src/HOL/UNITY/Detects.thy
author wenzelm
Thu Jul 23 22:13:42 2015 +0200 (2015-07-23)
changeset 60773 d09c66a0ea10
parent 58889 5b7a9633cfa8
child 61635 c657ee4f59b7
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     1 (*  Title:      HOL/UNITY/Detects.thy
     2     Author:     Tanja Vos, Cambridge University Computer Laboratory
     3     Copyright   2000  University of Cambridge
     4 
     5 Detects definition (Section 3.8 of Chandy & Misra) using LeadsTo
     6 *)
     7 
     8 section{*The Detects Relation*}
     9 
    10 theory Detects imports FP SubstAx begin
    11 
    12 definition Detects :: "['a set, 'a set] => 'a program set"  (infixl "Detects" 60)
    13   where "A Detects B = (Always (-A \<union> B)) \<inter> (B LeadsTo A)"
    14 
    15 definition Equality :: "['a set, 'a set] => 'a set"  (infixl "<==>" 60)
    16   where "A <==> B = (-A \<union> B) \<inter> (A \<union> -B)"
    17 
    18 
    19 (* Corollary from Sectiom 3.6.4 *)
    20 
    21 lemma Always_at_FP:
    22      "[|F \<in> A LeadsTo B; all_total F|] ==> F \<in> Always (-((FP F) \<inter> A \<inter> -B))"
    23 apply (rule LeadsTo_empty)
    24 apply (subgoal_tac "F \<in> (FP F \<inter> A \<inter> - B) LeadsTo (B \<inter> (FP F \<inter> -B))")
    25 apply (subgoal_tac [2] " (FP F \<inter> A \<inter> - B) = (A \<inter> (FP F \<inter> -B))")
    26 apply (subgoal_tac "(B \<inter> (FP F \<inter> -B)) = {}")
    27 apply auto
    28 apply (blast intro: PSP_Stable stable_imp_Stable stable_FP_Int)
    29 done
    30 
    31 
    32 lemma Detects_Trans: 
    33      "[| F \<in> A Detects B; F \<in> B Detects C |] ==> F \<in> A Detects C"
    34 apply (unfold Detects_def Int_def)
    35 apply (simp (no_asm))
    36 apply safe
    37 apply (rule_tac [2] LeadsTo_Trans, auto)
    38 apply (subgoal_tac "F \<in> Always ((-A \<union> B) \<inter> (-B \<union> C))")
    39  apply (blast intro: Always_weaken)
    40 apply (simp add: Always_Int_distrib)
    41 done
    42 
    43 lemma Detects_refl: "F \<in> A Detects A"
    44 apply (unfold Detects_def)
    45 apply (simp (no_asm) add: Un_commute Compl_partition subset_imp_LeadsTo)
    46 done
    47 
    48 lemma Detects_eq_Un: "(A<==>B) = (A \<inter> B) \<union> (-A \<inter> -B)"
    49 by (unfold Equality_def, blast)
    50 
    51 (*Not quite antisymmetry: sets A and B agree in all reachable states *)
    52 lemma Detects_antisym: 
    53      "[| F \<in> A Detects B;  F \<in> B Detects A|] ==> F \<in> Always (A <==> B)"
    54 apply (unfold Detects_def Equality_def)
    55 apply (simp add: Always_Int_I Un_commute)
    56 done
    57 
    58 
    59 (* Theorem from Section 3.8 *)
    60 
    61 lemma Detects_Always: 
    62      "[|F \<in> A Detects B; all_total F|] ==> F \<in> Always (-(FP F) \<union> (A <==> B))"
    63 apply (unfold Detects_def Equality_def)
    64 apply (simp add: Un_Int_distrib Always_Int_distrib)
    65 apply (blast dest: Always_at_FP intro: Always_weaken)
    66 done
    67 
    68 (* Theorem from exercise 11.1 Section 11.3.1 *)
    69 
    70 lemma Detects_Imp_LeadstoEQ: 
    71      "F \<in> A Detects B ==> F \<in> UNIV LeadsTo (A <==> B)"
    72 apply (unfold Detects_def Equality_def)
    73 apply (rule_tac B = B in LeadsTo_Diff)
    74  apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
    75 apply (blast intro: Always_LeadsTo_weaken)
    76 done
    77 
    78 
    79 end
    80