src/HOL/IMPP/EvenOdd.thy
 author bulwahn Fri Oct 21 11:17:14 2011 +0200 (2011-10-21) changeset 45231 d85a2fdc586c parent 41589 bbd861837ebc child 58648 3ccafeb9a1d1 permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
```     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 header {* 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 Misc
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```     9 begin
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```    10
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```    11 definition
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```    12   even :: "nat => bool" where
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```    13   "even n = (2 dvd n)"
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```    14
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```    15 axiomatization
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```    16   Even :: pname and
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```    17   Odd :: pname
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```    18 where
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```    19   Even_neq_Odd: "Even ~= Odd" and
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```    20   Arg_neq_Res:  "Arg  ~= Res"
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```    21
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```    22 definition
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```    23   evn :: com where
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```    24  "evn = (IF (%s. s<Arg> = 0)
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```    25          THEN Loc Res:==(%s. 0)
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```    26          ELSE(Loc Res:=CALL Odd(%s. s<Arg> - 1);;
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```    27               Loc Arg:=CALL Odd(%s. s<Arg> - 1);;
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```    28               Loc Res:==(%s. s<Res> * s<Arg>)))"
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```    29
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```    30 definition
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```    31   odd :: com where
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```    32  "odd = (IF (%s. s<Arg> = 0)
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```    33          THEN Loc Res:==(%s. 1)
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```    34          ELSE(Loc Res:=CALL Even (%s. s<Arg> - 1)))"
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```    35
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```    36 defs
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```    37   bodies_def: "bodies == [(Even,evn),(Odd,odd)]"
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```    38
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```    39 definition
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```    40   Z_eq_Arg_plus :: "nat => nat assn" ("Z=Arg+_" [50]50) where
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```    41   "Z=Arg+n = (%Z s.      Z =  s<Arg>+n)"
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```    42
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```    43 definition
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```    44   Res_ok :: "nat assn" where
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```    45   "Res_ok = (%Z s. even Z = (s<Res> = 0))"
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```    46
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```    47
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```    48 subsection "even"
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```    49
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```    50 lemma even_0 [simp]: "even 0"
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```    51 apply (unfold even_def)
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```    52 apply simp
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```    53 done
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```    54
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```    55 lemma not_even_1 [simp]: "even (Suc 0) = False"
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```    56 apply (unfold even_def)
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```    57 apply simp
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```    58 done
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```    59
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```    60 lemma even_step [simp]: "even (Suc (Suc n)) = even n"
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```    61 apply (unfold even_def)
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```    62 apply (subgoal_tac "Suc (Suc n) = n+2")
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```    63 prefer 2
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```    64 apply  simp
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```    65 apply (erule ssubst)
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```    66 apply (rule dvd_reduce)
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```    67 done
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```    68
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```    69
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```    70 subsection "Arg, Res"
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```    71
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```    72 declare Arg_neq_Res [simp] Arg_neq_Res [THEN not_sym, simp]
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```    73 declare Even_neq_Odd [simp] Even_neq_Odd [THEN not_sym, simp]
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```    74
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```    75 lemma Z_eq_Arg_plus_def2: "(Z=Arg+n) Z s = (Z = s<Arg>+n)"
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```    76 apply (unfold Z_eq_Arg_plus_def)
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```    77 apply (rule refl)
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```    78 done
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```    79
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```    80 lemma Res_ok_def2: "Res_ok Z s = (even Z = (s<Res> = 0))"
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```    81 apply (unfold Res_ok_def)
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```    82 apply (rule refl)
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```    83 done
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```    84
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```    85 lemmas Arg_Res_simps = Z_eq_Arg_plus_def2 Res_ok_def2
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```    86
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```    87 lemma body_Odd [simp]: "body Odd = Some odd"
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```    88 apply (unfold body_def bodies_def)
```
```    89 apply auto
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```    90 done
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```    91
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```    92 lemma body_Even [simp]: "body Even = Some evn"
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```    93 apply (unfold body_def bodies_def)
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```    94 apply auto
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```    95 done
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```    96
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```    97
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```    98 subsection "verification"
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```    99
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```   100 lemma Odd_lemma: "{{Z=Arg+0}. BODY Even .{Res_ok}}|-{Z=Arg+Suc 0}. odd .{Res_ok}"
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```   101 apply (unfold odd_def)
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```   102 apply (rule hoare_derivs.If)
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```   103 apply (rule hoare_derivs.Ass [THEN conseq1])
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```   104 apply  (clarsimp simp: Arg_Res_simps)
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```   105 apply (rule export_s)
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```   106 apply (rule hoare_derivs.Call [THEN conseq1])
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```   107 apply  (rule_tac P = "Z=Arg+Suc (Suc 0) " in conseq12)
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```   108 apply (rule single_asm)
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```   109 apply (auto simp: Arg_Res_simps)
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```   110 done
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```   111
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```   112 lemma Even_lemma: "{{Z=Arg+1}. BODY Odd .{Res_ok}}|-{Z=Arg+0}. evn .{Res_ok}"
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```   113 apply (unfold evn_def)
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```   114 apply (rule hoare_derivs.If)
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```   115 apply (rule hoare_derivs.Ass [THEN conseq1])
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```   116 apply  (clarsimp simp: Arg_Res_simps)
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```   117 apply (rule hoare_derivs.Comp)
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```   118 apply (rule_tac [2] hoare_derivs.Ass)
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```   119 apply clarsimp
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```   120 apply (rule_tac Q = "%Z s. ?P Z s & Res_ok Z s" in hoare_derivs.Comp)
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```   121 apply (rule export_s)
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```   122 apply  (rule_tac I1 = "%Z l. Z = l Arg & 0 < Z" and Q1 = "Res_ok" in Call_invariant [THEN conseq12])
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```   123 apply (rule single_asm [THEN conseq2])
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```   124 apply   (clarsimp simp: Arg_Res_simps)
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```   125 apply  (force simp: Arg_Res_simps)
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```   126 apply (rule export_s)
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```   127 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])
```
```   128 apply (rule single_asm [THEN conseq2])
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```   129 apply  (clarsimp simp: Arg_Res_simps)
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```   130 apply (force simp: Arg_Res_simps)
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```   131 done
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```   132
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```   133
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```   134 lemma Even_ok_N: "{}|-{Z=Arg+0}. BODY Even .{Res_ok}"
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```   135 apply (rule BodyN)
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```   136 apply (simp (no_asm))
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```   137 apply (rule Even_lemma [THEN hoare_derivs.cut])
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```   138 apply (rule BodyN)
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```   139 apply (simp (no_asm))
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```   140 apply (rule Odd_lemma [THEN thin])
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```   141 apply (simp (no_asm))
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```   142 done
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```   143
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```   144 lemma Even_ok_S: "{}|-{Z=Arg+0}. BODY Even .{Res_ok}"
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```   145 apply (rule conseq1)
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```   146 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)
```
```   147 apply    auto
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```   148 apply (rule hoare_derivs.insert)
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```   149 apply (rule Odd_lemma [THEN thin])
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```   150 apply  (simp (no_asm))
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```   151 apply (rule Even_lemma [THEN thin])
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```   152 apply (simp (no_asm))
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```   153 done
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```   154
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```   155 end
```