author huffman Wed, 31 Aug 2011 08:11:47 -0700 changeset 44628 bd17b7543af1 parent 44627 134c06282ae6 child 44629 1cd782f3458b
move lemmas from Topology_Euclidean_Space to Euclidean_Space
```--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Aug 31 07:51:55 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Aug 31 08:11:47 2011 -0700
@@ -7,7 +7,7 @@

theory Euclidean_Space
imports
-  Complex_Main
+  L2_Norm
"~~/src/HOL/Library/Inner_Product"
"~~/src/HOL/Library/Product_Vector"
begin
@@ -216,10 +216,20 @@
dot_basis if_distrib setsum_cases mult_commute)

+lemma euclidean_dist_l2:
+  fixes x y :: "'a::euclidean_space"
+  shows "dist x y = setL2 (\<lambda>i. dist (x \$\$ i) (y \$\$ i)) {..<DIM('a)}"
+  unfolding dist_norm norm_eq_sqrt_inner setL2_def
+  by (simp add: euclidean_inner power2_eq_square)
+
lemma component_le_norm: "\<bar>x\$\$i\<bar> \<le> norm (x::'a::euclidean_space)"
unfolding euclidean_component_def
by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp

+lemma dist_nth_le: "dist (x \$\$ i) (y \$\$ i) \<le> dist x (y::'a::euclidean_space)"
+  unfolding euclidean_dist_l2 [where 'a='a]
+  by (cases "i < DIM('a)", rule member_le_setL2, auto)
+
subsection {* Subclass relationships *}

instance euclidean_space \<subseteq> perfect_space```
```--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 31 07:51:55 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 31 08:11:47 2011 -0700
@@ -7,19 +7,9 @@
header {* Elementary topology in Euclidean space. *}

theory Topology_Euclidean_Space
-imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith L2_Norm
+imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith
begin

-(* to be moved elsewhere *)
-
-lemma euclidean_dist_l2:"dist x (y::'a::euclidean_space) = setL2 (\<lambda>i. dist(x\$\$i) (y\$\$i)) {..<DIM('a)}"
-  unfolding dist_norm norm_eq_sqrt_inner setL2_def apply(subst euclidean_inner)