author berghofe Sat, 10 Mar 2007 16:24:52 +0100 changeset 22431 28344ccffc35 parent 22430 6a56bf1b3a64 child 22432 1d00d26fee0d
Adapted to changes in definition of SUP.
 src/HOL/Library/Continuity.thy file | annotate | diff | comparison | revisions src/HOL/Library/Graphs.thy file | annotate | diff | comparison | revisions src/HOL/Library/Kleene_Algebras.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Library/Continuity.thy	Sat Mar 10 16:23:28 2007 +0100
+++ b/src/HOL/Library/Continuity.thy	Sat Mar 10 16:24:52 2007 +0100
@@ -16,7 +16,7 @@
"chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"

definition
-  continuous :: "('a::order \<Rightarrow> 'a::order) \<Rightarrow> bool" where
+  continuous :: "('a::comp_lat \<Rightarrow> 'a::comp_lat) \<Rightarrow> bool" where
"continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"

abbreviation
@@ -24,12 +24,12 @@
"bot \<equiv> Sup {}"

lemma SUP_nat_conv:
-  "(SUP n::nat. M n::'a::comp_lat) = sup (M 0) (SUP n. M(Suc n))"
+  "(SUP n. M n) = sup (M 0) (SUP n. M(Suc n))"
apply(rule order_antisym)
apply(rule SUP_leI)
apply(case_tac n)
apply simp
- apply (blast intro:le_SUPI le_supI2)
+ apply (fast intro:le_SUPI le_supI2)
apply(simp)
apply (blast intro:SUP_leI le_SUPI)
done
@@ -43,10 +43,10 @@
have "F B = sup (F A) (F B)"
proof -
have "sup A B = B" using `A <= B` by (simp add:sup_absorb2)
-    hence "F B = F(SUP i. ?C i)" by(simp add: SUP_nat_conv)
+    hence "F B = F(SUP i. ?C i)" by (subst SUP_nat_conv) simp
also have "\<dots> = (SUP i. F(?C i))"
using `chain ?C` `continuous F` by(simp add:continuous_def)
-    also have "\<dots> = sup (F A) (F B)" by(simp add: SUP_nat_conv)
+    also have "\<dots> = sup (F A) (F B)" by (subst SUP_nat_conv) simp
finally show ?thesis .
qed
thus "F A \<le> F B" by(subst le_iff_sup, simp)
@@ -80,7 +80,7 @@
qed
hence "F ?U = (SUP i. (F^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
-    also have "\<dots> \<le> ?U" by(blast intro:SUP_leI le_SUPI)
+    also have "\<dots> \<le> ?U" by(fast intro:SUP_leI le_SUPI)
finally show "F ?U \<le> ?U" .
qed
ultimately show ?thesis by (blast intro:order_antisym)```
```--- a/src/HOL/Library/Graphs.thy	Sat Mar 10 16:23:28 2007 +0100
+++ b/src/HOL/Library/Graphs.thy	Sat Mar 10 16:24:52 2007 +0100
@@ -245,7 +245,7 @@

lemma Sup_graph_eq:
"(SUP n. Graph (G n)) = Graph (\<Union>n. G n)"
-  unfolding SUP_def
+  unfolding SUPR_def
apply (rule order_antisym)
apply (rule Sup_least)
apply auto```
```--- a/src/HOL/Library/Kleene_Algebras.thy	Sat Mar 10 16:23:28 2007 +0100
+++ b/src/HOL/Library/Kleene_Algebras.thy	Sat Mar 10 16:24:52 2007 +0100
@@ -104,7 +104,7 @@
assumes "l \<le> M i"
shows "l \<le> (SUP i. M i)"
using prems
-  by (rule order_trans) (rule le_SUPI, rule refl)
+  by (rule order_trans) (rule le_SUPI [OF UNIV_I])

lemma zero_minimum[simp]: "(0::'a::pre_kleene) \<le> x"
@@ -117,21 +117,20 @@

have [simp]: "1 \<le> star a"
unfolding star_cont[of 1 a 1, simplified]
-    by (rule le_SUPI) (rule power_0[symmetric])
+    by (subst power_0[symmetric]) (rule le_SUPI [OF UNIV_I])

show "1 + a * star a \<le> star a"
apply (rule plus_leI, simp)
apply (simp add:star_cont[of a a 1, simplified])
apply (simp add:star_cont[of 1 a 1, simplified])
-    apply (intro SUP_leI le_SUPI)
-    by (rule power_Suc[symmetric])
+    apply (subst power_Suc[symmetric])
+    by (intro SUP_leI le_SUPI UNIV_I)

show "1 + star a * a \<le> star a"
apply (rule plus_leI, simp)
apply (simp add:star_cont[of 1 a a, simplified])
apply (simp add:star_cont[of 1 a 1, simplified])
-    apply (intro SUP_leI le_SUPI)