--- a/src/HOL/Multivariate_Analysis/Integration.thy Tue Dec 29 22:41:22 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Tue Dec 29 23:04:53 2015 +0100
@@ -4580,7 +4580,7 @@
by (auto intro!: triangle3)
qed
qed
- then obtain s where s: "i ----> s"
+ then obtain s where s: "i \<longlonglongrightarrow> s"
using convergent_eq_cauchy[symmetric] by blast
show ?thesis
unfolding integrable_on_def has_integral
@@ -10201,8 +10201,8 @@
fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow> real"
assumes f: "\<And>k. (f k has_integral x k) s"
assumes "\<And>k x. x \<in> s \<Longrightarrow> f k x \<le> f (Suc k) x"
- assumes "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) ----> g x"
- assumes "x ----> x'"
+ assumes "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
+ assumes "x \<longlonglongrightarrow> x'"
shows "(g has_integral x') s"
proof -
have x_eq: "x = (\<lambda>i. integral s (f i))"
@@ -10210,13 +10210,13 @@
then have x: "{integral s (f k) |k. True} = range x"
by auto
- have "g integrable_on s \<and> (\<lambda>k. integral s (f k)) ----> integral s g"
+ have "g integrable_on s \<and> (\<lambda>k. integral s (f k)) \<longlonglongrightarrow> integral s g"
proof (intro monotone_convergence_increasing allI ballI assms)
show "bounded {integral s (f k) |k. True}"
unfolding x by (rule convergent_imp_bounded) fact
qed (auto intro: f)
moreover then have "integral s g = x'"
- by (intro LIMSEQ_unique[OF _ \<open>x ----> x'\<close>]) (simp add: x_eq)
+ by (intro LIMSEQ_unique[OF _ \<open>x \<longlonglongrightarrow> x'\<close>]) (simp add: x_eq)
ultimately show ?thesis
by (simp add: has_integral_integral)
qed
@@ -12087,7 +12087,7 @@
show "Inf {f j x |j. k \<le> j} \<le> Inf {f j x |j. Suc k \<le> j}"
by (intro cInf_superset_mono) (auto simp: \<open>x\<in>s\<close>)
- show "(\<lambda>k::nat. Inf {f j x |j. k \<le> j}) ----> g x"
+ show "(\<lambda>k::nat. Inf {f j x |j. k \<le> j}) \<longlonglongrightarrow> g x"
proof (rule LIMSEQ_I, goal_cases)
case r: (1 r)
then have "0<r/2"
@@ -12183,8 +12183,8 @@
lemma has_integral_dominated_convergence:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s"
- "\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) ----> g x"
- and x: "y ----> x"
+ "\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
+ and x: "y \<longlonglongrightarrow> x"
shows "(g has_integral x) s"
proof -
have int_f: "\<And>k. (f k) integrable_on s"
@@ -12193,9 +12193,9 @@
by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+
moreover have "integral s g = x"
proof (rule LIMSEQ_unique)
- show "(\<lambda>i. integral s (f i)) ----> x"
+ show "(\<lambda>i. integral s (f i)) \<longlonglongrightarrow> x"
using integral_unique[OF assms(1)] x by simp
- show "(\<lambda>i. integral s (f i)) ----> integral s g"
+ show "(\<lambda>i. integral s (f i)) \<longlonglongrightarrow> integral s g"
by (intro dominated_convergence[OF int_f assms(2)]) fact+
qed
ultimately show ?thesis