src/HOL/Bali/TypeSafe.thy
author wenzelm
Mon Jan 28 18:50:23 2002 +0100 (2002-01-28)
changeset 12857 a4386cc9b1c3
parent 12854 00d4a435777f
child 12858 6214f03d6d27
permissions -rw-r--r--
tuned header;
     1 (*  Title:      HOL/Bali/TypeSafe.thy
     2     ID:         $Id$
     3     Author:     David von Oheimb
     4     Copyright   1997 Technische Universitaet Muenchen
     5 *)
     6 header {* The type soundness proof for Java *}
     7 
     8 
     9 theory TypeSafe = Eval + WellForm + Conform:
    10 
    11 section "result conformance"
    12 
    13 constdefs
    14   assign_conforms :: "st \<Rightarrow> (val \<Rightarrow> state \<Rightarrow> state) \<Rightarrow> ty \<Rightarrow> env_ \<Rightarrow> bool"
    15           ("_\<le>|_\<preceq>_\<Colon>\<preceq>_"                                        [71,71,71,71] 70)
    16  "s\<le>|f\<preceq>T\<Colon>\<preceq>E \<equiv>
    17   \<forall>s' w. Norm s'\<Colon>\<preceq>E \<longrightarrow> fst E,s'\<turnstile>w\<Colon>\<preceq>T \<longrightarrow> s\<le>|s' \<longrightarrow> assign f w (Norm s')\<Colon>\<preceq>E"
    18 
    19   rconf :: "prog \<Rightarrow> lenv \<Rightarrow> st \<Rightarrow> term \<Rightarrow> vals \<Rightarrow> tys \<Rightarrow> bool"
    20           ("_,_,_\<turnstile>_\<succ>_\<Colon>\<preceq>_"                               [71,71,71,71,71,71] 70)
    21   "G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T 
    22     \<equiv> case T of
    23         Inl T  \<Rightarrow> if (\<exists>vf. t=In2 vf)
    24                   then G,s\<turnstile>fst (the_In2 v)\<Colon>\<preceq>T \<and> s\<le>|snd (the_In2 v)\<preceq>T\<Colon>\<preceq>(G,L)
    25                   else G,s\<turnstile>the_In1 v\<Colon>\<preceq>T
    26       | Inr Ts \<Rightarrow> list_all2 (conf G s) (the_In3 v) Ts"
    27 
    28 lemma rconf_In1 [simp]: 
    29  "G,L,s\<turnstile>In1 ec\<succ>In1 v \<Colon>\<preceq>Inl T  =  G,s\<turnstile>v\<Colon>\<preceq>T"
    30 apply (unfold rconf_def)
    31 apply (simp (no_asm))
    32 done
    33 
    34 lemma rconf_In2 [simp]: 
    35  "G,L,s\<turnstile>In2 va\<succ>In2 vf\<Colon>\<preceq>Inl T  = (G,s\<turnstile>fst vf\<Colon>\<preceq>T \<and> s\<le>|snd vf\<preceq>T\<Colon>\<preceq>(G,L))"
    36 apply (unfold rconf_def)
    37 apply (simp (no_asm))
    38 done
    39 
    40 lemma rconf_In3 [simp]: 
    41  "G,L,s\<turnstile>In3 es\<succ>In3 vs\<Colon>\<preceq>Inr Ts = list_all2 (\<lambda>v T. G,s\<turnstile>v\<Colon>\<preceq>T) vs Ts"
    42 apply (unfold rconf_def)
    43 apply (simp (no_asm))
    44 done
    45 
    46 section "fits and conf"
    47 
    48 (* unused *)
    49 lemma conf_fits: "G,s\<turnstile>v\<Colon>\<preceq>T \<Longrightarrow> G,s\<turnstile>v fits T"
    50 apply (unfold fits_def)
    51 apply clarify
    52 apply (erule swap, simp (no_asm_use))
    53 apply (drule conf_RefTD)
    54 apply auto
    55 done
    56 
    57 lemma fits_conf: 
    58   "\<lbrakk>G,s\<turnstile>v\<Colon>\<preceq>T; G\<turnstile>T\<preceq>? T'; G,s\<turnstile>v fits T'; ws_prog G\<rbrakk> \<Longrightarrow> G,s\<turnstile>v\<Colon>\<preceq>T'"
    59 apply (auto dest!: fitsD cast_PrimT2 cast_RefT2)
    60 apply (force dest: conf_RefTD intro: conf_AddrI)
    61 done
    62 
    63 lemma fits_Array: 
    64  "\<lbrakk>G,s\<turnstile>v\<Colon>\<preceq>T; G\<turnstile>T'.[]\<preceq>T.[]; G,s\<turnstile>v fits T'; ws_prog G\<rbrakk> \<Longrightarrow> G,s\<turnstile>v\<Colon>\<preceq>T'"
    65 apply (auto dest!: fitsD widen_ArrayPrimT widen_ArrayRefT)
    66 apply (force dest: conf_RefTD intro: conf_AddrI)
    67 done
    68 
    69 
    70 
    71 section "gext"
    72 
    73 lemma halloc_gext: "\<And>s1 s2. G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2 \<Longrightarrow> snd s1\<le>|snd s2"
    74 apply (simp (no_asm_simp) only: split_tupled_all)
    75 apply (erule halloc.induct)
    76 apply  (auto dest!: new_AddrD)
    77 done
    78 
    79 lemma sxalloc_gext: "\<And>s1 s2. G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2 \<Longrightarrow> snd s1\<le>|snd s2"
    80 apply (simp (no_asm_simp) only: split_tupled_all)
    81 apply (erule sxalloc.induct)
    82 apply   (auto dest!: halloc_gext)
    83 done
    84 
    85 lemma eval_gext_lemma [rule_format (no_asm)]: 
    86  "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s') \<Longrightarrow> snd s\<le>|snd s' \<and> (case w of  
    87     In1 v \<Rightarrow> True  
    88   | In2 vf \<Rightarrow> normal s \<longrightarrow> (\<forall>v x s. s\<le>|snd (assign (snd vf) v (x,s)))  
    89   | In3 vs \<Rightarrow> True)"
    90 apply (erule eval_induct)
    91 prefer 24 
    92   apply (case_tac "inited C (globs s0)", clarsimp, erule thin_rl) (* Init *)
    93 apply (auto del: conjI  dest!: not_initedD gext_new sxalloc_gext halloc_gext
    94  simp  add: lvar_def fvar_def2 avar_def2 init_lvars_def2
    95  split del: split_if_asm split add: sum3.split)
    96 (* 6 subgoals *)
    97 apply force+
    98 done
    99 
   100 lemma evar_gext_f: 
   101   "G\<turnstile>Norm s1 \<midarrow>e=\<succ>vf \<rightarrow> s2 \<Longrightarrow> s\<le>|snd (assign (snd vf) v (x,s))"
   102 apply (drule eval_gext_lemma [THEN conjunct2])
   103 apply auto
   104 done
   105 
   106 lemmas eval_gext = eval_gext_lemma [THEN conjunct1]
   107 
   108 lemma eval_gext': "G\<turnstile>(x1,s1) \<midarrow>t\<succ>\<rightarrow> (w,x2,s2) \<Longrightarrow> s1\<le>|s2"
   109 apply (drule eval_gext)
   110 apply auto
   111 done
   112 
   113 lemma init_yields_initd: "G\<turnstile>Norm s1 \<midarrow>Init C\<rightarrow> s2 \<Longrightarrow> initd C s2"
   114 apply (erule eval_cases , auto split del: split_if_asm)
   115 apply (case_tac "inited C (globs s1)")
   116 apply  (clarsimp split del: split_if_asm)+
   117 apply (drule eval_gext')+
   118 apply (drule init_class_obj_inited)
   119 apply (erule inited_gext)
   120 apply (simp (no_asm_use))
   121 done
   122 
   123 
   124 section "Lemmas"
   125 
   126 lemma obj_ty_obj_class1: 
   127  "\<lbrakk>wf_prog G; is_type G (obj_ty obj)\<rbrakk> \<Longrightarrow> is_class G (obj_class obj)"
   128 apply (case_tac "tag obj")
   129 apply (auto simp add: obj_ty_def obj_class_def)
   130 done
   131 
   132 lemma oconf_init_obj: 
   133  "\<lbrakk>wf_prog G;  
   134  (case r of Heap a \<Rightarrow> is_type G (obj_ty obj) | Stat C \<Rightarrow> is_class G C)
   135 \<rbrakk> \<Longrightarrow> G,s\<turnstile>obj \<lparr>values:=init_vals (var_tys G (tag obj) r)\<rparr>\<Colon>\<preceq>\<surd>r"
   136 apply (auto intro!: oconf_init_obj_lemma unique_fields)
   137 done
   138 
   139 (*
   140 lemma obj_split: "P obj = (\<forall> oi vs. obj = \<lparr>tag=oi,values=vs\<rparr> \<longrightarrow> ?P \<lparr>tag=oi,values=vs\<rparr>)"
   141 apply auto
   142 apply (case_tac "obj")
   143 apply auto
   144 *)
   145 
   146 lemma conforms_newG: "\<lbrakk>globs s oref = None; (x, s)\<Colon>\<preceq>(G,L);   
   147   wf_prog G; case oref of Heap a \<Rightarrow> is_type G (obj_ty \<lparr>tag=oi,values=vs\<rparr>)  
   148                         | Stat C \<Rightarrow> is_class G C\<rbrakk> \<Longrightarrow>  
   149   (x, init_obj G oi oref s)\<Colon>\<preceq>(G, L)"
   150 apply (unfold init_obj_def)
   151 apply (auto elim!: conforms_gupd dest!: oconf_init_obj 
   152             simp add: obj.update_defs)
   153 done
   154 
   155 lemma conforms_init_class_obj: 
   156  "\<lbrakk>(x,s)\<Colon>\<preceq>(G, L); wf_prog G; class G C=Some y; \<not> inited C (globs s)\<rbrakk> \<Longrightarrow> 
   157   (x,init_class_obj G C s)\<Colon>\<preceq>(G, L)"
   158 apply (rule not_initedD [THEN conforms_newG])
   159 apply    (auto)
   160 done
   161 
   162 
   163 lemma fst_init_lvars[simp]: 
   164  "fst (init_lvars G C sig (invmode m e) a' pvs (x,s)) = 
   165   (if static m then x else (np a') x)"
   166 apply (simp (no_asm) add: init_lvars_def2)
   167 done
   168 
   169 
   170 lemma halloc_conforms: "\<And>s1. \<lbrakk>G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2; wf_prog G; s1\<Colon>\<preceq>(G, L); 
   171   is_type G (obj_ty \<lparr>tag=oi,values=fs\<rparr>)\<rbrakk> \<Longrightarrow> s2\<Colon>\<preceq>(G, L)"
   172 apply (simp (no_asm_simp) only: split_tupled_all)
   173 apply (case_tac "aa")
   174 apply  (auto elim!: halloc_elim_cases dest!: new_AddrD 
   175        intro!: conforms_newG [THEN conforms_xconf] conf_AddrI)
   176 done
   177 
   178 lemma halloc_type_sound: "\<And>s1. \<lbrakk>G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> (x,s); wf_prog G; s1\<Colon>\<preceq>(G, L);
   179   T = obj_ty \<lparr>tag=oi,values=fs\<rparr>; is_type G T\<rbrakk> \<Longrightarrow>  
   180   (x,s)\<Colon>\<preceq>(G, L) \<and> (x = None \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>T)"
   181 apply (auto elim!: halloc_conforms)
   182 apply (case_tac "aa")
   183 apply (subst obj_ty_eq)
   184 apply  (auto elim!: halloc_elim_cases dest!: new_AddrD intro!: conf_AddrI)
   185 done
   186 
   187 lemma sxalloc_type_sound: 
   188  "\<And>s1 s2. \<lbrakk>G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2; wf_prog G\<rbrakk> \<Longrightarrow> case fst s1 of  
   189   None \<Rightarrow> s2 = s1 | Some x \<Rightarrow>  
   190   (\<exists>a. fst s2 = Some(Xcpt (Loc a)) \<and> (\<forall>L. s1\<Colon>\<preceq>(G,L) \<longrightarrow> s2\<Colon>\<preceq>(G,L)))"
   191 apply (simp (no_asm_simp) only: split_tupled_all)
   192 apply (erule sxalloc.induct)
   193 apply   auto
   194 apply (rule halloc_conforms [THEN conforms_xconf])
   195 apply     (auto elim!: halloc_elim_cases dest!: new_AddrD intro!: conf_AddrI)
   196 done
   197 
   198 lemma wt_init_comp_ty: 
   199 "is_acc_type G (pid C) T \<Longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>init_comp_ty T\<Colon>\<surd>"
   200 apply (unfold init_comp_ty_def)
   201 apply (clarsimp simp add: accessible_in_RefT_simp 
   202                           is_acc_type_def is_acc_class_def)
   203 done
   204 
   205 
   206 declare fun_upd_same [simp]
   207 
   208 declare fun_upd_apply [simp del]
   209 
   210 
   211 constdefs
   212   DynT_prop::"[prog,inv_mode,qtname,ref_ty] \<Rightarrow> bool" 
   213                                               ("_\<turnstile>_\<rightarrow>_\<preceq>_"[71,71,71,71]70)
   214  "G\<turnstile>mode\<rightarrow>D\<preceq>t \<equiv> mode = IntVir \<longrightarrow> is_class G D \<and> 
   215                      (if (\<exists>T. t=ArrayT T) then D=Object else G\<turnstile>Class D\<preceq>RefT t)"
   216 
   217 lemma DynT_propI: 
   218  "\<lbrakk>(x,s)\<Colon>\<preceq>(G, L); G,s\<turnstile>a'\<Colon>\<preceq>RefT statT; wf_prog G; mode = IntVir \<longrightarrow> a' \<noteq> Null\<rbrakk> 
   219   \<Longrightarrow>  G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"
   220 proof (unfold DynT_prop_def)
   221   assume state_conform: "(x,s)\<Colon>\<preceq>(G, L)"
   222      and      statT_a': "G,s\<turnstile>a'\<Colon>\<preceq>RefT statT"
   223      and            wf: "wf_prog G"
   224      and          mode: "mode = IntVir \<longrightarrow> a' \<noteq> Null"
   225   let ?invCls = "(invocation_class mode s a' statT)"
   226   let ?IntVir = "mode = IntVir"
   227   let ?Concl  = "\<lambda>invCls. is_class G invCls \<and>
   228                           (if \<exists>T. statT = ArrayT T
   229                                   then invCls = Object
   230                                   else G\<turnstile>Class invCls\<preceq>RefT statT)"
   231   show "?IntVir \<longrightarrow> ?Concl ?invCls"
   232   proof  
   233     assume modeIntVir: ?IntVir 
   234     with mode have not_Null: "a' \<noteq> Null" ..
   235     from statT_a' not_Null state_conform 
   236     obtain a obj 
   237       where obj_props:  "a' = Addr a" "globs s (Inl a) = Some obj"   
   238                         "G\<turnstile>obj_ty obj\<preceq>RefT statT" "is_type G (obj_ty obj)"
   239       by (blast dest: conforms_RefTD)
   240     show "?Concl ?invCls"
   241     proof (cases "tag obj")
   242       case CInst
   243       with modeIntVir obj_props
   244       show ?thesis 
   245 	by (auto dest!: widen_Array2 split add: split_if)
   246     next
   247       case Arr
   248       from Arr obtain T where "obj_ty obj = T.[]" by (blast dest: obj_ty_Arr1)
   249       moreover from Arr have "obj_class obj = Object" 
   250 	by (blast dest: obj_class_Arr1)
   251       moreover note modeIntVir obj_props wf 
   252       ultimately show ?thesis by (auto dest!: widen_Array )
   253     qed
   254   qed
   255 qed
   256 
   257 lemma invocation_methd:
   258 "\<lbrakk>wf_prog G; statT \<noteq> NullT; 
   259  (\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC);
   260  (\<forall>     I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM);
   261  (\<forall>     T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM);
   262  G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT;  
   263  dynlookup G statT (invocation_class mode s a' statT) sig = Some m \<rbrakk> 
   264 \<Longrightarrow> methd G (invocation_declclass G mode s a' statT sig) sig = Some m"
   265 proof -
   266   assume         wf: "wf_prog G"
   267      and  not_NullT: "statT \<noteq> NullT"
   268      and statC_prop: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"
   269      and statI_prop: "(\<forall> I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM)"
   270      and statA_prop: "(\<forall>     T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM)"
   271      and  invC_prop: "G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"
   272      and  dynlookup: "dynlookup G statT (invocation_class mode s a' statT) sig 
   273                       = Some m"
   274   show ?thesis
   275   proof (cases statT)
   276     case NullT
   277     with not_NullT show ?thesis by simp
   278   next
   279     case IfaceT
   280     with statI_prop obtain I 
   281       where    statI: "statT = IfaceT I" and 
   282             is_iface: "is_iface G I"     and
   283           not_SuperM: "mode \<noteq> SuperM" by blast            
   284     
   285     show ?thesis 
   286     proof (cases mode)
   287       case Static
   288       with wf dynlookup statI is_iface 
   289       show ?thesis
   290 	by (auto simp add: invocation_declclass_def dynlookup_def 
   291                            dynimethd_def dynmethd_C_C 
   292 	            intro: dynmethd_declclass
   293 	            dest!: wf_imethdsD
   294                      dest: table_of_map_SomeI
   295                     split: split_if_asm)
   296     next	
   297       case SuperM
   298       with not_SuperM show ?thesis ..
   299     next
   300       case IntVir
   301       with wf dynlookup IfaceT invC_prop show ?thesis 
   302 	by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def
   303                            DynT_prop_def
   304 	            intro: methd_declclass dynmethd_declclass
   305 	            split: split_if_asm)
   306     qed
   307   next
   308     case ClassT
   309     show ?thesis
   310     proof (cases mode)
   311       case Static
   312       with wf ClassT dynlookup statC_prop 
   313       show ?thesis by (auto simp add: invocation_declclass_def dynlookup_def
   314                                intro: dynmethd_declclass)
   315     next
   316       case SuperM
   317       with wf ClassT dynlookup statC_prop 
   318       show ?thesis by (auto simp add: invocation_declclass_def dynlookup_def
   319                                intro: dynmethd_declclass)
   320     next
   321       case IntVir
   322       with wf ClassT dynlookup statC_prop invC_prop 
   323       show ?thesis
   324 	by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def
   325                            DynT_prop_def
   326 	            intro: dynmethd_declclass)
   327     qed
   328   next
   329     case ArrayT
   330     show ?thesis
   331     proof (cases mode)
   332       case Static
   333       with wf ArrayT dynlookup show ?thesis
   334 	by (auto simp add: invocation_declclass_def dynlookup_def 
   335                            dynimethd_def dynmethd_C_C
   336                     intro: dynmethd_declclass
   337                      dest: table_of_map_SomeI)
   338     next
   339       case SuperM
   340       with ArrayT statA_prop show ?thesis by blast
   341     next
   342       case IntVir
   343       with wf ArrayT dynlookup invC_prop show ?thesis
   344 	by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def
   345                            DynT_prop_def dynmethd_C_C
   346                     intro: dynmethd_declclass
   347                      dest: table_of_map_SomeI)
   348     qed
   349   qed
   350 qed
   351    
   352 declare split_paired_All [simp del] split_paired_Ex [simp del] 
   353 ML_setup {*
   354 simpset_ref() := simpset() delloop "split_all_tac";
   355 claset_ref () := claset () delSWrapper "split_all_tac"
   356 *}
   357 lemma DynT_mheadsD: 
   358 "\<lbrakk>G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT; 
   359   wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT; 
   360   sm \<in> mheads G C statT sig; 
   361   invC = invocation_class (invmode (mhd sm) e) s a' statT;
   362   declC =invocation_declclass G (invmode (mhd sm) e) s a' statT sig
   363  \<rbrakk> \<Longrightarrow> 
   364   \<exists> dm. 
   365   methd G declC sig = Some dm  \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm) \<and> 
   366   wf_mdecl G declC (sig, mthd dm) \<and>
   367   declC = declclass dm \<and>
   368   is_static dm = is_static sm \<and>  
   369   is_class G invC \<and> is_class G declC  \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and>  
   370   (if invmode (mhd sm) e = IntVir
   371       then (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)
   372       else (  (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
   373             \<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object)) \<and> 
   374            (declrefT sm) = ClassT (declclass dm))"
   375 proof -
   376   assume invC_prop: "G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT" 
   377      and        wf: "wf_prog G" 
   378      and      wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT"
   379      and        sm: "sm \<in> mheads G C statT sig" 
   380      and      invC: "invC = invocation_class (invmode (mhd sm) e) s a' statT"
   381      and     declC: "declC = 
   382                     invocation_declclass G (invmode (mhd sm) e) s a' statT sig"
   383   from wt_e wf have type_statT: "is_type G (RefT statT)"
   384     by (auto dest: ty_expr_is_type)
   385   from sm have not_Null: "statT \<noteq> NullT" by auto
   386   from type_statT 
   387   have wf_C: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"
   388     by (auto)
   389   from type_statT wt_e 
   390   have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> 
   391                                         invmode (mhd sm) e \<noteq> SuperM)"
   392     by (auto dest: invocationTypeExpr_noClassD)
   393   from wt_e
   394   have wf_A: "(\<forall>     T. statT = ArrayT T \<longrightarrow> invmode (mhd sm) e \<noteq> SuperM)"
   395     by (auto dest: invocationTypeExpr_noClassD)
   396   show ?thesis
   397   proof (cases "invmode (mhd sm) e = IntVir")
   398     case True
   399     with invC_prop not_Null
   400     have invC_prop': " is_class G invC \<and> 
   401                       (if (\<exists>T. statT=ArrayT T) then invC=Object 
   402                                               else G\<turnstile>Class invC\<preceq>RefT statT)"
   403       by (auto simp add: DynT_prop_def) 
   404     from True 
   405     have "\<not> is_static sm"
   406       by (simp add: invmode_IntVir_eq)
   407     with invC_prop' not_Null
   408     have "G,statT \<turnstile> invC valid_lookup_cls_for (is_static sm)"
   409       by (cases statT) auto
   410     with sm wf type_statT obtain dm where
   411            dm: "dynlookup G statT invC sig = Some dm" and
   412       resT_dm: "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm)"      and
   413        static: "is_static dm = is_static sm"
   414       by - (drule dynamic_mheadsD,auto)
   415     with declC invC not_Null 
   416     have declC': "declC = (declclass dm)" 
   417       by (auto simp add: invocation_declclass_def)
   418     with wf invC declC not_Null wf_C wf_I wf_A invC_prop dm 
   419     have dm': "methd G declC sig = Some dm"
   420       by - (drule invocation_methd,auto)
   421     from wf dm invC_prop' declC' type_statT 
   422     have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
   423       by (auto dest: dynlookup_declC)
   424     from wf dm' declC_prop declC' 
   425     have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
   426       by (auto dest: methd_wf_mdecl)
   427     from invC_prop' 
   428     have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)"
   429       by auto
   430     from True dm' resT_dm wf_dm invC_prop' declC_prop statC_prop declC' static
   431     show ?thesis by auto
   432   next
   433     case False
   434     with type_statT wf invC not_Null wf_I wf_A
   435     have invC_prop': "is_class G invC \<and>  
   436                      ((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>
   437                       (\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "
   438         by (case_tac "statT") (auto simp add: invocation_class_def 
   439                                        split: inv_mode.splits)
   440     with not_Null wf
   441     have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"
   442       by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_C_C
   443                                             dynimethd_def)
   444     from sm wf wt_e not_Null False invC_prop' obtain "dm" where
   445                     dm: "methd G invC sig = Some dm"          and
   446 	eq_declC_sm_dm:"declrefT sm = ClassT (declclass dm)"  and
   447 	     eq_mheads:"mhd sm=mhead (mthd dm) "
   448       by - (drule static_mheadsD, auto dest: accmethd_SomeD)
   449     then have static: "is_static dm = is_static sm" by - (case_tac "sm",auto)
   450     with declC invC dynlookup_static dm
   451     have declC': "declC = (declclass dm)"  
   452       by (auto simp add: invocation_declclass_def)
   453     from invC_prop' wf declC' dm 
   454     have dm': "methd G declC sig = Some dm"
   455       by (auto intro: methd_declclass)
   456     from wf dm invC_prop' declC' type_statT 
   457     have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
   458       by (auto dest: methd_declC )
   459     then have declC_prop1: "invC=Object \<longrightarrow> declC=Object"  by auto
   460     from wf dm' declC_prop declC' 
   461     have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
   462       by (auto dest: methd_wf_mdecl)
   463     from invC_prop' declC_prop declC_prop1
   464     have statC_prop: "(   (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
   465                        \<or>  (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object))" 
   466       by auto
   467     from False dm' wf_dm invC_prop' declC_prop statC_prop declC' 
   468          eq_declC_sm_dm eq_mheads static
   469     show ?thesis by auto
   470   qed
   471 qed
   472 
   473 (*
   474 lemma DynT_mheadsD: 
   475 "\<lbrakk>G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT; 
   476   wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT; 
   477   sm \<in> mheads G C statT sig; 
   478   invC = invocation_class (invmode (mhd sm) e) s a' statT;
   479   declC =invocation_declclass G (invmode (mhd sm) e) s a' statT sig
   480  \<rbrakk> \<Longrightarrow> 
   481   \<exists> dm. 
   482   methd G declC sig = Some dm  \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm) \<and> 
   483   wf_mdecl G declC (sig, mthd dm) \<and>  
   484   is_class G invC \<and> is_class G declC  \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and>  
   485   (if invmode (mhd sm) e = IntVir
   486       then (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)
   487       else (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>statC \<preceq>\<^sub>C declC) \<and> 
   488            (declrefT sm) = ClassT (declclass dm))"
   489 proof -
   490   assume invC_prop: "G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT" 
   491      and        wf: "wf_prog G" 
   492      and      wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT"
   493      and        sm: "sm \<in> mheads G C statT sig" 
   494      and      invC: "invC = invocation_class (invmode (mhd sm) e) s a' statT"
   495      and     declC: "declC = 
   496                     invocation_declclass G (invmode (mhd sm) e) s a' statT sig"
   497   from wt_e wf have type_statT: "is_type G (RefT statT)"
   498     by (auto dest: ty_expr_is_type)
   499   from sm have not_Null: "statT \<noteq> NullT" by auto
   500   from type_statT 
   501   have wf_C: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"
   502     by (auto)
   503   from type_statT wt_e 
   504   have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> 
   505                                         invmode (mhd sm) e \<noteq> SuperM)"
   506     by (auto dest: invocationTypeExpr_noClassD)
   507   from wt_e
   508   have wf_A: "(\<forall>     T. statT = ArrayT T \<longrightarrow> invmode (mhd sm) e \<noteq> SuperM)"
   509     by (auto dest: invocationTypeExpr_noClassD)
   510   show ?thesis
   511   proof (cases "invmode (mhd sm) e = IntVir")
   512     case True
   513     with invC_prop not_Null
   514     have invC_prop': "is_class G invC \<and>  
   515                       (if (\<exists>T. statT=ArrayT T) then invC=Object 
   516                                               else G\<turnstile>Class invC\<preceq>RefT statT)"
   517       by (auto simp add: DynT_prop_def) 
   518     with sm wf type_statT not_Null obtain dm where
   519            dm: "dynlookup G statT invC sig = Some dm" and
   520       resT_dm: "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm)"
   521       by - (clarify,drule dynamic_mheadsD,auto)
   522     with declC invC not_Null 
   523     have declC': "declC = (declclass dm)" 
   524       by (auto simp add: invocation_declclass_def)
   525     with wf invC declC not_Null wf_C wf_I wf_A invC_prop dm 
   526     have dm': "methd G declC sig = Some dm"
   527       by - (drule invocation_methd,auto)
   528     from wf dm invC_prop' declC' type_statT 
   529     have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
   530       by (auto dest: dynlookup_declC)
   531     from wf dm' declC_prop declC' 
   532     have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
   533       by (auto dest: methd_wf_mdecl)
   534     from invC_prop' 
   535     have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)"
   536       by auto
   537     from True dm' resT_dm wf_dm invC_prop' declC_prop statC_prop
   538     show ?thesis by auto
   539   next
   540     case False
   541     
   542     with type_statT wf invC not_Null wf_I wf_A
   543     have invC_prop': "is_class G invC \<and>  
   544                      ((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>
   545                       (\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "
   546         
   547         by (case_tac "statT") (auto simp add: invocation_class_def 
   548                                        split: inv_mode.splits)
   549     with not_Null 
   550     have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"
   551       by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_def 
   552                                             dynimethd_def)
   553     from sm wf wt_e not_Null False invC_prop' obtain "dm" where
   554                     dm: "methd G invC sig = Some dm"          and
   555 	eq_declC_sm_dm:"declrefT sm = ClassT (declclass dm)"  and
   556 	     eq_mheads:"mhd sm=mhead (mthd dm) "
   557       by - (drule static_mheadsD, auto dest: accmethd_SomeD)
   558     with declC invC dynlookup_static dm
   559     have declC': "declC = (declclass dm)"  
   560       by (auto simp add: invocation_declclass_def)
   561     from invC_prop' wf declC' dm 
   562     have dm': "methd G declC sig = Some dm"
   563       by (auto intro: methd_declclass)
   564     from wf dm invC_prop' declC' type_statT 
   565     have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
   566       by (auto dest: methd_declC )   
   567     from wf dm' declC_prop declC' 
   568     have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
   569       by (auto dest: methd_wf_mdecl)
   570     from invC_prop' declC_prop
   571     have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>statC \<preceq>\<^sub>C declC)" 
   572       by auto
   573     from False dm' wf_dm invC_prop' declC_prop statC_prop 
   574          eq_declC_sm_dm eq_mheads
   575     show ?thesis by auto
   576   qed
   577 qed	
   578 *)
   579 
   580 declare split_paired_All [simp del] split_paired_Ex [simp del] 
   581 declare split_if     [split del] split_if_asm     [split del] 
   582         option.split [split del] option.split_asm [split del]
   583 ML_setup {*
   584 simpset_ref() := simpset() delloop "split_all_tac";
   585 claset_ref () := claset () delSWrapper "split_all_tac"
   586 *}
   587 
   588 lemma DynT_conf: "\<lbrakk>G\<turnstile>invocation_class mode s a' statT \<preceq>\<^sub>C declC; wf_prog G;
   589  isrtype G (statT);
   590  G,s\<turnstile>a'\<Colon>\<preceq>RefT statT; mode = IntVir \<longrightarrow> a' \<noteq> Null;  
   591   mode \<noteq> IntVir \<longrightarrow>    (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
   592                     \<or>  (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object)\<rbrakk> 
   593  \<Longrightarrow>G,s\<turnstile>a'\<Colon>\<preceq> Class declC"
   594 apply (case_tac "mode = IntVir")
   595 apply (drule conf_RefTD)
   596 apply (force intro!: conf_AddrI 
   597             simp add: obj_class_def split add: obj_tag.split_asm)
   598 apply  clarsimp
   599 apply  safe
   600 apply    (erule (1) widen.subcls [THEN conf_widen])
   601 apply    (erule wf_ws_prog)
   602 
   603 apply    (frule widen_Object) apply (erule wf_ws_prog)
   604 apply    (erule (1) conf_widen) apply (erule wf_ws_prog)
   605 done
   606 
   607 
   608 lemma Ass_lemma: 
   609  "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w, f)\<rightarrow> Norm s1; G\<turnstile>Norm s1 \<midarrow>e-\<succ>v\<rightarrow> Norm s2; G,s2\<turnstile>v\<Colon>\<preceq>T'; 
   610    s1\<le>|s2 \<longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L)
   611   \<rbrakk> \<Longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L) \<and> 
   612         (\<lambda>(x',s'). x' = None \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T') (assign f v (Norm s2))"
   613 apply (drule_tac x = "None" and s = "s2" and v = "v" in evar_gext_f)
   614 apply (drule eval_gext', clarsimp)
   615 apply (erule conf_gext)
   616 apply simp
   617 done
   618 
   619 lemma Throw_lemma: "\<lbrakk>G\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable; wf_prog G; (x1,s1)\<Colon>\<preceq>(G, L);  
   620     x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq> Class tn\<rbrakk> \<Longrightarrow> (throw a' x1, s1)\<Colon>\<preceq>(G, L)"
   621 apply (auto split add: split_abrupt_if simp add: throw_def2)
   622 apply (erule conforms_xconf)
   623 apply (frule conf_RefTD)
   624 apply (auto elim: widen.subcls [THEN conf_widen])
   625 done
   626 
   627 lemma Try_lemma: "\<lbrakk>G\<turnstile>obj_ty (the (globs s1' (Heap a)))\<preceq> Class tn; 
   628  (Some (Xcpt (Loc a)), s1')\<Colon>\<preceq>(G, L); wf_prog G\<rbrakk> 
   629  \<Longrightarrow> Norm (lupd(vn\<mapsto>Addr a) s1')\<Colon>\<preceq>(G, L(vn\<mapsto>Class tn))"
   630 apply (rule conforms_allocL)
   631 apply  (erule conforms_NormI)
   632 apply (drule conforms_XcptLocD [THEN conf_RefTD],rule HOL.refl)
   633 apply (auto intro!: conf_AddrI)
   634 done
   635 
   636 lemma Fin_lemma: 
   637 "\<lbrakk>G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> (x2,s2); wf_prog G; (Some a, s1)\<Colon>\<preceq>(G, L); (x2,s2)\<Colon>\<preceq>(G, L)\<rbrakk> 
   638 \<Longrightarrow>  (abrupt_if True (Some a) x2, s2)\<Colon>\<preceq>(G, L)"
   639 apply (force elim: eval_gext' conforms_xgext split add: split_abrupt_if)
   640 done
   641 
   642 lemma FVar_lemma1: "\<lbrakk>table_of (DeclConcepts.fields G Ca) (fn, C) = Some f ; 
   643   x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq> Class Ca; wf_prog G; G\<turnstile>Ca\<preceq>\<^sub>C C; C \<noteq> Object; 
   644   class G C = Some y; (x2,s2)\<Colon>\<preceq>(G, L); s1\<le>|s2; inited C (globs s1); 
   645   (if static f then id else np a) x2 = None\<rbrakk> 
   646  \<Longrightarrow>  
   647   \<exists>obj. globs s2 (if static f then Inr C else Inl (the_Addr a)) = Some obj \<and> 
   648   var_tys G (tag obj)  (if static f then Inr C else Inl(the_Addr a)) 
   649           (Inl(fn,C)) = Some (type f)"
   650 apply (drule initedD)
   651 apply (frule subcls_is_class2, simp (no_asm_simp))
   652 apply (case_tac "static f")
   653 apply  clarsimp
   654 apply  (drule (1) rev_gext_objD, clarsimp)
   655 apply  (frule fields_declC, erule wf_ws_prog, simp (no_asm_simp))
   656 apply  (rule var_tys_Some_eq [THEN iffD2])
   657 apply  clarsimp
   658 apply  (erule fields_table_SomeI, simp (no_asm))
   659 apply clarsimp
   660 apply (drule conf_RefTD, clarsimp, rule var_tys_Some_eq [THEN iffD2])
   661 apply (auto dest!: widen_Array split add: obj_tag.split)
   662 apply (rotate_tac -1) (* to front: tag obja = CInst pid_field_type to enable
   663                          conditional rewrite *)
   664 apply (rule fields_table_SomeI)
   665 apply (auto elim!: fields_mono subcls_is_class2)
   666 done
   667 
   668 lemma FVar_lemma: 
   669 "\<lbrakk>((v, f), Norm s2') = fvar C (static field) fn a (x2, s2); G\<turnstile>Ca\<preceq>\<^sub>C C;  
   670   table_of (DeclConcepts.fields G Ca) (fn, C) = Some field; wf_prog G;   
   671   x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq>Class Ca; C \<noteq> Object; class G C = Some y;   
   672   (x2, s2)\<Colon>\<preceq>(G, L); s1\<le>|s2; inited C (globs s1)\<rbrakk> \<Longrightarrow>  
   673   G,s2'\<turnstile>v\<Colon>\<preceq>type field \<and> s2'\<le>|f\<preceq>type field\<Colon>\<preceq>(G, L)"
   674 apply (unfold assign_conforms_def)
   675 apply (drule sym)
   676 apply (clarsimp simp add: fvar_def2)
   677 apply (drule (9) FVar_lemma1)
   678 apply (clarsimp)
   679 apply (drule (2) conforms_globsD [THEN oconf_lconf, THEN lconfD])
   680 apply clarsimp
   681 apply (drule (1) rev_gext_objD)
   682 apply (auto elim!: conforms_upd_gobj)
   683 done
   684 
   685 
   686 lemma AVar_lemma1: "\<lbrakk>globs s (Inl a) = Some obj;tag obj=Arr ty i; 
   687  the_Intg i' in_bounds i; wf_prog G; G\<turnstile>ty.[]\<preceq>Tb.[]; Norm s\<Colon>\<preceq>(G, L)
   688 \<rbrakk> \<Longrightarrow> G,s\<turnstile>the ((values obj) (Inr (the_Intg i')))\<Colon>\<preceq>Tb"
   689 apply (erule widen_Array_Array [THEN conf_widen])
   690 apply  (erule_tac [2] wf_ws_prog)
   691 apply (drule (1) conforms_globsD [THEN oconf_lconf, THEN lconfD])
   692 defer apply assumption
   693 apply (force intro: var_tys_Some_eq [THEN iffD2])
   694 done
   695 
   696 lemma obj_split: "\<And> obj. \<exists> t vs. obj = \<lparr>tag=t,values=vs\<rparr>"
   697 proof record_split
   698   fix tag values more
   699   show "\<exists>t vs. \<lparr>tag = tag, values = values, \<dots> = more\<rparr> = \<lparr>tag = t, values = vs\<rparr>"
   700     by auto
   701 qed
   702  
   703 lemma AVar_lemma: "\<lbrakk>wf_prog G; G\<turnstile>(x1, s1) \<midarrow>e2-\<succ>i\<rightarrow> (x2, s2);  
   704   ((v,f), Norm s2') = avar G i a (x2, s2); x1 = None \<longrightarrow> G,s1\<turnstile>a\<Colon>\<preceq>Ta.[];  
   705   (x2, s2)\<Colon>\<preceq>(G, L); s1\<le>|s2\<rbrakk> \<Longrightarrow> G,s2'\<turnstile>v\<Colon>\<preceq>Ta \<and> s2'\<le>|f\<preceq>Ta\<Colon>\<preceq>(G, L)"
   706 apply (unfold assign_conforms_def)
   707 apply (drule sym)
   708 apply (clarsimp simp add: avar_def2)
   709 apply (drule (1) conf_gext)
   710 apply (drule conf_RefTD, clarsimp)
   711 apply (subgoal_tac "\<exists> t vs. obj = \<lparr>tag=t,values=vs\<rparr>")
   712 defer
   713 apply   (rule obj_split)
   714 apply clarify
   715 apply (frule obj_ty_widenD)
   716 apply (auto dest!: widen_Class)
   717 apply  (force dest: AVar_lemma1)
   718 apply (auto split add: split_if)
   719 apply (force elim!: fits_Array dest: gext_objD 
   720        intro: var_tys_Some_eq [THEN iffD2] conforms_upd_gobj)
   721 done
   722 
   723 
   724 section "Call"
   725 lemma conforms_init_lvars_lemma: "\<lbrakk>wf_prog G;  
   726   wf_mhead G P sig mh; 
   727   Ball (set lvars) (split (\<lambda>vn. is_type G)); 
   728   list_all2 (conf G s) pvs pTsa; G\<turnstile>pTsa[\<preceq>](parTs sig)\<rbrakk> \<Longrightarrow>  
   729   G,s\<turnstile>init_vals (table_of lvars)(pars mh[\<mapsto>]pvs)
   730       [\<Colon>\<preceq>]table_of lvars(pars mh[\<mapsto>]parTs sig)"
   731 apply (unfold wf_mhead_def)
   732 apply clarify
   733 apply (erule (2) Ball_set_table [THEN lconf_init_vals, THEN lconf_ext_list])
   734 apply (drule wf_ws_prog)
   735 apply (erule (2) conf_list_widen)
   736 done
   737 
   738 
   739 lemma lconf_map_lname [simp]: 
   740   "G,s\<turnstile>(lname_case l1 l2)[\<Colon>\<preceq>](lname_case L1 L2)
   741    =
   742   (G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>(\<lambda>x::unit . l2)[\<Colon>\<preceq>](\<lambda>x::unit. L2))"
   743 apply (unfold lconf_def)
   744 apply safe
   745 apply (case_tac "n")
   746 apply (force split add: lname.split)+
   747 done
   748 
   749 lemma lconf_map_ename [simp]:
   750   "G,s\<turnstile>(ename_case l1 l2)[\<Colon>\<preceq>](ename_case L1 L2)
   751    =
   752   (G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>(\<lambda>x::unit. l2)[\<Colon>\<preceq>](\<lambda>x::unit. L2))"
   753 apply (unfold lconf_def)
   754 apply safe
   755 apply (force split add: ename.split)+
   756 done
   757 
   758 
   759 lemma defval_conf1 [rule_format (no_asm), elim]: 
   760   "is_type G T \<longrightarrow> (\<exists>v\<in>Some (default_val T): G,s\<turnstile>v\<Colon>\<preceq>T)"
   761 apply (unfold conf_def)
   762 apply (induct "T")
   763 apply (auto intro: prim_ty.induct)
   764 done
   765 
   766 
   767 lemma conforms_init_lvars: 
   768 "\<lbrakk>wf_mhead G (pid declC) sig (mhead (mthd dm)); wf_prog G;  
   769   list_all2 (conf G s) pvs pTsa; G\<turnstile>pTsa[\<preceq>](parTs sig);  
   770   (x, s)\<Colon>\<preceq>(G, L); 
   771   methd G declC sig = Some dm;  
   772   isrtype G statT;
   773   G\<turnstile>invC\<preceq>\<^sub>C declC; 
   774   G,s\<turnstile>a'\<Colon>\<preceq>RefT statT;  
   775   invmode (mhd sm) e = IntVir \<longrightarrow> a' \<noteq> Null; 
   776   invmode (mhd sm) e \<noteq> IntVir \<longrightarrow>  
   777        (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
   778     \<or>  (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object);
   779   invC  = invocation_class (invmode (mhd sm) e) s a' statT;
   780   declC = invocation_declclass G (invmode (mhd sm) e) s a' statT sig;
   781   Ball (set (lcls (mbody (mthd dm)))) 
   782        (split (\<lambda>vn. is_type G)) 
   783  \<rbrakk> \<Longrightarrow>  
   784   init_lvars G declC sig (invmode (mhd sm) e) a'  
   785   pvs (x,s)\<Colon>\<preceq>(G,\<lambda> k. 
   786                 (case k of
   787                    EName e \<Rightarrow> (case e of 
   788                                  VNam v 
   789                                   \<Rightarrow> (table_of (lcls (mbody (mthd dm)))
   790                                         (pars (mthd dm)[\<mapsto>]parTs sig)) v
   791                                | Res \<Rightarrow> Some (resTy (mthd dm)))
   792                  | This \<Rightarrow> if (static (mthd sm)) 
   793                               then None else Some (Class declC)))"
   794 apply (simp add: init_lvars_def2)
   795 apply (rule conforms_set_locals)
   796 apply  (simp (no_asm_simp) split add: split_if)
   797 apply (drule  (4) DynT_conf)
   798 apply clarsimp
   799 (* apply intro *)
   800 apply (drule (4) conforms_init_lvars_lemma)
   801 apply (case_tac "dm",simp)
   802 apply (rule conjI)
   803 apply (unfold lconf_def, clarify)
   804 apply (rule defval_conf1)
   805 apply (clarsimp simp add: wf_mhead_def is_acc_type_def)
   806 apply (case_tac "is_static sm")
   807 apply simp_all
   808 done
   809 
   810 
   811 lemma Call_type_sound: "\<lbrakk>wf_prog G; G\<turnstile>(x1, s1) \<midarrow>ps\<doteq>\<succ>pvs\<rightarrow> (x2, s2);  
   812  declC 
   813  = invocation_declclass G (invmode (mhd esm) e) s2 a' statT \<lparr>name=mn,parTs=pTs\<rparr>;
   814 s2'=init_lvars G declC \<lparr>name=mn,parTs=pTs\<rparr> (invmode (mhd esm) e) a' pvs (x2,s2);
   815  G\<turnstile>s2' \<midarrow>Methd declC \<lparr>name=mn,parTs=pTs\<rparr>-\<succ>v\<rightarrow> (x3, s3);    
   816  \<forall>L. s2'\<Colon>\<preceq>(G, L) 
   817      \<longrightarrow> (\<forall>T. \<lparr>prg=G,cls=declC,lcl=L\<rparr>\<turnstile> Methd declC \<lparr>name=mn,parTs=pTs\<rparr>\<Colon>-T 
   818      \<longrightarrow> (x3, s3)\<Colon>\<preceq>(G, L) \<and> (x3 = None \<longrightarrow> G,s3\<turnstile>v\<Colon>\<preceq>T));  
   819  Norm s0\<Colon>\<preceq>(G, L); 
   820  \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>ps\<Colon>\<doteq>pTsa;  
   821  max_spec G C statT \<lparr>name=mn,parTs=pTsa\<rparr> = {(esm, pTs)}; 
   822  (x1, s1)\<Colon>\<preceq>(G, L); 
   823  x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq>RefT statT; (x2, s2)\<Colon>\<preceq>(G, L);  
   824  x2 = None \<longrightarrow> list_all2 (conf G s2) pvs pTsa;
   825  sm=(mhd esm)\<rbrakk> \<Longrightarrow>     
   826  (x3, set_locals (locals s2) s3)\<Colon>\<preceq>(G, L) \<and> 
   827  (x3 = None \<longrightarrow> G,s3\<turnstile>v\<Colon>\<preceq>resTy sm)"
   828 apply clarify
   829 apply (case_tac "x2")
   830 defer
   831 apply  (clarsimp split add: split_if_asm simp add: init_lvars_def2)
   832 apply (case_tac "a' = Null \<and> \<not> (static (mhd esm)) \<and> e \<noteq> Super")
   833 apply  (clarsimp simp add: init_lvars_def2)
   834 apply clarsimp
   835 apply (drule eval_gext')
   836 apply (frule (1) conf_gext)
   837 apply (drule max_spec2mheads, clarsimp)
   838 apply (subgoal_tac "invmode (mhd esm) e = IntVir \<longrightarrow> a' \<noteq> Null")
   839 defer  
   840 apply  (clarsimp simp add: invmode_IntVir_eq)
   841 apply (frule (6) DynT_mheadsD [OF DynT_propI,of _ "s2"],(rule HOL.refl)+)
   842 apply clarify
   843 apply (drule wf_mdeclD1, clarsimp) 
   844 apply (frule  ty_expr_is_type) apply simp
   845 apply (frule (2) conforms_init_lvars)
   846 apply   simp
   847 apply   assumption+
   848 apply   simp
   849 apply   assumption+
   850 apply   clarsimp
   851 apply   (rule HOL.refl)
   852 apply   simp
   853 apply   (rule Ball_weaken)
   854 apply     assumption
   855 apply     (force simp add: is_acc_type_def)
   856 apply (tactic "smp_tac 1 1")
   857 apply (frule (2) wt_MethdI, clarsimp)
   858 apply (subgoal_tac "is_static dm = (static (mthd esm))") 
   859 apply   (simp only:)
   860 apply   (tactic "smp_tac 1 1")
   861 apply   (rule conjI)
   862 apply     (erule  conforms_return)
   863 apply     blast
   864 
   865 apply     (force dest!: eval_gext del: impCE simp add: init_lvars_def2)
   866 apply     clarsimp
   867 apply     (drule (2) widen_trans, erule (1) conf_widen)
   868 apply     (erule wf_ws_prog)
   869 
   870 apply   auto
   871 done
   872 
   873 
   874 subsection "accessibility"
   875 
   876 theorem dynamic_field_access_ok:
   877   (assumes wf: "wf_prog G" and
   878        eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2" and
   879      not_Null: "a\<noteq>Null" and
   880     conform_a: "G,(store s2)\<turnstile>a\<Colon>\<preceq> Class statC" and
   881    conform_s2: "s2\<Colon>\<preceq>(G, L)" and 
   882     normal_s2: "normal s2" and
   883          wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>,dt\<Turnstile>e\<Colon>-Class statC" and
   884             f: "accfield G accC statC fn = Some f" and
   885          dynC: "if stat then dynC=statC  
   886                         else dynC=obj_class (lookup_obj (store s2) a)"
   887   ) "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f) \<and> 
   888      G\<turnstile>Field fn f in dynC dyn_accessible_from accC"
   889 proof (cases "stat")
   890   case True
   891   with dynC 
   892   have dynC': "dynC=statC" by simp
   893   with f
   894   have "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f)"
   895     by (auto simp add: accfield_def Let_def intro!: table_of_remap_SomeD)
   896   with dynC' f
   897   show ?thesis
   898     by (auto intro!: static_to_dynamic_accessible_from
   899          dest: accfield_accessibleD accessible_from_commonD)
   900 next
   901   case False
   902   with wf conform_a not_Null conform_s2 dynC
   903   obtain subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
   904     "is_class G dynC"
   905     by (auto dest!: conforms_RefTD [of _ _ _ _ "(fst s2)" L]
   906               dest: obj_ty_obj_class1
   907           simp add: obj_ty_obj_class )
   908   with wf f
   909   have "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f)"
   910     by (auto simp add: accfield_def Let_def
   911                  dest: fields_mono
   912                 dest!: table_of_remap_SomeD)
   913   moreover
   914   from f subclseq
   915   have "G\<turnstile>Field fn f in dynC dyn_accessible_from accC"
   916     by (auto intro!: static_to_dynamic_accessible_from 
   917                dest: accfield_accessibleD)
   918   ultimately show ?thesis
   919     by blast
   920 qed
   921 
   922 lemma call_access_ok: 
   923 (assumes invC_prop: "G\<turnstile>invmode (mhd statM) e\<rightarrow>invC\<preceq>statT" 
   924      and        wf: "wf_prog G" 
   925      and      wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT"
   926      and     statM: "statM \<in> mheads G accC statT sig" 
   927      and      invC: "invC = invocation_class (invmode (mhd statM) e) s a statT"
   928 )"\<exists> dynM. dynlookup G statT invC sig = Some dynM \<and>
   929   G\<turnstile>Methd sig dynM in invC dyn_accessible_from accC"
   930 proof -
   931   from wt_e wf have type_statT: "is_type G (RefT statT)"
   932     by (auto dest: ty_expr_is_type)
   933   from statM have not_Null: "statT \<noteq> NullT" by auto
   934   from type_statT wt_e 
   935   have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> 
   936                                         invmode (mhd statM) e \<noteq> SuperM)"
   937     by (auto dest: invocationTypeExpr_noClassD)
   938   from wt_e
   939   have wf_A: "(\<forall>     T. statT = ArrayT T \<longrightarrow> invmode (mhd statM) e \<noteq> SuperM)"
   940     by (auto dest: invocationTypeExpr_noClassD)
   941   show ?thesis
   942   proof (cases "invmode (mhd statM) e = IntVir")
   943     case True
   944     with invC_prop not_Null
   945     have invC_prop': "is_class G invC \<and>  
   946                       (if (\<exists>T. statT=ArrayT T) then invC=Object 
   947                                               else G\<turnstile>Class invC\<preceq>RefT statT)"
   948       by (auto simp add: DynT_prop_def)
   949     with True not_Null
   950     have "G,statT \<turnstile> invC valid_lookup_cls_for is_static statM"
   951      by (cases statT) (auto simp add: invmode_def 
   952                          split: split_if split_if_asm) (*  was deleted above *)
   953     with statM type_statT wf 
   954     show ?thesis
   955       by - (rule dynlookup_access,auto)
   956   next
   957     case False
   958     with type_statT wf invC not_Null wf_I wf_A
   959     have invC_prop': " is_class G invC \<and>
   960                       ((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>
   961                       (\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "
   962         by (case_tac "statT") (auto simp add: invocation_class_def 
   963                                        split: inv_mode.splits)
   964     with not_Null wf
   965     have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"
   966       by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_C_C
   967                                             dynimethd_def)
   968    from statM wf wt_e not_Null False invC_prop' obtain dynM where
   969                 "accmethd G accC invC sig = Some dynM" 
   970      by (auto dest!: static_mheadsD)
   971    from invC_prop' False not_Null wf_I
   972    have "G,statT \<turnstile> invC valid_lookup_cls_for is_static statM"
   973      by (cases statT) (auto simp add: invmode_def
   974                         split: split_if split_if_asm) (*  was deleted above *)
   975    with statM type_statT wf 
   976     show ?thesis
   977       by - (rule dynlookup_access,auto)
   978   qed
   979 qed
   980 
   981 section "main proof of type safety"
   982 
   983 ML {*
   984 val forward_hyp_tac = EVERY' [smp_tac 1,
   985 	FIRST'[mp_tac,etac exI,smp_tac 2,smp_tac 1,EVERY'[etac impE,etac exI]],
   986 	REPEAT o (etac conjE)];
   987 val typD_tac = eresolve_tac (thms "wt_elim_cases") THEN_ALL_NEW 
   988 	EVERY' [full_simp_tac (simpset() setloop (K no_tac)), 
   989          clarify_tac(claset() addSEs[])]
   990 *}
   991 
   992 lemma conforms_locals [rule_format]: 
   993   "(a,b)\<Colon>\<preceq>(G, L) \<longrightarrow> L x = Some T \<longrightarrow> G,b\<turnstile>the (locals b x)\<Colon>\<preceq>T"
   994 apply (force simp: conforms_def Let_def lconf_def)
   995 done
   996 
   997 lemma eval_type_sound [rule_format (no_asm)]: 
   998  "wf_prog G \<Longrightarrow> G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1) \<Longrightarrow> (\<forall>L. s0\<Colon>\<preceq>(G,L) \<longrightarrow>    
   999   (\<forall>C T. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T \<longrightarrow> s1\<Colon>\<preceq>(G,L) \<and>  
  1000   (let (x,s) = s1 in x = None \<longrightarrow> G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T)))"
  1001 apply (erule eval_induct)
  1002 
  1003 (* 29 subgoals *)
  1004 (* Xcpt, Inst, Methd, Nil, Skip, Expr, Comp *)
  1005 apply         (simp_all (no_asm_use) add: Let_def body_def)
  1006 apply       (tactic "ALLGOALS (EVERY'[Clarify_tac, TRY o typD_tac, 
  1007                      TRY o forward_hyp_tac])")
  1008 apply      (tactic"ALLGOALS(EVERY'[asm_simp_tac(simpset()),TRY o Clarify_tac])")
  1009 
  1010 (* 20 subgoals *)
  1011 
  1012 (* Break *)
  1013 apply (erule conforms_absorb)
  1014 
  1015 (* Cons *)
  1016 apply (erule_tac V = "G\<turnstile>Norm s0 \<midarrow>?ea\<succ>\<rightarrow> ?vs1" in thin_rl)
  1017 apply (frule eval_gext')
  1018 apply force
  1019 
  1020 (* LVar *)
  1021 apply (force elim: conforms_localD [THEN lconfD] conforms_lupd 
  1022        simp add: assign_conforms_def lvar_def)
  1023 
  1024 (* Cast *)
  1025 apply (force dest: fits_conf)
  1026 
  1027 (* Lit *)
  1028 apply (rule conf_litval)
  1029 apply (simp add: empty_dt_def)
  1030 
  1031 (* Super *)
  1032 apply (rule conf_widen)
  1033 apply   (erule (1) subcls_direct [THEN widen.subcls])
  1034 apply  (erule (1) conforms_localD [THEN lconfD])
  1035 apply (erule wf_ws_prog)
  1036 
  1037 (* Acc *)
  1038 apply fast
  1039 
  1040 (* Body *)
  1041 apply (rule conjI)
  1042 apply (rule conforms_absorb)
  1043 apply (fast)
  1044 apply (fast intro: conforms_locals)
  1045 
  1046 (* Cond *)
  1047 apply (simp split: split_if_asm)
  1048 apply  (tactic "forward_hyp_tac 1", force)
  1049 apply (tactic "forward_hyp_tac 1", force)
  1050 
  1051 (* If *)
  1052 apply (force split add: split_if_asm)
  1053 
  1054 (* Loop *)
  1055 apply (drule (1) wt.Loop)
  1056 apply (clarsimp split: split_if_asm)
  1057 apply (fast intro: conforms_absorb)
  1058 
  1059 (* Fin *)
  1060 apply (case_tac "x1", force)
  1061 apply (drule spec, erule impE, erule conforms_NormI)
  1062 apply (erule impE)
  1063 apply   blast
  1064 apply (clarsimp)
  1065 apply (erule (3) Fin_lemma)
  1066 
  1067 (* Throw *)
  1068 apply (erule (3) Throw_lemma)
  1069 
  1070 (* NewC *)
  1071 apply (clarsimp simp add: is_acc_class_def)
  1072 apply (drule (1) halloc_type_sound,blast, rule HOL.refl, simp, simp)
  1073 
  1074 (* NewA *)
  1075 apply (tactic "smp_tac 1 1",frule wt_init_comp_ty,erule impE,blast)
  1076 apply (tactic "forward_hyp_tac 1")
  1077 apply (case_tac "check_neg i' ab")
  1078 apply  (clarsimp simp add: is_acc_type_def)
  1079 apply  (drule (2) halloc_type_sound, rule HOL.refl, simp, simp)
  1080 apply force
  1081 
  1082 (* Level 34, 6 subgoals *)
  1083 
  1084 (* Init *)
  1085 apply (case_tac "inited C (globs s0)")
  1086 apply  (clarsimp)
  1087 apply (clarsimp)
  1088 apply (frule (1) wf_prog_cdecl)
  1089 apply (drule spec, erule impE, erule (3) conforms_init_class_obj)
  1090 apply (drule_tac "psi" = "class G C = ?x" in asm_rl,erule impE,
  1091       force dest!: wf_cdecl_supD split add: split_if simp add: is_acc_class_def)
  1092 apply (drule spec, erule impE, erule conforms_set_locals, rule lconf_empty)
  1093 apply (erule impE) apply (rule exI) apply (erule wf_cdecl_wt_init)
  1094 apply (drule (1) conforms_return, force dest: eval_gext', assumption)
  1095 
  1096 
  1097 (* Ass *)
  1098 apply (tactic "forward_hyp_tac 1")
  1099 apply (rename_tac x1 s1 x2 s2 v va w L C Ta T', case_tac x1)
  1100 prefer 2 apply force
  1101 apply (case_tac x2)
  1102 prefer 2 apply force
  1103 apply (simp, drule conjunct2)
  1104 apply (drule (1) conf_widen)
  1105 apply  (erule wf_ws_prog)
  1106 apply (erule (2) Ass_lemma)
  1107 apply (clarsimp simp add: assign_conforms_def)
  1108 
  1109 (* Try *)
  1110 apply (drule (1) sxalloc_type_sound, simp (no_asm_use))
  1111 apply (case_tac a)
  1112 apply  clarsimp
  1113 apply clarsimp
  1114 apply (tactic "smp_tac 1 1")
  1115 apply (simp split add: split_if_asm)
  1116 apply (fast dest: conforms_deallocL Try_lemma)
  1117 
  1118 (* FVar *)
  1119 
  1120 apply (frule accfield_fields)
  1121 apply (frule ty_expr_is_type [THEN type_is_class],simp)
  1122 apply simp
  1123 apply (frule wf_ws_prog)
  1124 apply (frule (1) fields_declC,simp)
  1125 apply clarsimp 
  1126 (*b y EVERY'[datac cfield_defpl_is_class 2, Clarsimp_tac] 1; not useful here*)
  1127 apply (tactic "smp_tac 1 1")
  1128 apply (tactic "forward_hyp_tac 1")
  1129 apply (rule conjI, force split add: split_if simp add: fvar_def2)
  1130 apply (drule init_yields_initd, frule eval_gext')
  1131 apply clarsimp
  1132 apply (case_tac "C=Object")
  1133 apply  clarsimp
  1134 apply (erule (9) FVar_lemma)
  1135 
  1136 (* AVar *)
  1137 apply (tactic "forward_hyp_tac 1")
  1138 apply (erule_tac V = "G\<turnstile>Norm s0 \<midarrow>?e1-\<succ>?a'\<rightarrow> (?x1 1, ?s1)" in thin_rl, 
  1139          frule eval_gext')
  1140 apply (rule conjI)
  1141 apply  (clarsimp simp add: avar_def2)
  1142 apply clarsimp
  1143 apply (erule (5) AVar_lemma)
  1144 
  1145 (* Call *)
  1146 apply (tactic "forward_hyp_tac 1")
  1147 apply (rule Call_type_sound)
  1148 apply auto
  1149 done
  1150 
  1151 
  1152 declare fun_upd_apply [simp]
  1153 declare split_paired_All [simp] split_paired_Ex [simp]
  1154 declare split_if     [split] split_if_asm     [split] 
  1155         option.split [split] option.split_asm [split]
  1156 ML_setup {* 
  1157 simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac);
  1158 claset_ref()  := claset () addSbefore ("split_all_tac", split_all_tac)
  1159 *}
  1160 
  1161 theorem eval_ts: 
  1162  "\<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>v \<rightarrow> (x',s'); wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-T\<rbrakk> 
  1163 \<Longrightarrow>  (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T)"
  1164 apply (drule (3) eval_type_sound)
  1165 apply (unfold Let_def)
  1166 apply clarsimp
  1167 done
  1168 
  1169 theorem evals_ts: 
  1170 "\<lbrakk>G\<turnstile>s \<midarrow>es\<doteq>\<succ>vs\<rightarrow> (x',s'); wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>es\<Colon>\<doteq>Ts\<rbrakk> 
  1171 \<Longrightarrow>  (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> list_all2 (conf G s') vs Ts)"
  1172 apply (drule (3) eval_type_sound)
  1173 apply (unfold Let_def)
  1174 apply clarsimp
  1175 done
  1176 
  1177 theorem evar_ts: 
  1178 "\<lbrakk>G\<turnstile>s \<midarrow>v=\<succ>vf\<rightarrow> (x',s'); wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>v\<Colon>=T\<rbrakk> \<Longrightarrow>  
  1179   (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> G,L,s'\<turnstile>In2 v\<succ>In2 vf\<Colon>\<preceq>Inl T)"
  1180 apply (drule (3) eval_type_sound)
  1181 apply (unfold Let_def)
  1182 apply clarsimp
  1183 done
  1184 
  1185 theorem exec_ts: 
  1186 "\<lbrakk>G\<turnstile>s \<midarrow>s0\<rightarrow> s'; wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>s0\<Colon>\<surd>\<rbrakk> \<Longrightarrow> s'\<Colon>\<preceq>(G,L)"
  1187 apply (drule (3) eval_type_sound)
  1188 apply (unfold Let_def)
  1189 apply clarsimp
  1190 done
  1191 
  1192 (*
  1193 theorem dyn_methods_understood: 
  1194  "\<And>s. \<lbrakk>wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>{t,md,IntVir}e..mn({pTs'}ps)\<Colon>-rT;  
  1195   s\<Colon>\<preceq>(G,L); G\<turnstile>s \<midarrow>e-\<succ>a'\<rightarrow> Norm s'; a' \<noteq> Null\<rbrakk> \<Longrightarrow>  
  1196   \<exists>a obj. a'=Addr a \<and> heap s' a = Some obj \<and> 
  1197   cmethd G (obj_class obj) (mn, pTs') \<noteq> None"
  1198 apply (erule wt_elim_cases)
  1199 apply (drule max_spec2mheads)
  1200 apply (drule (3) eval_ts)
  1201 apply (clarsimp split del: split_if split_if_asm)
  1202 apply (drule (2) DynT_propI)
  1203 apply  (simp (no_asm_simp))
  1204 apply (tactic *) (* {* exhaust_cmethd_tac "the (cmethd G (target (invmode m e) s' a' md) (mn, pTs'))" 1 *} *)(*)
  1205 apply (drule (4) DynT_mheadsD [THEN conjunct1], rule HOL.refl)
  1206 apply (drule conf_RefTD)
  1207 apply clarsimp
  1208 done 
  1209 *)
  1210 
  1211 end