# HG changeset patch
# User huffman
# Date 1244048291 25200
# Node ID c12b25b7f015c42a8d92f67a5fa74464c04ba910
# Parent  f4c079225845a8956af902926521b94ebde48a27
replace class open with class topo

diff -r f4c079225845 -r c12b25b7f015 src/HOL/Complex.thy
--- a/src/HOL/Complex.thy	Wed Jun 03 08:46:13 2009 -0700
+++ b/src/HOL/Complex.thy	Wed Jun 03 09:58:11 2009 -0700
@@ -281,8 +281,8 @@
 definition dist_complex_def:
   "dist x y = cmod (x - y)"
 
-definition open_complex_def:
-  "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y::complex. dist y x < e \<longrightarrow> y \<in> S)"
+definition topo_complex_def:
+  "topo = {S::complex set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
 
 lemmas cmod_def = complex_norm_def
 
@@ -290,7 +290,7 @@
   by (simp add: complex_norm_def)
 
 instance proof
-  fix r :: real and x y :: complex and S :: "complex set"
+  fix r :: real and x y :: complex
   show "0 \<le> norm x"
     by (induct x) simp
   show "(norm x = 0) = (x = 0)"
@@ -308,8 +308,8 @@
     by (rule complex_sgn_def)
   show "dist x y = cmod (x - y)"
     by (rule dist_complex_def)
-  show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-    by (rule open_complex_def)
+  show "topo = {S::complex set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
+    by (rule topo_complex_def)
 qed
 
 end
diff -r f4c079225845 -r c12b25b7f015 src/HOL/Library/Euclidean_Space.thy
--- a/src/HOL/Library/Euclidean_Space.thy	Wed Jun 03 08:46:13 2009 -0700
+++ b/src/HOL/Library/Euclidean_Space.thy	Wed Jun 03 09:58:11 2009 -0700
@@ -506,8 +506,8 @@
 definition dist_vector_def:
   "dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
 
-definition open_vector_def:
-  "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y::'a ^ 'b. dist y x < e \<longrightarrow> y \<in> S)"
+definition topo_vector_def:
+  "topo = {S::('a ^ 'b) set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
 
 instance proof
   fix x y :: "'a ^ 'b"
@@ -522,9 +522,8 @@
     apply (simp add: setL2_mono dist_triangle2)
     done
 next
-  fix S :: "('a ^ 'b) set"
-  show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-    by (rule open_vector_def)
+  show "topo = {S::('a ^ 'b) set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
+    by (rule topo_vector_def)
 qed
 
 end
diff -r f4c079225845 -r c12b25b7f015 src/HOL/Library/Inner_Product.thy
--- a/src/HOL/Library/Inner_Product.thy	Wed Jun 03 08:46:13 2009 -0700
+++ b/src/HOL/Library/Inner_Product.thy	Wed Jun 03 09:58:11 2009 -0700
@@ -10,7 +10,7 @@
 
 subsection {* Real inner product spaces *}
 
-class real_inner = real_vector + sgn_div_norm + dist_norm + open_dist +
+class real_inner = real_vector + sgn_div_norm + dist_norm + topo_dist +
   fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
   assumes inner_commute: "inner x y = inner y x"
   and inner_left_distrib: "inner (x + y) z = inner x z + inner y z"
diff -r f4c079225845 -r c12b25b7f015 src/HOL/Library/Product_Vector.thy
--- a/src/HOL/Library/Product_Vector.thy	Wed Jun 03 08:46:13 2009 -0700
+++ b/src/HOL/Library/Product_Vector.thy	Wed Jun 03 09:58:11 2009 -0700
@@ -45,28 +45,29 @@
   "*" :: (topological_space, topological_space) topological_space
 begin
 
-definition open_prod_def:
-  "open S = (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
+definition topo_prod_def:
+  "topo = {S. \<forall>x\<in>S. \<exists>A\<in>topo. \<exists>B\<in>topo. x \<in> A \<times> B \<and> A \<times> B \<subseteq> S}"
 
 instance proof
-  show "open (UNIV :: ('a \<times> 'b) set)"
-    unfolding open_prod_def by (fast intro: open_UNIV)
+  show "(UNIV :: ('a \<times> 'b) set) \<in> topo"
+    unfolding topo_prod_def by (auto intro: topo_UNIV)
 next
   fix S T :: "('a \<times> 'b) set"
-  assume "open S" "open T" thus "open (S \<inter> T)"
-    unfolding open_prod_def
+  assume "S \<in> topo" "T \<in> topo" thus "S \<inter> T \<in> topo"
+    unfolding topo_prod_def
     apply clarify
     apply (drule (1) bspec)+
     apply (clarify, rename_tac Sa Ta Sb Tb)
-    apply (rule_tac x="Sa \<inter> Ta" in exI)
-    apply (rule_tac x="Sb \<inter> Tb" in exI)
-    apply (simp add: open_Int)
+    apply (rule_tac x="Sa \<inter> Ta" in rev_bexI)
+    apply (simp add: topo_Int)
+    apply (rule_tac x="Sb \<inter> Tb" in rev_bexI)
+    apply (simp add: topo_Int)
     apply fast
     done
 next
   fix T :: "('a \<times> 'b) set set"
-  assume "\<forall>A\<in>T. open A" thus "open (\<Union>T)"
-    unfolding open_prod_def by fast
+  assume "T \<subseteq> topo" thus "\<Union>T \<in> topo"
+    unfolding topo_prod_def Bex_def by fast
 qed
 
 end
@@ -103,10 +104,9 @@
   (* FIXME: long proof! *)
   (* Maybe it would be easier to define topological spaces *)
   (* in terms of neighborhoods instead of open sets? *)
-  fix S :: "('a \<times> 'b) set"
-  show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-    unfolding open_prod_def open_dist
-    apply safe
+  show "topo = {S::('a \<times> 'b) set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
+    unfolding topo_prod_def topo_dist
+    apply (safe, rename_tac S a b)
     apply (drule (1) bspec)
     apply clarify
     apply (drule (1) bspec)+
@@ -121,19 +121,18 @@
     apply (drule spec, erule mp)
     apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
 
+    apply (rename_tac S a b)
     apply (drule (1) bspec)
     apply clarify
     apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
     apply clarify
-    apply (rule_tac x="{y. dist y a < r}" in exI)
-    apply (rule_tac x="{y. dist y b < s}" in exI)
-    apply (rule conjI)
+    apply (rule_tac x="{y. dist y a < r}" in rev_bexI)
     apply clarify
     apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
     apply clarify
     apply (rule le_less_trans [OF dist_triangle])
     apply (erule less_le_trans [OF add_strict_right_mono], simp)
-    apply (rule conjI)
+    apply (rule_tac x="{y. dist y b < s}" in rev_bexI)
     apply clarify
     apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
     apply clarify
diff -r f4c079225845 -r c12b25b7f015 src/HOL/RealVector.thy
--- a/src/HOL/RealVector.thy	Wed Jun 03 08:46:13 2009 -0700
+++ b/src/HOL/RealVector.thy	Wed Jun 03 09:58:11 2009 -0700
@@ -418,13 +418,13 @@
 
 subsection {* Topological spaces *}
 
-class "open" =
-  fixes "open" :: "'a set \<Rightarrow> bool"
+class topo =
+  fixes topo :: "'a set set"
 
-class topological_space = "open" +
-  assumes open_UNIV: "open UNIV"
-  assumes open_Int: "open A \<Longrightarrow> open B \<Longrightarrow> open (A \<inter> B)"
-  assumes open_Union: "\<forall>A\<in>T. open A \<Longrightarrow> open (\<Union>T)"
+class topological_space = topo +
+  assumes topo_UNIV: "UNIV \<in> topo"
+  assumes topo_Int: "A \<in> topo \<Longrightarrow> B \<in> topo \<Longrightarrow> A \<inter> B \<in> topo"
+  assumes topo_Union: "T \<subseteq> topo \<Longrightarrow> \<Union>T \<in> topo"
 
 
 subsection {* Metric spaces *}
@@ -432,10 +432,10 @@
 class dist =
   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
 
-class open_dist = "open" + dist +
-  assumes open_dist: "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
+class topo_dist = topo + dist +
+  assumes topo_dist: "topo = {S. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
 
-class metric_space = open_dist +
+class metric_space = topo_dist +
   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
 begin
@@ -470,21 +470,20 @@
 proof
   have "\<exists>e::real. 0 < e"
     by (fast intro: zero_less_one)
-  then show "open UNIV"
-    unfolding open_dist by simp
+  then show "UNIV \<in> topo"
+    unfolding topo_dist by simp
 next
-  fix A B assume "open A" "open B"
-  then show "open (A \<inter> B)"
-    unfolding open_dist
+  fix A B assume "A \<in> topo" "B \<in> topo"
+  then show "A \<inter> B \<in> topo"
+    unfolding topo_dist
     apply clarify
     apply (drule (1) bspec)+
     apply (clarify, rename_tac r s)
     apply (rule_tac x="min r s" in exI, simp)
     done
 next
-  fix T assume "\<forall>A\<in>T. open A"
-  then show "open (\<Union>T)"
-    unfolding open_dist by fast
+  fix T assume "T \<subseteq> topo" thus "\<Union>T \<in> topo"
+    unfolding topo_dist by fast
 qed
 
 end
@@ -501,7 +500,7 @@
 class dist_norm = dist + norm + minus +
   assumes dist_norm: "dist x y = norm (x - y)"
 
-class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
+class real_normed_vector = real_vector + sgn_div_norm + dist_norm + topo_dist +
   assumes norm_ge_zero [simp]: "0 \<le> norm x"
   and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
@@ -538,14 +537,14 @@
 definition dist_real_def:
   "dist x y = \<bar>x - y\<bar>"
 
-definition open_real_def:
-  "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y::real. dist y x < e \<longrightarrow> y \<in> S)"
+definition topo_real_def:
+  "topo = {S::real set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
 
 instance
 apply (intro_classes, unfold real_norm_def real_scaleR_def)
 apply (rule dist_real_def)
-apply (rule open_real_def)
 apply (simp add: real_sgn_def)
+apply (rule topo_real_def)
 apply (rule abs_ge_zero)
 apply (rule abs_eq_0)
 apply (rule abs_triangle_ineq)