--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Sep 24 23:51:32 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Sep 24 15:03:49 2013 -0700
@@ -1485,9 +1485,6 @@
text{* The expected monotonicity property. *}
-lemma Lim_within_empty: "(f ---> l) (at x within {})"
- unfolding tendsto_def eventually_at_filter by simp
-
lemma Lim_Un:
assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
shows "(f ---> l) (at x within (S \<union> T))"
@@ -1551,7 +1548,7 @@
shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
proof (cases "{x<..} \<inter> I = {}")
case True
- then show ?thesis by (simp add: Lim_within_empty)
+ then show ?thesis by simp
next
case False
show ?thesis
--- a/src/HOL/Topological_Spaces.thy Tue Sep 24 23:51:32 2013 +0200
+++ b/src/HOL/Topological_Spaces.thy Tue Sep 24 15:03:49 2013 -0700
@@ -490,7 +490,7 @@
unfolding le_filter_def eventually_Sup by simp }
{ show "Inf {} = (top::'a filter)"
by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
- (metis (full_types) Topological_Spaces.top_filter_def always_eventually eventually_top) }
+ (metis (full_types) top_filter_def always_eventually eventually_top) }
{ show "Sup {} = (bot::'a filter)"
by (auto simp: bot_filter_def Sup_filter_def) }
qed
@@ -754,6 +754,9 @@
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
+lemma at_within_empty [simp]: "at a within {} = bot"
+ unfolding at_within_def by simp
+
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
unfolding trivial_limit_def eventually_at_topological
by (safe, case_tac "S = {a}", simp, fast, fast)