author huffman Thu, 12 Sep 2013 09:03:52 -0700 changeset 53593 a7bcbb5a17d8 parent 53590 b6dc5403cad1 child 53594 8a9fb53294f4
```--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Fri Sep 13 00:55:44 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Thu Sep 12 09:03:52 2013 -0700
@@ -322,7 +322,6 @@
shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
by (simp add: vec_eq_iff setsum_right_distrib)

-(* TODO: use setsum_norm_allsubsets_bound *)
lemma setsum_norm_allsubsets_bound_cart:
fixes f:: "'a \<Rightarrow> real ^'n"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"```
```--- a/src/HOL/Multivariate_Analysis/Path_Connected.thy	Fri Sep 13 00:55:44 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy	Thu Sep 12 09:03:52 2013 -0700
@@ -587,7 +587,7 @@
qed

lemma open_path_component:
-  fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
+  fixes s :: "'a::real_normed_vector set"
assumes "open s"
shows "open {y. path_component s x y}"
unfolding open_contains_ball
@@ -620,7 +620,7 @@
qed

lemma open_non_path_component:
-  fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
+  fixes s :: "'a::real_normed_vector set"
assumes "open s"
shows "open(s - {y. path_component s x y})"
unfolding open_contains_ball
@@ -648,7 +648,7 @@
qed

lemma connected_open_path_connected:
-  fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
+  fixes s :: "'a::real_normed_vector set"
assumes "open s" "connected s"
shows "path_connected s"
unfolding path_connected_component_set```