--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Wed Aug 24 15:06:13 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Wed Aug 24 15:32:40 2011 -0700
@@ -8,8 +8,6 @@
imports
Euclidean_Space
"~~/src/HOL/Library/Infinite_Set"
- L2_Norm
- "~~/src/HOL/Library/Convex"
begin
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
@@ -63,7 +61,7 @@
*)
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
- by (simp add: setL2_def power2_eq_square)
+ by (simp add: power2_eq_square)
lemma norm_cauchy_schwarz:
shows "inner x y <= norm x * norm y"
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 24 15:06:13 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 24 15:32:40 2011 -0700
@@ -7,7 +7,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith
+imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith L2_Norm
begin
(* to be moved elsewhere *)
@@ -20,7 +20,7 @@
apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
apply(rule member_le_setL2) by auto
-subsection {* General notion of a topologies as values *}
+subsection {* General notion of a topology as a value *}
definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"