# HG changeset patch
# User huffman
# Date 1243636353 25200
# Node ID fc09ec06b89b956bf59657797fb57ed7ab7b75b1
# Parent  9983f648f9bbaba759b8845da272e33cbde67817
instance ^ :: (metric_space, finite) metric_space

diff -r 9983f648f9bb -r fc09ec06b89b src/HOL/Library/Euclidean_Space.thy
--- a/src/HOL/Library/Euclidean_Space.thy	Fri May 29 14:09:58 2009 -0700
+++ b/src/HOL/Library/Euclidean_Space.thy	Fri May 29 15:32:33 2009 -0700
@@ -498,6 +498,30 @@
   apply simp
   done
 
+subsection {* Metric *}
+
+instantiation "^" :: (metric_space, finite) metric_space
+begin
+
+definition dist_vector_def:
+  "dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
+
+instance proof
+  fix x y :: "'a ^ 'b"
+  show "dist x y = 0 \<longleftrightarrow> x = y"
+    unfolding dist_vector_def
+    by (simp add: setL2_eq_0_iff Cart_eq)
+next
+  fix x y z :: "'a ^ 'b"
+  show "dist x y \<le> dist x z + dist y z"
+    unfolding dist_vector_def
+    apply (rule order_trans [OF _ setL2_triangle_ineq])
+    apply (simp add: setL2_mono dist_triangle2)
+    done
+qed
+
+end
+
 subsection {* Norms *}
 
 instantiation "^" :: (real_normed_vector, finite) real_normed_vector
@@ -509,9 +533,6 @@
 definition vector_sgn_def:
   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
 
-definition dist_vector_def:
-  "dist (x::'a^'b) y = norm (x - y)"
-
 instance proof
   fix a :: real and x y :: "'a ^ 'b"
   show "0 \<le> norm x"
@@ -531,7 +552,8 @@
   show "sgn x = scaleR (inverse (norm x)) x"
     by (rule vector_sgn_def)
   show "dist x y = norm (x - y)"
-    by (rule dist_vector_def)
+    unfolding dist_vector_def vector_norm_def
+    by (simp add: dist_norm)
 qed
 
 end
@@ -949,6 +971,11 @@
   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
   using norm_ge_zero[of "x - y"] by auto
 
+lemma vector_dist_norm:
+  fixes x y :: "real ^ _"
+  shows "dist x y = norm (x - y)"
+  by (rule dist_norm)
+
 use "normarith.ML"
 
 method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
@@ -2566,7 +2593,7 @@
 qed
 
 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
-  by (metis dist_vector_def fstcart_sub[symmetric] norm_fstcart)
+  unfolding dist_norm by (metis fstcart_sub[symmetric] norm_fstcart)
 
 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
 proof-
@@ -2581,7 +2608,7 @@
 qed
 
 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
-  by (metis dist_vector_def sndcart_sub[symmetric] norm_sndcart)
+  unfolding dist_norm by (metis sndcart_sub[symmetric] norm_sndcart)
 
 lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n::finite) (x2::'a::{times,comm_monoid_add}^'m::finite)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
   by (simp add: dot_def setsum_UNIV_sum pastecart_def)
diff -r 9983f648f9bb -r fc09ec06b89b src/HOL/Library/Topology_Euclidean_Space.thy
--- a/src/HOL/Library/Topology_Euclidean_Space.thy	Fri May 29 14:09:58 2009 -0700
+++ b/src/HOL/Library/Topology_Euclidean_Space.thy	Fri May 29 15:32:33 2009 -0700
@@ -1314,7 +1314,7 @@
     { fix x
       have "netord net x y \<longrightarrow> dist (h (f x)) (h l) < e"
 	using y(2) b unfolding dist_norm using linear_sub[of h "f x" l] `linear h`
-	apply auto by (metis b(1) b(2) dist_vector_def dist_commute less_le_not_le linorder_not_le mult_imp_div_pos_le real_mult_commute xt1(7)) (* FIXME: VERY slow! *)
+	apply auto by (metis b(1) b(2) less_le_not_le linorder_not_le mult_imp_div_pos_le real_mult_commute xt1(7)) (* FIXME: VERY slow! *)
     }
     hence " (\<exists>y. (\<exists>x. netord net x y) \<and> (\<forall>x. netord net x y \<longrightarrow> dist (h (f x)) (h l) < e))" using y
       by(rule_tac x="y" in exI) auto
@@ -2353,7 +2353,7 @@
   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
   { fix n::nat assume "n\<ge>N"
     hence "norm (s n) \<le> norm (s N) + 1" using N apply(erule_tac x=n in allE) unfolding dist_norm
-      using norm_triangle_sub[of "s N" "s n"] by (auto, metis dist_vector_def dist_commute le_add_right_mono norm_triangle_sub real_less_def)
+      using norm_triangle_sub[of "s N" "s n"] by (auto, metis norm_minus_commute le_add_right_mono norm_triangle_sub real_less_def)
   }
   hence "\<forall>n\<ge>N. norm (s n) \<le> norm (s N) + 1" by auto
   moreover
diff -r 9983f648f9bb -r fc09ec06b89b src/HOL/Library/normarith.ML
--- a/src/HOL/Library/normarith.ML	Fri May 29 14:09:58 2009 -0700
+++ b/src/HOL/Library/normarith.ML	Fri May 29 15:32:33 2009 -0700
@@ -391,7 +391,7 @@
 
   fun init_conv ctxt = 
    Simplifier.rewrite (Simplifier.context ctxt 
-     (HOL_basic_ss addsimps ([@{thm vec_0}, @{thm vec_1}, @{thm dist_vector_def}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_0}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
+     (HOL_basic_ss addsimps ([@{thm vec_0}, @{thm vec_1}, @{thm vector_dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_0}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
    then_conv field_comp_conv 
    then_conv nnf_conv