author haftmann Fri, 17 Apr 2009 16:41:30 +0200 changeset 30950 1435a8f01a41 parent 30949 37f887b55e7f child 30951 a6e26a248f03
power operation on relations now with syntax ^^
```--- a/src/HOL/Library/Continuity.thy	Fri Apr 17 15:57:26 2009 +0200
+++ b/src/HOL/Library/Continuity.thy	Fri Apr 17 16:41:30 2009 +0200
@@ -48,25 +48,25 @@
qed

lemma continuous_lfp:
- assumes "continuous F" shows "lfp F = (SUP i. (F^i) bot)"
+ assumes "continuous F" shows "lfp F = (SUP i. (F^^i) bot)"
proof -
note mono = continuous_mono[OF `continuous F`]
-  { fix i have "(F^i) bot \<le> lfp F"
+  { fix i have "(F^^i) bot \<le> lfp F"
proof (induct i)
-      show "(F^0) bot \<le> lfp F" by simp
+      show "(F^^0) bot \<le> lfp F" by simp
next
case (Suc i)
-      have "(F^(Suc i)) bot = F((F^i) bot)" by simp
+      have "(F^^(Suc i)) bot = F((F^^i) bot)" by simp
also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
finally show ?case .
qed }
-  hence "(SUP i. (F^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
-  moreover have "lfp F \<le> (SUP i. (F^i) bot)" (is "_ \<le> ?U")
+  hence "(SUP i. (F^^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
+  moreover have "lfp F \<le> (SUP i. (F^^i) bot)" (is "_ \<le> ?U")
proof (rule lfp_lowerbound)
-    have "chain(%i. (F^i) bot)"
+    have "chain(%i. (F^^i) bot)"
proof -
-      { fix i have "(F^i) bot \<le> (F^(Suc i)) bot"
+      { fix i have "(F^^i) bot \<le> (F^^(Suc i)) bot"
proof (induct i)
case 0 show ?case by simp
next
@@ -74,7 +74,7 @@
qed }
qed
-    hence "F ?U = (SUP i. (F^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
+    hence "F ?U = (SUP i. (F^^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
also have "\<dots> \<le> ?U" by(fast intro:SUP_leI le_SUPI)
finally show "F ?U \<le> ?U" .
qed
@@ -193,7 +193,7 @@

definition
up_iterate :: "('a set => 'a set) => nat => 'a set" where
-  "up_iterate f n = (f^n) {}"
+  "up_iterate f n = (f^^n) {}"

lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"