src/HOL/IMPP/EvenOdd.thy
author wenzelm
Wed Mar 25 10:44:57 2015 +0100 (2015-03-25)
changeset 59807 22bc39064290
parent 58889 5b7a9633cfa8
child 62145 5b946c81dfbf
permissions -rw-r--r--
prefer local fixes;
     1 (*  Title:      HOL/IMPP/EvenOdd.thy
     2     Author:     David von Oheimb, TUM
     3 *)
     4 
     5 section {* Example of mutually recursive procedures verified with Hoare logic *}
     6 
     7 theory EvenOdd
     8 imports Main Misc
     9 begin
    10 
    11 axiomatization
    12   Even :: pname and
    13   Odd :: pname
    14 where
    15   Even_neq_Odd: "Even ~= Odd" and
    16   Arg_neq_Res:  "Arg  ~= Res"
    17 
    18 definition
    19   evn :: com where
    20  "evn = (IF (%s. s<Arg> = 0)
    21          THEN Loc Res:==(%s. 0)
    22          ELSE(Loc Res:=CALL Odd(%s. s<Arg> - 1);;
    23               Loc Arg:=CALL Odd(%s. s<Arg> - 1);;
    24               Loc Res:==(%s. s<Res> * s<Arg>)))"
    25 
    26 definition
    27   odd :: com where
    28  "odd = (IF (%s. s<Arg> = 0)
    29          THEN Loc Res:==(%s. 1)
    30          ELSE(Loc Res:=CALL Even (%s. s<Arg> - 1)))"
    31 
    32 defs
    33   bodies_def: "bodies == [(Even,evn),(Odd,odd)]"
    34 
    35 definition
    36   Z_eq_Arg_plus :: "nat => nat assn" ("Z=Arg+_" [50]50) where
    37   "Z=Arg+n = (%Z s.      Z =  s<Arg>+n)"
    38 
    39 definition
    40   Res_ok :: "nat assn" where
    41   "Res_ok = (%Z s. even Z = (s<Res> = 0))"
    42 
    43 
    44 subsection "Arg, Res"
    45 
    46 declare Arg_neq_Res [simp] Arg_neq_Res [THEN not_sym, simp]
    47 declare Even_neq_Odd [simp] Even_neq_Odd [THEN not_sym, simp]
    48 
    49 lemma Z_eq_Arg_plus_def2: "(Z=Arg+n) Z s = (Z = s<Arg>+n)"
    50 apply (unfold Z_eq_Arg_plus_def)
    51 apply (rule refl)
    52 done
    53 
    54 lemma Res_ok_def2: "Res_ok Z s = (even Z = (s<Res> = 0))"
    55 apply (unfold Res_ok_def)
    56 apply (rule refl)
    57 done
    58 
    59 lemmas Arg_Res_simps = Z_eq_Arg_plus_def2 Res_ok_def2
    60 
    61 lemma body_Odd [simp]: "body Odd = Some odd"
    62 apply (unfold body_def bodies_def)
    63 apply auto
    64 done
    65 
    66 lemma body_Even [simp]: "body Even = Some evn"
    67 apply (unfold body_def bodies_def)
    68 apply auto
    69 done
    70 
    71 
    72 subsection "verification"
    73 
    74 lemma Odd_lemma: "{{Z=Arg+0}. BODY Even .{Res_ok}}|-{Z=Arg+Suc 0}. odd .{Res_ok}"
    75 apply (unfold odd_def)
    76 apply (rule hoare_derivs.If)
    77 apply (rule hoare_derivs.Ass [THEN conseq1])
    78 apply  (clarsimp simp: Arg_Res_simps)
    79 apply (rule export_s)
    80 apply (rule hoare_derivs.Call [THEN conseq1])
    81 apply  (rule_tac P = "Z=Arg+Suc (Suc 0) " in conseq12)
    82 apply (rule single_asm)
    83 apply (auto simp: Arg_Res_simps)
    84 done
    85 
    86 lemma Even_lemma: "{{Z=Arg+1}. BODY Odd .{Res_ok}}|-{Z=Arg+0}. evn .{Res_ok}"
    87 apply (unfold evn_def)
    88 apply (rule hoare_derivs.If)
    89 apply (rule hoare_derivs.Ass [THEN conseq1])
    90 apply  (clarsimp simp: Arg_Res_simps)
    91 apply (rule hoare_derivs.Comp)
    92 apply (rule_tac [2] hoare_derivs.Ass)
    93 apply clarsimp
    94 apply (rule_tac Q = "%Z s. P Z s & Res_ok Z s" and P = P for P in hoare_derivs.Comp)
    95 apply (rule export_s)
    96 apply  (rule_tac I1 = "%Z l. Z = l Arg & 0 < Z" and Q1 = "Res_ok" in Call_invariant [THEN conseq12])
    97 apply (rule single_asm [THEN conseq2])
    98 apply   (clarsimp simp: Arg_Res_simps)
    99 apply  (force simp: Arg_Res_simps)
   100 apply (rule export_s)
   101 apply (rule_tac I1 = "%Z l. even Z = (l Res = 0) " and Q1 = "%Z s. even Z = (s<Arg> = 0) " in Call_invariant [THEN conseq12])
   102 apply (rule single_asm [THEN conseq2])
   103 apply  (clarsimp simp: Arg_Res_simps)
   104 apply (force simp: Arg_Res_simps)
   105 done
   106 
   107 
   108 lemma Even_ok_N: "{}|-{Z=Arg+0}. BODY Even .{Res_ok}"
   109 apply (rule BodyN)
   110 apply (simp (no_asm))
   111 apply (rule Even_lemma [THEN hoare_derivs.cut])
   112 apply (rule BodyN)
   113 apply (simp (no_asm))
   114 apply (rule Odd_lemma [THEN thin])
   115 apply (simp (no_asm))
   116 done
   117 
   118 lemma Even_ok_S: "{}|-{Z=Arg+0}. BODY Even .{Res_ok}"
   119 apply (rule conseq1)
   120 apply  (rule_tac Procs = "{Odd, Even}" and pn = "Even" and P = "%pn. Z=Arg+ (if pn = Odd then 1 else 0) " and Q = "%pn. Res_ok" in Body1)
   121 apply    auto
   122 apply (rule hoare_derivs.insert)
   123 apply (rule Odd_lemma [THEN thin])
   124 apply  (simp (no_asm))
   125 apply (rule Even_lemma [THEN thin])
   126 apply (simp (no_asm))
   127 done
   128 
   129 end