src/HOL/UNITY/Detects.thy
changeset 13805 3786b2fd6808
parent 13798 4c1a53627500
child 13812 91713a1915ee
     1.1 --- a/src/HOL/UNITY/Detects.thy	Mon Feb 03 11:45:05 2003 +0100
     1.2 +++ b/src/HOL/UNITY/Detects.thy	Tue Feb 04 18:12:40 2003 +0100
     1.3 @@ -15,47 +15,47 @@
     1.4     op_Equality :: "['a set, 'a set] => 'a set"          (infixl "<==>" 60)
     1.5     
     1.6  defs
     1.7 -  Detects_def:  "A Detects B == (Always (-A Un B)) Int (B LeadsTo A)"
     1.8 -  Equality_def: "A <==> B == (-A Un B) Int (A Un -B)"
     1.9 +  Detects_def:  "A Detects B == (Always (-A \<union> B)) \<inter> (B LeadsTo A)"
    1.10 +  Equality_def: "A <==> B == (-A \<union> B) \<inter> (A \<union> -B)"
    1.11  
    1.12  
    1.13  (* Corollary from Sectiom 3.6.4 *)
    1.14  
    1.15 -lemma Always_at_FP: "F: A LeadsTo B ==> F : Always (-((FP F) Int A Int -B))"
    1.16 +lemma Always_at_FP: "F \<in> A LeadsTo B ==> F \<in> Always (-((FP F) \<inter> A \<inter> -B))"
    1.17  apply (rule LeadsTo_empty)
    1.18 -apply (subgoal_tac "F : (FP F Int A Int - B) LeadsTo (B Int (FP F Int -B))")
    1.19 -apply (subgoal_tac [2] " (FP F Int A Int - B) = (A Int (FP F Int -B))")
    1.20 -apply (subgoal_tac "(B Int (FP F Int -B)) = {}")
    1.21 +apply (subgoal_tac "F \<in> (FP F \<inter> A \<inter> - B) LeadsTo (B \<inter> (FP F \<inter> -B))")
    1.22 +apply (subgoal_tac [2] " (FP F \<inter> A \<inter> - B) = (A \<inter> (FP F \<inter> -B))")
    1.23 +apply (subgoal_tac "(B \<inter> (FP F \<inter> -B)) = {}")
    1.24  apply auto
    1.25  apply (blast intro: PSP_Stable stable_imp_Stable stable_FP_Int)
    1.26  done
    1.27  
    1.28  
    1.29  lemma Detects_Trans: 
    1.30 -     "[| F : A Detects B; F : B Detects C |] ==> F : A Detects C"
    1.31 +     "[| F \<in> A Detects B; F \<in> B Detects C |] ==> F \<in> A Detects C"
    1.32  apply (unfold Detects_def Int_def)
    1.33  apply (simp (no_asm))
    1.34  apply safe
    1.35  apply (rule_tac [2] LeadsTo_Trans)
    1.36  apply auto
    1.37 -apply (subgoal_tac "F : Always ((-A Un B) Int (-B Un C))")
    1.38 +apply (subgoal_tac "F \<in> Always ((-A \<union> B) \<inter> (-B \<union> C))")
    1.39   apply (blast intro: Always_weaken)
    1.40  apply (simp add: Always_Int_distrib)
    1.41  done
    1.42  
    1.43 -lemma Detects_refl: "F : A Detects A"
    1.44 +lemma Detects_refl: "F \<in> A Detects A"
    1.45  apply (unfold Detects_def)
    1.46  apply (simp (no_asm) add: Un_commute Compl_partition subset_imp_LeadsTo)
    1.47  done
    1.48  
    1.49 -lemma Detects_eq_Un: "(A<==>B) = (A Int B) Un (-A Int -B)"
    1.50 +lemma Detects_eq_Un: "(A<==>B) = (A \<inter> B) \<union> (-A \<inter> -B)"
    1.51  apply (unfold Equality_def)
    1.52  apply blast
    1.53  done
    1.54  
    1.55  (*Not quite antisymmetry: sets A and B agree in all reachable states *)
    1.56  lemma Detects_antisym: 
    1.57 -     "[| F : A Detects B;  F : B Detects A|] ==> F : Always (A <==> B)"
    1.58 +     "[| F \<in> A Detects B;  F \<in> B Detects A|] ==> F \<in> Always (A <==> B)"
    1.59  apply (unfold Detects_def Equality_def)
    1.60  apply (simp add: Always_Int_I Un_commute)
    1.61  done
    1.62 @@ -64,7 +64,7 @@
    1.63  (* Theorem from Section 3.8 *)
    1.64  
    1.65  lemma Detects_Always: 
    1.66 -     "F : A Detects B ==> F : Always ((-(FP F)) Un (A <==> B))"
    1.67 +     "F \<in> A Detects B ==> F \<in> Always ((-(FP F)) \<union> (A <==> B))"
    1.68  apply (unfold Detects_def Equality_def)
    1.69  apply (simp (no_asm) add: Un_Int_distrib Always_Int_distrib)
    1.70  apply (blast dest: Always_at_FP intro: Always_weaken)
    1.71 @@ -73,11 +73,11 @@
    1.72  (* Theorem from exercise 11.1 Section 11.3.1 *)
    1.73  
    1.74  lemma Detects_Imp_LeadstoEQ: 
    1.75 -     "F : A Detects B ==> F : UNIV LeadsTo (A <==> B)"
    1.76 +     "F \<in> A Detects B ==> F \<in> UNIV LeadsTo (A <==> B)"
    1.77  apply (unfold Detects_def Equality_def)
    1.78  apply (rule_tac B = "B" in LeadsTo_Diff)
    1.79 -prefer 2 apply (blast intro: Always_LeadsTo_weaken)
    1.80 -apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
    1.81 + apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
    1.82 +apply (blast intro: Always_LeadsTo_weaken)
    1.83  done
    1.84  
    1.85