src/HOL/UNITY/Detects.thy

--- a/src/HOL/UNITY/Detects.thy	Mon Feb 03 11:45:05 2003 +0100
+++ b/src/HOL/UNITY/Detects.thy	Tue Feb 04 18:12:40 2003 +0100
@@ -15,47 +15,47 @@
    op_Equality :: "['a set, 'a set] => 'a set"          (infixl "<==>" 60)
    
 defs
-  Detects_def:  "A Detects B == (Always (-A Un B)) Int (B LeadsTo A)"
-  Equality_def: "A <==> B == (-A Un B) Int (A Un -B)"
+  Detects_def:  "A Detects B == (Always (-A \<union> B)) \<inter> (B LeadsTo A)"
+  Equality_def: "A <==> B == (-A \<union> B) \<inter> (A \<union> -B)"
 
 
 (* Corollary from Sectiom 3.6.4 *)
 
-lemma Always_at_FP: "F: A LeadsTo B ==> F : Always (-((FP F) Int A Int -B))"
+lemma Always_at_FP: "F \<in> A LeadsTo B ==> F \<in> Always (-((FP F) \<inter> A \<inter> -B))"
 apply (rule LeadsTo_empty)
-apply (subgoal_tac "F : (FP F Int A Int - B) LeadsTo (B Int (FP F Int -B))")
-apply (subgoal_tac [2] " (FP F Int A Int - B) = (A Int (FP F Int -B))")
-apply (subgoal_tac "(B Int (FP F Int -B)) = {}")
+apply (subgoal_tac "F \<in> (FP F \<inter> A \<inter> - B) LeadsTo (B \<inter> (FP F \<inter> -B))")
+apply (subgoal_tac [2] " (FP F \<inter> A \<inter> - B) = (A \<inter> (FP F \<inter> -B))")
+apply (subgoal_tac "(B \<inter> (FP F \<inter> -B)) = {}")
 apply auto
 apply (blast intro: PSP_Stable stable_imp_Stable stable_FP_Int)
 done
 
 
 lemma Detects_Trans: 
-     "[| F : A Detects B; F : B Detects C |] ==> F : A Detects C"
+     "[| F \<in> A Detects B; F \<in> B Detects C |] ==> F \<in> A Detects C"
 apply (unfold Detects_def Int_def)
 apply (simp (no_asm))
 apply safe
 apply (rule_tac [2] LeadsTo_Trans)
 apply auto
-apply (subgoal_tac "F : Always ((-A Un B) Int (-B Un C))")
+apply (subgoal_tac "F \<in> Always ((-A \<union> B) \<inter> (-B \<union> C))")
  apply (blast intro: Always_weaken)
 apply (simp add: Always_Int_distrib)
 done
 
-lemma Detects_refl: "F : A Detects A"
+lemma Detects_refl: "F \<in> A Detects A"
 apply (unfold Detects_def)
 apply (simp (no_asm) add: Un_commute Compl_partition subset_imp_LeadsTo)
 done
 
-lemma Detects_eq_Un: "(A<==>B) = (A Int B) Un (-A Int -B)"
+lemma Detects_eq_Un: "(A<==>B) = (A \<inter> B) \<union> (-A \<inter> -B)"
 apply (unfold Equality_def)
 apply blast
 done
 
 (*Not quite antisymmetry: sets A and B agree in all reachable states *)
 lemma Detects_antisym: 
-     "[| F : A Detects B;  F : B Detects A|] ==> F : Always (A <==> B)"
+     "[| F \<in> A Detects B;  F \<in> B Detects A|] ==> F \<in> Always (A <==> B)"
 apply (unfold Detects_def Equality_def)
 apply (simp add: Always_Int_I Un_commute)
 done
@@ -64,7 +64,7 @@
 (* Theorem from Section 3.8 *)
 
 lemma Detects_Always: 
-     "F : A Detects B ==> F : Always ((-(FP F)) Un (A <==> B))"
+     "F \<in> A Detects B ==> F \<in> Always ((-(FP F)) \<union> (A <==> B))"
 apply (unfold Detects_def Equality_def)
 apply (simp (no_asm) add: Un_Int_distrib Always_Int_distrib)
 apply (blast dest: Always_at_FP intro: Always_weaken)
@@ -73,11 +73,11 @@
 (* Theorem from exercise 11.1 Section 11.3.1 *)
 
 lemma Detects_Imp_LeadstoEQ: 
-     "F : A Detects B ==> F : UNIV LeadsTo (A <==> B)"
+     "F \<in> A Detects B ==> F \<in> UNIV LeadsTo (A <==> B)"
 apply (unfold Detects_def Equality_def)
 apply (rule_tac B = "B" in LeadsTo_Diff)
-prefer 2 apply (blast intro: Always_LeadsTo_weaken)
-apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
+ apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
+apply (blast intro: Always_LeadsTo_weaken)
 done