src/HOL/Library/Continuity.thy
 changeset 30952 7ab2716dd93b parent 30950 1435a8f01a41 child 30971 7fbebf75b3ef
```     1.1 --- a/src/HOL/Library/Continuity.thy	Fri Apr 17 16:41:31 2009 +0200
1.2 +++ b/src/HOL/Library/Continuity.thy	Mon Apr 20 09:32:07 2009 +0200
1.3 @@ -5,7 +5,7 @@
1.4  header {* Continuity and iterations (of set transformers) *}
1.5
1.6  theory Continuity
1.7 -imports Relation_Power Main
1.8 +imports Transitive_Closure Main
1.9  begin
1.10
1.11  subsection {* Continuity for complete lattices *}
1.12 @@ -48,25 +48,25 @@
1.13  qed
1.14
1.15  lemma continuous_lfp:
1.16 - assumes "continuous F" shows "lfp F = (SUP i. (F^^i) bot)"
1.17 + assumes "continuous F" shows "lfp F = (SUP i. (F o^ i) bot)"
1.18  proof -
1.19    note mono = continuous_mono[OF `continuous F`]
1.20 -  { fix i have "(F^^i) bot \<le> lfp F"
1.21 +  { fix i have "(F o^ i) bot \<le> lfp F"
1.22      proof (induct i)
1.23 -      show "(F^^0) bot \<le> lfp F" by simp
1.24 +      show "(F o^ 0) bot \<le> lfp F" by simp
1.25      next
1.26        case (Suc i)
1.27 -      have "(F^^(Suc i)) bot = F((F^^i) bot)" by simp
1.28 +      have "(F o^ Suc i) bot = F((F o^ i) bot)" by simp
1.29        also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
1.30        also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
1.31        finally show ?case .
1.32      qed }
1.33 -  hence "(SUP i. (F^^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
1.34 -  moreover have "lfp F \<le> (SUP i. (F^^i) bot)" (is "_ \<le> ?U")
1.35 +  hence "(SUP i. (F o^ i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
1.36 +  moreover have "lfp F \<le> (SUP i. (F o^ i) bot)" (is "_ \<le> ?U")
1.37    proof (rule lfp_lowerbound)
1.38 -    have "chain(%i. (F^^i) bot)"
1.39 +    have "chain(%i. (F o^ i) bot)"
1.40      proof -
1.41 -      { fix i have "(F^^i) bot \<le> (F^^(Suc i)) bot"
1.42 +      { fix i have "(F o^ i) bot \<le> (F o^ (Suc i)) bot"
1.43  	proof (induct i)
1.44  	  case 0 show ?case by simp
1.45  	next
1.46 @@ -74,7 +74,7 @@
1.47  	qed }
1.48        thus ?thesis by(auto simp add:chain_def)
1.49      qed
1.50 -    hence "F ?U = (SUP i. (F^^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
1.51 +    hence "F ?U = (SUP i. (F o^ (i+1)) bot)" using `continuous F` by (simp add:continuous_def)
1.52      also have "\<dots> \<le> ?U" by(fast intro:SUP_leI le_SUPI)
1.53      finally show "F ?U \<le> ?U" .
1.54    qed
1.55 @@ -193,7 +193,7 @@
1.56
1.57  definition
1.58    up_iterate :: "('a set => 'a set) => nat => 'a set" where
1.59 -  "up_iterate f n = (f^^n) {}"
1.60 +  "up_iterate f n = (f o^ n) {}"
1.61
1.62  lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"