
diff -r f8c2def19f84 -r fa56622bb7bc src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Aug 23 15:46:53 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Aug 23 16:17:22 2011 -0700
@@ -921,7 +921,69 @@
   thus ?lhs unfolding affine_dependent_explicit using assms by auto
 qed
 
-subsection {* A general lemma. *}
+subsection {* Connectedness of convex sets *}
+
+lemma connected_real_lemma:
+  fixes f :: "real \<Rightarrow> 'a::metric_space"
+  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
+  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
+  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
+  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
+  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
+  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
+proof-
+  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
+  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
+  have Sub: "\<exists>y. isUb UNIV ?S y"
+    apply (rule exI[where x= b])
+    using ab fb e12 by (auto simp add: isUb_def setle_def)
+  from reals_complete[OF Se Sub] obtain l where
+    l: "isLub UNIV ?S l"by blast
+  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
+    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
+    by (metis linorder_linear)
+  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
+    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
+    by (metis linorder_linear not_le)
+    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
+    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
+    have "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
+    then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
+    {assume le2: "f l \<in> e2"
+      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
+      hence lap: "l - a > 0" using alb by arith
+      from e2[rule_format, OF le2] obtain e where
+        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
+      from dst[OF alb e(1)] obtain d where
+        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
+      let ?d' = "min (d/2) ((l - a)/2)"
+      have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
+        by (simp add: min_max.less_infI2)
+      then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
+      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
+      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
+      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
+      moreover
+      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
+      ultimately have False using e12 alb d' by auto}
+    moreover
+    {assume le1: "f l \<in> e1"
+    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
+      hence blp: "b - l > 0" using alb by arith
+      from e1[rule_format, OF le1] obtain e where
+        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
+      from dst[OF alb e(1)] obtain d where
+        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
+      have "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
+      then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
+      then obtain d' where d': "d' > 0" "d' < d" by metis
+      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
+      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
+      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
+      with l d' have False
+        by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
+    ultimately show ?thesis using alb by metis
+qed
 
 lemma convex_connected:
   fixes s :: "'a::real_normed_vector set"




