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