author hoelzl Thu, 03 Mar 2011 10:55:41 +0100 changeset 41874 a3035d56171d parent 41873 250468a1bd7a child 41875 e3cd0dce9b1a child 41879 7f9c48c17d2a
finally remove upper_bound_finite_set
```--- a/src/HOL/Multivariate_Analysis/Integration.thy	Thu Mar 03 15:59:44 2011 +1100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Thu Mar 03 10:55:41 2011 +0100
@@ -4470,24 +4470,6 @@

subsection {* monotone convergence (bounded interval first). *}

-lemma upper_bound_finite_set:
-  assumes fS: "finite S"
-  shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<le> a"
-proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case by simp
-next
-  case (2 x F)
-  from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<le> a" by blast
-  let ?a = "max a (f x)"
-  have m: "a \<le> ?a" "f x \<le> ?a" by simp_all
-  {fix y assume y: "y \<in> insert x F"
-    {assume "y = x" hence "f y \<le> ?a" using m by simp}
-    moreover
-    {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<le> ?a" by (simp add: max_def)}
-    ultimately have "f y \<le> ?a" using y by blast}
-  then show ?case by blast
-qed
-
lemma monotone_convergence_interval: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>k. (f k) integrable_on {a..b}"
"\<forall>k. \<forall>x\<in>{a..b}.(f k x) \<le> (f (Suc k) x)"```