src/HOL/Bali/Trans.thy
author haftmann
Tue Oct 07 16:07:50 2008 +0200 (2008-10-07)
changeset 28524 644b62cf678f
parent 23747 b07cff284683
child 30952 7ab2716dd93b
permissions -rw-r--r--
arbitrary is undefined
wenzelm@12857
     1
(*  Title:      HOL/Bali/Trans.thy
schirmer@12854
     2
    ID:         $Id$
schirmer@13688
     3
    Author:     David von Oheimb and Norbert Schirmer
schirmer@12854
     4
schirmer@12854
     5
Operational transition (small-step) semantics of the 
schirmer@12854
     6
execution of Java expressions and statements
schirmer@12854
     7
schirmer@12854
     8
PRELIMINARY!!!!!!!!
schirmer@12854
     9
*)
schirmer@12854
    10
haftmann@16417
    11
theory Trans imports Evaln begin
schirmer@13337
    12
schirmer@13337
    13
constdefs groundVar:: "var \<Rightarrow> bool"
schirmer@13337
    14
"groundVar v \<equiv> (case v of
schirmer@13337
    15
                   LVar ln \<Rightarrow> True
schirmer@13337
    16
                 | {accC,statDeclC,stat}e..fn \<Rightarrow> \<exists> a. e=Lit a
schirmer@13337
    17
                 | e1.[e2] \<Rightarrow> \<exists> a i. e1= Lit a \<and> e2 = Lit i
schirmer@13337
    18
                 | InsInitV c v \<Rightarrow> False)"
schirmer@13337
    19
schirmer@13337
    20
lemma groundVar_cases [consumes 1, case_names LVar FVar AVar]:
schirmer@13337
    21
  assumes ground: "groundVar v" and
schirmer@13337
    22
          LVar: "\<And> ln. \<lbrakk>v=LVar ln\<rbrakk> \<Longrightarrow> P" and
schirmer@13337
    23
          FVar: "\<And> accC statDeclC stat a fn. 
schirmer@13337
    24
                    \<lbrakk>v={accC,statDeclC,stat}(Lit a)..fn\<rbrakk> \<Longrightarrow> P" and
schirmer@13337
    25
          AVar: "\<And> a i. \<lbrakk>v=(Lit a).[Lit i]\<rbrakk> \<Longrightarrow> P"
schirmer@13337
    26
  shows "P"
schirmer@13337
    27
proof -
schirmer@13337
    28
  from ground LVar FVar AVar
schirmer@13337
    29
  show ?thesis
schirmer@13337
    30
    apply (cases v)
schirmer@13337
    31
    apply (simp add: groundVar_def)
schirmer@13337
    32
    apply (simp add: groundVar_def,blast)
schirmer@13337
    33
    apply (simp add: groundVar_def,blast)
schirmer@13337
    34
    apply (simp add: groundVar_def)
schirmer@13337
    35
    done
schirmer@13337
    36
qed
schirmer@13337
    37
schirmer@13337
    38
constdefs groundExprs:: "expr list \<Rightarrow> bool"
schirmer@13337
    39
"groundExprs es \<equiv> list_all (\<lambda> e. \<exists> v. e=Lit v) es"
schirmer@13337
    40
  
schirmer@13337
    41
consts the_val:: "expr \<Rightarrow> val"
schirmer@13337
    42
primrec
schirmer@13337
    43
"the_val (Lit v) = v"
schirmer@13337
    44
schirmer@13337
    45
consts the_var:: "prog \<Rightarrow> state \<Rightarrow> var \<Rightarrow> (vvar \<times> state)"
schirmer@13337
    46
primrec
schirmer@13337
    47
"the_var G s (LVar ln)                    =(lvar ln (store s),s)"
schirmer@13337
    48
the_var_FVar_def:
schirmer@13337
    49
"the_var G s ({accC,statDeclC,stat}a..fn) =fvar statDeclC stat fn (the_val a) s"
schirmer@13337
    50
the_var_AVar_def:
schirmer@13337
    51
"the_var G s(a.[i])                       =avar G (the_val i) (the_val a) s"
schirmer@13337
    52
schirmer@13337
    53
lemma the_var_FVar_simp[simp]:
schirmer@13337
    54
"the_var G s ({accC,statDeclC,stat}(Lit a)..fn) = fvar statDeclC stat fn a s"
schirmer@13337
    55
by (simp)
schirmer@13337
    56
declare the_var_FVar_def [simp del]
schirmer@13337
    57
schirmer@13337
    58
lemma the_var_AVar_simp:
schirmer@13337
    59
"the_var G s ((Lit a).[Lit i]) = avar G i a s"
schirmer@13337
    60
by (simp)
schirmer@13337
    61
declare the_var_AVar_def [simp del]
schirmer@12854
    62
berghofe@21765
    63
syntax (xsymbols)
schirmer@13337
    64
  Ref  :: "loc \<Rightarrow> expr"
schirmer@13337
    65
  SKIP :: "expr"
schirmer@12854
    66
schirmer@12854
    67
translations
schirmer@12854
    68
  "Ref a" == "Lit (Addr a)"
schirmer@13337
    69
  "SKIP"  == "Lit Unit"
schirmer@12854
    70
berghofe@23747
    71
inductive
berghofe@21765
    72
  step :: "[prog,term \<times> state,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>1 _"[61,82,82] 81)
berghofe@21765
    73
  for G :: prog
berghofe@21765
    74
where
schirmer@12854
    75
schirmer@12854
    76
(* evaluation of expression *)
schirmer@12854
    77
  (* cf. 15.5 *)
berghofe@21765
    78
  Abrupt:       "\<lbrakk>\<forall>v. t \<noteq> \<langle>Lit v\<rangle>;
schirmer@13337
    79
                  \<forall> t. t \<noteq> \<langle>l\<bullet> Skip\<rangle>;
schirmer@13337
    80
                  \<forall> C vn c.  t \<noteq> \<langle>Try Skip Catch(C vn) c\<rangle>;
schirmer@13337
    81
                  \<forall> x c. t \<noteq> \<langle>Skip Finally c\<rangle> \<and> xc \<noteq> Xcpt x;
schirmer@13337
    82
                  \<forall> a c. t \<noteq> \<langle>FinA a c\<rangle>\<rbrakk> 
schirmer@13337
    83
                \<Longrightarrow> 
haftmann@28524
    84
                  G\<turnstile>(t,Some xc,s) \<mapsto>1 (\<langle>Lit undefined\<rangle>,Some xc,s)"
schirmer@12854
    85
berghofe@21765
    86
| InsInitE: "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,Norm s) \<mapsto>1 (\<langle>c'\<rangle>, s')\<rbrakk>
berghofe@21765
    87
             \<Longrightarrow> 
berghofe@21765
    88
             G\<turnstile>(\<langle>InsInitE c e\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE c' e\<rangle>, s')"
schirmer@12854
    89
schirmer@13337
    90
(* SeqE: "G\<turnstile>(\<langle>Seq Skip e\<rangle>,Norm s) \<mapsto>1 (\<langle>e\<rangle>, Norm s)" *)
schirmer@13337
    91
(* Specialised rules to evaluate: 
schirmer@13337
    92
   InsInitE Skip (NewC C), InisInitE Skip (NewA T[e]) *)
schirmer@13337
    93
 
schirmer@13337
    94
  (* cf. 15.8.1 *)
berghofe@21765
    95
| NewC: "G\<turnstile>(\<langle>NewC C\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE (Init C) (NewC C)\<rangle>, Norm s)"
berghofe@21765
    96
| NewCInited: "\<lbrakk>G\<turnstile> Norm s \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s'\<rbrakk> 
berghofe@21765
    97
               \<Longrightarrow> 
berghofe@21765
    98
               G\<turnstile>(\<langle>InsInitE Skip (NewC C)\<rangle>,Norm s) \<mapsto>1 (\<langle>Ref a\<rangle>, s')"
schirmer@12854
    99
schirmer@12854
   100
schirmer@12854
   101
schirmer@13337
   102
(* Alternative when rule SeqE is present 
schirmer@13337
   103
NewCInited: "\<lbrakk>inited C (globs s); 
schirmer@13337
   104
              G\<turnstile> Norm s \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s'\<rbrakk> 
schirmer@13337
   105
             \<Longrightarrow> 
schirmer@13337
   106
              G\<turnstile>(\<langle>NewC C\<rangle>,Norm s) \<mapsto>1 (\<langle>Ref a\<rangle>, s')"
schirmer@13337
   107
schirmer@13337
   108
NewC:
schirmer@13337
   109
     "\<lbrakk>\<not> inited C (globs s)\<rbrakk> 
schirmer@13337
   110
     \<Longrightarrow> 
schirmer@13337
   111
      G\<turnstile>(\<langle>NewC C\<rangle>,Norm s) \<mapsto>1 (\<langle>Seq (Init C) (NewC C)\<rangle>, Norm s)"
schirmer@13337
   112
schirmer@13337
   113
*)
schirmer@13337
   114
  (* cf. 15.9.1 *)
berghofe@21765
   115
| NewA: 
schirmer@13337
   116
   "G\<turnstile>(\<langle>New T[e]\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE (init_comp_ty T) (New T[e])\<rangle>,Norm s)"
berghofe@21765
   117
| InsInitNewAIdx: 
schirmer@13337
   118
   "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>, s')\<rbrakk>
schirmer@13337
   119
    \<Longrightarrow>  
schirmer@13337
   120
    G\<turnstile>(\<langle>InsInitE Skip (New T[e])\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE Skip (New T[e'])\<rangle>,s')"
berghofe@21765
   121
| InsInitNewA: 
schirmer@13337
   122
   "\<lbrakk>G\<turnstile>abupd (check_neg i) (Norm s) \<midarrow>halloc (Arr T (the_Intg i))\<succ>a\<rightarrow> s' \<rbrakk>
schirmer@13337
   123
    \<Longrightarrow>
schirmer@13337
   124
    G\<turnstile>(\<langle>InsInitE Skip (New T[Lit i])\<rangle>,Norm s) \<mapsto>1 (\<langle>Ref a\<rangle>,s')"
schirmer@13337
   125
 
schirmer@13337
   126
  (* cf. 15.15 *)
berghofe@21765
   127
| CastE:	
schirmer@13337
   128
   "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
schirmer@13337
   129
    \<Longrightarrow>
schirmer@13337
   130
    G\<turnstile>(\<langle>Cast T e\<rangle>,None,s) \<mapsto>1 (\<langle>Cast T e'\<rangle>,s')" 
berghofe@21765
   131
| Cast:	
schirmer@13337
   132
   "\<lbrakk>s' = abupd (raise_if (\<not>G,s\<turnstile>v fits T)  ClassCast) (Norm s)\<rbrakk> 
schirmer@13337
   133
    \<Longrightarrow> 
schirmer@13337
   134
    G\<turnstile>(\<langle>Cast T (Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
schirmer@13337
   135
  (* can be written without abupd, since we know Norm s *)
schirmer@12854
   136
schirmer@12854
   137
berghofe@21765
   138
| InstE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'::expr\<rangle>,s')\<rbrakk> 
schirmer@13337
   139
        \<Longrightarrow> 
schirmer@13337
   140
        G\<turnstile>(\<langle>e InstOf T\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')" 
berghofe@21765
   141
| Inst:  "\<lbrakk>b = (v\<noteq>Null \<and> G,s\<turnstile>v fits RefT T)\<rbrakk> 
berghofe@21765
   142
          \<Longrightarrow> 
berghofe@21765
   143
          G\<turnstile>(\<langle>(Lit v) InstOf T\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (Bool b)\<rangle>,s')"
schirmer@13337
   144
schirmer@13337
   145
  (* cf. 15.7.1 *)
schirmer@13337
   146
(*Lit				"G\<turnstile>(Lit v,None,s) \<mapsto>1 (Lit v,None,s)"*)
schirmer@13337
   147
berghofe@21765
   148
| UnOpE:  "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s') \<rbrakk>
schirmer@13337
   149
           \<Longrightarrow> 
berghofe@21765
   150
           G\<turnstile>(\<langle>UnOp unop e\<rangle>,Norm s) \<mapsto>1 (\<langle>UnOp unop e'\<rangle>,s')"
berghofe@21765
   151
| UnOp:   "G\<turnstile>(\<langle>UnOp unop (Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (eval_unop unop v)\<rangle>,Norm s)"
berghofe@21765
   152
berghofe@21765
   153
| BinOpE1:  "\<lbrakk>G\<turnstile>(\<langle>e1\<rangle>,Norm s) \<mapsto>1 (\<langle>e1'\<rangle>,s') \<rbrakk>
berghofe@21765
   154
             \<Longrightarrow> 
berghofe@21765
   155
             G\<turnstile>(\<langle>BinOp binop e1 e2\<rangle>,Norm s) \<mapsto>1 (\<langle>BinOp binop e1' e2\<rangle>,s')"
berghofe@21765
   156
| BinOpE2:  "\<lbrakk>need_second_arg binop v1; G\<turnstile>(\<langle>e2\<rangle>,Norm s) \<mapsto>1 (\<langle>e2'\<rangle>,s') \<rbrakk>
schirmer@13384
   157
             \<Longrightarrow> 
schirmer@13384
   158
             G\<turnstile>(\<langle>BinOp binop (Lit v1) e2\<rangle>,Norm s) 
berghofe@21765
   159
              \<mapsto>1 (\<langle>BinOp binop (Lit v1) e2'\<rangle>,s')"
berghofe@21765
   160
| BinOpTerm:  "\<lbrakk>\<not> need_second_arg binop v1\<rbrakk>
berghofe@21765
   161
               \<Longrightarrow> 
berghofe@21765
   162
               G\<turnstile>(\<langle>BinOp binop (Lit v1) e2\<rangle>,Norm s) 
berghofe@21765
   163
                \<mapsto>1 (\<langle>Lit v1\<rangle>,Norm s)"
berghofe@21765
   164
| BinOp:    "G\<turnstile>(\<langle>BinOp binop (Lit v1) (Lit v2)\<rangle>,Norm s) 
berghofe@21765
   165
              \<mapsto>1 (\<langle>Lit (eval_binop binop v1 v2)\<rangle>,Norm s)"
schirmer@13384
   166
(* Maybe its more convenient to add: need_second_arg as precondition to BinOp 
schirmer@13384
   167
   to make the choice between BinOpTerm and BinOp deterministic *)
schirmer@13384
   168
   
berghofe@21765
   169
| Super: "G\<turnstile>(\<langle>Super\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (val_this s)\<rangle>,Norm s)"
schirmer@13337
   170
berghofe@21765
   171
| AccVA: "\<lbrakk>G\<turnstile>(\<langle>va\<rangle>,Norm s) \<mapsto>1 (\<langle>va'\<rangle>,s') \<rbrakk>
berghofe@21765
   172
          \<Longrightarrow> 
berghofe@21765
   173
          G\<turnstile>(\<langle>Acc va\<rangle>,Norm s) \<mapsto>1 (\<langle>Acc va'\<rangle>,s')"
berghofe@21765
   174
| Acc:  "\<lbrakk>groundVar va; ((v,vf),s') = the_var G (Norm s) va\<rbrakk>
berghofe@21765
   175
         \<Longrightarrow>  
berghofe@21765
   176
         G\<turnstile>(\<langle>Acc va\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
schirmer@13337
   177
schirmer@13337
   178
(*
schirmer@13337
   179
AccLVar: "G\<turnstile>(\<langle>Acc (LVar vn)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (fst (lvar vn s))\<rangle>,Norm s)"
schirmer@13337
   180
AccFVar: "\<lbrakk>((v,vf),s') = fvar statDeclC stat fn a (Norm s)\<rbrakk>
schirmer@13337
   181
          \<Longrightarrow>  
schirmer@13337
   182
          G\<turnstile>(\<langle>Acc ({accC,statDeclC,stat}(Lit a)..fn)\<rangle>,Norm s) 
schirmer@13337
   183
           \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
schirmer@13337
   184
AccAVar: "\<lbrakk>((v,vf),s') = avar G i a (Norm s)\<rbrakk>
schirmer@13337
   185
          \<Longrightarrow>  
schirmer@13337
   186
          G\<turnstile>(\<langle>Acc ((Lit a).[Lit i])\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
schirmer@13337
   187
*) 
berghofe@21765
   188
| AssVA:  "\<lbrakk>G\<turnstile>(\<langle>va\<rangle>,Norm s) \<mapsto>1 (\<langle>va'\<rangle>,s')\<rbrakk> 
berghofe@21765
   189
           \<Longrightarrow> 
berghofe@21765
   190
           G\<turnstile>(\<langle>va:=e\<rangle>,Norm s) \<mapsto>1 (\<langle>va':=e\<rangle>,s')"
berghofe@21765
   191
| AssE:   "\<lbrakk>groundVar va; G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
berghofe@21765
   192
           \<Longrightarrow> 
berghofe@21765
   193
           G\<turnstile>(\<langle>va:=e\<rangle>,Norm s) \<mapsto>1 (\<langle>va:=e'\<rangle>,s')"
berghofe@21765
   194
| Ass:    "\<lbrakk>groundVar va; ((w,f),s') = the_var G (Norm s) va\<rbrakk> 
berghofe@21765
   195
           \<Longrightarrow> 
berghofe@21765
   196
           G\<turnstile>(\<langle>va:=(Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,assign f v s')"
schirmer@13337
   197
berghofe@21765
   198
| CondC: "\<lbrakk>G\<turnstile>(\<langle>e0\<rangle>,Norm s) \<mapsto>1 (\<langle>e0'\<rangle>,s')\<rbrakk> 
berghofe@21765
   199
          \<Longrightarrow> 
berghofe@21765
   200
          G\<turnstile>(\<langle>e0? e1:e2\<rangle>,Norm s) \<mapsto>1 (\<langle>e0'? e1:e2\<rangle>,s')"
berghofe@21765
   201
| Cond:  "G\<turnstile>(\<langle>Lit b? e1:e2\<rangle>,Norm s) \<mapsto>1 (\<langle>if the_Bool b then e1 else e2\<rangle>,Norm s)"
schirmer@13337
   202
schirmer@13337
   203
berghofe@21765
   204
| CallTarget: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
berghofe@21765
   205
               \<Longrightarrow>
berghofe@21765
   206
               G\<turnstile>(\<langle>{accC,statT,mode}e\<cdot>mn({pTs}args)\<rangle>,Norm s) 
berghofe@21765
   207
                \<mapsto>1 (\<langle>{accC,statT,mode}e'\<cdot>mn({pTs}args)\<rangle>,s')"
berghofe@21765
   208
| CallArgs:   "\<lbrakk>G\<turnstile>(\<langle>args\<rangle>,Norm s) \<mapsto>1 (\<langle>args'\<rangle>,s')\<rbrakk> 
berghofe@21765
   209
               \<Longrightarrow>
berghofe@21765
   210
               G\<turnstile>(\<langle>{accC,statT,mode}Lit a\<cdot>mn({pTs}args)\<rangle>,Norm s) 
berghofe@21765
   211
                \<mapsto>1 (\<langle>{accC,statT,mode}Lit a\<cdot>mn({pTs}args')\<rangle>,s')"
berghofe@21765
   212
| Call:       "\<lbrakk>groundExprs args; vs = map the_val args;
berghofe@21765
   213
                D = invocation_declclass G mode s a statT \<lparr>name=mn,parTs=pTs\<rparr>;
berghofe@21765
   214
                s'=init_lvars G D \<lparr>name=mn,parTs=pTs\<rparr> mode a' vs (Norm s)\<rbrakk> 
berghofe@21765
   215
               \<Longrightarrow> 
berghofe@21765
   216
               G\<turnstile>(\<langle>{accC,statT,mode}Lit a\<cdot>mn({pTs}args)\<rangle>,Norm s) 
berghofe@21765
   217
                \<mapsto>1 (\<langle>Callee (locals s) (Methd D \<lparr>name=mn,parTs=pTs\<rparr>)\<rangle>,s')"
schirmer@13337
   218
           
berghofe@21765
   219
| Callee:     "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'::expr\<rangle>,s')\<rbrakk>
berghofe@21765
   220
               \<Longrightarrow> 
berghofe@21765
   221
               G\<turnstile>(\<langle>Callee lcls_caller e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')"
schirmer@13337
   222
berghofe@21765
   223
| CalleeRet:   "G\<turnstile>(\<langle>Callee lcls_caller (Lit v)\<rangle>,Norm s) 
berghofe@21765
   224
                 \<mapsto>1 (\<langle>Lit v\<rangle>,(set_lvars lcls_caller (Norm s)))"
schirmer@13337
   225
berghofe@21765
   226
| Methd: "G\<turnstile>(\<langle>Methd D sig\<rangle>,Norm s) \<mapsto>1 (\<langle>body G D sig\<rangle>,Norm s)"
schirmer@13337
   227
berghofe@21765
   228
| Body: "G\<turnstile>(\<langle>Body D c\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE (Init D) (Body D c)\<rangle>,Norm s)"
schirmer@13337
   229
berghofe@21765
   230
| InsInitBody: 
schirmer@13337
   231
    "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,Norm s) \<mapsto>1 (\<langle>c'\<rangle>,s')\<rbrakk>
schirmer@13337
   232
     \<Longrightarrow> 
schirmer@13337
   233
     G\<turnstile>(\<langle>InsInitE Skip (Body D c)\<rangle>,Norm s) \<mapsto>1(\<langle>InsInitE Skip (Body D c')\<rangle>,s')"
berghofe@21765
   234
| InsInitBodyRet: 
schirmer@13337
   235
     "G\<turnstile>(\<langle>InsInitE Skip (Body D Skip)\<rangle>,Norm s)
schirmer@13337
   236
       \<mapsto>1 (\<langle>Lit (the ((locals s) Result))\<rangle>,abupd (absorb Ret) (Norm s))"
schirmer@13337
   237
schirmer@13337
   238
(*   LVar: "G\<turnstile>(LVar vn,Norm s)" is already evaluated *)
schirmer@13337
   239
  
berghofe@21765
   240
| FVar: "\<lbrakk>\<not> inited statDeclC (globs s)\<rbrakk>
berghofe@21765
   241
         \<Longrightarrow> 
berghofe@21765
   242
         G\<turnstile>(\<langle>{accC,statDeclC,stat}e..fn\<rangle>,Norm s) 
berghofe@21765
   243
          \<mapsto>1 (\<langle>InsInitV (Init statDeclC) ({accC,statDeclC,stat}e..fn)\<rangle>,Norm s)"
berghofe@21765
   244
| InsInitFVarE:
schirmer@13337
   245
      "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk>
schirmer@13337
   246
       \<Longrightarrow>
schirmer@13337
   247
       G\<turnstile>(\<langle>InsInitV Skip ({accC,statDeclC,stat}e..fn)\<rangle>,Norm s) 
schirmer@13337
   248
        \<mapsto>1 (\<langle>InsInitV Skip ({accC,statDeclC,stat}e'..fn)\<rangle>,s')"
berghofe@21765
   249
| InsInitFVar:
schirmer@13337
   250
      "G\<turnstile>(\<langle>InsInitV Skip ({accC,statDeclC,stat}Lit a..fn)\<rangle>,Norm s) 
schirmer@13337
   251
        \<mapsto>1 (\<langle>{accC,statDeclC,stat}Lit a..fn\<rangle>,Norm s)"
schirmer@13337
   252
--  {* Notice, that we do not have literal values for @{text vars}. 
schirmer@13337
   253
The rules for accessing variables (@{text Acc}) and assigning to variables 
schirmer@13337
   254
(@{text Ass}), test this with the predicate @{text groundVar}.  After 
schirmer@13337
   255
initialisation is done and the @{text FVar} is evaluated, we can't just 
schirmer@13337
   256
throw away the @{text InsInitFVar} term and return a literal value, as in the 
schirmer@13337
   257
cases of @{text New}  or @{text NewC}. Instead we just return the evaluated 
schirmer@13337
   258
@{text FVar} and test for initialisation in the rule @{text FVar}. 
schirmer@13337
   259
*}
schirmer@13337
   260
schirmer@13337
   261
berghofe@21765
   262
| AVarE1: "\<lbrakk>G\<turnstile>(\<langle>e1\<rangle>,Norm s) \<mapsto>1 (\<langle>e1'\<rangle>,s')\<rbrakk> 
berghofe@21765
   263
           \<Longrightarrow> 
berghofe@21765
   264
           G\<turnstile>(\<langle>e1.[e2]\<rangle>,Norm s) \<mapsto>1 (\<langle>e1'.[e2]\<rangle>,s')"
schirmer@13337
   265
berghofe@21765
   266
| AVarE2: "G\<turnstile>(\<langle>e2\<rangle>,Norm s) \<mapsto>1 (\<langle>e2'\<rangle>,s') 
berghofe@21765
   267
           \<Longrightarrow> 
berghofe@21765
   268
           G\<turnstile>(\<langle>Lit a.[e2]\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit a.[e2']\<rangle>,s')"
schirmer@13337
   269
schirmer@13337
   270
(* AVar: \<langle>(Lit a).[Lit i]\<rangle> is fully evaluated *)
schirmer@13337
   271
schirmer@13337
   272
(* evaluation of expression lists *)
schirmer@13337
   273
schirmer@13337
   274
  -- {* @{text Nil}  is fully evaluated *}
schirmer@13337
   275
berghofe@21765
   276
| ConsHd: "\<lbrakk>G\<turnstile>(\<langle>e::expr\<rangle>,Norm s) \<mapsto>1 (\<langle>e'::expr\<rangle>,s')\<rbrakk> 
berghofe@21765
   277
           \<Longrightarrow>
berghofe@21765
   278
           G\<turnstile>(\<langle>e#es\<rangle>,Norm s) \<mapsto>1 (\<langle>e'#es\<rangle>,s')"
schirmer@13337
   279
  
berghofe@21765
   280
| ConsTl: "\<lbrakk>G\<turnstile>(\<langle>es\<rangle>,Norm s) \<mapsto>1 (\<langle>es'\<rangle>,s')\<rbrakk> 
berghofe@21765
   281
           \<Longrightarrow>
berghofe@21765
   282
           G\<turnstile>(\<langle>(Lit v)#es\<rangle>,Norm s) \<mapsto>1 (\<langle>(Lit v)#es'\<rangle>,s')"
schirmer@12854
   283
schirmer@12854
   284
(* execution of statements *)
schirmer@12854
   285
schirmer@12854
   286
  (* cf. 14.5 *)
berghofe@21765
   287
| Skip: "G\<turnstile>(\<langle>Skip\<rangle>,Norm s) \<mapsto>1 (\<langle>SKIP\<rangle>,Norm s)"
schirmer@12854
   288
berghofe@21765
   289
| ExprE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
berghofe@21765
   290
          \<Longrightarrow> 
berghofe@21765
   291
          G\<turnstile>(\<langle>Expr e\<rangle>,Norm s) \<mapsto>1 (\<langle>Expr e'\<rangle>,s')"
berghofe@21765
   292
| Expr:  "G\<turnstile>(\<langle>Expr (Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,Norm s)"
schirmer@12854
   293
schirmer@12854
   294
berghofe@21765
   295
| LabC: "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,Norm s) \<mapsto>1 (\<langle>c'\<rangle>,s')\<rbrakk> 
berghofe@21765
   296
         \<Longrightarrow>  
berghofe@21765
   297
         G\<turnstile>(\<langle>l\<bullet> c\<rangle>,Norm s) \<mapsto>1 (\<langle>l\<bullet> c'\<rangle>,s')"
berghofe@21765
   298
| Lab:  "G\<turnstile>(\<langle>l\<bullet> Skip\<rangle>,s) \<mapsto>1 (\<langle>Skip\<rangle>, abupd (absorb l) s)"
schirmer@13337
   299
schirmer@13337
   300
  (* cf. 14.2 *)
berghofe@21765
   301
| CompC1: "\<lbrakk>G\<turnstile>(\<langle>c1\<rangle>,Norm s) \<mapsto>1 (\<langle>c1'\<rangle>,s')\<rbrakk> 
berghofe@21765
   302
           \<Longrightarrow> 
berghofe@21765
   303
           G\<turnstile>(\<langle>c1;; c2\<rangle>,Norm s) \<mapsto>1 (\<langle>c1';; c2\<rangle>,s')"
schirmer@13337
   304
berghofe@21765
   305
| Comp:   "G\<turnstile>(\<langle>Skip;; c2\<rangle>,Norm s) \<mapsto>1 (\<langle>c2\<rangle>,Norm s)"
schirmer@12854
   306
schirmer@12854
   307
  (* cf. 14.8.2 *)
berghofe@21765
   308
| IfE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle> ,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
berghofe@21765
   309
        \<Longrightarrow>
berghofe@21765
   310
        G\<turnstile>(\<langle>If(e) s1 Else s2\<rangle>,Norm s) \<mapsto>1 (\<langle>If(e') s1 Else s2\<rangle>,s')"
berghofe@21765
   311
| If:  "G\<turnstile>(\<langle>If(Lit v) s1 Else s2\<rangle>,Norm s) 
berghofe@21765
   312
         \<mapsto>1 (\<langle>if the_Bool v then s1 else s2\<rangle>,Norm s)"
schirmer@12854
   313
schirmer@12854
   314
  (* cf. 14.10, 14.10.1 *)
berghofe@21765
   315
| Loop: "G\<turnstile>(\<langle>l\<bullet> While(e) c\<rangle>,Norm s) 
berghofe@21765
   316
          \<mapsto>1 (\<langle>If(e) (Cont l\<bullet>c;; l\<bullet> While(e) c) Else Skip\<rangle>,Norm s)"
schirmer@13337
   317
berghofe@21765
   318
| Jmp: "G\<turnstile>(\<langle>Jmp j\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,(Some (Jump j), s))"
schirmer@13337
   319
berghofe@21765
   320
| ThrowE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
berghofe@21765
   321
           \<Longrightarrow>
berghofe@21765
   322
           G\<turnstile>(\<langle>Throw e\<rangle>,Norm s) \<mapsto>1 (\<langle>Throw e'\<rangle>,s')"
berghofe@21765
   323
| Throw:  "G\<turnstile>(\<langle>Throw (Lit a)\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,abupd (throw a) (Norm s))"
schirmer@13337
   324
berghofe@21765
   325
| TryC1: "\<lbrakk>G\<turnstile>(\<langle>c1\<rangle>,Norm s) \<mapsto>1 (\<langle>c1'\<rangle>,s')\<rbrakk> 
berghofe@21765
   326
          \<Longrightarrow>
berghofe@21765
   327
          G\<turnstile>(\<langle>Try c1 Catch(C vn) c2\<rangle>, Norm s) \<mapsto>1 (\<langle>Try c1' Catch(C vn) c2\<rangle>,s')"
berghofe@21765
   328
| Try:   "\<lbrakk>G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'\<rbrakk>
berghofe@21765
   329
          \<Longrightarrow>
berghofe@21765
   330
          G\<turnstile>(\<langle>Try Skip Catch(C vn) c2\<rangle>, s) 
berghofe@21765
   331
           \<mapsto>1 (if G,s'\<turnstile>catch C then (\<langle>c2\<rangle>,new_xcpt_var vn s')
berghofe@21765
   332
                                else (\<langle>Skip\<rangle>,s'))"
schirmer@13337
   333
berghofe@21765
   334
| FinC1: "\<lbrakk>G\<turnstile>(\<langle>c1\<rangle>,Norm s) \<mapsto>1 (\<langle>c1'\<rangle>,s')\<rbrakk> 
berghofe@21765
   335
          \<Longrightarrow>
berghofe@21765
   336
          G\<turnstile>(\<langle>c1 Finally c2\<rangle>,Norm s) \<mapsto>1 (\<langle>c1' Finally c2\<rangle>,s')"
schirmer@13337
   337
berghofe@21765
   338
| Fin:    "G\<turnstile>(\<langle>Skip Finally c2\<rangle>,(a,s)) \<mapsto>1 (\<langle>FinA a c2\<rangle>,Norm s)"
schirmer@13337
   339
berghofe@21765
   340
| FinAC: "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,s) \<mapsto>1 (\<langle>c'\<rangle>,s')\<rbrakk>
berghofe@21765
   341
          \<Longrightarrow>
berghofe@21765
   342
          G\<turnstile>(\<langle>FinA a c\<rangle>,s) \<mapsto>1 (\<langle>FinA a c'\<rangle>,s')"
berghofe@21765
   343
| FinA: "G\<turnstile>(\<langle>FinA a Skip\<rangle>,s) \<mapsto>1 (\<langle>Skip\<rangle>,abupd (abrupt_if (a\<noteq>None) a) s)"
schirmer@13337
   344
schirmer@13337
   345
berghofe@21765
   346
| Init1: "\<lbrakk>inited C (globs s)\<rbrakk> 
berghofe@21765
   347
          \<Longrightarrow> 
berghofe@21765
   348
          G\<turnstile>(\<langle>Init C\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,Norm s)"
berghofe@21765
   349
| Init: "\<lbrakk>the (class G C)=c; \<not> inited C (globs s)\<rbrakk>  
berghofe@21765
   350
         \<Longrightarrow> 
berghofe@21765
   351
         G\<turnstile>(\<langle>Init C\<rangle>,Norm s) 
berghofe@21765
   352
          \<mapsto>1 (\<langle>(if C = Object then Skip else (Init (super c)));;
berghofe@21765
   353
                Expr (Callee (locals s) (InsInitE (init c) SKIP))\<rangle>
berghofe@21765
   354
               ,Norm (init_class_obj G C s))"
schirmer@13337
   355
-- {* @{text InsInitE} is just used as trick to embed the statement 
schirmer@13337
   356
@{text "init c"} into an expression*} 
berghofe@21765
   357
| InsInitESKIP:
berghofe@21765
   358
    "G\<turnstile>(\<langle>InsInitE Skip SKIP\<rangle>,Norm s) \<mapsto>1 (\<langle>SKIP\<rangle>,Norm s)"
berghofe@21765
   359
berghofe@21765
   360
abbreviation
berghofe@21765
   361
  stepn:: "[prog, term \<times> state,nat,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>_ _"[61,82,82] 81)
berghofe@21765
   362
  where "G\<turnstile>p \<mapsto>n p' \<equiv> (p,p') \<in> {(x, y). step G x y}^n"
berghofe@21765
   363
berghofe@21765
   364
abbreviation
berghofe@21765
   365
  steptr:: "[prog,term \<times> state,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>* _"[61,82,82] 81)
berghofe@21765
   366
  where "G\<turnstile>p \<mapsto>* p' \<equiv> (p,p') \<in> {(x, y). step G x y}\<^sup>*"
schirmer@13337
   367
         
schirmer@13337
   368
(* Equivalenzen:
schirmer@13337
   369
  Bigstep zu Smallstep komplett.
schirmer@13337
   370
  Smallstep zu Bigstep, nur wenn nicht die Ausdrücke Callee, FinA ,\<dots>
schirmer@12925
   371
*)
schirmer@12854
   372
schirmer@13337
   373
lemma rtrancl_imp_rel_pow: "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R^n"
schirmer@13337
   374
proof -
schirmer@13337
   375
  assume "p \<in> R\<^sup>*"
schirmer@13337
   376
  moreover obtain x y where p: "p = (x,y)" by (cases p)
schirmer@13337
   377
  ultimately have "(x,y) \<in> R\<^sup>*" by hypsubst
schirmer@13337
   378
  hence "\<exists>n. (x,y) \<in> R^n"
schirmer@13337
   379
  proof induct
schirmer@13337
   380
    fix a have "(a,a) \<in> R^0" by simp
schirmer@13337
   381
    thus "\<exists>n. (a,a) \<in> R ^ n" ..
schirmer@13337
   382
  next
schirmer@13337
   383
    fix a b c assume "\<exists>n. (a,b) \<in> R ^ n"
schirmer@13337
   384
    then obtain n where "(a,b) \<in> R^n" ..
schirmer@13337
   385
    moreover assume "(b,c) \<in> R"
schirmer@13337
   386
    ultimately have "(a,c) \<in> R^(Suc n)" by auto
schirmer@13337
   387
    thus "\<exists>n. (a,c) \<in> R^n" ..
schirmer@13337
   388
  qed
schirmer@13337
   389
  with p show ?thesis by hypsubst
schirmer@13337
   390
qed  
schirmer@13337
   391
schirmer@13337
   392
(*
schirmer@13337
   393
lemma imp_eval_trans:
schirmer@13337
   394
  assumes eval: "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)" 
schirmer@13337
   395
    shows trans: "G\<turnstile>(t,s0) \<mapsto>* (\<langle>Lit v\<rangle>,s1)"
schirmer@13337
   396
*)
schirmer@13337
   397
(* Jetzt muss man bei trans natürlich wieder unterscheiden: Stmt, Expr, Var!
schirmer@13337
   398
   Sowas blödes:
schirmer@13337
   399
   Am besten den Terminus ground auf Var,Stmt,Expr hochziehen und dann
schirmer@13337
   400
   the_vals definieren\<dots>
schirmer@13337
   401
  G\<turnstile>(t,s0) \<mapsto>* (t',s1) \<and> the_vals t' = v
schirmer@13337
   402
*)
schirmer@13337
   403
schirmer@13337
   404
schirmer@13337
   405
end