   682   show "\<exists>x y. u *s xa + u *s ya + (v *s xb + v *s yb) = x + y \<and> x \<in> s \<and> y \<in> t"
   683     apply(rule_tac x="u *s xa + v *s xb" in exI) apply(rule_tac x="u *s ya + v *s yb" in exI)
   684     using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
   685     using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
   686     using uv xy by auto
   687 qed
   688 
   689 lemma convex_differences: 
   690   assumes "convex s" "convex t"
   691   shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
   692 proof-
   693   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
   694     apply(rule_tac x=xa in exI) apply(rule_tac x="-y" in exI) apply simp
   695     apply(rule_tac x=xa in exI) apply(rule_tac x=xb in exI) by simp
   696   thus ?thesis using convex_sums[OF assms(1)  convex_negations[OF assms(2)]] by auto
   697 qed
   698 
   699 lemma convex_translation: assumes "convex s" shows "convex ((\<lambda>x. a + x) ` s)"
   700 proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
   701   thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
   702 
   703 lemma convex_affinity: assumes "convex (s::(real^'n) set)" shows "convex ((\<lambda>x. a + c *s x) ` s)"
   704 proof- have "(\<lambda>x. a + c *s x) ` s = op + a ` op *s c ` s" by auto
   705   thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
   706 
   707 lemma convex_linear_image: assumes c:"convex s" and l:"linear f" shows "convex(f ` s)"
   708 proof(auto simp add: convex_def)
   709   fix x y assume xy:"x \<in> s" "y \<in> s"
   710   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   711   show "u *s f x + v *s f y \<in> f ` s" unfolding image_iff
   712     apply(rule_tac x="u *s x + v *s y" in bexI)
   713     unfolding linear_add[OF l] linear_cmul[OF l] 
   714     using c[unfolded convex_def] xy uv by auto
   715 qed
   716 
   717 subsection {* Balls, being convex, are connected. *}
   718 
   719 lemma convex_ball: "convex (ball x e)" 
   720 proof(auto simp add: convex_def)
   721   fix y z assume yz:"dist x y < e" "dist x z < e"
   722   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   723   have "dist x (u *s y + v *s z) \<le> u * dist x y + v * dist x z" using uv yz
   724     using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
   725   thus "dist x (u *s y + v *s z) < e" using real_convex_bound_lt[OF yz uv] by auto 
   726 qed
   727 
   728 lemma convex_cball: "convex(cball x e)"
   729 proof(auto simp add: convex_def Ball_def mem_cball)
   730   fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
   731   fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
   732   have "dist x (u *s y + v *s z) \<le> u * dist x y + v * dist x z" using uv yz
   733     using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
   734   thus "dist x (u *s y + v *s z) \<le> e" using real_convex_bound_le[OF yz uv] by auto 
   735 qed
   736 
   737 lemma connected_ball: "connected(ball (x::real^_) e)" (* FIXME: generalize *)
   738   using convex_connected convex_ball by auto
   739 
   740 lemma connected_cball: "connected(cball (x::real^_) e)" (* FIXME: generalize *)
   741   using convex_connected convex_cball by auto
   742 
   743 subsection {* Convex hull. *}
   744 
   745 lemma convex_convex_hull: "convex(convex hull s)"
   746   unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
   747   unfolding mem_def by auto
   748 
   749 lemma convex_hull_eq: "(convex hull s = s) \<longleftrightarrow> convex s" apply(rule hull_eq[unfolded mem_def])
   750   using convex_Inter[unfolded Ball_def mem_def] by auto
   751 
   752 lemma bounded_convex_hull: assumes "bounded s" shows "bounded(convex hull s)"
   753 proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
   754   show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
   755     unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
   756     unfolding subset_eq mem_cball dist_norm using B by auto qed
   757 
   758 lemma finite_imp_bounded_convex_hull:
   759   "finite s \<Longrightarrow> bounded(convex hull s)"
   760   using bounded_convex_hull finite_imp_bounded by auto
   761 
   762 subsection {* Stepping theorems for convex hulls of finite sets. *}
   763 
   764 lemma convex_hull_empty[simp]: "convex hull {} = {}"
   765   apply(rule hull_unique) unfolding mem_def by auto
   766 
   767 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
   768   apply(rule hull_unique) unfolding mem_def by auto
   769 
   770 lemma convex_hull_insert:
   771   assumes "s \<noteq> {}"
   772   shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
   773                                     b \<in> (convex hull s) \<and> (x = u *s a + v *s b)}" (is "?xyz = ?hull")
   774  apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
   775  fix x assume x:"x = a \<or> x \<in> s"
   776  thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
   777    apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
   778 next
   779   fix x assume "x\<in>?hull"
   780   then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *s a + v *s b" by auto
   781   have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
   782     using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
   783   thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
   784     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
   785 next
   786   show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
   787     fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
   788     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 *s a + v1 *s b1" by auto
   789     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 *s a + v2 *s b2" by auto
   790     have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
   791     have "\<exists>b \<in> convex hull s. u *s x + v *s y = (u * u1) *s a + (v * u2) *s a + (b - (u * u1) *s b - (v * u2) *s b)"
   792     proof(cases "u * v1 + v * v2 = 0")
   793       have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
   794       case True hence **:"u * v1 = 0" "v * v2 = 0" apply- apply(rule_tac [!] ccontr)
   795 	using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by auto
   796       hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
   797       thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: **) 
   798     next
   799       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)
   800       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
   801       also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
   802       case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
   803 	apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
   804 	using as(1,2) obt1(1,2) obt2(1,2) by auto 
   805       thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
   806 	apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *s b1 + ((v * v2) / (u * v1 + v * v2)) *s b2" in bexI) defer
   807 	apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
   808 	unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff by auto
   809     qed note * = this
   810     have u1:"u1 \<le> 1" apply(rule ccontr) unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
   811     have u2:"u2 \<le> 1" apply(rule ccontr) unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
   812     have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
   813       apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
   814     also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
   815     finally 
   816     show "u *s x + v *s y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
   817       apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
   818       using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add:field_simps)
   819   qed
   820 qed
   821 
   822 
   823 subsection {* Explicit expression for convex hull. *}
   824 
   825 lemma convex_hull_indexed:
   826   "convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
   827                             (setsum u {1..k} = 1) \<and>
   828                             (setsum (\<lambda>i. u i *s x i) {1..k} = y)}" (is "?xyz = ?hull")
   829   apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
   830   apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
   831 proof-
   832   fix x assume "x\<in>s"
   833   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
   834 next
   835   fix t assume as:"s \<subseteq> t" "convex t"
   836   show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
   837     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 *s y i) = x"
   838     show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
   839       using assm(1,2) as(1) by auto qed
   840 next
   841   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"
   842   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 *s x1 i) = x" by auto
   843   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 *s x2 i) = y" by auto
   844   have *:"\<And>P x1 x2 s1 s2 i.(if P i then s1 else s2) *s (if P i then x1 else x2) = (if P i then s1 *s x1 else s2 *s x2)"
   845     "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
   846     prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
   847   have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
   848   show "u *s x + v *s y \<in> ?hull" apply(rule)
   849     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)
   850     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)
   851     unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def
   852     unfolding vector_smult_assoc[THEN sym] setsum_cmul setsum_right_distrib[THEN sym] proof-
   853     fix i assume i:"i \<in> {1..k1+k2}"
   854     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"
   855     proof(cases "i\<in>{1..k1}")
   856       case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
   857     next def j \<equiv> "i - k1"
   858       case False with i have "j \<in> {1..k2}" unfolding j_def by auto
   859       thus ?thesis unfolding j_def[symmetric] using False
   860 	using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
   861   qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
   862 qed
   863 
   864 lemma convex_hull_finite:
   865   assumes "finite (s::(real^'n)set)"
   866   shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
   867          setsum u s = 1 \<and> setsum (\<lambda>x. u x *s x) s = y}" (is "?HULL = ?set")
   868 proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
   869   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 *s x) = x" 
   870     apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
   871     unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
   872 next
   873   fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
   874   fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
   875   fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
   876   { fix x assume "x\<in>s"
   877     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)
   878       by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
   879   moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
   880     unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
   881   moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
   882     unfolding vector_sadd_rdistrib and setsum_addf and vector_smult_assoc[THEN sym] and setsum_cmul by auto
   883   ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
   884     apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
   885 next
   886   fix t assume t:"s \<subseteq> t" "convex t" 
   887   fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
   888   thus "(\<Sum>x\<in>s. u x *s x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
   889     using assms and t(1) by auto
   890 qed
   891 
   892 subsection {* Another formulation from Lars Schewe. *}
   893 
   894 lemma convex_hull_explicit:
   895   "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
   896              (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}" (is "?lhs = ?rhs")
   897 proof-
   898   { fix x assume "x\<in>?lhs"
   899     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 *s y i) = x"
   900       unfolding convex_hull_indexed by auto
   901 
   902     have fin:"finite {1..k}" by auto
   903     have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
   904     { fix j assume "j\<in>{1..k}"
   905       hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
   906 	using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
   907 	apply(rule setsum_nonneg) using obt(1) by auto } 
   908     moreover
   909     have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
   910       unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
   911     moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *s v) = x"
   912       using setsum_image_gen[OF fin, of "\<lambda>i. u i *s y i" y, THEN sym]
   913       unfolding setsum_vmul[OF fin']  using obt(3) by auto
   914     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 *s v) = x"
   915       apply(rule_tac x="y ` {1..k}" in exI)
   916       apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
   917     hence "x\<in>?rhs" by auto  }
   918   moreover
   919   { fix y assume "y\<in>?rhs"
   920     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 *s v) = y" by auto
   921 
   922     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
   923     
   924     { fix i::nat assume "i\<in>{1..card s}"
   925       hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
   926       hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
   927     moreover have *:"finite {1..card s}" by auto
   928     { fix y assume "y\<in>s"
   929       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
   930       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
   931       hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
   932       hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y" "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *s f x) = u y *s y" by auto   }
   933 
   934     hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *s f i) = y"
   935       unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *s f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
   936       unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *s f x)" "\<lambda>v. u v *s v"]
   937       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
   938     
   939     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 *s x i) = y"
   940       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
   941     hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
   942   ultimately show ?thesis unfolding expand_set_eq by blast
   943 qed
   944 
   945 subsection {* A stepping theorem for that expansion. *}
   946 
   947 lemma convex_hull_finite_step:
   948   assumes "finite (s::(real^'n) set)"
   949   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 *s x) (insert a s) = y)
   950      \<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 *s x) s = y - v *s a)" (is "?lhs = ?rhs")
   951 proof(rule, case_tac[!] "a\<in>s")
   952   assume "a\<in>s" hence *:"insert a s = s" by auto
   953   assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
   954 next
   955   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 *s x) = y" by auto
   956   assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
   957     apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
   958 next
   959   assume "a\<in>s" hence *:"insert a s = s" by auto
   960   have fin:"finite (insert a s)" using assms by auto
   961   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 *s x) = y - v *s a" by auto
   962   show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding vector_sadd_rdistrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
   963     unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
   964 next
   965   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 *s x) = y - v *s a" by auto
   966   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) *s x) = (\<Sum>x\<in>s. u x *s x)"
   967     apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
   968   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
   969 qed
   970 
   971 subsection {* Hence some special cases. *}
   972 
   973 lemma convex_hull_2:
   974   "convex hull {a,b} = {u *s a + v *s b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
   975 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
   976 show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
   977   apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
   978   apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
   979 
   980 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *s (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
   981   unfolding convex_hull_2 unfolding Collect_def 
   982 proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
   983   fix x show "(\<exists>v u. x = v *s a + u *s b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *s (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
   984     unfolding * apply auto apply(rule_tac[!] x=u in exI) by auto qed
   985 
   986 lemma convex_hull_3:
   987   "convex hull {a::real^'n,b,c} = { u *s a + v *s b + w *s c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
   988 proof-
   989   have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
   990   have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
   991          "\<And>x y z ::real^'n. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: ring_simps)
   992   show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
   993     unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
   994     apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
   995     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
   996 
   997 lemma convex_hull_3_alt:
   998   "convex hull {a,b,c} = {a + u *s (b - a) + v *s (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
   999 proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
  1000   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
  1001     apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by simp qed
  1002 
  1003 subsection {* Relations among closure notions and corresponding hulls. *}
  1004 
  1005 lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
  1006   unfolding subspace_def affine_def by auto
  1007 
  1008 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
  1009   unfolding affine_def convex_def by auto
  1010 
  1011 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
  1012   using subspace_imp_affine affine_imp_convex by auto
  1013 
  1014 lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
  1015   unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
  1016   using subspace_imp_affine  by auto
  1017 
  1018 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
  1019   unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
  1020   using subspace_imp_convex by auto
  1021 
  1022 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
  1023   unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
  1024   using affine_imp_convex by auto
  1025 
  1026 lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
  1027   unfolding affine_dependent_def dependent_def 
  1028   using affine_hull_subset_span by auto
  1029 
  1030 lemma dependent_imp_affine_dependent:
  1031   assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
  1032   shows "affine_dependent (insert a s)"
  1033 proof-
  1034   from assms(1)[unfolded dependent_explicit] obtain S u v 
  1035     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 *s v) = 0" by auto
  1036   def t \<equiv> "(\<lambda>x. x + a) ` S"
  1037 
  1038   have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
  1039   have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
  1040   have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
  1041 
  1042   hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
  1043   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
  1044     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1045   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
  1046     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
  1047   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"
  1048     apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
  1049   moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *s x) = (\<Sum>x\<in>t. Q x *s x)"
  1050     apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
  1051   have "(\<Sum>x\<in>t. u (x - a)) *s a = (\<Sum>v\<in>t. u (v - a) *s v)" 
  1052     unfolding setsum_vmul[OF fin(1)] unfolding t_def and setsum_reindex[OF inj] and o_def
  1053     using obt(5) by (auto simp add: setsum_addf)
  1054   hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *s v) = 0"
  1055     unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *  vector_smult_lneg) 
  1056   ultimately show ?thesis unfolding affine_dependent_explicit
  1057     apply(rule_tac x="insert a t" in exI) by auto 
  1058 qed
  1059 
  1060 lemma convex_cone:
  1061   "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 *s x) \<in> s)" (is "?lhs = ?rhs")
  1062 proof-
  1063   { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
  1064     hence "2 *s x \<in>s" "2 *s y \<in> s" unfolding cone_def by auto
  1065     hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
  1066       apply(erule_tac x="2*s x" in ballE) apply(erule_tac x="2*s y" in ballE)
  1067       apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
  1068   thus ?thesis unfolding convex_def cone_def by blast
  1069 qed
  1070 
  1071 lemma affine_dependent_biggerset: fixes s::"(real^'n::finite) set"
  1072   assumes "finite s" "card s \<ge> CARD('n) + 2"
  1073   shows "affine_dependent s"
  1074 proof-
  1075   have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
  1076   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1077   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1078     apply(rule card_image) unfolding inj_on_def by auto
  1079   also have "\<dots> > CARD('n)" using assms(2)
  1080     unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1081   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1082     apply(rule dependent_imp_affine_dependent)
  1083     apply(rule dependent_biggerset) by auto qed
  1084 
  1085 lemma affine_dependent_biggerset_general:
  1086   assumes "finite (s::(real^'n::finite) set)" "card s \<ge> dim s + 2"
  1087   shows "affine_dependent s"
  1088 proof-
  1089   from assms(2) have "s \<noteq> {}" by auto
  1090   then obtain a where "a\<in>s" by auto
  1091   have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
  1092   have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
  1093     apply(rule card_image) unfolding inj_on_def by auto
  1094   have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
  1095     apply(rule subset_le_dim) unfolding subset_eq
  1096     using `a\<in>s` by (auto simp add:span_superset span_sub)
  1097   also have "\<dots> < dim s + 1" by auto
  1098   also have "\<dots> \<le> card (s - {a})" using assms
  1099     using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
  1100   finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
  1101     apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
  1102 
  1103 subsection {* Caratheodory's theorem. *}
  1104 
  1105 lemma convex_hull_caratheodory: fixes p::"(real^'n::finite) set"
  1106   shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and>
  1107   (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
  1108   unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
  1109 proof(rule,rule)
  1110   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 *s v) = y"
  1111   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 *s v) = y"
  1112   then obtain N where "?P N" by auto
  1113   hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
  1114   then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
  1115   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 *s v) = y" by auto
  1116 
  1117   have "card s \<le> CARD('n) + 1" proof(rule ccontr, simp only: not_le)
  1118     assume "CARD('n) + 1 < card s"
  1119     hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
  1120     then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *s v) = 0"
  1121       using affine_dependent_explicit_finite[OF obt(1)] by auto
  1122     def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
  1123     have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
  1124       assume as:"\<forall>x\<in>s. 0 \<le> w x"
  1125       hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
  1126       hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
  1127 	using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
  1128       thus False using wv(1) by auto
  1129     qed hence "i\<noteq>{}" unfolding i_def by auto
  1130 
  1131     hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
  1132       using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
  1133     have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
  1134       fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
  1135       show"0 \<le> u v + t * w v" proof(cases "w v < 0")
  1136 	case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
  1137 	  using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
  1138 	case True hence "t \<le> u v / (- w v)" using `v\<in>s`
  1139 	  unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
  1140 	thus ?thesis unfolding real_0_le_add_iff
  1141 	  using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
  1142       qed qed
  1143 
  1144     obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
  1145       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
  1146     hence a:"a\<in>s" "u a + t * w a = 0" by auto
  1147     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 
  1148     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
  1149       unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
  1150     moreover have "(\<Sum>v\<in>s. u v *s v + (t * w v) *s v) - (u a *s a + (t * w a) *s a) = y" 
  1151       unfolding setsum_addf obt(6) vector_smult_assoc[THEN sym] setsum_cmul wv(4)
  1152       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]]
  1153       by (simp add: vector_smult_lneg)
  1154     ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
  1155       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: *)
  1156     thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
  1157   thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1
  1158     \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y" using obt by auto
  1159 qed auto
  1160 
  1161 lemma caratheodory:
  1162  "convex hull p = {x::real^'n::finite. \<exists>s. finite s \<and> s \<subseteq> p \<and>
  1163       card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s}"
  1164   unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
  1165   fix x assume "x \<in> convex hull p"
  1166   then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1"
  1167      "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"unfolding convex_hull_caratheodory by auto
  1168   thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
  1169     apply(rule_tac x=s in exI) using hull_subset[of s convex]
  1170   using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
  1171 next
  1172   fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
  1173   then obtain s where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1" "x \<in> convex hull s" by auto
  1174   thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
  1175 qed
  1176 
  1177 subsection {* Openness and compactness are preserved by convex hull operation. *}
  1178 
  1179 lemma open_convex_hull:
  1180   assumes "open s"
  1181   shows "open(convex hull s)"
  1182   unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10) 
  1183 proof(rule, rule) fix a
  1184   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 *s v) = a"
  1185   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 *s v) = a" by auto
  1186 
  1187   from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
  1188     using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
  1189   have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
  1190 
  1191   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 *s v) = y}"
  1192     apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
  1193   proof-
  1194     show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
  1195       using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
  1196   next  fix y assume "y \<in> cball a (Min i)"
  1197     hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
  1198     { fix x assume "x\<in>t"
  1199       hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
  1200       hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
  1201       moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
  1202       ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by auto }
  1203     moreover
  1204     have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
  1205     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
  1206       unfolding setsum_reindex[OF *] o_def using obt(4) by auto
  1207     moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *s v) = y"
  1208       unfolding setsum_reindex[OF *] o_def using obt(4,5)
  1209       by (simp add: setsum_addf setsum_subtractf setsum_vmul[OF obt(1), THEN sym]) 
  1210     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 *s v) = y"
  1211       apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
  1212       using obt(1, 3) by auto
  1213   qed
  1214 qed
  1215 
  1216 lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
  1217 unfolding open_vector_def all_1
  1218 by (auto simp add: dest_vec1_def)
  1219 
  1220 lemma tendsto_dest_vec1: "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
  1221   unfolding tendsto_def
  1222   apply clarify
  1223   apply (drule_tac x="dest_vec1 -` S" in spec)
  1224   apply (simp add: open_dest_vec1_vimage)
  1225   done
  1226 
  1227 lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
  1228   unfolding continuous_def by (rule tendsto_dest_vec1)
  1229 
  1230 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
  1231 by (induct x) simp
  1232 
  1233 (* TODO: move *)
  1234 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
  1235 unfolding compact_def
  1236 apply clarify
  1237 apply (drule_tac x="fst \<circ> f" in spec)
  1238 apply (drule mp, simp add: mem_Times_iff)
  1239 apply (clarify, rename_tac l1 r1)
  1240 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
  1241 apply (drule mp, simp add: mem_Times_iff)
  1242 apply (clarify, rename_tac l2 r2)
  1243 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  1244 apply (rule_tac x="r1 \<circ> r2" in exI)
  1245 apply (rule conjI, simp add: subseq_def)
  1246 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
  1247 apply (drule (1) tendsto_Pair) back
  1248 apply (simp add: o_def)
  1249 done
  1250 
  1251 (* TODO: move *)
  1252 lemma compact_real_interval:
  1253   fixes a b :: real shows "compact {a..b}"
  1254 proof -
  1255   have "continuous_on {vec1 a .. vec1 b} dest_vec1"
  1256     unfolding continuous_on
  1257     by (simp add: tendsto_dest_vec1 Lim_at_within Lim_ident_at)
  1258   moreover have "compact {vec1 a .. vec1 b}" by (rule compact_interval)
  1259   ultimately have "compact (dest_vec1 ` {vec1 a .. vec1 b})"
  1260     by (rule compact_continuous_image)
  1261   also have "dest_vec1 ` {vec1 a .. vec1 b} = {a..b}"
  1262     by (auto simp add: image_def Bex_def exists_vec1)
  1263   finally show ?thesis .
  1264 qed
  1265 
  1266 lemma compact_convex_combinations:
  1267   fixes s t :: "(real ^ 'n::finite) set"
  1268   assumes "compact s" "compact t"
  1269   shows "compact { (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
  1270 proof-
  1271   let ?X = "{0..1} \<times> s \<times> t"
  1272   let ?h = "(\<lambda>z. (1 - fst z) *s fst (snd z) + fst z *s snd (snd z))"
  1273   have *:"{ (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
  1274     apply(rule set_ext) unfolding image_iff mem_Collect_eq
  1275     apply rule apply auto
  1276     apply (rule_tac x=u in rev_bexI, simp)
  1277     apply (erule rev_bexI, erule rev_bexI, simp)
  1278     by auto
  1279   { fix u::"real" fix x y assume as:"0 \<le> u" "u \<le> 1" "x \<in> s" "y \<in> t"
  1280     hence "continuous (at (u, x, y))
  1281            (\<lambda>z. fst (snd z) - fst z *s fst (snd z) + fst z *s snd (snd z))"
  1282       apply (auto intro!: continuous_add continuous_sub continuous_mul)
  1283       unfolding continuous_at
  1284       by (safe intro!: tendsto_fst tendsto_snd Lim_at_id [unfolded id_def])
  1285   }
  1286   hence "continuous_on ({0..1} \<times> s \<times> t)
  1287      (\<lambda>z. (1 - fst z) *s fst (snd z) + fst z *s snd (snd z))"
  1288     apply(rule_tac continuous_at_imp_continuous_on) by auto
  1289  thus ?thesis unfolding * apply(rule compact_continuous_image)
  1290     defer apply(rule compact_Times) defer apply(rule compact_Times)
  1291     using compact_real_interval assms by auto
  1292 qed
  1293 
  1294 lemma compact_convex_hull: fixes s::"(real^'n::finite) set"
  1295   assumes "compact s"  shows "compact(convex hull s)"
  1296 proof(cases "s={}")
  1297   case True thus ?thesis using compact_empty by simp
  1298 next
  1299   case False then obtain w where "w\<in>s" by auto
  1300   show ?thesis unfolding caratheodory[of s]
  1301   proof(induct "CARD('n) + 1")
  1302     have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
  1303       using compact_empty by (auto simp add: convex_hull_empty)
  1304     case 0 thus ?case unfolding * by simp
  1305   next
  1306     case (Suc n)
  1307     show ?case proof(cases "n=0")
  1308       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"
  1309 	unfolding expand_set_eq and mem_Collect_eq proof(rule, rule)
  1310 	fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1311 	then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  1312 	show "x\<in>s" proof(cases "card t = 0")
  1313 	  case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by(simp add: convex_hull_empty)
  1314 	next
  1315 	  case False hence "card t = Suc 0" using t(3) `n=0` by auto
  1316 	  then obtain a where "t = {a}" unfolding card_Suc_eq by auto
  1317 	  thus ?thesis using t(2,4) by (simp add: convex_hull_singleton)
  1318 	qed
  1319       next
  1320 	fix x assume "x\<in>s"
  1321 	thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1322 	  apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
  1323       qed thus ?thesis using assms by simp
  1324     next
  1325       case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
  1326 	{ (1 - u) *s x + u *s y | x y u. 
  1327 	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}}"
  1328 	unfolding expand_set_eq and mem_Collect_eq proof(rule,rule)
  1329 	fix x assume "\<exists>u v c. x = (1 - c) *s u + c *s v \<and>
  1330           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)"
  1331 	then obtain u v c t where obt:"x = (1 - c) *s u + c *s v"
  1332           "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
  1333 	moreover have "(1 - c) *s u + c *s v \<in> convex hull insert u t"
  1334 	  apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
  1335 	  using obt(7) and hull_mono[of t "insert u t"] by auto
  1336 	ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1337 	  apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
  1338       next
  1339 	fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
  1340 	then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
  1341 	let ?P = "\<exists>u v c. x = (1 - c) *s u + c *s v \<and>
  1342           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)"
  1343 	show ?P proof(cases "card t = Suc n")
  1344 	  case False hence "card t \<le> n" using t(3) by auto
  1345 	  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
  1346 	    by(auto intro!: exI[where x=t])
  1347 	next
  1348 	  case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
  1349 	  show ?P proof(cases "u={}")
  1350 	    case True hence "x=a" using t(4)[unfolded au] by auto
  1351 	    show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
  1352 	      using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
  1353 	  next
  1354 	    case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *s a + vx *s b"
  1355 	      using t(4)[unfolded au convex_hull_insert[OF False]] by auto
  1356 	    have *:"1 - vx = ux" using obt(3) by auto
  1357 	    show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
  1358 	      using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
  1359 	      by(auto intro!: exI[where x=u])
  1360 	  qed
  1361 	qed
  1362       qed
  1363       thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
  1364     qed
  1365   qed 
  1366 qed
  1367 
  1368 lemma finite_imp_compact_convex_hull:
  1369  "finite s \<Longrightarrow> compact(convex hull s)"
  1370   apply(drule finite_imp_compact, drule compact_convex_hull) by assumption
  1371 
  1372 subsection {* Extremal points of a simplex are some vertices. *}
  1373 
  1374 lemma dist_increases_online:
  1375   fixes a b d :: "real ^ 'n::finite"
  1376   assumes "d \<noteq> 0"
  1377   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
  1378 proof(cases "a \<bullet> d - b \<bullet> d > 0")
  1379   case True hence "0 < d \<bullet> d + (a \<bullet> d * 2 - b \<bullet> d * 2)" 
  1380     apply(rule_tac add_pos_pos) using assms by auto
  1381   thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and real_vector_norm_def and real_sqrt_less_iff
  1382     by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
  1383 next
  1384   case False hence "0 < d \<bullet> d + (b \<bullet> d * 2 - a \<bullet> d * 2)" 
  1385     apply(rule_tac add_pos_nonneg) using assms by auto
  1386   thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and real_vector_norm_def and real_sqrt_less_iff
  1387     by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
  1388 qed
  1389 
  1390 lemma norm_increases_online:
  1391  "(d::real^'n::finite) \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
  1392   using dist_increases_online[of d a 0] unfolding dist_norm by auto
  1393 
  1394 lemma simplex_furthest_lt:
  1395   fixes s::"(real^'n::finite) set" assumes "finite s"
  1396   shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
  1397 proof(induct_tac rule: finite_induct[of s])
  1398   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))"
  1399   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))"
  1400   proof(rule,rule,cases "s = {}")
  1401     case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
  1402     obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *s x + v *s b"
  1403       using y(1)[unfolded convex_hull_insert[OF False]] by auto
  1404     show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
  1405     proof(cases "y\<in>convex hull s")
  1406       case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
  1407 	using as(3)[THEN bspec[where x=y]] and y(2) by auto
  1408       thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
  1409     next
  1410       case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
  1411 	assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
  1412 	thus ?thesis using False and obt(4) by auto
  1413       next
  1414 	assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
  1415 	thus ?thesis using y(2) by auto
  1416       next
  1417 	assume "u\<noteq>0" "v\<noteq>0"
  1418 	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
  1419 	have "x\<noteq>b" proof(rule ccontr) 
  1420 	  assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
  1421 	    using obt(3) by(auto simp add: vector_sadd_rdistrib[THEN sym])
  1422 	  thus False using obt(4) and False by simp qed
  1423 	hence *:"w *s (x - b) \<noteq> 0" using w(1) by auto
  1424 	show ?thesis using dist_increases_online[OF *, of a y]
  1425  	proof(erule_tac disjE)
  1426 	  assume "dist a y < dist a (y + w *s (x - b))"
  1427 	  hence "norm (y - a) < norm ((u + w) *s x + (v - w) *s b - a)"
  1428 	    unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: ring_simps)
  1429 	  moreover have "(u + w) *s x + (v - w) *s b \<in> convex hull insert x s"
  1430 	    unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  1431 	    apply(rule_tac x="u + w" in exI) apply rule defer 
  1432 	    apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  1433 	  ultimately show ?thesis by auto
  1434 	next
  1435 	  assume "dist a y < dist a (y - w *s (x - b))"
  1436 	  hence "norm (y - a) < norm ((u - w) *s x + (v + w) *s b - a)"
  1437 	    unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: ring_simps)
  1438 	  moreover have "(u - w) *s x + (v + w) *s b \<in> convex hull insert x s"
  1439 	    unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
  1440 	    apply(rule_tac x="u - w" in exI) apply rule defer 
  1441 	    apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
  1442 	  ultimately show ?thesis by auto
  1443 	qed
  1444       qed auto
  1445     qed
  1446   qed auto
  1447 qed (auto simp add: assms)
  1448 
  1449 lemma simplex_furthest_le:
  1450   assumes "finite s" "s \<noteq> {}"
  1451   shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
  1452 proof-
  1453   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  1454   then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
  1455     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
  1456     unfolding dist_commute[of a] unfolding dist_norm by auto
  1457   thus ?thesis proof(cases "x\<in>s")
  1458     case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
  1459       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
  1460     thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
  1461   qed auto
  1462 qed
  1463 
  1464 lemma simplex_furthest_le_exists:
  1465   "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
  1466   using simplex_furthest_le[of s] by (cases "s={}")auto
  1467 
  1468 lemma simplex_extremal_le:
  1469   assumes "finite s" "s \<noteq> {}"
  1470   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)"
  1471 proof-
  1472   have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
  1473   then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
  1474     "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
  1475     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
  1476   thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
  1477     assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
  1478       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
  1479     thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
  1480   next
  1481     assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
  1482       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
  1483     thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
  1484       by (auto simp add: norm_minus_commute)
  1485   qed auto
  1486 qed 
  1487 
  1488 lemma simplex_extremal_le_exists:
  1489   "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
  1490   \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
  1491   using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
  1492 
  1493 subsection {* Closest point of a convex set is unique, with a continuous projection. *}
  1494 
  1495 definition
  1496   closest_point :: "(real ^ 'n::finite) set \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 'n" where
  1497  "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
  1498 
  1499 lemma closest_point_exists:
  1500   assumes "closed s" "s \<noteq> {}"
  1501   shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
  1502   unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
  1503   using distance_attains_inf[OF assms(1,2), of a] by auto
  1504 
  1505 lemma closest_point_in_set:
  1506   "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
  1507   by(meson closest_point_exists)
  1508 
  1509 lemma closest_point_le:
  1510   "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
  1511   using closest_point_exists[of s] by auto
  1512 
  1513 lemma closest_point_self:
  1514   assumes "x \<in> s"  shows "closest_point s x = x"
  1515   unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
  1516   using assms by auto
  1517 
  1518 lemma closest_point_refl:
  1519  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
  1520   using closest_point_in_set[of s x] closest_point_self[of x s] by auto
  1521 
  1522 lemma closer_points_lemma: fixes y::"real^'n::finite"
  1523   assumes "y \<bullet> z > 0"
  1524   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *s z - y) < norm y"
  1525 proof- have z:"z \<bullet> z > 0" unfolding dot_pos_lt using assms by auto
  1526   thus ?thesis using assms apply(rule_tac x="(y \<bullet> z) / (z \<bullet> z)" in exI) apply(rule) defer proof(rule+)
  1527     fix v assume "0<v" "v \<le> y \<bullet> z / (z \<bullet> z)"
  1528     thus "norm (v *s z - y) < norm y" unfolding norm_lt using z and assms
  1529       by (simp add: field_simps dot_sym  mult_strict_left_mono[OF _ `0<v`])
  1530   qed(rule divide_pos_pos, auto) qed
  1531 
  1532 lemma closer_point_lemma:
  1533   fixes x y z :: "real ^ 'n::finite"
  1534   assumes "(y - x) \<bullet> (z - x) > 0"
  1535   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *s (z - x)) y < dist x y"
  1536 proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *s (z - x) - (y - x)) < norm (y - x)"
  1537     using closer_points_lemma[OF assms] by auto
  1538   show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
  1539     unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
  1540 
  1541 lemma any_closest_point_dot:
  1542   assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  1543   shows "(a - x) \<bullet> (y - x) \<le> 0"
  1544 proof(rule ccontr) assume "\<not> (a - x) \<bullet> (y - x) \<le> 0"
  1545   then obtain u where u:"u>0" "u\<le>1" "dist (x + u *s (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
  1546   let ?z = "(1 - u) *s x + u *s y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
  1547   thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute field_simps) qed
  1548 
  1549 lemma any_closest_point_unique:
  1550   assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
  1551   "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
  1552   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)]
  1553   unfolding norm_pths(1) and norm_le_square by auto
  1554 
  1555 lemma closest_point_unique:
  1556   assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
  1557   shows "x = closest_point s a"
  1558   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
  1559   using closest_point_exists[OF assms(2)] and assms(3) by auto
  1560 
  1561 lemma closest_point_dot:
  1562   assumes "convex s" "closed s" "x \<in> s"
  1563   shows "(a - closest_point s a) \<bullet> (x - closest_point s a) \<le> 0"
  1564   apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
  1565   using closest_point_exists[OF assms(2)] and assms(3) by auto
  1566 
  1567 lemma closest_point_lt:
  1568   assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
  1569   shows "dist a (closest_point s a) < dist a x"
  1570   apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
  1571   apply(rule closest_point_unique[OF assms(1-3), of a])
  1572   using closest_point_le[OF assms(2), of _ a] by fastsimp
  1573 
  1574 lemma closest_point_lipschitz:
  1575   assumes "convex s" "closed s" "s \<noteq> {}"
  1576   shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
  1577 proof-
  1578   have "(x - closest_point s x) \<bullet> (closest_point s y - closest_point s x) \<le> 0"
  1579        "(y - closest_point s y) \<bullet> (closest_point s x - closest_point s y) \<le> 0"
  1580     apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
  1581     using closest_point_exists[OF assms(2-3)] by auto
  1582   thus ?thesis unfolding dist_norm and norm_le
  1583     using dot_pos_le[of "(x - closest_point s x) - (y - closest_point s y)"]
  1584     by (auto simp add: dot_sym dot_ladd dot_radd) qed
  1585 
  1586 lemma continuous_at_closest_point:
  1587   assumes "convex s" "closed s" "s \<noteq> {}"
  1588   shows "continuous (at x) (closest_point s)"
  1589   unfolding continuous_at_eps_delta 
  1590   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
  1591 
  1592 lemma continuous_on_closest_point:
  1593   assumes "convex s" "closed s" "s \<noteq> {}"
  1594   shows "continuous_on t (closest_point s)"
  1595   apply(rule continuous_at_imp_continuous_on) using continuous_at_closest_point[OF assms] by auto
  1596 
  1597 subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
  1598 
  1599 lemma supporting_hyperplane_closed_point:
  1600   assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
  1601   shows "\<exists>a b. \<exists>y\<in>s. a \<bullet> z < b \<and> (a \<bullet> y = b) \<and> (\<forall>x\<in>s. a \<bullet> x \<ge> b)"
  1602 proof-
  1603   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
  1604   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="(y - z) \<bullet> y" in exI, rule_tac x=y in bexI)
  1605     apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
  1606     show "(y - z) \<bullet> z < (y - z) \<bullet> y" apply(subst diff_less_iff(1)[THEN sym])
  1607       unfolding dot_rsub[THEN sym] and dot_pos_lt using `y\<in>s` `z\<notin>s` by auto
  1608   next
  1609     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) *s y + u *s x)"
  1610       using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
  1611     assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x" then obtain v where
  1612       "v>0" "v\<le>1" "dist (y + v *s (x - y)) z < dist y z" using closer_point_lemma[of z y x] by auto
  1613     thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute field_simps)
  1614   qed auto
  1615 qed
  1616 
  1617 lemma separating_hyperplane_closed_point:
  1618   assumes "convex s" "closed s" "z \<notin> s"
  1619   shows "\<exists>a b. a \<bullet> z < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
  1620 proof(cases "s={}")
  1621   case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
  1622     using less_le_trans[OF _ dot_pos_le[of z]] by auto
  1623 next
  1624   case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
  1625     using distance_attains_inf[OF assms(2) False] by auto
  1626   show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="(y - z) \<bullet> z + (norm(y - z))\<twosuperior> / 2" in exI)
  1627     apply rule defer apply rule proof-
  1628     fix x assume "x\<in>s"
  1629     have "\<not> 0 < (z - y) \<bullet> (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
  1630       assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *s (x - y)) z < dist y z"
  1631       then obtain u where "u>0" "u\<le>1" "dist (y + u *s (x - y)) z < dist y z" by auto
  1632       thus False using y[THEN bspec[where x="y + u *s (x - y)"]]
  1633 	using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
  1634 	using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute field_simps) qed
  1635     moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
  1636     hence "0 < (y - z) \<bullet> (y - z)" unfolding norm_pow_2 by simp
  1637     ultimately show "(y - z) \<bullet> z + (norm (y - z))\<twosuperior> / 2 < (y - z) \<bullet> x"
  1638       unfolding norm_pow_2 and dlo_simps(3) by (auto simp add: field_simps dot_sym)
  1639   qed(insert `y\<in>s` `z\<notin>s`, auto)
  1640 qed
  1641 
  1642 lemma separating_hyperplane_closed_0:
  1643   assumes "convex (s::(real^'n::finite) set)" "closed s" "0 \<notin> s"
  1644   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
  1645   proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
  1646   case True have "norm ((basis a)::real^'n::finite) = 1" 
  1647     using norm_basis and dimindex_ge_1 by auto
  1648   thus ?thesis apply(rule_tac x="basis a" in exI, rule_tac x=1 in exI) using True by auto
  1649 next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
  1650     apply - apply(erule exE)+ unfolding dot_rzero apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
  1651 
  1652 subsection {* Now set-to-set for closed/compact sets. *}
  1653 
  1654 lemma separating_hyperplane_closed_compact:
  1655   assumes "convex (s::(real^'n::finite) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
  1656   shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
  1657 proof(cases "s={}")
  1658   case True
  1659   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
  1660   obtain z::"real^'n" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
  1661   hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
  1662   then obtain a b where ab:"a \<bullet> z < b" "\<forall>x\<in>t. b < a \<bullet> x"
  1663     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
  1664   thus ?thesis using True by auto
  1665 next
  1666   case False then obtain y where "y\<in>s" by auto
  1667   obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < a \<bullet> x"
  1668     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
  1669     using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
  1670   hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + a \<bullet> y < a \<bullet> x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by auto
  1671   def k \<equiv> "rsup ((\<lambda>x. a \<bullet> x) ` t)"
  1672   show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
  1673     apply(rule,rule) defer apply(rule) unfolding dot_lneg and neg_less_iff_less proof-
  1674     from ab have "((\<lambda>x. a \<bullet> x) ` t) *<= (a \<bullet> y - b)"
  1675       apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
  1676     hence k:"isLub UNIV ((\<lambda>x. a \<bullet> x) ` t) k" unfolding k_def apply(rule_tac rsup) using assms(5) by auto
  1677     fix x assume "x\<in>t" thus "a \<bullet> x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "a \<bullet> x"] by auto
  1678   next
  1679     fix x assume "x\<in>s" 
  1680     hence "k \<le> a \<bullet> x - b" unfolding k_def apply(rule_tac rsup_le) using assms(5)
  1681       unfolding setle_def
  1682       using ab[THEN bspec[where x=x]] by auto
  1683     thus "k + b / 2 < a \<bullet> x" using `0 < b` by auto
  1684   qed
  1685 qed
  1686 
  1687 lemma separating_hyperplane_compact_closed:
  1688   assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
  1689   shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
  1690 proof- obtain a b where "(\<forall>x\<in>t. a \<bullet> x < b) \<and> (\<forall>x\<in>s. b < a \<bullet> x)"
  1691     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
  1692   thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
  1693 
  1694 subsection {* General case without assuming closure and getting non-strict separation. *}
  1695 
  1696 lemma separating_hyperplane_set_0:
  1697   assumes "convex s" "(0::real^'n::finite) \<notin> s"
  1698   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> a \<bullet> x)"
  1699 proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> c \<bullet> x}"
  1700   have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
  1701     apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
  1702     defer apply(rule,rule,erule conjE) proof-
  1703     fix f assume as:"f \<subseteq> ?k ` s" "finite f"
  1704     obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
  1705     then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < a \<bullet> x"
  1706       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
  1707       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
  1708       using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
  1709     hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> y \<bullet> x)" apply(rule_tac x="inverse(norm a) *s a" in exI)
  1710        using hull_subset[of c convex] unfolding subset_eq and dot_rmult
  1711        apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
  1712        by(auto simp add: dot_sym elim!: ballE) 
  1713     thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
  1714   qed(insert closed_halfspace_ge, auto)
  1715   then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
  1716   thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: dot_sym) qed
  1717 
  1718 lemma separating_hyperplane_sets:
  1719   assumes "convex s" "convex (t::(real^'n::finite) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
  1720   shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. a \<bullet> x \<le> b) \<and> (\<forall>x\<in>t. a \<bullet> x \<ge> b)"
  1721 proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
  1722   obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> a \<bullet> x"  using assms(3-5) by auto 
  1723   hence "\<forall>x\<in>t. \<forall>y\<in>s. a \<bullet> y \<le> a \<bullet> x" apply- apply(rule, rule) apply(erule_tac x="x - y" in ballE) by auto
  1724   thus ?thesis apply(rule_tac x=a in exI, rule_tac x="rsup ((\<lambda>x. a \<bullet> x) ` s)" in exI) using `a\<noteq>0`
  1725     apply(rule) apply(rule,rule) apply(rule rsup[THEN isLubD2]) prefer 4 apply(rule,rule rsup_le) unfolding setle_def
  1726     prefer 4 using assms(3-5) by blast+ qed
  1727 
  1728 subsection {* More convexity generalities. *}
  1729 
  1730 lemma convex_closure: assumes "convex s" shows "convex(closure s)"
  1731   unfolding convex_def Ball_def closure_sequential
  1732   apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
  1733   apply(rule_tac x="\<lambda>n. u *s xb n + v *s xc n" in exI) apply(rule,rule)
  1734   apply(rule assms[unfolded convex_def, rule_format]) prefer 6
  1735   apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
  1736 
  1737 lemma convex_interior: assumes "convex s" shows "convex(interior s)"
  1738   unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
  1739   fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
  1740   fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
  1741   show "\<exists>e>0. ball ((1 - u) *s x + u *s y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
  1742     apply rule unfolding subset_eq defer apply rule proof-
  1743     fix z assume "z \<in> ball ((1 - u) *s x + u *s y) (min d e)"
  1744     hence "(1- u) *s (z - u *s (y - x)) + u *s (z + (1 - u) *s (y - x)) \<in> s"
  1745       apply(rule_tac assms[unfolded convex_alt, rule_format])
  1746       using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: ring_simps)
  1747     thus "z \<in> s" using u by (auto simp add: ring_simps) qed(insert u ed(3-4), auto) qed
  1748 
  1749 lemma convex_hull_eq_empty: "convex hull s = {} \<longleftrightarrow> s = {}"
  1750   using hull_subset[of s convex] convex_hull_empty by auto
  1751 
  1752 subsection {* Moving and scaling convex hulls. *}
  1753 
  1754 lemma convex_hull_translation_lemma:
  1755   "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
  1756   apply(rule hull_minimal, rule image_mono, rule hull_subset) unfolding mem_def
  1757   using convex_translation[OF convex_convex_hull, of a s] by assumption
  1758 
  1759 lemma convex_hull_bilemma: fixes neg
  1760   assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
  1761   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)
  1762   \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
  1763   using assms by(metis subset_antisym) 
  1764 
  1765 lemma convex_hull_translation:
  1766   "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
  1767   apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
  1768 
  1769 lemma convex_hull_scaling_lemma:
  1770  "(convex hull ((\<lambda>x. c *s x) ` s)) \<subseteq> (\<lambda>x. c *s x) ` (convex hull s)"
  1771   apply(rule hull_minimal, rule image_mono, rule hull_subset)
  1772   unfolding mem_def by(rule convex_scaling, rule convex_convex_hull)
  1773 
  1774 lemma convex_hull_scaling:
  1775   "convex hull ((\<lambda>x. c *s x) ` s) = (\<lambda>x. c *s x) ` (convex hull s)"
  1776   apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
  1777   unfolding image_image vector_smult_assoc by(auto simp add:image_constant_conv convex_hull_eq_empty)
  1778 
  1779 lemma convex_hull_affinity:
  1780   "convex hull ((\<lambda>x. a + c *s x) ` s) = (\<lambda>x. a + c *s x) ` (convex hull s)"
  1781   unfolding image_image[THEN sym] convex_hull_scaling convex_hull_translation  ..
  1782 
  1783 subsection {* Convex set as intersection of halfspaces. *}
  1784 
  1785 lemma convex_halfspace_intersection:
  1786   assumes "closed s" "convex s"
  1787   shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. a \<bullet> x \<le> b})}"
  1788   apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
  1789   fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. a \<bullet> x \<le> b}) \<longrightarrow> x \<in> xa"
  1790   hence "\<forall>a b. s \<subseteq> {x. a \<bullet> x \<le> b} \<longrightarrow> x \<in> {x. a \<bullet> x \<le> b}" by blast
  1791   thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
  1792     apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
  1793 qed auto
  1794 
  1795 subsection {* Radon's theorem (from Lars Schewe). *}
  1796 
  1797 lemma radon_ex_lemma:
  1798   assumes "finite c" "affine_dependent c"
  1799   shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) c = 0"
  1800 proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
  1801   thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult vector_smult_lzero
  1802     and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
  1803 
  1804 lemma radon_s_lemma:
  1805   assumes "finite s" "setsum f s = (0::real)"
  1806   shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
  1807 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
  1808   show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  1809     using assms(2) by assumption qed
  1810 
  1811 lemma radon_v_lemma:
  1812   assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::real^'n)"
  1813   shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
  1814 proof-
  1815   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 
  1816   show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
  1817     using assms(2) by assumption qed
  1818 
  1819 lemma radon_partition:
  1820   assumes "finite c" "affine_dependent c"
  1821   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
  1822   obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *s v) = 0" using radon_ex_lemma[OF assms] by auto
  1823   have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
  1824   def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *s setsum (\<lambda>x. u x *s x) {x\<in>c. u x > 0}"
  1825   have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
  1826     case False hence "u v < 0" by auto
  1827     thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
  1828       case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
  1829     next
  1830       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
  1831       thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
  1832   qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
  1833 
  1834   hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding real_less_def apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
  1835   moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
  1836     "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *s x) = (\<Sum>x\<in>c. u x *s x)"
  1837     using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
  1838   hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
  1839    "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *s x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *s x)" 
  1840     unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
  1841   moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
  1842     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
  1843 
  1844   ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
  1845     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)
  1846     using assms(1) unfolding vector_smult_assoc[THEN sym] setsum_cmul and z_def
  1847     by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
  1848   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
  1849     apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
  1850   hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
  1851     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)
  1852     using assms(1) unfolding vector_smult_assoc[THEN sym] setsum_cmul and z_def using *
  1853     by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
  1854   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
  1855 qed
  1856 
  1857 lemma radon: assumes "affine_dependent c"
  1858   obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
  1859 proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
  1860   hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
  1861   from radon_partition[OF *] guess m .. then guess p ..
  1862   thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
  1863 
  1864 subsection {* Helly's theorem. *}
  1865 
  1866 lemma helly_induct: fixes f::"(real^'n::finite) set set"
  1867   assumes "f hassize n" "n \<ge> CARD('n) + 1"
  1868   "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  1869   shows "\<Inter> f \<noteq> {}"
  1870   using assms unfolding hassize_def apply(erule_tac conjE) proof(induct n arbitrary: f)
  1871 case (Suc n)
  1872 show "\<Inter> f \<noteq> {}" apply(cases "n = CARD('n)") apply(rule Suc(4)[rule_format])
  1873   unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6) proof-
  1874   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
  1875     apply(rule, rule Suc(1)[rule_format])  unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6)
  1876     defer apply(rule Suc(3)[rule_format]) defer apply(rule Suc(4)[rule_format]) using Suc(2,5) by auto
  1877   then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
  1878   show ?thesis proof(cases "inj_on X f")
  1879     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
  1880     hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
  1881     show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
  1882       apply(rule, rule X[rule_format]) using X st by auto
  1883   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> {}"
  1884       using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
  1885       unfolding card_image[OF True] and Suc(6) using Suc(2,5) and ng by auto
  1886     have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
  1887     then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
  1888     hence "f \<union> (g \<union> h) = f" by auto
  1889     hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
  1890       unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
  1891     have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
  1892     have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
  1893       apply(rule_tac [!] hull_minimal) using Suc(3) gh(3-4)  unfolding mem_def unfolding subset_eq
  1894       apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
  1895       fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
  1896       thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
  1897       fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
  1898       thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
  1899     qed(auto)
  1900     thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
  1901 qed(insert dimindex_ge_1, auto) qed(auto)
  1902 
  1903 lemma helly: fixes f::"(real^'n::finite) set set"
  1904   assumes "finite f" "card f \<ge> CARD('n) + 1" "\<forall>s\<in>f. convex s"
  1905           "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
  1906   shows "\<Inter> f \<noteq>{}"
  1907   apply(rule helly_induct) unfolding hassize_def using assms by auto
  1908 
  1909 subsection {* Convex hull is "preserved" by a linear function. *}
  1910 
  1911 lemma convex_hull_linear_image:
  1912   assumes "linear f"
  1913   shows "f ` (convex hull s) = convex hull (f ` s)"
  1914   apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
  1915   apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
  1916   apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
  1917 proof- show "convex {x. f x \<in> convex hull f ` s}" 
  1918   unfolding convex_def by(auto simp add: linear_cmul[OF assms]  linear_add[OF assms]
  1919     convex_convex_hull[unfolded convex_def, rule_format]) next
  1920   show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
  1921     unfolding convex_def by (auto simp add: linear_cmul[OF assms, THEN sym]  linear_add[OF assms, THEN sym])
  1922 qed auto
  1923 
  1924 lemma in_convex_hull_linear_image:
  1925   assumes "linear f" "x \<in> convex hull s" shows "(f x) \<in> convex hull (f ` s)"
  1926 using convex_hull_linear_image[OF assms(1)] assms(2) by auto
  1927 
  1928 subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
  1929 
  1930 lemma compact_frontier_line_lemma:
  1931   fixes s :: "(real ^ _) set"
  1932   assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
  1933   obtains u where "0 \<le> u" "(u *s x) \<in> frontier s" "\<forall>v>u. (v *s x) \<notin> s"
  1934 proof-
  1935   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
  1936   let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *s x)}"
  1937   have A:"?A = (\<lambda>u. dest_vec1 u *s x) ` {0 .. vec1 (b / norm x)}"
  1938     unfolding image_image[of "\<lambda>u. u *s x" "\<lambda>x. dest_vec1 x", THEN sym]
  1939     unfolding dest_vec1_inverval vec1_dest_vec1 by auto
  1940   have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
  1941     apply(rule, rule continuous_vmul)
  1942     apply (rule continuous_dest_vec1)
  1943     apply(rule continuous_at_id) by(rule compact_interval)
  1944   moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *s x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
  1945     unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
  1946   ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *s x"
  1947     "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
  1948 
  1949   have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
  1950   { fix v assume as:"v > u" "v *s x \<in> s"
  1951     hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
  1952       using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] and norm_mul by auto
  1953     hence "norm (v *s x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
  1954       apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
  1955       using as(1) `u\<ge>0` by(auto simp add:field_simps) 
  1956     hence False unfolding obt(3) unfolding norm_mul using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
  1957   } note u_max = this
  1958 
  1959   have "u *s x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *s x" in bexI) unfolding obt(3)[THEN sym]
  1960     prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *s x" in exI) apply(rule, rule) proof-
  1961     fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *s x \<in> s"
  1962     hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
  1963     thus False using u_max[OF _ as] by auto
  1964   qed(insert `y\<in>s`, auto simp add: dist_norm obt(3))
  1965   thus ?thesis apply(rule_tac that[of u]) apply(rule obt(1), assumption)
  1966     apply(rule,rule,rule ccontr) apply(rule u_max) by auto qed
  1967 
  1968 lemma starlike_compact_projective:
  1969   assumes "compact s" "cball (0::real^'n::finite) 1 \<subseteq> s "
  1970   "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *s x) \<in> (s - frontier s )"
  1971   shows "s homeomorphic (cball (0::real^'n::finite) 1)"
  1972 proof-
  1973   have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
  1974   def pi \<equiv> "\<lambda>x::real^'n. inverse (norm x) *s x"
  1975   have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
  1976     using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
  1977   have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
  1978 
  1979   have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
  1980     apply rule unfolding pi_def
  1981     apply (rule continuous_mul)
  1982     apply (rule continuous_at_inv[unfolded o_def])
  1983     apply (rule continuous_at_norm)
  1984     apply simp
  1985     apply (rule continuous_at_id)
  1986     done
  1987   def sphere \<equiv> "{x::real^'n. norm x = 1}"
  1988   have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *s x) = pi x" unfolding pi_def sphere_def by auto
  1989 
  1990   have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
  1991   have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *s x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
  1992     fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
  1993     hence "x\<noteq>0" using `0\<notin>frontier s` by auto
  1994     obtain v where v:"0 \<le> v" "v *s x \<in> frontier s" "\<forall>w>v. w *s x \<notin> s"
  1995       using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
  1996     have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
  1997       assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
  1998       assume "v>1" thus False using assms(3)[THEN bspec[where x="v *s x"], THEN spec[where x="inverse v"]]
  1999 	using v and x and fs unfolding inverse_less_1_iff by auto qed
  2000     show "u *s x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
  2001       assume "u\<le>1" thus "u *s x \<in> s" apply(cases "u=1")
  2002 	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
  2003 
  2004   have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
  2005     apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
  2006     apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_ext,rule) 
  2007     unfolding inj_on_def prefer 3 apply(rule,rule,rule)
  2008   proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
  2009     thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
  2010   next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
  2011     then obtain u where "0 \<le> u" "u *s x \<in> frontier s" "\<forall>v>u. v *s x \<notin> s"
  2012       using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
  2013     thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *s x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
  2014   next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
  2015     hence xys:"x\<in>s" "y\<in>s" using fs by auto
  2016     from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
  2017     from nor have x:"x = norm x *s ((inverse (norm y)) *s y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
  2018     from nor have y:"y = norm y *s ((inverse (norm x)) *s x)" unfolding as(3)[unfolded pi_def] by auto 
  2019     have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
  2020       unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
  2021     hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
  2022       using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
  2023       using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
  2024       using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
  2025     thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
  2026   qed(insert `0 \<notin> frontier s`, auto)
  2027   then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
  2028     "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
  2029   
  2030   have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
  2031     apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
  2032 
  2033   { fix x assume as:"x \<in> cball (0::real^'n) 1"
  2034     have "norm x *s surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
  2035       case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
  2036       thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
  2037 	apply(rule_tac fs[unfolded subset_eq, rule_format])
  2038 	unfolding surf(5)[THEN sym] by auto
  2039     next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
  2040 	unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
  2041 
  2042   { fix x assume "x\<in>s"
  2043     hence "x \<in> (\<lambda>x. norm x *s surf (pi x)) ` cball 0 1" proof(cases "x=0")
  2044       case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
  2045     next let ?a = "inverse (norm (surf (pi x)))"
  2046       case False hence invn:"inverse (norm x) \<noteq> 0" by auto
  2047       from False have pix:"pi x\<in>sphere" using pi(1) by auto
  2048       hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
  2049       hence **:"norm x *s (?a *s surf (pi x)) = x" apply(rule_tac vector_mul_lcancel_imp[OF invn]) unfolding pi_def by auto
  2050       hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
  2051 	apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
  2052       have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
  2053       hence "norm x = norm ((?a * norm x) *s surf (pi x))"
  2054 	unfolding norm_mul abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
  2055       moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *s surf (pi x))" 
  2056 	unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
  2057       moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
  2058       hence "dist 0 (inverse (norm (surf (pi x))) *s x) \<le> 1" unfolding dist_norm
  2059 	using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
  2060 	using False `x\<in>s` by(auto simp add:field_simps)
  2061       ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *s x" in bexI)
  2062 	apply(subst injpi[THEN sym]) unfolding norm_mul abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
  2063 	unfolding pi(2)[OF `?a > 0`] by auto
  2064     qed } note hom2 = this
  2065 
  2066   show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *s surf (pi x)"])
  2067     apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
  2068     prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
  2069     fix x::"real^'n" assume as:"x \<in> cball 0 1"
  2070     thus "continuous (at x) (\<lambda>x. norm x *s surf (pi x))" proof(cases "x=0")
  2071       case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
  2072 	using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
  2073     next guess a using UNIV_witness[where 'a = 'n] ..
  2074       obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
  2075       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)
  2076 	unfolding Ball_def mem_cball dist_norm by (auto simp add: norm_basis[unfolded One_nat_def])
  2077       case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
  2078 	apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
  2079 	unfolding norm_0 vector_smult_lzero dist_norm diff_0_right norm_mul abs_norm_cancel proof-
  2080 	fix e and x::"real^'n" assume as:"norm x < e / B" "0 < norm x" "0<e"
  2081 	hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
  2082 	hence "norm (surf (pi x)) \<le> B" using B fs by auto
  2083 	hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
  2084 	also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
  2085 	also have "\<dots> = e" using `B>0` by auto
  2086 	finally show "norm x * norm (surf (pi x)) < e" by assumption
  2087       qed(insert `B>0`, auto) qed
  2088   next { fix x assume as:"surf (pi x) = 0"
  2089       have "x = 0" proof(rule ccontr)
  2090 	assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
  2091 	hence "surf (pi x) \<in> frontier s" using surf(5) by auto
  2092 	thus False using `0\<notin>frontier s` unfolding as by simp qed
  2093     } note surf_0 = this
  2094     show "inj_on (\<lambda>x. norm x *s surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
  2095       fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *s surf (pi x) = norm y *s surf (pi y)"
  2096       thus "x=y" proof(cases "x=0 \<or> y=0") 
  2097 	case True thus ?thesis using as by(auto elim: surf_0) next
  2098 	case False
  2099 	hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
  2100 	  using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
  2101 	moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
  2102 	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 
  2103 	moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
  2104 	ultimately show ?thesis using injpi by auto qed qed
  2105   qed auto qed
  2106 
  2107 lemma homeomorphic_convex_compact_lemma: fixes s::"(real^'n::finite) set"
  2108   assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
  2109   shows "s homeomorphic (cball (0::real^'n) 1)"
  2110   apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
  2111   fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
  2112   hence "u *s x \<in> interior s" unfolding interior_def mem_Collect_eq
  2113     apply(rule_tac x="ball (u *s x) (1 - u)" in exI) apply(rule, rule open_ball)
  2114     unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
  2115     fix y assume "dist (u *s x) y < 1 - u"
  2116     hence "inverse (1 - u) *s (y - u *s x) \<in> s"
  2117       using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
  2118       unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_mul      
  2119       apply (rule mult_left_le_imp_le[of "1 - u"])
  2120       unfolding class_semiring.mul_a using `u<1` by auto
  2121     thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *s (y - u *s x)" x "1 - u" u]
  2122       using as unfolding vector_smult_assoc by auto qed auto
  2123   thus "u *s x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
  2124 
  2125 lemma homeomorphic_convex_compact_cball: fixes e::real and s::"(real^'n::finite) set"
  2126   assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
  2127   shows "s homeomorphic (cball (b::real^'n::finite) e)"
  2128 proof- obtain a where "a\<in>interior s" using assms(3) by auto
  2129   then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
  2130   let ?d = "inverse d" and ?n = "0::real^'n"
  2131   have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *s (x - a)) ` s"
  2132     apply(rule, rule_tac x="d *s x + a" in image_eqI) defer
  2133     apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
  2134     by(auto simp add: mult_right_le_one_le)
  2135   hence "(\<lambda>x. inverse d *s (x - a)) ` s homeomorphic cball ?n 1"
  2136     using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *s -a + ?d *s x) ` s", OF convex_affinity compact_affinity]
  2137     using assms(1,2) by(auto simp add: uminus_add_conv_diff)
  2138   thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
  2139     apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *s -a"]])
  2140     using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff) qed
  2141 
  2142 lemma homeomorphic_convex_compact: fixes s::"(real^'n::finite) set" and t::"(real^'n) set"
  2143   assumes "convex s" "compact s" "interior s \<noteq> {}"
  2144           "convex t" "compact t" "interior t \<noteq> {}"
  2145   shows "s homeomorphic t"
  2146   using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
  2147 
  2148 subsection {* Epigraphs of convex functions. *}
  2149 
  2150 definition "epigraph s (f::real^'n \<Rightarrow> real) = {xy. fstcart xy \<in> s \<and> f(fstcart xy) \<le> dest_vec1 (sndcart xy)}"
  2151 
  2152 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
  2153 
  2154 lemma convex_epigraph: 
  2155   "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
  2156   unfolding convex_def convex_on_def unfolding Ball_def forall_pastecart epigraph_def
  2157   unfolding mem_Collect_eq fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
  2158   unfolding Ball_def[symmetric] unfolding dest_vec1_add dest_vec1_cmul
  2159   apply(subst forall_dest_vec1[THEN sym])+ by(meson real_le_refl real_le_trans add_mono mult_left_mono) 
  2160 
  2161 lemma convex_epigraphI: assumes "convex_on s f" "convex s"
  2162   shows "convex(epigraph s f)" using assms unfolding convex_epigraph by auto
  2163 
  2164 lemma convex_epigraph_convex: "convex s \<Longrightarrow> (convex_on s f \<longleftrightarrow> convex(epigraph s f))"
  2165   using convex_epigraph by auto
  2166 
  2167 subsection {* Use this to derive general bound property of convex function. *}
  2168 
  2169 lemma forall_of_pastecart:
  2170   "(\<forall>p. P (\<lambda>x. fstcart (p x)) (\<lambda>x. sndcart (p x))) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
  2171   apply(erule_tac x="\<lambda>a. pastecart (x a) (y a)" in allE) unfolding o_def by auto
  2172 
  2173 lemma forall_of_pastecart':
  2174   "(\<forall>p. P (fstcart p) (sndcart p)) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
  2175   apply(erule_tac x="pastecart x y" in allE) unfolding o_def by auto
  2176 
  2177 lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
  2178   apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto 
  2179 
  2180 lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
  2181   apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule 
  2182   apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto
  2183 
  2184 lemma convex_on:
  2185   assumes "convex s"
  2186   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>
  2187    f (setsum (\<lambda>i. u i *s x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
  2188   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
  2189   unfolding sndcart_setsum[OF finite_atLeastAtMost] fstcart_setsum[OF finite_atLeastAtMost] dest_vec1_setsum[OF finite_atLeastAtMost]
  2190   unfolding fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
  2191   unfolding dest_vec1_add dest_vec1_cmul apply(subst forall_of_pastecart)+ apply(subst forall_of_dest_vec1)+ apply rule
  2192   using assms[unfolded convex] apply simp apply(rule,rule,rule)
  2193   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
  2194   apply(rule_tac j="\<Sum>i = 1..k. u i * f (x i)" in real_le_trans)
  2195   defer apply(rule setsum_mono) apply(erule conjE)+ apply(erule_tac x=i in allE)apply(rule mult_left_mono)
  2196   using assms[unfolded convex] by auto
  2197 
  2198 subsection {* Convexity of general and special intervals. *}
  2199 
  2200 lemma is_interval_convex: assumes "is_interval s" shows "convex s"
  2201   unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
  2202   fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
  2203   hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
  2204   { fix a b assume "\<not> b \<le> u * a + v * b"
  2205     hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
  2206     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)
  2207     hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
  2208   } moreover
  2209   { fix a b assume "\<not> u * a + v * b \<le> a"
  2210     hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
  2211     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: ring_simps)
  2212     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
  2213   ultimately show "u *s x + v *s y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
  2214     using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
  2215 
  2216 lemma is_interval_connected:
  2217   fixes s :: "(real ^ _) set"
  2218   shows "is_interval s \<Longrightarrow> connected s"
  2219   using is_interval_convex convex_connected by auto
  2220 
  2221 lemma convex_interval: "convex {a .. b}" "convex {a<..<b::real^'n::finite}"
  2222   apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
  2223 
  2224 subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
  2225 
  2226 lemma is_interval_1:
  2227   "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)"
  2228   unfolding is_interval_def dest_vec1_def forall_1 by auto
  2229 
  2230 lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
  2231   apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
  2232   apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
  2233   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"
  2234   hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
  2235   let ?halfl = "{z. basis 1 \<bullet> z < dest_vec1 x} " and ?halfr = "{z. basis 1 \<bullet> z > dest_vec1 x} "
  2236   { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
  2237     using as(6) `y\<in>s` by (auto simp add: basis_component field_simps dest_vec1_eq[unfolded dest_vec1_def One_nat_def] dest_vec1_def) }
  2238   moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: basis_component field_simps dest_vec1_def) 
  2239   hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
  2240   ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
  2241     apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
  2242     apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt) apply(rule, rule, rule ccontr)
  2243     by(auto simp add: basis_component field_simps) qed 
  2244 
  2245 lemma is_interval_convex_1:
  2246   "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
  2247   using is_interval_convex convex_connected is_interval_connected_1 by auto
  2248 
  2249 lemma convex_connected_1:
  2250   "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
  2251   using is_interval_convex convex_connected is_interval_connected_1 by auto
  2252 
  2253 subsection {* Another intermediate value theorem formulation. *}
  2254 
  2255 lemma ivt_increasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2256   assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f a)$k \<le> y" "y \<le> (f b)$k"
  2257   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2258 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
  2259     using assms(1) by(auto simp add: vector_less_eq_def dest_vec1_def)
  2260   thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
  2261     using connected_continuous_image[OF assms(2) convex_connected[OF convex_interval(1)]]
  2262     using assms by(auto intro!: imageI) qed
  2263 
  2264 lemma ivt_increasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2265   assumes "dest_vec1 a \<le> dest_vec1 b"
  2266   "\<forall>x\<in>{a .. b}. continuous (at x) f" "f a$k \<le> y" "y \<le> f b$k"
  2267   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2268   apply(rule ivt_increasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
  2269 
  2270 lemma ivt_decreasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2271   assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f b)$k \<le> y" "y \<le> (f a)$k"
  2272   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2273   apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
  2274   apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
  2275   by(auto simp add:vector_uminus_component)
  2276 
  2277 lemma ivt_decreasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
  2278   assumes "dest_vec1 a \<le> dest_vec1 b" "\<forall>x\<in>{a .. b}. continuous (at x) f" "f b$k \<le> y" "y \<le> f a$k"
  2279   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
  2280   apply(rule ivt_decreasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
  2281 
  2282 subsection {* A bound within a convex hull, and so an interval. *}
  2283 
  2284 lemma convex_on_convex_hull_bound:
  2285   assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
  2286   shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
  2287   fix x assume "x\<in>convex hull s"
  2288   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 *s v i) = x"
  2289     unfolding convex_hull_indexed mem_Collect_eq by auto
  2290   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"]
  2291     unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
  2292     using assms(2) obt(1) by auto
  2293   thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
  2294     unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
  2295 
  2296 lemma unit_interval_convex_hull:
  2297   "{0::real^'n::finite .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
  2298 proof- have 01:"{0,1} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
  2299   { fix n x assume "x\<in>{0::real^'n .. 1}" "n \<le> CARD('n)" "card {i. x$i \<noteq> 0} \<le> n" 
  2300   hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
  2301     case 0 hence "x = 0" apply(subst Cart_eq) apply rule by auto
  2302     thus "x\<in>convex hull ?points" using 01 by auto
  2303   next
  2304     case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. x$i \<noteq> 0} = {}")
  2305       case True hence "x = 0" unfolding Cart_eq by auto
  2306       thus "x\<in>convex hull ?points" using 01 by auto
  2307     next
  2308       case False def xi \<equiv> "Min ((\<lambda>i. x$i) ` {i. x$i \<noteq> 0})"
  2309       have "xi \<in> (\<lambda>i. x$i) ` {i. x$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
  2310       then obtain i where i':"x$i = xi" "x$i \<noteq> 0" by auto
  2311       have i:"\<And>j. x$j > 0 \<Longrightarrow> x$i \<le> x$j"
  2312 	unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
  2313 	defer apply(rule_tac x=j in bexI) using i' by auto
  2314       have i01:"x$i \<le> 1" "x$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i] using i'(2) `x$i \<noteq> 0`
  2315 	by(auto simp add: Cart_lambda_beta) 
  2316       show ?thesis proof(cases "x$i=1")
  2317 	case True have "\<forall>j\<in>{i. x$i \<noteq> 0}. x$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq proof-
  2318 	  fix j assume "x $ j \<noteq> 0" "x $ j \<noteq> 1"
  2319 	  hence j:"x$j \<in> {0<..<1}" using Suc(2) by(auto simp add: vector_less_eq_def elim!:allE[where x=j])
  2320 	  hence "x$j \<in> op $ x ` {i. x $ i \<noteq> 0}" by auto 
  2321 	  hence "x$j \<ge> x$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
  2322 	  thus False using True Suc(2) j by(auto simp add: vector_less_eq_def elim!:ballE[where x=j]) qed
  2323         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
  2324 	  by(auto simp add: Cart_lambda_beta)
  2325       next let ?y = "\<lambda>j. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
  2326 	case False hence *:"x = x$i *s (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *s (\<chi> j. ?y j)" unfolding Cart_eq
  2327 	  by(auto simp add: Cart_lambda_beta vector_add_component vector_smult_component vector_minus_component field_simps)
  2328 	{ fix j have "x$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $ j - x $ i) / (1 - x $ i)" "(x $ j - x $ i) / (1 - x $ i) \<le> 1"
  2329 	    apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
  2330 	    using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps Cart_lambda_beta) 
  2331 	  hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
  2332 	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)
  2333 	hence "{j. x$j \<noteq> 0} \<noteq> {j. ((\<chi> j. ?y j)::real^'n::finite) $ j \<noteq> 0}" by auto
  2334 	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)  
  2335 	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
  2336 	ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
  2337 	  apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
  2338 	  unfolding mem_interval using i01 Suc(3) by (auto simp add: Cart_lambda_beta)
  2339       qed qed qed } note * = this
  2340   show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
  2341     apply(rule_tac n2="CARD('n)" in *) prefer 3 apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
  2342     unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
  2343     by(auto simp add: vector_less_eq_def mem_def[of _ convex]) qed
  2344 
  2345 subsection {* And this is a finite set of vertices. *}
  2346 
  2347 lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. 1::real^'n::finite} = convex hull s"
  2348   apply(rule that[of "{x::real^'n::finite. \<forall>i. x$i=0 \<or> x$i=1}"])
  2349   apply(rule finite_subset[of _ "(\<lambda>s. (\<chi> i. if i\<in>s then 1::real else 0)::real^'n::finite) ` UNIV"])
  2350   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
  2351   fix x::"real^'n" assume as:"\<forall>i. x $ i = 0 \<or> x $ i = 1"
  2352   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}"])
  2353     unfolding Cart_eq using as by(auto simp add:Cart_lambda_beta) qed auto
  2354 
  2355 subsection {* Hence any cube (could do any nonempty interval). *}
  2356 
  2357 lemma cube_convex_hull:
  2358   assumes "0 < d" obtains s::"(real^'n::finite) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" proof-
  2359   let ?d = "(\<chi> i. d)::real^'n"
  2360   have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *s y) ` {0 .. 1}" apply(rule set_ext, rule)
  2361     unfolding image_iff defer apply(erule bexE) proof-
  2362     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
  2363     { 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]]
  2364 	by(auto simp add: vector_component)
  2365       hence "1 \<ge> inverse d * (x $ i - y $ i)" "1 \<ge> inverse d * (y $ i - x $ i)"
  2366 	apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
  2367 	using assms by(auto simp add: field_simps right_inverse) 
  2368       hence "inverse d * (x $ i * 2) \<le> 2 + inverse d * (y $ i * 2)"
  2369             "inverse d * (y $ i * 2) \<le> 2 + inverse d * (x $ i * 2)" by(auto simp add:field_simps) }
  2370     hence "inverse (2 * d) *s (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
  2371       by(auto simp add: Cart_eq vector_component_simps field_simps)
  2372     thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *s z" apply- apply(rule_tac x="inverse (2 * d) *s (y - (x - ?d))" in bexI) 
  2373       using assms by(auto simp add: Cart_eq vector_less_eq_def Cart_lambda_beta)
  2374   next
  2375     fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *s z" 
  2376     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)
  2377       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
  2378       using assms by(auto simp add: vector_component_simps Cart_eq)
  2379     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
  2380       apply(erule_tac x=i in allE) using assms by(auto simp add:  vector_component_simps Cart_eq) qed
  2381   obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
  2382   thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *s y)` s"]) unfolding * and convex_hull_affinity by auto qed
  2383 
  2384 subsection {* Bounded convex function on open set is continuous. *}
  2385 
  2386 lemma convex_on_bounded_continuous:
  2387   assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
  2388   shows "continuous_on s f"
  2389   apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
  2390   fix x e assume "x\<in>s" "(0::real) < e"
  2391   def B \<equiv> "abs b + 1"
  2392   have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
  2393     unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
  2394   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
  2395   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
  2396     apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
  2397     fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)" 
  2398     show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
  2399       case False def t \<equiv> "k / norm (y - x)"
  2400       have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
  2401       have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
  2402 	apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute) 
  2403       { def w \<equiv> "x + t *s (y - x)"
  2404 	have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  2405 	  unfolding t_def using `k>0` by(auto simp add: norm_mul simp del: vector_ssub_ldistrib) 
  2406 	have "(1 / t) *s x + - x + ((t - 1) / t) *s x = (1 / t - 1 + (t - 1) / t) *s x" by auto 
  2407 	also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps simp del:vector_sadd_rdistrib)
  2408 	finally have w:"(1 / t) *s w + ((t - 1) / t) *s x = y" unfolding w_def using False and `t>0` by auto
  2409 	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) 
  2410 	hence "(f w - f x) / t < e"
  2411 	  using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps) 
  2412 	hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
  2413 	  using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
  2414 	  using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
  2415       moreover 
  2416       { def w \<equiv> "x - t *s (y - x)"
  2417 	have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
  2418 	  unfolding t_def using `k>0` by(auto simp add: norm_mul simp del: vector_ssub_ldistrib) 
  2419 	have "(1 / (1 + t)) *s x + (t / (1 + t)) *s x = (1 / (1 + t) + t / (1 + t)) *s x" by auto
  2420 	also have "\<dots>=x" using `t>0` by (auto simp add:field_simps simp del:vector_sadd_rdistrib)
  2421 	finally have w:"(1 / (1+t)) *s w + (t / (1 + t)) *s y = x" unfolding w_def using False and `t>0` by auto 
  2422 	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) 
  2423 	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) 
  2424 	have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y" 
  2425 	  using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
  2426 	  using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
  2427 	also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding real_divide_def by (auto simp add:field_simps)
  2428 	also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
  2429 	finally have "f x - f y < e" by auto }
  2430       ultimately show ?thesis by auto 
  2431     qed(insert `0<e`, auto) 
  2432   qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
  2433 
  2434 subsection {* Upper bound on a ball implies upper and lower bounds. *}
  2435 
  2436 lemma convex_bounds_lemma:
  2437   assumes "convex_on (cball x e) f"  "\<forall>y \<in> cball x e. f y \<le> b"
  2438   shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
  2439   apply(rule) proof(cases "0 \<le> e") case True
  2440   fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *s x - y"
  2441   have *:"x - (2 *s x - y) = y - x" by vector
  2442   have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
  2443   have "(1 / 2) *s y + (1 / 2) *s z = x" unfolding z_def by auto
  2444   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"]
  2445     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
  2446 next case False fix y assume "y\<in>cball x e" 
  2447   hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
  2448   thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
  2449 
  2450 subsection {* Hence a convex function on an open set is continuous. *}
  2451 
  2452 lemma convex_on_continuous:
  2453   assumes "open (s::(real^'n::finite) set)" "convex_on s f" 
  2454   shows "continuous_on s f"
  2455   unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
  2456   note dimge1 = dimindex_ge_1[where 'a='n]
  2457   fix x assume "x\<in>s"
  2458   then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
  2459   def d \<equiv> "e / real CARD('n)"
  2460   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
  2461   let ?d = "(\<chi> i. d)::real^'n"
  2462   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
  2463   have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:vector_component_simps)
  2464   hence "c\<noteq>{}" apply(rule_tac ccontr) using c by(auto simp add:convex_hull_empty)
  2465   def k \<equiv> "Max (f ` c)"
  2466   have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
  2467     apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
  2468     fix z assume z:"z\<in>{x - ?d..x + ?d}"
  2469     have e:"e = setsum (\<lambda>i. d) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
  2470       by (metis card_enum field_simps d_def not_one_le_zero of_nat_le_iff real_eq_of_nat real_of_nat_1)
  2471     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
  2472       using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:field_simps vector_component_simps) qed
  2473   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
  2474     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
  2475   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
  2476   hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
  2477   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
  2478   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
  2479     fix y assume y:"y\<in>cball x d"
  2480     { fix i::'n have "x $ i - d \<le> y $ i"  "y $ i \<le> x $ i + d" 
  2481 	using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add: vector_component)  }
  2482     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
  2483       by(auto simp add: vector_component_simps) qed
  2484   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
  2485     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball) by auto
  2486   thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball] using `d>0` by auto qed
  2487 
  2488 subsection {* Line segments, starlike sets etc.                                         *)
  2489 (* Use the same overloading tricks as for intervals, so that                 *)
  2490 (* segment[a,b] is closed and segment(a,b) is open relative to affine hull. *}
  2491 
  2492 definition "midpoint a b = (inverse (2::real)) *s (a + b)"
  2493 
  2494 definition "open_segment a b = {(1 - u) *s a + u *s b | u::real.  0 < u \<and> u < 1}"
  2495 
  2496 definition "closed_segment a b = {(1 - u) *s a + u *s b | u::real. 0 \<le> u \<and> u \<le> 1}"
  2497 
  2498 definition "between = (\<lambda> (a,b). closed_segment a b)"
  2499 
  2500 lemmas segment = open_segment_def closed_segment_def
  2501 
  2502 definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
  2503 
  2504 lemma midpoint_refl: "midpoint x x = x"
  2505   unfolding midpoint_def unfolding vector_add_ldistrib unfolding vector_sadd_rdistrib[THEN sym] by auto
  2506 
  2507 lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by auto
  2508 
  2509 lemma dist_midpoint:
  2510   "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
  2511   "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
  2512   "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
  2513   "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
  2514 proof-
  2515   have *: "\<And>x y::real^'n::finite. 2 *s x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
  2516   have **:"\<And>x y::real^'n::finite. 2 *s x =   y \<Longrightarrow> norm x = (norm y) / 2" by auto
  2517   show ?t1 unfolding midpoint_def dist_norm apply (rule **) by(auto,vector)
  2518   show ?t2 unfolding midpoint_def dist_norm apply (rule *)  by(auto,vector)
  2519   show ?t3 unfolding midpoint_def dist_norm apply (rule *)  by(auto,vector)
  2520   show ?t4 unfolding midpoint_def dist_norm apply (rule **) by(auto,vector) qed
  2521 
  2522 lemma midpoint_eq_endpoint:
  2523   "midpoint a b = a \<longleftrightarrow> a = (b::real^'n::finite)"
  2524   "midpoint a b = b \<longleftrightarrow> a = b"
  2525   unfolding dist_eq_0_iff[where 'a="real^'n", THEN sym] dist_midpoint by auto
  2526 
  2527 lemma convex_contains_segment:
  2528   "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. closed_segment a b \<subseteq> s)"
  2529   unfolding convex_alt closed_segment_def by auto
  2530 
  2531 lemma convex_imp_starlike:
  2532   "convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
  2533   unfolding convex_contains_segment starlike_def by auto
  2534 
  2535 lemma segment_convex_hull:
  2536  "closed_segment a b = convex hull {a,b}" proof-
  2537   have *:"\<And>x. {x} \<noteq> {}" by auto
  2538   have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
  2539   show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_ext)
  2540     unfolding mem_Collect_eq apply(rule,erule exE) 
  2541     apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
  2542     apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
  2543 
  2544 lemma convex_segment: "convex (closed_segment a b)"
  2545   unfolding segment_convex_hull by(rule convex_convex_hull)
  2546 
  2547 lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
  2548   unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto
  2549 
  2550 lemma segment_furthest_le:
  2551   assumes "x \<in> closed_segment a b" shows "norm(y - x) \<le> norm(y - a) \<or>  norm(y - x) \<le> norm(y - b)" proof-
  2552   obtain z where "z\<in>{a, b}" "norm (x - y) \<le> norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
  2553     using assms[unfolded segment_convex_hull] by auto
  2554   thus ?thesis by(auto simp add:norm_minus_commute) qed
  2555 
  2556 lemma segment_bound:
  2557   assumes "x \<in> closed_segment a b"
  2558   shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
  2559   using segment_furthest_le[OF assms, of a]
  2560   using segment_furthest_le[OF assms, of b]
  2561   by (auto simp add:norm_minus_commute) 
  2562 
  2563 lemma segment_refl:"closed_segment a a = {a}" unfolding segment by auto
  2564 
  2565 lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
  2566   unfolding between_def mem_def by auto
  2567 
  2568 lemma between:"between (a,b) (x::real^'n::finite) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
  2569 proof(cases "a = b")
  2570   case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
  2571     by(auto simp add:segment_refl dist_commute) next
  2572   case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto 
  2573   have *:"\<And>u. a - ((1 - u) *s a + u *s b) = u *s (a - b)" by auto
  2574   show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
  2575     apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
  2576       fix u assume as:"x = (1 - u) *s a + u *s b" "0 \<le> u" "u \<le> 1" 
  2577       hence *:"a - x = u *s (a - b)" "x - b = (1 - u) *s (a - b)"
  2578 	unfolding as(1) by(auto simp add:field_simps)
  2579       show "norm (a - x) *s (x - b) = norm (x - b) *s (a - x)"
  2580 	unfolding norm_minus_commute[of x a] * norm_mul Cart_eq using as(2,3)
  2581 	by(auto simp add: vector_component_simps field_simps)
  2582     next assume as:"dist a b = dist a x + dist x b"
  2583       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 
  2584       thus "\<exists>u. x = (1 - u) *s a + u *s b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
  2585 	unfolding dist_norm Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
  2586 	  fix i::'n have "((1 - norm (a - x) / norm (a - b)) *s a + (norm (a - x) / norm (a - b)) *s b) $ i =
  2587 	    ((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"
  2588 	    using Fal by(auto simp add:vector_component_simps field_simps)
  2589 	  also have "\<dots> = x$i" apply(rule divide_eq_imp[OF Fal])
  2590 	    unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
  2591 	    by(auto simp add:field_simps vector_component_simps)
  2592 	  finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *s a + (norm (a - x) / norm (a - b)) *s b) $ i" by auto
  2593 	qed(insert Fal2, auto) qed qed
  2594 
  2595 lemma between_midpoint: fixes a::"real^'n::finite" shows
  2596   "between (a,b) (midpoint a b)" (is ?t1) 
  2597   "between (b,a) (midpoint a b)" (is ?t2)
  2598 proof- have *:"\<And>x y z. x = (1/2::real) *s z \<Longrightarrow> y = (1/2) *s z \<Longrightarrow> norm z = norm x + norm y" by auto
  2599   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
  2600     by(auto simp add:field_simps Cart_eq vector_component_simps) qed
  2601 
  2602 lemma between_mem_convex_hull:
  2603   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
  2604   unfolding between_mem_segment segment_convex_hull ..
  2605 
  2606 subsection {* Shrinking towards the interior of a convex set. *}
  2607 
  2608 lemma mem_interior_convex_shrink:
  2609   assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
  2610   shows "x - e *s (x - c) \<in> interior s"
  2611 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  2612   show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
  2613     apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
  2614     fix y assume as:"dist (x - e *s (x - c)) y < e * d"
  2615     have *:"y = (1 - (1 - e)) *s ((1 / e) *s y - ((1 - e) / e) *s x) + (1 - e) *s x" using `e>0` by auto
  2616     have "dist c ((1 / e) *s y - ((1 - e) / e) *s x) = abs(1/e) * norm (e *s c - y + (1 - e) *s x)"
  2617       unfolding dist_norm unfolding norm_mul[THEN sym] apply(rule norm_eqI) using `e>0`
  2618       by(auto simp add:vector_component_simps Cart_eq field_simps) 
  2619     also have "\<dots> = abs(1/e) * norm (x - e *s (x - c) - y)" by(auto intro!:norm_eqI)
  2620     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
  2621       by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
  2622     finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
  2623       apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
  2624   qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
  2625 
  2626 lemma mem_interior_closure_convex_shrink:
  2627   assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
  2628   shows "x - e *s (x - c) \<in> interior s"
  2629 proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
  2630   have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
  2631     case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
  2632     case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
  2633     show ?thesis proof(cases "e=1")
  2634       case True obtain y where "y\<in>s" "y \<noteq> x" "dist y x < 1"
  2635 	using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
  2636       thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
  2637       case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
  2638 	using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
  2639       then obtain y where "y\<in>s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
  2640 	using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
  2641       thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
  2642   then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
  2643   def z \<equiv> "c + ((1 - e) / e) *s (x - y)"
  2644   have *:"x - e *s (x - c) = y - e *s (y - z)" unfolding z_def using `e>0` by auto
  2645   have "z\<in>interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
  2646     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
  2647     by(auto simp del:vector_ssub_ldistrib simp add:field_simps norm_minus_commute) 
  2648   thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink) 
  2649     using assms(1,4-5) `y\<in>s` by auto qed
  2650 
  2651 subsection {* Some obvious but surprisingly hard simplex lemmas. *}
  2652 
  2653 lemma simplex:
  2654   assumes "finite s" "0 \<notin> s"
  2655   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 *s x) s = y)}"
  2656   unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_ext, rule) unfolding mem_Collect_eq
  2657   apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
  2658   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)
  2659   unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
  2660 
  2661 lemma std_simplex:
  2662   "convex hull (insert 0 { basis i | i. i\<in>UNIV}) =
  2663         {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")
  2664 proof- let ?D = "UNIV::'n set"
  2665   have "0\<notin>?p" by(auto simp add: basis_nonzero)
  2666   have "{(basis i)::real^'n |i. i \<in> ?D} = basis ` ?D" by auto
  2667   note sumbas = this  setsum_reindex[OF basis_inj, unfolded o_def]
  2668   show ?thesis unfolding simplex[OF finite_stdbasis `0\<notin>?p`] apply(rule set_ext) unfolding mem_Collect_eq apply rule
  2669     apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
  2670     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 *s x) = x"
  2671     have *:"\<forall>i. u (basis i) = x$i" using as(3) unfolding sumbas and basis_expansion_unique by auto
  2672     hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $ x) ?D" unfolding sumbas by(rule_tac setsum_cong, auto)
  2673     show " (\<forall>i. 0 \<le> x $ i) \<and> setsum (op $ x) ?D \<le> 1" apply - proof(rule,rule)
  2674       fix i::'n show "0 \<le> x$i" unfolding *[rule_format,of i,THEN sym] apply(rule_tac as(1)[rule_format]) by auto
  2675     qed(insert as(2)[unfolded **], auto)
  2676   next fix x::"real^'n" assume as:"\<forall>i. 0 \<le> x $ i" "setsum (op $ x) ?D \<le> 1"
  2677     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 *s x) = x"
  2678       apply(rule_tac x="\<lambda>y. y \<bullet> x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE) using as(1) apply(erule_tac x=i in allE) 
  2679       unfolding sumbas using as(2) and basis_expansion_unique by(auto simp add:dot_basis) qed qed 
  2680 
  2681 lemma interior_std_simplex:
  2682   "interior (convex hull (insert 0 { basis i| i. i\<in>UNIV})) =
  2683   {x::real^'n::finite. (\<forall>i. 0 < x$i) \<and> setsum (\<lambda>i. x$i) UNIV < 1 }"
  2684   apply(rule set_ext) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
  2685   unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
  2686   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"
  2687   show "(\<forall>xa. 0 < x $ xa) \<and> setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
  2688     fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *s basis i"]] and `e>0`
  2689       unfolding dist_norm by(auto simp add: norm_basis vector_component_simps basis_component elim:allE[where x=i])
  2690   next guess a using UNIV_witness[where 'a='n] ..
  2691     have **:"dist x (x + (e / 2) *s basis a) < e" using  `e>0` and norm_basis[of a]
  2692       unfolding dist_norm by(auto simp add: vector_component_simps basis_component intro!: mult_strict_left_mono_comm)
  2693     have "\<And>i. (x + (e / 2) *s basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
  2694     hence *:"setsum (op $ (x + (e / 2) *s basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto) 
  2695     have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *s basis a)) UNIV" unfolding * setsum_addf
  2696       using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
  2697     also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
  2698     finally show "setsum (op $ x) UNIV < 1" by auto qed
  2699 next
  2700   fix x::"real^'n::finite" assume as:"\<forall>i. 0 < x $ i" "setsum (op $ x) UNIV < 1"
  2701   guess a using UNIV_witness[where 'a='b] ..
  2702   let ?d = "(1 - setsum (op $ x) UNIV) / real (CARD('n))"
  2703   have "Min ((op $ x) ` UNIV) > 0" apply(rule Min_grI) using as(1) dimindex_ge_1 by auto
  2704   moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) using dimindex_ge_1 by(auto simp add: Suc_le_eq)
  2705   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"
  2706     apply(rule_tac x="min (Min ((op $ x) ` UNIV)) ?D" in exI) apply rule defer apply(rule,rule) proof-
  2707     fix y assume y:"dist x y < min (Min (op $ x ` UNIV)) ?d"
  2708     have "setsum (op $ y) UNIV \<le> setsum (\<lambda>i. x$i + ?d) UNIV" proof(rule setsum_mono)
  2709       fix i::'n have "abs (y$i - x$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
  2710 	using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add:vector_component_simps norm_minus_commute)
  2711       thus "y $ i \<le> x $ i + ?d" by auto qed
  2712     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)
  2713     finally show "(\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1" apply- proof(rule,rule)
  2714       fix i::'n have "norm (x - y) < x$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
  2715 	using Min_gr_iff[of "op $ x ` dimset x"] dimindex_ge_1 by auto
  2716       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)
  2717     qed auto qed auto qed
  2718 
  2719 lemma interior_std_simplex_nonempty: obtains a::"real^'n::finite" where
  2720   "a \<in> interior(convex hull (insert 0 {basis i | i . i \<in> UNIV}))" proof-
  2721   let ?D = "UNIV::'n set" let ?a = "setsum (\<lambda>b. inverse (2 * real CARD('n)) *s b) {(basis i) | i. i \<in> ?D}"
  2722   have *:"{basis i | i. i \<in> ?D} = basis ` ?D" by auto
  2723   { fix i have "?a $ i = inverse (2 * real CARD('n))"
  2724     unfolding setsum_component vector_smult_component and * and setsum_reindex[OF basis_inj] and o_def
  2725     apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real CARD('n)) else 0) ?D"]) apply(rule setsum_cong2)
  2726       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]) }
  2727   note ** = this
  2728   show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof(rule,rule)
  2729     fix i::'n show "0 < ?a $ i" unfolding ** using dimindex_ge_1 by(auto simp add: Suc_le_eq) next
  2730     have "setsum (op $ ?a) ?D = setsum (\<lambda>i. inverse (2 * real CARD('n))) ?D" by(rule setsum_cong2, rule **) 
  2731     also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat real_divide_def[THEN sym] by (auto simp add:field_simps)
  2732     finally show "setsum (op $ ?a) ?D < 1" by auto qed qed
  2733 
  2734 subsection {* Paths. *}
  2735 
  2736 definition "path (g::real^1 \<Rightarrow> real^'n::finite) \<longleftrightarrow> continuous_on {0 .. 1} g"
  2737 
  2738 definition "pathstart (g::real^1 \<Rightarrow> real^'n) = g 0"
  2739 
  2740 definition "pathfinish (g::real^1 \<Rightarrow> real^'n) = g 1"
  2741 
  2742 definition "path_image (g::real^1 \<Rightarrow> real^'n) = g ` {0 .. 1}"
  2743 
  2744 definition "reversepath (g::real^1 \<Rightarrow> real^'n) = (\<lambda>x. g(1 - x))"
  2745 
  2746 definition joinpaths:: "(real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n)" (infixr "+++" 75)
  2747   where "joinpaths g1 g2 = (\<lambda>x. if dest_vec1 x \<le> ((1 / 2)::real) then g1 (2 *s x) else g2(2 *s x - 1))"
  2748 definition "simple_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
  2749   (\<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)"
  2750 
  2751 definition "injective_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
  2752   (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)"
  2753 
  2754 subsection {* Some lemmas about these concepts. *}
  2755 
  2756 lemma injective_imp_simple_path:
  2757   "injective_path g \<Longrightarrow> simple_path g"
  2758   unfolding injective_path_def simple_path_def by auto
  2759 
  2760 lemma path_image_nonempty: "path_image g \<noteq> {}"
  2761   unfolding path_image_def image_is_empty interval_eq_empty by auto 
  2762 
  2763 lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g"
  2764   unfolding pathstart_def path_image_def apply(rule imageI)
  2765   unfolding mem_interval_1 vec_1[THEN sym] dest_vec1_vec by auto
  2766 
  2767 lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g"
  2768   unfolding pathfinish_def path_image_def apply(rule imageI)
  2769   unfolding mem_interval_1 vec_1[THEN sym] dest_vec1_vec by auto
  2770 
  2771 lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"
  2772   unfolding path_def path_image_def apply(rule connected_continuous_image, assumption)
  2773   by(rule convex_connected, rule convex_interval)
  2774 
  2775 lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"
  2776   unfolding path_def path_image_def apply(rule compact_continuous_image, assumption)
  2777   by(rule compact_interval)
  2778 
  2779 lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g"
  2780   unfolding reversepath_def by auto
  2781 
  2782 lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g"
  2783   unfolding pathstart_def reversepath_def pathfinish_def by auto
  2784 
  2785 lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"
  2786   unfolding pathstart_def reversepath_def pathfinish_def by auto
  2787 
  2788 lemma pathstart_join[simp]: "pathstart(g1 +++ g2) = pathstart g1"
  2789   unfolding pathstart_def joinpaths_def pathfinish_def by auto
  2790 
  2791 lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2" proof-
  2792   have "2 *s 1 - 1 = (1::real^1)" unfolding Cart_eq by(auto simp add:vector_component_simps)
  2793   thus ?thesis unfolding pathstart_def joinpaths_def pathfinish_def
  2794     unfolding vec_1[THEN sym] dest_vec1_vec by auto qed
  2795 
  2796 lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g" proof-
  2797   have *:"\<And>g. path_image(reversepath g) \<subseteq> path_image g"
  2798     unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply(rule,rule,erule bexE)  
  2799     apply(rule_tac x="1 - xa" in bexI) by(auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
  2800   show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
  2801 
  2802 lemma path_reversepath[simp]: "path(reversepath g) \<longleftrightarrow> path g" proof-
  2803   have *:"\<And>g. path g \<Longrightarrow> path(reversepath g)" unfolding path_def reversepath_def
  2804     apply(rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
  2805     apply(rule continuous_on_sub, rule continuous_on_const, rule continuous_on_id)
  2806     apply(rule continuous_on_subset[of "{0..1}"], assumption)
  2807     by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
  2808   show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
  2809 
  2810 lemmas reversepath_simps = path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
  2811 
  2812 lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow>  path g1 \<and> path g2"
  2813   unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof-
  2814   assume as:"continuous_on {0..1} (g1 +++ g2)"
  2815   have *:"g1 = (\<lambda>x. g1 (2 *s x)) \<circ> (\<lambda>x. (1/2) *s x)" 
  2816          "g2 = (\<lambda>x. g2 (2 *s x - 1)) \<circ> (\<lambda>x. (1/2) *s (x + 1))" unfolding o_def by auto
  2817   have "op *s (1 / 2) ` {0::real^1..1} \<subseteq> {0..1}"  "(\<lambda>x. (1 / 2) *s (x + 1)) ` {(0::real^1)..1} \<subseteq> {0..1}"
  2818     unfolding image_smult_interval by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
  2819   thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule
  2820     apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose)
  2821     apply (rule continuous_on_cmul, rule continuous_on_add, rule continuous_on_id, rule continuous_on_const) defer
  2822     apply (rule continuous_on_cmul, rule continuous_on_id) apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3
  2823     apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption)
  2824     apply(rule) defer apply rule proof-
  2825     fix x assume "x \<in> op *s (1 / 2) ` {0::real^1..1}"
  2826     hence "dest_vec1 x \<le> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
  2827     thus "(g1 +++ g2) x = g1 (2 *s x)" unfolding joinpaths_def by auto next
  2828     fix x assume "x \<in> (\<lambda>x. (1 / 2) *s (x + 1)) ` {0::real^1..1}"
  2829     hence "dest_vec1 x \<ge> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
  2830     thus "(g1 +++ g2) x = g2 (2 *s x - 1)" proof(cases "dest_vec1 x = 1 / 2")
  2831       case True hence "x = (1/2) *s 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
  2832       thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by auto
  2833     qed (auto simp add:le_less joinpaths_def) qed
  2834 next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"
  2835   have *:"{0 .. 1::real^1} = {0.. (1/2)*s 1} \<union> {(1/2) *s 1 .. 1}" by(auto simp add: vector_component_simps) 
  2836   have **:"op *s 2 ` {0..(1 / 2) *s 1} = {0..1::real^1}" apply(rule set_ext, rule) unfolding image_iff 
  2837     defer apply(rule_tac x="(1/2)*s x" in bexI) by(auto simp add: vector_component_simps)
  2838   have ***:"(\<lambda>x. 2 *s x - 1) ` {(1 / 2) *s 1..1} = {0..1::real^1}"
  2839     unfolding image_affinity_interval[of _ "- 1", unfolded diff_def[symmetric]] and interval_eq_empty_1
  2840     by(auto simp add: vector_component_simps)
  2841   have ****:"\<And>x::real^1. x $ 1 * 2 = 1 \<longleftrightarrow> x = (1/2) *s 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
  2842   show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply(rule closed_interval)+ proof-
  2843     show "continuous_on {0..(1 / 2) *s 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *s x)"]) defer
  2844       unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply(rule continuous_on_cmul, rule continuous_on_id)
  2845       unfolding ** apply(rule as(1)) unfolding joinpaths_def by(auto simp add: vector_component_simps) next
  2846     show "continuous_on {(1/2)*s1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *s x - 1)"]) defer
  2847       apply(rule continuous_on_compose) apply(rule continuous_on_sub, rule continuous_on_cmul, rule continuous_on_id, rule continuous_on_const)
  2848       unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def]
  2849       by(auto simp add: vector_component_simps ****) qed qed
  2850 
  2851 lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof
  2852   fix x assume "x \<in> path_image (g1 +++ g2)"
  2853   then obtain y where y:"y\<in>{0..1}" "x = (if dest_vec1 y \<le> 1 / 2 then g1 (2 *s y) else g2 (2 *s y - 1))"
  2854     unfolding path_image_def image_iff joinpaths_def by auto
  2855   thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "dest_vec1 y \<le> 1/2")
  2856     apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1)
  2857     by(auto intro!: imageI simp add: vector_component_simps) qed
  2858 
  2859 lemma subset_path_image_join:
  2860   assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s" shows "path_image(g1 +++ g2) \<subseteq> s"
  2861   using path_image_join_subset[of g1 g2] and assms by auto
  2862 
  2863 lemma path_image_join:
  2864   assumes "path g1" "path g2" "pathfinish g1 = pathstart g2"
  2865   shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"
  2866 apply(rule, rule path_image_join_subset, rule) unfolding Un_iff proof(erule disjE)
  2867   fix x assume "x \<in> path_image g1"
  2868   then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto
  2869   thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
  2870     apply(rule_tac x="(1/2) *s y" in bexI) by(auto simp add: vector_component_simps) next
  2871   fix x assume "x \<in> path_image g2"
  2872   then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto
  2873   moreover have *:"y $ 1 = 0 \<Longrightarrow> y = 0" unfolding Cart_eq by auto
  2874   ultimately show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
  2875     apply(rule_tac x="(1/2) *s (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]
  2876     by(auto simp add: vector_component_simps) qed 
  2877 
  2878 lemma not_in_path_image_join:
  2879   assumes "x \<notin> path_image g1" "x \<notin> path_image g2" shows "x \<notin> path_image(g1 +++ g2)"
  2880   using assms and path_image_join_subset[of g1 g2] by auto
  2881 
  2882 lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)"
  2883   using assms unfolding simple_path_def reversepath_def apply- apply(rule ballI)+
  2884   apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
  2885   unfolding mem_interval_1 by(auto simp add:vector_component_simps)
  2886 
  2887 lemma simple_path_join_loop:
  2888   assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
  2889   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
  2890   shows "simple_path(g1 +++ g2)"
  2891 unfolding simple_path_def proof((rule ballI)+, rule impI) let ?g = "g1 +++ g2"
  2892   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
  2893   fix x y::"real^1" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
  2894   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)
  2895     assume as:"x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2"
  2896     hence "g1 (2 *s x) = g1 (2 *s y)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
  2897     moreover have "2 *s x \<in> {0..1}" "2 *s y \<in> {0..1}" using xy(1,2) as
  2898       unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
  2899     ultimately show ?thesis using inj(1)[of "2*s x" "2*s y"] by auto
  2900   next assume as:"x $ 1 > 1 / 2" "y $ 1 > 1 / 2"
  2901     hence "g2 (2 *s x - 1) = g2 (2 *s y - 1)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
  2902     moreover have "2 *s x - 1 \<in> {0..1}" "2 *s y - 1 \<in> {0..1}" using xy(1,2) as
  2903       unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
  2904     ultimately show ?thesis using inj(2)[of "2*s x - 1" "2*s y - 1"] by auto
  2905   next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2"
  2906     hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
  2907       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  2908     moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
  2909       using inj(2)[of "2 *s y - 1" 0] and xy(2)[unfolded mem_interval_1]
  2910       apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
  2911     ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
  2912     hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)[unfolded mem_interval_1]
  2913       using inj(1)[of "2 *s x" 0] by(auto simp add:vector_component_simps)
  2914     moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
  2915       unfolding joinpaths_def pathfinish_def using as(2) and xy(2)[unfolded mem_interval_1]
  2916       using inj(2)[of "2 *s y - 1" 1] by (auto simp add:vector_component_simps Cart_eq)
  2917     ultimately show ?thesis by auto 
  2918   next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2"
  2919     hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
  2920       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  2921     moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
  2922       using inj(2)[of "2 *s x - 1" 0] and xy(1)[unfolded mem_interval_1]
  2923       apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
  2924     ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
  2925     hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)[unfolded mem_interval_1]
  2926       using inj(1)[of "2 *s y" 0] by(auto simp add:vector_component_simps)
  2927     moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
  2928       unfolding joinpaths_def pathfinish_def using as(1) and xy(1)[unfolded mem_interval_1]
  2929       using inj(2)[of "2 *s x - 1" 1] by(auto simp add:vector_component_simps Cart_eq)
  2930     ultimately show ?thesis by auto qed qed
  2931 
  2932 lemma injective_path_join:
  2933   assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
  2934   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
  2935   shows "injective_path(g1 +++ g2)"
  2936   unfolding injective_path_def proof(rule,rule,rule) let ?g = "g1 +++ g2"
  2937   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
  2938   fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
  2939   show "x = y" proof(cases "x$1 \<le> 1/2", case_tac[!] "y$1 \<le> 1/2", unfold not_le)
  2940     assume "x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2" thus ?thesis using inj(1)[of "2*s x" "2*s y"] and xy
  2941       unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
  2942   next assume "x $ 1 > 1 / 2" "y $ 1 > 1 / 2" thus ?thesis using inj(2)[of "2*s x - 1" "2*s y - 1"] and xy
  2943       unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
  2944   next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2" 
  2945     hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
  2946       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  2947     hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto
  2948     thus ?thesis using as and inj(1)[of "2 *s x" 1] inj(2)[of "2 *s y - 1" 0] and xy(1,2)
  2949       unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
  2950       by(auto simp add:vector_component_simps Cart_eq forall_1)
  2951   next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2" 
  2952     hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
  2953       using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
  2954     hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto
  2955     thus ?thesis using as and inj(2)[of "2 *s x - 1" 0] inj(1)[of "2 *s y" 1] and xy(1,2)
  2956       unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
  2957       by(auto simp add:vector_component_simps forall_1 Cart_eq) qed qed
  2958 
  2959 lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
  2960  
  2961 subsection {* Reparametrizing a closed curve to start at some chosen point. *}
  2962 
  2963 definition "shiftpath a (f::real^1 \<Rightarrow> real^'n) =
  2964   (\<lambda>x. if dest_vec1 (a + x) \<le> 1 then f(a + x) else f(a + x - 1))"
  2965 
  2966 lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"
  2967   unfolding pathstart_def shiftpath_def by auto
  2968 
  2969 (** move this **)
  2970 declare forall_1[simp] ex_1[simp]
  2971 
  2972 lemma pathfinish_shiftpath: assumes "0 \<le> a" "pathfinish g = pathstart g"
  2973   shows "pathfinish(shiftpath a g) = g a"
  2974   using assms unfolding pathstart_def pathfinish_def shiftpath_def
  2975   by(auto simp add: vector_component_simps)
  2976 
  2977 lemma endpoints_shiftpath:
  2978   assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}" 
  2979   shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"
  2980   using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)
  2981 
  2982 lemma closed_shiftpath:
  2983   assumes "pathfinish g = pathstart g" "a \<in> {0..1}"
  2984   shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)"
  2985   using endpoints_shiftpath[OF assms] by auto
  2986 
  2987 lemma path_shiftpath:
  2988   assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
  2989   shows "path(shiftpath a g)" proof-
  2990   have *:"{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by(auto simp add: vector_component_simps)
  2991   have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
  2992     using assms(2)[unfolded pathfinish_def pathstart_def] by auto
  2993   show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union)
  2994     apply(rule closed_interval)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
  2995     apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
  2996     apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+
  2997     apply(rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
  2998     using assms(3) and ** by(auto simp add:vector_component_simps field_simps Cart_eq) qed
  2999 
  3000 lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" 
  3001   shows "shiftpath (1 - a) (shiftpath a g) x = g x"
  3002   using assms unfolding pathfinish_def pathstart_def shiftpath_def 
  3003   by(auto simp add: vector_component_simps)
  3004 
  3005 lemma path_image_shiftpath:
  3006   assumes "a \<in> {0..1}" "pathfinish g = pathstart g"
  3007   shows "path_image(shiftpath a g) = path_image g" proof-
  3008   { 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)" 
  3009     hence "\<exists>y\<in>{0..1} \<inter> {x. a $ 1 + x $ 1 \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x")
  3010       case False thus ?thesis apply(rule_tac x="1 + x - a" in bexI)
  3011 	using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
  3012 	by(auto simp add:vector_component_simps field_simps atomize_not) next
  3013       case True thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
  3014 	by(auto simp add:vector_component_simps field_simps) qed }
  3015   thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def 
  3016     by(auto simp add:vector_component_simps image_iff) qed
  3017 
  3018 subsection {* Special case of straight-line paths. *}
  3019 
  3020 definition
  3021   linepath :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 1 \<Rightarrow> real ^ 'n" where
  3022   "linepath a b = (\<lambda>x. (1 - dest_vec1 x) *s a + dest_vec1 x *s b)"
  3023 
  3024 lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
  3025   unfolding pathstart_def linepath_def by auto
  3026 
  3027 lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b"
  3028   unfolding pathfinish_def linepath_def by auto
  3029 
  3030 lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
  3031   unfolding linepath_def
  3032   by (intro continuous_intros continuous_dest_vec1)
  3033 
  3034 lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
  3035   using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
  3036 
  3037 lemma path_linepath[intro]: "path(linepath a b)"
  3038   unfolding path_def by(rule continuous_on_linepath)
  3039 
  3040 lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
  3041   unfolding path_image_def segment linepath_def apply (rule set_ext, rule) defer
  3042   unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *s 1" in bexI)
  3043   by(auto simp add:vector_component_simps)
  3044 
  3045 lemma reversepath_linepath[simp]:  "reversepath(linepath a b) = linepath b a"
  3046   unfolding reversepath_def linepath_def by(rule ext, auto simp add:vector_component_simps)
  3047 
  3048 lemma injective_path_linepath: assumes "a \<noteq> b" shows "injective_path(linepath a b)" proof- 
  3049   { obtain i where i:"a$i \<noteq> b$i" using assms[unfolded Cart_eq] by auto
  3050     fix x y::"real^1" assume "x $ 1 *s b + y $ 1 *s a = x $ 1 *s a + y $ 1 *s b"
  3051     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)
  3052     hence "x = y" unfolding mult_cancel_right Cart_eq using i(1) by(auto simp add:field_simps) }
  3053   thus ?thesis unfolding injective_path_def linepath_def by(auto simp add:vector_component_simps field_simps) qed
  3054 
  3055 lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath)
  3056 
  3057 subsection {* Bounding a point away from a path. *}
  3058 
  3059 lemma not_on_path_ball: assumes "path g" "z \<notin> path_image g"
  3060   shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof-
  3061   obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y"
  3062     using distance_attains_inf[OF _ path_image_nonempty, of g z]
  3063     using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
  3064   thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed
  3065 
  3066 lemma not_on_path_cball: assumes "path g" "z \<notin> path_image g"
  3067   shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof-
  3068   obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto
  3069   moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto
  3070   ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed
  3071 
  3072 subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
  3073 
  3074 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)"
  3075 
  3076 lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def 
  3077 
  3078 lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s"
  3079   using assms unfolding path_defs by auto
  3080 
  3081 lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x"
  3082   unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms 
  3083   by(auto intro!:continuous_on_intros)    
  3084 
  3085 lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
  3086   by(auto intro!: path_component_mem path_component_refl) 
  3087 
  3088 lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
  3089   using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI) 
  3090   by(auto simp add: reversepath_simps)
  3091 
  3092 lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z"
  3093   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)
  3094 
  3095 lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow>  path_component s x y \<Longrightarrow> path_component t x y"
  3096   unfolding path_component_def by auto
  3097 
  3098 subsection {* Can also consider it as a set, as the name suggests. *}
  3099 
  3100 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 )}"
  3101   apply(rule set_ext) unfolding mem_Collect_eq unfolding mem_def path_component_def by auto
  3102 
  3103 lemma mem_path_component_set:"x \<in> path_component s y \<longleftrightarrow> path_component s y x" unfolding mem_def by auto
  3104 
  3105 lemma path_component_subset: "(path_component s x) \<subseteq> s"
  3106   apply(rule, rule path_component_mem(2)) by(auto simp add:mem_def)
  3107 
  3108 lemma path_component_eq_empty: "path_component s x = {} \<longleftrightarrow> x \<notin> s"
  3109   apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_path_component_set
  3110   apply(drule path_component_mem(1)) using path_component_refl by auto
  3111 
  3112 subsection {* Path connectedness of a space. *}
  3113 
  3114 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)"
  3115 
  3116 lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
  3117   unfolding path_connected_def path_component_def by auto
  3118 
  3119 lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component s x = s)" 
  3120   unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset) 
  3121   unfolding subset_eq mem_path_component_set Ball_def mem_def by auto
  3122 
  3123 subsection {* Some useful lemmas about path-connectedness. *}
  3124 
  3125 lemma convex_imp_path_connected: assumes "convex s" shows "path_connected s"
  3126   unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI)
  3127   unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto
  3128 
  3129 lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s"
  3130   unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof-
  3131   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> {}"
  3132   then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto
  3133   then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
  3134     using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
  3135   have *:"connected {0..1::real^1}" by(auto intro!: convex_connected convex_interval)
  3136   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
  3137   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 
  3138   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)
  3139   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}"]
  3140     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]
  3141     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed
  3142 
  3143 lemma open_path_component: assumes "open s" shows "open(path_component s x)"
  3144   unfolding open_contains_ball proof
  3145   fix y assume as:"y \<in> path_component s x"
  3146   hence "y\<in>s" apply- apply(rule path_component_mem(2)) unfolding mem_def by auto
  3147   then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
  3148   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-
  3149     fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer 
  3150       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`
  3151       using as[unfolded mem_def] by auto qed qed
  3152 
  3153 lemma open_non_path_component: assumes "open s" shows "open(s - path_component s x)" unfolding open_contains_ball proof
  3154   fix y assume as:"y\<in>s - path_component s x" 
  3155   then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
  3156   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)
  3157     fix z assume "z\<in>ball y e" "\<not> z \<notin> path_component s x" 
  3158     hence "y \<in> path_component s x" unfolding not_not mem_path_component_set using `e>0` 
  3159       apply- apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)])
  3160       apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto
  3161     thus False using as by auto qed(insert e(2), auto) qed
  3162 
  3163 lemma connected_open_path_connected: assumes "open s" "connected s" shows "path_connected s"
  3164   unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule)
  3165   fix x y assume "x \<in> s" "y \<in> s" show "y \<in> path_component s x" proof(rule ccontr)
  3166     assume "y \<notin> path_component s x" moreover
  3167     have "path_component s x \<inter> s \<noteq> {}" using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto
  3168     ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
  3169     using assms(2)[unfolded connected_def not_ex, rule_format, of"path_component s x" "s - path_component s x"] by auto
  3170 qed qed
  3171 
  3172 lemma path_connected_continuous_image:
  3173   assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)"
  3174   unfolding path_connected_def proof(rule,rule)
  3175   fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s"
  3176   then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
  3177   guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] ..
  3178   thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"
  3179     unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs
  3180     using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed
  3181 
  3182 lemma homeomorphic_path_connectedness:
  3183   "s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"
  3184   unfolding homeomorphic_def homeomorphism_def apply(erule exE|erule conjE)+ apply rule
  3185   apply(drule_tac f=f in path_connected_continuous_image) prefer 3
  3186   apply(drule_tac f=g in path_connected_continuous_image) by auto
  3187 
  3188 lemma path_connected_empty: "path_connected {}"
  3189   unfolding path_connected_def by auto
  3190 
  3191 lemma path_connected_singleton: "path_connected {a}"
  3192   unfolding path_connected_def apply(rule,rule)
  3193   apply(rule_tac x="linepath a a" in exI) by(auto simp add:segment)
  3194 
  3195 lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
  3196   shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule)
  3197   fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t" 
  3198   from assms(3) obtain z where "z \<in> s \<inter> t" by auto
  3199   thus "path_component (s \<union> t) x y" using as using assms(1-2)[unfolded path_connected_component] apply- 
  3200     apply(erule_tac[!] UnE)+ apply(rule_tac[2-3] path_component_trans[of _ _ z])
  3201     by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed
  3202 
  3203 subsection {* sphere is path-connected. *}
  3204 
  3205 lemma path_connected_punctured_universe:
  3206  assumes "2 \<le> CARD('n::finite)" shows "path_connected((UNIV::(real^'n::finite) set) - {a})" proof-
  3207   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
  3208   let ?U = "UNIV::(real^'n) set" let ?u = "?U - {0}"
  3209   let ?basis = "\<lambda>k. basis (\<psi> k)"
  3210   let ?A = "\<lambda>k. {x::real^'n. \<exists>i\<in>{1..k}. (basis (\<psi> i)) \<bullet> x \<noteq> 0}"
  3211   have "\<forall>k\<in>{2..CARD('n)}. path_connected (?A k)" proof
  3212     have *:"\<And>k. ?A (Suc k) = {x. ?basis (Suc k) \<bullet> x < 0} \<union> {x. ?basis (Suc k) \<bullet> x > 0} \<union> ?A k" apply(rule set_ext,rule) defer
  3213       apply(erule UnE)+  unfolding mem_Collect_eq apply(rule_tac[1-2] x="Suc k" in bexI)
  3214       by(auto elim!: ballE simp add: not_less le_Suc_eq)
  3215     fix k assume "k \<in> {2..CARD('n)}" thus "path_connected (?A k)" proof(induct k)
  3216       case (Suc k) show ?case proof(cases "k = 1")
  3217 	case False from Suc have d:"k \<in> {1..CARD('n)}" "Suc k \<in> {1..CARD('n)}" by auto
  3218 	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
  3219 	hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < ?basis (Suc k) \<bullet> x} \<inter> (?A k)" 
  3220           "?basis k - ?basis (Suc k) \<in> {x. 0 > ?basis (Suc k) \<bullet> x} \<inter> ({x. 0 < ?basis (Suc k) \<bullet> x} \<union> (?A k))" using d
  3221 	  by(auto simp add: dot_basis vector_component_simps intro!:bexI[where x=k])
  3222 	show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un) 
  3223 	  prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt)
  3224 	  apply(rule Suc(1)) apply(rule_tac[2-3] ccontr) using d ** False by auto
  3225       next case True hence d:"1\<in>{1..CARD('n)}" "2\<in>{1..CARD('n)}" using Suc(2) by auto
  3226 	have ***:"Suc 1 = 2" by auto
  3227 	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
  3228 	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
  3229 	thus ?thesis unfolding * True unfolding ** neq_iff bex_disj_distrib apply -
  3230 	  apply(rule path_connected_Un, rule_tac[1-2] path_connected_Un) defer 3 apply(rule_tac[1-4] convex_imp_path_connected) 
  3231 	  apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I)
  3232 	  apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I)
  3233 	  apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I)
  3234 	  using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps dot_basis)
  3235   qed qed auto qed note lem = this
  3236 
  3237   have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. basis (\<psi> i) \<bullet> x \<noteq> 0) \<longleftrightarrow> (\<exists>i. basis i \<bullet> x \<noteq> 0)"
  3238     apply rule apply(erule bexE) apply(rule_tac x="\<psi> i" in exI) defer apply(erule exE) proof- 
  3239     fix x::"real^'n" and i assume as:"basis i \<bullet> x \<noteq> 0"
  3240     have "i\<in>\<psi> ` {1..CARD('n)}" using \<psi>[unfolded bij_betw_def, THEN conjunct2] by auto
  3241     then obtain j where "j\<in>{1..CARD('n)}" "\<psi> j = i" by auto
  3242     thus "\<exists>i\<in>{1..CARD('n)}. basis (\<psi> i) \<bullet> x \<noteq> 0" apply(rule_tac x=j in bexI) using as by auto qed auto
  3243   have *:"?U - {a} = (\<lambda>x. x + a) ` {x. x \<noteq> 0}" apply(rule set_ext) unfolding image_iff 
  3244     apply rule apply(rule_tac x="x - a" in bexI) by auto
  3245   have **:"\<And>x::real^'n. x\<noteq>0 \<longleftrightarrow> (\<exists>i. basis i \<bullet> x \<noteq> 0)" unfolding Cart_eq by(auto simp add: dot_basis)
  3246   show ?thesis unfolding * apply(rule path_connected_continuous_image) apply(rule continuous_on_intros)+ 
  3247     unfolding ** apply(rule lem[THEN bspec[where x="CARD('n)"], unfolded ***]) using assms by auto qed
  3248 
  3249 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")
  3250   case True thus ?thesis proof(cases "r=0") 
  3251     case False hence "{x::real^'n. norm(x - a) = r} = {}" using True by auto
  3252     thus ?thesis using path_connected_empty by auto
  3253   qed(auto intro!:path_connected_singleton) next
  3254   case False hence *:"{x::real^'n. norm(x - a) = r} = (\<lambda>x. a + r *s x) ` {x. norm x = 1}" unfolding not_le apply -apply(rule set_ext,rule)
  3255     unfolding image_iff apply(rule_tac x="(1/r) *s (x - a)" in bexI) unfolding mem_Collect_eq norm_mul by auto
  3256   have ***:"\<And>xa. (if xa = 0 then 0 else 1) \<noteq> 1 \<Longrightarrow> xa = 0" apply(rule ccontr) by auto
  3257   have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *s x) ` (UNIV - {0})" apply(rule set_ext,rule)
  3258     unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq norm_mul by(auto intro!: ***) 
  3259   have "continuous_on (UNIV - {0}) (\<lambda>x::real^'n. 1 / norm x)" unfolding o_def continuous_on_eq_continuous_within
  3260     apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
  3261     apply(rule continuous_at_norm[unfolded o_def]) by auto
  3262   thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
  3263     by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
  3264 
  3265 lemma connected_sphere: "2 \<le> CARD('n) \<Longrightarrow> connected {x::real^'n::finite. norm(x - a) = r}"
  3266   using path_connected_sphere path_connected_imp_connected by auto
  3267  
  3268 (** In continuous_at_vec1_norm : Use \<And> instead of \<forall>. **)
  3269 
  3270 end