src/HOL/UNITY/Detects.thy
author paulson
Fri Jan 24 14:06:49 2003 +0100 (2003-01-24)
changeset 13785 e2fcd88be55d
parent 8334 7896bcbd8641
child 13798 4c1a53627500
permissions -rw-r--r--
Partial conversion of UNITY to Isar new-style theories
     1 (*  Title:      HOL/UNITY/Detects
     2     ID:         $Id$
     3     Author:     Tanja Vos, Cambridge University Computer Laboratory
     4     Copyright   2000  University of Cambridge
     5 
     6 Detects definition (Section 3.8 of Chandy & Misra) using LeadsTo
     7 *)
     8 
     9 theory Detects = FP + SubstAx:
    10 
    11 consts
    12    op_Detects  :: "['a set, 'a set] => 'a program set"  (infixl "Detects" 60)
    13    op_Equality :: "['a set, 'a set] => 'a set"          (infixl "<==>" 60)
    14    
    15 defs
    16   Detects_def:  "A Detects B == (Always (-A Un B)) Int (B LeadsTo A)"
    17   Equality_def: "A <==> B == (-A Un B) Int (A Un -B)"
    18 
    19 
    20 (* Corollary from Sectiom 3.6.4 *)
    21 
    22 lemma Always_at_FP: "F: A LeadsTo B ==> F : Always (-((FP F) Int A Int -B))"
    23 apply (rule LeadsTo_empty)
    24 apply (subgoal_tac "F : (FP F Int A Int - B) LeadsTo (B Int (FP F Int -B))")
    25 apply (subgoal_tac [2] " (FP F Int A Int - B) = (A Int (FP F Int -B))")
    26 apply (subgoal_tac "(B Int (FP F Int -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 : A Detects B; F : B Detects C |] ==> F : 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)
    38 apply auto
    39 apply (subgoal_tac "F : Always ((-A Un B) Int (-B Un C))")
    40  apply (blast intro: Always_weaken)
    41 apply (simp add: Always_Int_distrib)
    42 done
    43 
    44 lemma Detects_refl: "F : A Detects A"
    45 apply (unfold Detects_def)
    46 apply (simp (no_asm) add: Un_commute Compl_partition subset_imp_LeadsTo)
    47 done
    48 
    49 lemma Detects_eq_Un: "(A<==>B) = (A Int B) Un (-A Int -B)"
    50 apply (unfold Equality_def)
    51 apply blast
    52 done
    53 
    54 (*Not quite antisymmetry: sets A and B agree in all reachable states *)
    55 lemma Detects_antisym: 
    56      "[| F : A Detects B;  F : B Detects A|] ==> F : Always (A <==> B)"
    57 apply (unfold Detects_def Equality_def)
    58 apply (simp add: Always_Int_I Un_commute)
    59 done
    60 
    61 
    62 (* Theorem from Section 3.8 *)
    63 
    64 lemma Detects_Always: 
    65      "F : A Detects B ==> F : Always ((-(FP F)) Un (A <==> B))"
    66 apply (unfold Detects_def Equality_def)
    67 apply (simp (no_asm) add: Un_Int_distrib Always_Int_distrib)
    68 apply (blast dest: Always_at_FP intro: Always_weaken)
    69 done
    70 
    71 (* Theorem from exercise 11.1 Section 11.3.1 *)
    72 
    73 lemma Detects_Imp_LeadstoEQ: 
    74      "F : A Detects B ==> F : UNIV LeadsTo (A <==> B)"
    75 apply (unfold Detects_def Equality_def)
    76 apply (rule_tac B = "B" in LeadsTo_Diff)
    77 prefer 2 apply (blast intro: Always_LeadsTo_weaken)
    78 apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
    79 done
    80 
    81 
    82 end
    83