src/HOL/Bali/WellForm.thy
author wenzelm
Sat Oct 17 14:43:18 2009 +0200 (2009-10-17)
changeset 32960 69916a850301
parent 30235 58d147683393
child 34915 7894c7dab132
permissions -rw-r--r--
eliminated hard tabulators, guessing at each author's individual tab-width;
tuned headers;
     1 (*  Title:      HOL/Bali/WellForm.thy
     2     ID:         $Id$
     3     Author:     David von Oheimb and Norbert Schirmer
     4 *)
     5 
     6 header {* Well-formedness of Java programs *}
     7 theory WellForm imports DefiniteAssignment begin
     8 
     9 text {*
    10 For static checks on expressions and statements, see WellType.thy
    11 
    12 improvements over Java Specification 1.0 (cf. 8.4.6.3, 8.4.6.4, 9.4.1):
    13 \begin{itemize}
    14 \item a method implementing or overwriting another method may have a result 
    15       type that widens to the result type of the other method 
    16       (instead of identical type)
    17 \item if a method hides another method (both methods have to be static!)
    18   there are no restrictions to the result type 
    19   since the methods have to be static and there is no dynamic binding of 
    20   static methods
    21 \item if an interface inherits more than one method with the same signature, the
    22   methods need not have identical return types
    23 \end{itemize}
    24 simplifications:
    25 \begin{itemize}
    26 \item Object and standard exceptions are assumed to be declared like normal 
    27       classes
    28 \end{itemize}
    29 *}
    30 
    31 section "well-formed field declarations"
    32 text  {* well-formed field declaration (common part for classes and interfaces),
    33         cf. 8.3 and (9.3) *}
    34 
    35 constdefs
    36   wf_fdecl :: "prog \<Rightarrow> pname \<Rightarrow> fdecl \<Rightarrow> bool"
    37  "wf_fdecl G P \<equiv> \<lambda>(fn,f). is_acc_type G P (type f)"
    38 
    39 lemma wf_fdecl_def2: "\<And>fd. wf_fdecl G P fd = is_acc_type G P (type (snd fd))"
    40 apply (unfold wf_fdecl_def)
    41 apply simp
    42 done
    43 
    44 
    45 
    46 section "well-formed method declarations"
    47   (*well-formed method declaration,cf. 8.4, 8.4.1, 8.4.3, 8.4.5, 14.3.2, (9.4)*)
    48   (* cf. 14.15, 15.7.2, for scope issues cf. 8.4.1 and 14.3.2 *)
    49 
    50 text {*
    51 A method head is wellformed if:
    52 \begin{itemize}
    53 \item the signature and the method head agree in the number of parameters
    54 \item all types of the parameters are visible
    55 \item the result type is visible
    56 \item the parameter names are unique
    57 \end{itemize} 
    58 *}
    59 constdefs
    60   wf_mhead :: "prog \<Rightarrow> pname \<Rightarrow> sig \<Rightarrow> mhead \<Rightarrow> bool"
    61  "wf_mhead G P \<equiv> \<lambda> sig mh. length (parTs sig) = length (pars mh) \<and>
    62                             \<spacespace> ( \<forall>T\<in>set (parTs sig). is_acc_type G P T) \<and> 
    63                             is_acc_type G P (resTy mh) \<and>
    64                             \<spacespace> distinct (pars mh)"
    65 
    66 
    67 text {*
    68 A method declaration is wellformed if:
    69 \begin{itemize}
    70 \item the method head is wellformed
    71 \item the names of the local variables are unique
    72 \item the types of the local variables must be accessible
    73 \item the local variables don't shadow the parameters
    74 \item the class of the method is defined
    75 \item the body statement is welltyped with respect to the
    76       modified environment of local names, were the local variables, 
    77       the parameters the special result variable (Res) and This are assoziated
    78       with there types. 
    79 \end{itemize}
    80 *}
    81 
    82 constdefs callee_lcl:: "qtname \<Rightarrow> sig \<Rightarrow> methd \<Rightarrow> lenv"
    83 "callee_lcl C sig m 
    84  \<equiv> \<lambda> k. (case k of
    85             EName e 
    86             \<Rightarrow> (case e of 
    87                   VNam v 
    88                   \<Rightarrow>(table_of (lcls (mbody m))((pars m)[\<mapsto>](parTs sig))) v
    89                 | Res \<Rightarrow> Some (resTy m))
    90           | This \<Rightarrow> if is_static m then None else Some (Class C))"
    91 
    92 constdefs parameters :: "methd \<Rightarrow> lname set"
    93 "parameters m \<equiv>  set (map (EName \<circ> VNam) (pars m)) 
    94                   \<union> (if (static m) then {} else {This})"
    95 
    96 constdefs
    97   wf_mdecl :: "prog \<Rightarrow> qtname \<Rightarrow> mdecl \<Rightarrow> bool"
    98  "wf_mdecl G C \<equiv> 
    99       \<lambda>(sig,m).
   100           wf_mhead G (pid C) sig (mhead m) \<and> 
   101           unique (lcls (mbody m)) \<and> 
   102           (\<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T) \<and> 
   103           (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None) \<and>
   104           jumpNestingOkS {Ret} (stmt (mbody m)) \<and> 
   105           is_class G C \<and>
   106           \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>(stmt (mbody m))\<Colon>\<surd> \<and>
   107           (\<exists> A. \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr> 
   108                 \<turnstile> parameters m \<guillemotright>\<langle>stmt (mbody m)\<rangle>\<guillemotright> A 
   109                \<and> Result \<in> nrm A)"
   110 
   111 lemma callee_lcl_VNam_simp [simp]:
   112 "callee_lcl C sig m (EName (VNam v)) 
   113   = (table_of (lcls (mbody m))((pars m)[\<mapsto>](parTs sig))) v"
   114 by (simp add: callee_lcl_def)
   115  
   116 lemma callee_lcl_Res_simp [simp]:
   117 "callee_lcl C sig m (EName Res) = Some (resTy m)" 
   118 by (simp add: callee_lcl_def)
   119 
   120 lemma callee_lcl_This_simp [simp]:
   121 "callee_lcl C sig m (This) = (if is_static m then None else Some (Class C))" 
   122 by (simp add: callee_lcl_def)
   123 
   124 lemma callee_lcl_This_static_simp:
   125 "is_static m \<Longrightarrow> callee_lcl C sig m (This) = None"
   126 by simp
   127 
   128 lemma callee_lcl_This_not_static_simp:
   129 "\<not> is_static m \<Longrightarrow> callee_lcl C sig m (This) = Some (Class C)"
   130 by simp
   131 
   132 lemma wf_mheadI: 
   133 "\<lbrakk>length (parTs sig) = length (pars m); \<forall>T\<in>set (parTs sig). is_acc_type G P T;
   134   is_acc_type G P (resTy m); distinct (pars m)\<rbrakk> \<Longrightarrow>  
   135   wf_mhead G P sig m"
   136 apply (unfold wf_mhead_def)
   137 apply (simp (no_asm_simp))
   138 done
   139 
   140 lemma wf_mdeclI: "\<lbrakk>  
   141   wf_mhead G (pid C) sig (mhead m); unique (lcls (mbody m));  
   142   (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None); 
   143   \<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T;
   144   jumpNestingOkS {Ret} (stmt (mbody m));
   145   is_class G C;
   146   \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>(stmt (mbody m))\<Colon>\<surd>;
   147   (\<exists> A. \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr> \<turnstile> parameters m \<guillemotright>\<langle>stmt (mbody m)\<rangle>\<guillemotright> A
   148         \<and> Result \<in> nrm A)
   149   \<rbrakk> \<Longrightarrow>  
   150   wf_mdecl G C (sig,m)"
   151 apply (unfold wf_mdecl_def)
   152 apply simp
   153 done
   154 
   155 lemma wf_mdeclE [consumes 1]:  
   156   "\<lbrakk>wf_mdecl G C (sig,m); 
   157     \<lbrakk>wf_mhead G (pid C) sig (mhead m); unique (lcls (mbody m));  
   158      \<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None; 
   159      \<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T;
   160      jumpNestingOkS {Ret} (stmt (mbody m));
   161      is_class G C;
   162      \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>(stmt (mbody m))\<Colon>\<surd>;
   163    (\<exists> A. \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile> parameters m \<guillemotright>\<langle>stmt (mbody m)\<rangle>\<guillemotright> A
   164         \<and> Result \<in> nrm A)
   165     \<rbrakk> \<Longrightarrow> P
   166   \<rbrakk> \<Longrightarrow> P"
   167 by (unfold wf_mdecl_def) simp
   168 
   169 
   170 lemma wf_mdeclD1: 
   171 "wf_mdecl G C (sig,m) \<Longrightarrow>  
   172    wf_mhead G (pid C) sig (mhead m) \<and> unique (lcls (mbody m)) \<and>  
   173   (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None) \<and> 
   174   (\<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T)"
   175 apply (unfold wf_mdecl_def)
   176 apply simp
   177 done
   178 
   179 lemma wf_mdecl_bodyD: 
   180 "wf_mdecl G C (sig,m) \<Longrightarrow>  
   181  (\<exists>T. \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>Body C (stmt (mbody m))\<Colon>-T \<and> 
   182       G\<turnstile>T\<preceq>(resTy m))"
   183 apply (unfold wf_mdecl_def)
   184 apply clarify
   185 apply (rule_tac x="(resTy m)" in exI)
   186 apply (unfold wf_mhead_def)
   187 apply (auto simp add: wf_mhead_def is_acc_type_def intro: wt.Body )
   188 done
   189 
   190 
   191 (*
   192 lemma static_Object_methodsE [elim!]: 
   193  "\<lbrakk>wf_mdecl G Object (sig, m);static m\<rbrakk> \<Longrightarrow> R"
   194 apply (unfold wf_mdecl_def)
   195 apply auto
   196 done
   197 *)
   198 
   199 lemma rT_is_acc_type: 
   200   "wf_mhead G P sig m \<Longrightarrow> is_acc_type G P (resTy m)"
   201 apply (unfold wf_mhead_def)
   202 apply auto
   203 done
   204 
   205 section "well-formed interface declarations"
   206   (* well-formed interface declaration, cf. 9.1, 9.1.2.1, 9.1.3, 9.4 *)
   207 
   208 text {*
   209 A interface declaration is wellformed if:
   210 \begin{itemize}
   211 \item the interface hierarchy is wellstructured
   212 \item there is no class with the same name
   213 \item the method heads are wellformed and not static and have Public access
   214 \item the methods are uniquely named
   215 \item all superinterfaces are accessible
   216 \item the result type of a method overriding a method of Object widens to the
   217       result type of the overridden method.
   218       Shadowing static methods is forbidden.
   219 \item the result type of a method overriding a set of methods defined in the
   220       superinterfaces widens to each of the corresponding result types
   221 \end{itemize}
   222 *}
   223 constdefs
   224   wf_idecl :: "prog  \<Rightarrow> idecl \<Rightarrow> bool"
   225  "wf_idecl G \<equiv> 
   226     \<lambda>(I,i). 
   227         ws_idecl G I (isuperIfs i) \<and> 
   228         \<not>is_class G I \<and>
   229         (\<forall>(sig,mh)\<in>set (imethods i). wf_mhead G (pid I) sig mh \<and> 
   230                                      \<not>is_static mh \<and>
   231                                       accmodi mh = Public) \<and>
   232         unique (imethods i) \<and>
   233         (\<forall> J\<in>set (isuperIfs i). is_acc_iface G (pid I) J) \<and>
   234         (table_of (imethods i)
   235           hiding (methd G Object)
   236           under  (\<lambda> new old. accmodi old \<noteq> Private)
   237           entails (\<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old \<and> 
   238                              is_static new = is_static old)) \<and> 
   239         (Option.set \<circ> table_of (imethods i) 
   240                hidings Un_tables((\<lambda>J.(imethds G J))`set (isuperIfs i))
   241                entails (\<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old))"
   242 
   243 lemma wf_idecl_mhead: "\<lbrakk>wf_idecl G (I,i); (sig,mh)\<in>set (imethods i)\<rbrakk> \<Longrightarrow>  
   244   wf_mhead G (pid I) sig mh \<and> \<not>is_static mh \<and> accmodi mh = Public"
   245 apply (unfold wf_idecl_def)
   246 apply auto
   247 done
   248 
   249 lemma wf_idecl_hidings: 
   250 "wf_idecl G (I, i) \<Longrightarrow> 
   251   (\<lambda>s. Option.set (table_of (imethods i) s)) 
   252   hidings Un_tables ((\<lambda>J. imethds G J) ` set (isuperIfs i))  
   253   entails \<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old"
   254 apply (unfold wf_idecl_def o_def)
   255 apply simp
   256 done
   257 
   258 lemma wf_idecl_hiding:
   259 "wf_idecl G (I, i) \<Longrightarrow> 
   260  (table_of (imethods i)
   261            hiding (methd G Object)
   262            under  (\<lambda> new old. accmodi old \<noteq> Private)
   263            entails (\<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old \<and> 
   264                               is_static new = is_static old))"
   265 apply (unfold wf_idecl_def)
   266 apply simp
   267 done
   268 
   269 lemma wf_idecl_supD: 
   270 "\<lbrakk>wf_idecl G (I,i); J \<in> set (isuperIfs i)\<rbrakk> 
   271  \<Longrightarrow> is_acc_iface G (pid I) J \<and> (J, I) \<notin> (subint1 G)^+"
   272 apply (unfold wf_idecl_def ws_idecl_def)
   273 apply auto
   274 done
   275 
   276 section "well-formed class declarations"
   277   (* well-formed class declaration, cf. 8.1, 8.1.2.1, 8.1.2.2, 8.1.3, 8.1.4 and
   278    class method declaration, cf. 8.4.3.3, 8.4.6.1, 8.4.6.2, 8.4.6.3, 8.4.6.4 *)
   279 
   280 text {*
   281 A class declaration is wellformed if:
   282 \begin{itemize}
   283 \item there is no interface with the same name
   284 \item all superinterfaces are accessible and for all methods implementing 
   285       an interface method the result type widens to the result type of 
   286       the interface method, the method is not static and offers at least 
   287       as much access 
   288       (this actually means that the method has Public access, since all 
   289       interface methods have public access)
   290 \item all field declarations are wellformed and the field names are unique
   291 \item all method declarations are wellformed and the method names are unique
   292 \item the initialization statement is welltyped
   293 \item the classhierarchy is wellstructured
   294 \item Unless the class is Object:
   295       \begin{itemize}
   296       \item the superclass is accessible
   297       \item for all methods overriding another method (of a superclass )the
   298             result type widens to the result type of the overridden method,
   299             the access modifier of the new method provides at least as much
   300             access as the overwritten one.
   301       \item for all methods hiding a method (of a superclass) the hidden 
   302             method must be static and offer at least as much access rights.
   303             Remark: In contrast to the Java Language Specification we don't
   304             restrict the result types of the method
   305             (as in case of overriding), because there seems to be no reason,
   306             since there is no dynamic binding of static methods.
   307             (cf. 8.4.6.3 vs. 15.12.1).
   308             Stricly speaking the restrictions on the access rights aren't 
   309             necessary to, since the static type and the access rights 
   310             together determine which method is to be called statically. 
   311             But if a class gains more then one static method with the 
   312             same signature due to inheritance, it is confusing when the 
   313             method selection depends on the access rights only: 
   314             e.g.
   315               Class C declares static public method foo().
   316               Class D is subclass of C and declares static method foo()
   317               with default package access.
   318               D.foo() ? if this call is in the same package as D then
   319                         foo of class D is called, otherwise foo of class C.
   320       \end{itemize}
   321 
   322 \end{itemize}
   323 *}
   324 (* to Table *)
   325 constdefs entails:: "('a,'b) table \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"
   326                                  ("_ entails _" 20)
   327 "t entails P \<equiv> \<forall>k. \<forall> x \<in> t k: P x"
   328 
   329 lemma entailsD:
   330  "\<lbrakk>t entails P; t k = Some x\<rbrakk> \<Longrightarrow> P x"
   331 by (simp add: entails_def)
   332 
   333 lemma empty_entails[simp]: "empty entails P"
   334 by (simp add: entails_def)
   335 
   336 constdefs
   337  wf_cdecl :: "prog \<Rightarrow> cdecl \<Rightarrow> bool"
   338 "wf_cdecl G \<equiv> 
   339    \<lambda>(C,c).
   340       \<not>is_iface G C \<and>
   341       (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and>
   342         (\<forall>s. \<forall> im \<in> imethds G I s.
   343             (\<exists> cm \<in> methd  G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and>
   344                                      \<not> is_static cm \<and>
   345                                      accmodi im \<le> accmodi cm))) \<and>
   346       (\<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f) \<and> unique (cfields c) \<and> 
   347       (\<forall>m\<in>set (methods c). wf_mdecl G C m) \<and> unique (methods c) \<and>
   348       jumpNestingOkS {} (init c) \<and>
   349       (\<exists> A. \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<guillemotright> A) \<and>
   350       \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd> \<and> ws_cdecl G C (super c) \<and>
   351       (C \<noteq> Object \<longrightarrow> 
   352             (is_acc_class G (pid C) (super c) \<and>
   353             (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
   354              entails (\<lambda> new. \<forall> old sig. 
   355                        (G,sig\<turnstile>new overrides\<^sub>S old 
   356                         \<longrightarrow> (G\<turnstile>resTy new\<preceq>resTy old \<and>
   357                              accmodi old \<le> accmodi new \<and>
   358                              \<not>is_static old)) \<and>
   359                        (G,sig\<turnstile>new hides old 
   360                          \<longrightarrow> (accmodi old \<le> accmodi new \<and>
   361                               is_static old)))) 
   362             ))"
   363 
   364 (*
   365 constdefs
   366  wf_cdecl :: "prog \<Rightarrow> cdecl \<Rightarrow> bool"
   367 "wf_cdecl G \<equiv> 
   368    \<lambda>(C,c).
   369       \<not>is_iface G C \<and>
   370       (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and>
   371         (\<forall>s. \<forall> im \<in> imethds G I s.
   372             (\<exists> cm \<in> methd  G C s: G\<turnstile>resTy (mthd cm)\<preceq>resTy (mthd im) \<and>
   373                                      \<not> is_static cm \<and>
   374                                      accmodi im \<le> accmodi cm))) \<and>
   375       (\<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f) \<and> unique (cfields c) \<and> 
   376       (\<forall>m\<in>set (methods c). wf_mdecl G C m) \<and> unique (methods c) \<and> 
   377       \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd> \<and> ws_cdecl G C (super c) \<and>
   378       (C \<noteq> Object \<longrightarrow> 
   379             (is_acc_class G (pid C) (super c) \<and>
   380             (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
   381               hiding methd G (super c)
   382               under (\<lambda> new old. G\<turnstile>new overrides old)
   383               entails (\<lambda> new old. 
   384                            (G\<turnstile>resTy (mthd new)\<preceq>resTy (mthd old) \<and>
   385                             accmodi old \<le> accmodi new \<and>
   386                            \<not> is_static old)))  \<and>
   387             (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
   388               hiding methd G (super c)
   389               under (\<lambda> new old. G\<turnstile>new hides old)
   390               entails (\<lambda> new old. is_static old \<and> 
   391                                   accmodi old \<le> accmodi new))  \<and>
   392             (table_of (cfields c) hiding accfield G C (super c)
   393               entails (\<lambda> newF oldF. accmodi oldF \<le> access newF))))"
   394 *)
   395 
   396 lemma wf_cdeclE [consumes 1]: 
   397  "\<lbrakk>wf_cdecl G (C,c);
   398    \<lbrakk>\<not>is_iface G C;
   399     (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and>
   400         (\<forall>s. \<forall> im \<in> imethds G I s.
   401             (\<exists> cm \<in> methd  G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and>
   402                                      \<not> is_static cm \<and>
   403                                      accmodi im \<le> accmodi cm))); 
   404       \<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f; unique (cfields c); 
   405       \<forall>m\<in>set (methods c). wf_mdecl G C m; unique (methods c);
   406       jumpNestingOkS {} (init c);
   407       \<exists> A. \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<guillemotright> A;
   408       \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd>; 
   409       ws_cdecl G C (super c); 
   410       (C \<noteq> Object \<longrightarrow> 
   411             (is_acc_class G (pid C) (super c) \<and>
   412             (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
   413              entails (\<lambda> new. \<forall> old sig. 
   414                        (G,sig\<turnstile>new overrides\<^sub>S old 
   415                         \<longrightarrow> (G\<turnstile>resTy new\<preceq>resTy old \<and>
   416                              accmodi old \<le> accmodi new \<and>
   417                              \<not>is_static old)) \<and>
   418                        (G,sig\<turnstile>new hides old 
   419                          \<longrightarrow> (accmodi old \<le> accmodi new \<and>
   420                               is_static old)))) 
   421             ))\<rbrakk> \<Longrightarrow> P
   422   \<rbrakk> \<Longrightarrow> P"
   423 by (unfold wf_cdecl_def) simp
   424 
   425 lemma wf_cdecl_unique: 
   426 "wf_cdecl G (C,c) \<Longrightarrow> unique (cfields c) \<and> unique (methods c)"
   427 apply (unfold wf_cdecl_def)
   428 apply auto
   429 done
   430 
   431 lemma wf_cdecl_fdecl: 
   432 "\<lbrakk>wf_cdecl G (C,c); f\<in>set (cfields c)\<rbrakk> \<Longrightarrow> wf_fdecl G (pid C) f"
   433 apply (unfold wf_cdecl_def)
   434 apply auto
   435 done
   436 
   437 lemma wf_cdecl_mdecl: 
   438 "\<lbrakk>wf_cdecl G (C,c); m\<in>set (methods c)\<rbrakk> \<Longrightarrow> wf_mdecl G C m"
   439 apply (unfold wf_cdecl_def)
   440 apply auto
   441 done
   442 
   443 lemma wf_cdecl_impD: 
   444 "\<lbrakk>wf_cdecl G (C,c); I\<in>set (superIfs c)\<rbrakk> 
   445 \<Longrightarrow> is_acc_iface G (pid C) I \<and>  
   446     (\<forall>s. \<forall>im \<in> imethds G I s.  
   447         (\<exists>cm \<in> methd G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and> \<not>is_static cm \<and>
   448                                    accmodi im \<le> accmodi cm))"
   449 apply (unfold wf_cdecl_def)
   450 apply auto
   451 done
   452 
   453 lemma wf_cdecl_supD: 
   454 "\<lbrakk>wf_cdecl G (C,c); C \<noteq> Object\<rbrakk> \<Longrightarrow>  
   455   is_acc_class G (pid C) (super c) \<and> (super c,C) \<notin> (subcls1 G)^+ \<and> 
   456    (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
   457     entails (\<lambda> new. \<forall> old sig. 
   458                  (G,sig\<turnstile>new overrides\<^sub>S old 
   459                   \<longrightarrow> (G\<turnstile>resTy new\<preceq>resTy old \<and>
   460                        accmodi old \<le> accmodi new \<and>
   461                        \<not>is_static old)) \<and>
   462                  (G,sig\<turnstile>new hides old 
   463                    \<longrightarrow> (accmodi old \<le> accmodi new \<and>
   464                         is_static old))))"
   465 apply (unfold wf_cdecl_def ws_cdecl_def)
   466 apply auto
   467 done
   468 
   469 
   470 lemma wf_cdecl_overrides_SomeD:
   471 "\<lbrakk>wf_cdecl G (C,c); C \<noteq> Object; table_of (methods c) sig = Some newM;
   472   G,sig\<turnstile>(C,newM) overrides\<^sub>S old
   473 \<rbrakk> \<Longrightarrow>  G\<turnstile>resTy newM\<preceq>resTy old \<and>
   474        accmodi old \<le> accmodi newM \<and>
   475        \<not> is_static old" 
   476 apply (drule (1) wf_cdecl_supD)
   477 apply (clarify)
   478 apply (drule entailsD)
   479 apply   (blast intro: table_of_map_SomeI)
   480 apply (drule_tac x="old" in spec)
   481 apply (auto dest: overrides_eq_sigD simp add: msig_def)
   482 done
   483 
   484 lemma wf_cdecl_hides_SomeD:
   485 "\<lbrakk>wf_cdecl G (C,c); C \<noteq> Object; table_of (methods c) sig = Some newM;
   486   G,sig\<turnstile>(C,newM) hides old
   487 \<rbrakk> \<Longrightarrow>  accmodi old \<le> access newM \<and>
   488        is_static old" 
   489 apply (drule (1) wf_cdecl_supD)
   490 apply (clarify)
   491 apply (drule entailsD)
   492 apply   (blast intro: table_of_map_SomeI)
   493 apply (drule_tac x="old" in spec)
   494 apply (auto dest: hides_eq_sigD simp add: msig_def)
   495 done
   496 
   497 lemma wf_cdecl_wt_init: 
   498  "wf_cdecl G (C, c) \<Longrightarrow> \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>init c\<Colon>\<surd>"
   499 apply (unfold wf_cdecl_def)
   500 apply auto
   501 done
   502 
   503 
   504 section "well-formed programs"
   505   (* well-formed program, cf. 8.1, 9.1 *)
   506 
   507 text {*
   508 A program declaration is wellformed if:
   509 \begin{itemize}
   510 \item the class ObjectC of Object is defined
   511 \item every method of Object has an access modifier distinct from Package. 
   512       This is
   513       necessary since every interface automatically inherits from Object.  
   514       We must know, that every time a Object method is "overriden" by an 
   515       interface method this is also overriden by the class implementing the
   516       the interface (see @{text "implement_dynmethd and class_mheadsD"})
   517 \item all standard Exceptions are defined
   518 \item all defined interfaces are wellformed
   519 \item all defined classes are wellformed
   520 \end{itemize}
   521 *}
   522 constdefs
   523   wf_prog  :: "prog \<Rightarrow> bool"
   524  "wf_prog G \<equiv> let is = ifaces G; cs = classes G in
   525                  ObjectC \<in> set cs \<and> 
   526                 (\<forall> m\<in>set Object_mdecls. accmodi m \<noteq> Package) \<and>
   527                 (\<forall>xn. SXcptC xn \<in> set cs) \<and>
   528                 (\<forall>i\<in>set is. wf_idecl G i) \<and> unique is \<and>
   529                 (\<forall>c\<in>set cs. wf_cdecl G c) \<and> unique cs"
   530 
   531 lemma wf_prog_idecl: "\<lbrakk>iface G I = Some i; wf_prog G\<rbrakk> \<Longrightarrow> wf_idecl G (I,i)"
   532 apply (unfold wf_prog_def Let_def)
   533 apply simp
   534 apply (fast dest: map_of_SomeD)
   535 done
   536 
   537 lemma wf_prog_cdecl: "\<lbrakk>class G C = Some c; wf_prog G\<rbrakk> \<Longrightarrow> wf_cdecl G (C,c)"
   538 apply (unfold wf_prog_def Let_def)
   539 apply simp
   540 apply (fast dest: map_of_SomeD)
   541 done
   542 
   543 lemma wf_prog_Object_mdecls:
   544 "wf_prog G \<Longrightarrow> (\<forall> m\<in>set Object_mdecls. accmodi m \<noteq> Package)"
   545 apply (unfold wf_prog_def Let_def)
   546 apply simp
   547 done
   548 
   549 lemma wf_prog_acc_superD:
   550  "\<lbrakk>wf_prog G; class G C = Some c; C \<noteq> Object \<rbrakk> 
   551   \<Longrightarrow> is_acc_class G (pid C) (super c)"
   552 by (auto dest: wf_prog_cdecl wf_cdecl_supD)
   553 
   554 lemma wf_ws_prog [elim!,simp]: "wf_prog G \<Longrightarrow> ws_prog G"
   555 apply (unfold wf_prog_def Let_def)
   556 apply (rule ws_progI)
   557 apply  (simp_all (no_asm))
   558 apply  (auto simp add: is_acc_class_def is_acc_iface_def 
   559              dest!: wf_idecl_supD wf_cdecl_supD )+
   560 done
   561 
   562 lemma class_Object [simp]: 
   563 "wf_prog G \<Longrightarrow> 
   564   class G Object = Some \<lparr>access=Public,cfields=[],methods=Object_mdecls,
   565                                   init=Skip,super=undefined,superIfs=[]\<rparr>"
   566 apply (unfold wf_prog_def Let_def ObjectC_def)
   567 apply (fast dest!: map_of_SomeI)
   568 done
   569 
   570 lemma methd_Object[simp]: "wf_prog G \<Longrightarrow> methd G Object =  
   571   table_of (map (\<lambda>(s,m). (s, Object, m)) Object_mdecls)"
   572 apply (subst methd_rec)
   573 apply (auto simp add: Let_def)
   574 done
   575 
   576 lemma wf_prog_Object_methd:
   577 "\<lbrakk>wf_prog G; methd G Object sig = Some m\<rbrakk> \<Longrightarrow> accmodi m \<noteq> Package"
   578 by (auto dest!: wf_prog_Object_mdecls) (auto dest!: map_of_SomeD) 
   579 
   580 lemma wf_prog_Object_is_public[intro]:
   581  "wf_prog G \<Longrightarrow> is_public G Object"
   582 by (auto simp add: is_public_def dest: class_Object)
   583 
   584 lemma class_SXcpt [simp]: 
   585 "wf_prog G \<Longrightarrow> 
   586   class G (SXcpt xn) = Some \<lparr>access=Public,cfields=[],methods=SXcpt_mdecls,
   587                                    init=Skip,
   588                                    super=if xn = Throwable then Object 
   589                                                            else SXcpt Throwable,
   590                                    superIfs=[]\<rparr>"
   591 apply (unfold wf_prog_def Let_def SXcptC_def)
   592 apply (fast dest!: map_of_SomeI)
   593 done
   594 
   595 lemma wf_ObjectC [simp]: 
   596         "wf_cdecl G ObjectC = (\<not>is_iface G Object \<and> Ball (set Object_mdecls)
   597   (wf_mdecl G Object) \<and> unique Object_mdecls)"
   598 apply (unfold wf_cdecl_def ws_cdecl_def ObjectC_def)
   599 apply (auto intro: da.Skip)
   600 done
   601 
   602 lemma Object_is_class [simp,elim!]: "wf_prog G \<Longrightarrow> is_class G Object"
   603 apply (simp (no_asm_simp))
   604 done
   605  
   606 lemma Object_is_acc_class [simp,elim!]: "wf_prog G \<Longrightarrow> is_acc_class G S Object"
   607 apply (simp (no_asm_simp) add: is_acc_class_def is_public_def
   608                                accessible_in_RefT_simp)
   609 done
   610 
   611 lemma SXcpt_is_class [simp,elim!]: "wf_prog G \<Longrightarrow> is_class G (SXcpt xn)"
   612 apply (simp (no_asm_simp))
   613 done
   614 
   615 lemma SXcpt_is_acc_class [simp,elim!]: 
   616 "wf_prog G \<Longrightarrow> is_acc_class G S (SXcpt xn)"
   617 apply (simp (no_asm_simp) add: is_acc_class_def is_public_def
   618                                accessible_in_RefT_simp)
   619 done
   620 
   621 lemma fields_Object [simp]: "wf_prog G \<Longrightarrow> DeclConcepts.fields G Object = []"
   622 by (force intro: fields_emptyI)
   623 
   624 lemma accfield_Object [simp]: 
   625  "wf_prog G \<Longrightarrow> accfield G S Object = empty"
   626 apply (unfold accfield_def)
   627 apply (simp (no_asm_simp) add: Let_def)
   628 done
   629 
   630 lemma fields_Throwable [simp]: 
   631  "wf_prog G \<Longrightarrow> DeclConcepts.fields G (SXcpt Throwable) = []"
   632 by (force intro: fields_emptyI)
   633 
   634 lemma fields_SXcpt [simp]: "wf_prog G \<Longrightarrow> DeclConcepts.fields G (SXcpt xn) = []"
   635 apply (case_tac "xn = Throwable")
   636 apply  (simp (no_asm_simp))
   637 by (force intro: fields_emptyI)
   638 
   639 lemmas widen_trans = ws_widen_trans [OF _ _ wf_ws_prog, elim]
   640 lemma widen_trans2 [elim]: "\<lbrakk>G\<turnstile>U\<preceq>T; G\<turnstile>S\<preceq>U; wf_prog G\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T"
   641 apply (erule (2) widen_trans)
   642 done
   643 
   644 lemma Xcpt_subcls_Throwable [simp]: 
   645 "wf_prog G \<Longrightarrow> G\<turnstile>SXcpt xn\<preceq>\<^sub>C SXcpt Throwable"
   646 apply (rule SXcpt_subcls_Throwable_lemma)
   647 apply auto
   648 done
   649 
   650 lemma unique_fields: 
   651  "\<lbrakk>is_class G C; wf_prog G\<rbrakk> \<Longrightarrow> unique (DeclConcepts.fields G C)"
   652 apply (erule ws_unique_fields)
   653 apply  (erule wf_ws_prog)
   654 apply (erule (1) wf_prog_cdecl [THEN wf_cdecl_unique [THEN conjunct1]])
   655 done
   656 
   657 lemma fields_mono: 
   658 "\<lbrakk>table_of (DeclConcepts.fields G C) fn = Some f; G\<turnstile>D\<preceq>\<^sub>C C; 
   659   is_class G D; wf_prog G\<rbrakk> 
   660    \<Longrightarrow> table_of (DeclConcepts.fields G D) fn = Some f"
   661 apply (rule map_of_SomeI)
   662 apply  (erule (1) unique_fields)
   663 apply (erule (1) map_of_SomeD [THEN fields_mono_lemma])
   664 apply (erule wf_ws_prog)
   665 done
   666 
   667 
   668 lemma fields_is_type [elim]: 
   669 "\<lbrakk>table_of (DeclConcepts.fields G C) m = Some f; wf_prog G; is_class G C\<rbrakk> \<Longrightarrow> 
   670       is_type G (type f)"
   671 apply (frule wf_ws_prog)
   672 apply (force dest: fields_declC [THEN conjunct1] 
   673                    wf_prog_cdecl [THEN wf_cdecl_fdecl]
   674              simp add: wf_fdecl_def2 is_acc_type_def)
   675 done
   676 
   677 lemma imethds_wf_mhead [rule_format (no_asm)]: 
   678 "\<lbrakk>m \<in> imethds G I sig; wf_prog G; is_iface G I\<rbrakk> \<Longrightarrow>  
   679   wf_mhead G (pid (decliface m)) sig (mthd m) \<and> 
   680   \<not> is_static m \<and> accmodi m = Public"
   681 apply (frule wf_ws_prog)
   682 apply (drule (2) imethds_declI [THEN conjunct1])
   683 apply clarify
   684 apply (frule_tac I="(decliface m)" in wf_prog_idecl,assumption)
   685 apply (drule wf_idecl_mhead)
   686 apply (erule map_of_SomeD)
   687 apply (cases m, simp)
   688 done
   689 
   690 lemma methd_wf_mdecl: 
   691  "\<lbrakk>methd G C sig = Some m; wf_prog G; class G C = Some y\<rbrakk> \<Longrightarrow>  
   692   G\<turnstile>C\<preceq>\<^sub>C (declclass m) \<and> is_class G (declclass m) \<and> 
   693   wf_mdecl G (declclass m) (sig,(mthd m))"
   694 apply (frule wf_ws_prog)
   695 apply (drule (1) methd_declC)
   696 apply  fast
   697 apply clarsimp
   698 apply (frule (1) wf_prog_cdecl, erule wf_cdecl_mdecl, erule map_of_SomeD)
   699 done
   700 
   701 (*
   702 This lemma doesn't hold!
   703 lemma methd_rT_is_acc_type: 
   704 "\<lbrakk>wf_prog G;methd G C C sig = Some (D,m);
   705     class G C = Some y\<rbrakk>
   706 \<Longrightarrow> is_acc_type G (pid C) (resTy m)"
   707 The result Type is only visible in the scope of defining class D 
   708 "is_vis_type G (pid D) (resTy m)" but not necessarily in scope of class C!
   709 (The same is true for the type of pramaters of a method)
   710 *)
   711 
   712 
   713 lemma methd_rT_is_type: 
   714 "\<lbrakk>wf_prog G;methd G C sig = Some m;
   715     class G C = Some y\<rbrakk>
   716 \<Longrightarrow> is_type G (resTy m)"
   717 apply (drule (2) methd_wf_mdecl)
   718 apply clarify
   719 apply (drule wf_mdeclD1)
   720 apply clarify
   721 apply (drule rT_is_acc_type)
   722 apply (cases m, simp add: is_acc_type_def)
   723 done
   724 
   725 lemma accmethd_rT_is_type:
   726 "\<lbrakk>wf_prog G;accmethd G S C sig = Some m;
   727     class G C = Some y\<rbrakk>
   728 \<Longrightarrow> is_type G (resTy m)"
   729 by (auto simp add: accmethd_def  
   730          intro: methd_rT_is_type)
   731 
   732 lemma methd_Object_SomeD:
   733 "\<lbrakk>wf_prog G;methd G Object sig = Some m\<rbrakk> 
   734  \<Longrightarrow> declclass m = Object"
   735 by (auto dest: class_Object simp add: methd_rec )
   736 
   737 lemma wf_imethdsD: 
   738  "\<lbrakk>im \<in> imethds G I sig;wf_prog G; is_iface G I\<rbrakk> 
   739  \<Longrightarrow> \<not>is_static im \<and> accmodi im = Public"
   740 proof -
   741   assume asm: "wf_prog G" "is_iface G I" "im \<in> imethds G I sig"
   742   have "wf_prog G \<longrightarrow> 
   743          (\<forall> i im. iface G I = Some i \<longrightarrow> im \<in> imethds G I sig
   744                   \<longrightarrow> \<not>is_static im \<and> accmodi im = Public)" (is "?P G I")
   745   proof (rule iface_rec.induct,intro allI impI)
   746     fix G I i im
   747     assume hyp: "\<forall> J i. J \<in> set (isuperIfs i) \<and> ws_prog G \<and> iface G I = Some i
   748                  \<longrightarrow> ?P G J"
   749     assume wf: "wf_prog G" and if_I: "iface G I = Some i" and 
   750            im: "im \<in> imethds G I sig" 
   751     show "\<not>is_static im \<and> accmodi im = Public" 
   752     proof -
   753       let ?inherited = "Un_tables (imethds G ` set (isuperIfs i))"
   754       let ?new = "(Option.set \<circ> table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)))"
   755       from if_I wf im have imethds:"im \<in> (?inherited \<oplus>\<oplus> ?new) sig"
   756         by (simp add: imethds_rec)
   757       from wf if_I have 
   758         wf_supI: "\<forall> J. J \<in> set (isuperIfs i) \<longrightarrow> (\<exists> j. iface G J = Some j)"
   759         by (blast dest: wf_prog_idecl wf_idecl_supD is_acc_ifaceD)
   760       from wf if_I have
   761         "\<forall> im \<in> set (imethods i). \<not> is_static im \<and> accmodi im = Public"
   762         by (auto dest!: wf_prog_idecl wf_idecl_mhead)
   763       then have new_ok: "\<forall> im. table_of (imethods i) sig = Some im 
   764                          \<longrightarrow>  \<not> is_static im \<and> accmodi im = Public"
   765         by (auto dest!: table_of_Some_in_set)
   766       show ?thesis
   767         proof (cases "?new sig = {}")
   768           case True
   769           from True wf wf_supI if_I imethds hyp 
   770           show ?thesis by (auto simp del:  split_paired_All)  
   771         next
   772           case False
   773           from False wf wf_supI if_I imethds new_ok hyp 
   774           show ?thesis by (auto dest: wf_idecl_hidings hidings_entailsD)
   775         qed
   776       qed
   777     qed
   778   with asm show ?thesis by (auto simp del: split_paired_All)
   779 qed
   780 
   781 lemma wf_prog_hidesD:
   782   assumes hides: "G \<turnstile>new hides old" and wf: "wf_prog G"
   783   shows
   784    "accmodi old \<le> accmodi new \<and>
   785     is_static old"
   786 proof -
   787   from hides 
   788   obtain c where 
   789     clsNew: "class G (declclass new) = Some c" and
   790     neqObj: "declclass new \<noteq> Object"
   791     by (auto dest: hidesD declared_in_classD)
   792   with hides obtain newM oldM where
   793     newM: "table_of (methods c) (msig new) = Some newM" and 
   794      new: "new = (declclass new,(msig new),newM)" and
   795      old: "old = (declclass old,(msig old),oldM)" and
   796           "msig new = msig old"
   797     by (cases new,cases old) 
   798        (auto dest: hidesD 
   799          simp add: cdeclaredmethd_def declared_in_def)
   800   with hides 
   801   have hides':
   802         "G,(msig new)\<turnstile>(declclass new,newM) hides (declclass old,oldM)"
   803     by auto
   804   from clsNew wf 
   805   have "wf_cdecl G (declclass new,c)" by (blast intro: wf_prog_cdecl)
   806   note wf_cdecl_hides_SomeD [OF this neqObj newM hides']
   807   with new old 
   808   show ?thesis
   809     by (cases new, cases old) auto
   810 qed
   811 
   812 text {* Compare this lemma about static  
   813 overriding @{term "G \<turnstile>new overrides\<^sub>S old"} with the definition of 
   814 dynamic overriding @{term "G \<turnstile>new overrides old"}. 
   815 Conforming result types and restrictions on the access modifiers of the old 
   816 and the new method are not part of the predicate for static overriding. But
   817 they are enshured in a wellfromed program.  Dynamic overriding has 
   818 no restrictions on the access modifiers but enforces confrom result types 
   819 as precondition. But with some efford we can guarantee the access modifier
   820 restriction for dynamic overriding, too. See lemma 
   821 @{text wf_prog_dyn_override_prop}.
   822 *}
   823 lemma wf_prog_stat_overridesD:
   824   assumes stat_override: "G \<turnstile>new overrides\<^sub>S old" and wf: "wf_prog G"
   825   shows
   826    "G\<turnstile>resTy new\<preceq>resTy old \<and>
   827     accmodi old \<le> accmodi new \<and>
   828     \<not> is_static old"
   829 proof -
   830   from stat_override 
   831   obtain c where 
   832     clsNew: "class G (declclass new) = Some c" and
   833     neqObj: "declclass new \<noteq> Object"
   834     by (auto dest: stat_overrides_commonD declared_in_classD)
   835   with stat_override obtain newM oldM where
   836     newM: "table_of (methods c) (msig new) = Some newM" and 
   837      new: "new = (declclass new,(msig new),newM)" and
   838      old: "old = (declclass old,(msig old),oldM)" and
   839           "msig new = msig old"
   840     by (cases new,cases old) 
   841        (auto dest: stat_overrides_commonD 
   842          simp add: cdeclaredmethd_def declared_in_def)
   843   with stat_override 
   844   have stat_override':
   845         "G,(msig new)\<turnstile>(declclass new,newM) overrides\<^sub>S (declclass old,oldM)"
   846     by auto
   847   from clsNew wf 
   848   have "wf_cdecl G (declclass new,c)" by (blast intro: wf_prog_cdecl)
   849   note wf_cdecl_overrides_SomeD [OF this neqObj newM stat_override']
   850   with new old 
   851   show ?thesis
   852     by (cases new, cases old) auto
   853 qed
   854     
   855 lemma static_to_dynamic_overriding: 
   856   assumes stat_override: "G\<turnstile>new overrides\<^sub>S old" and wf : "wf_prog G"
   857   shows "G\<turnstile>new overrides old"
   858 proof -
   859   from stat_override 
   860   show ?thesis (is "?Overrides new old")
   861   proof (induct)
   862     case (Direct new old superNew)
   863     then have stat_override:"G\<turnstile>new overrides\<^sub>S old" 
   864       by (rule stat_overridesR.Direct)
   865     from stat_override wf
   866     have resTy_widen: "G\<turnstile>resTy new\<preceq>resTy old" and
   867       not_static_old: "\<not> is_static old" 
   868       by (auto dest: wf_prog_stat_overridesD)  
   869     have not_private_new: "accmodi new \<noteq> Private"
   870     proof -
   871       from stat_override 
   872       have "accmodi old \<noteq> Private"
   873         by (rule no_Private_stat_override)
   874       moreover
   875       from stat_override wf
   876       have "accmodi old \<le> accmodi new"
   877         by (auto dest: wf_prog_stat_overridesD)
   878       ultimately
   879       show ?thesis
   880         by (auto dest: acc_modi_bottom)
   881     qed
   882     with Direct resTy_widen not_static_old 
   883     show "?Overrides new old" 
   884       by (auto intro: overridesR.Direct stat_override_declclasses_relation) 
   885   next
   886     case (Indirect new inter old)
   887     then show "?Overrides new old" 
   888       by (blast intro: overridesR.Indirect) 
   889   qed
   890 qed
   891 
   892 lemma non_Package_instance_method_inheritance:
   893   assumes old_inheritable: "G\<turnstile>Method old inheritable_in (pid C)" and
   894               accmodi_old: "accmodi old \<noteq> Package" and 
   895           instance_method: "\<not> is_static old" and
   896                    subcls: "G\<turnstile>C \<prec>\<^sub>C declclass old" and
   897              old_declared: "G\<turnstile>Method old declared_in (declclass old)" and
   898                        wf: "wf_prog G"
   899   shows "G\<turnstile>Method old member_of C \<or>
   900    (\<exists> new. G\<turnstile> new overrides\<^sub>S old \<and> G\<turnstile>Method new member_of C)"
   901 proof -
   902   from wf have ws: "ws_prog G" by auto
   903   from old_declared have iscls_declC_old: "is_class G (declclass old)"
   904     by (auto simp add: declared_in_def cdeclaredmethd_def)
   905   from subcls have  iscls_C: "is_class G C"
   906     by (blast dest:  subcls_is_class)
   907   from iscls_C ws old_inheritable subcls 
   908   show ?thesis (is "?P C old")
   909   proof (induct rule: ws_class_induct')
   910     case Object
   911     assume "G\<turnstile>Object\<prec>\<^sub>C declclass old"
   912     then show "?P Object old"
   913       by blast
   914   next
   915     case (Subcls C c)
   916     assume cls_C: "class G C = Some c" and 
   917        neq_C_Obj: "C \<noteq> Object" and
   918              hyp: "\<lbrakk>G \<turnstile>Method old inheritable_in pid (super c); 
   919                    G\<turnstile>super c\<prec>\<^sub>C declclass old\<rbrakk> \<Longrightarrow> ?P (super c) old" and
   920      inheritable: "G \<turnstile>Method old inheritable_in pid C" and
   921          subclsC: "G\<turnstile>C\<prec>\<^sub>C declclass old"
   922     from cls_C neq_C_Obj  
   923     have super: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 super c" 
   924       by (rule subcls1I)
   925     from wf cls_C neq_C_Obj
   926     have accessible_super: "G\<turnstile>(Class (super c)) accessible_in (pid C)" 
   927       by (auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_classD)
   928     {
   929       fix old
   930       assume    member_super: "G\<turnstile>Method old member_of (super c)"
   931       assume     inheritable: "G \<turnstile>Method old inheritable_in pid C"
   932       assume instance_method: "\<not> is_static old"
   933       from member_super
   934       have old_declared: "G\<turnstile>Method old declared_in (declclass old)"
   935        by (cases old) (auto dest: member_of_declC)
   936       have "?P C old"
   937       proof (cases "G\<turnstile>mid (msig old) undeclared_in C")
   938         case True
   939         with inheritable super accessible_super member_super
   940         have "G\<turnstile>Method old member_of C"
   941           by (cases old) (auto intro: members.Inherited)
   942         then show ?thesis
   943           by auto
   944       next
   945         case False
   946         then obtain new_member where
   947              "G\<turnstile>new_member declared_in C" and
   948              "mid (msig old) = memberid new_member"
   949           by (auto dest: not_undeclared_declared)
   950         then obtain new where
   951                   new: "G\<turnstile>Method new declared_in C" and
   952                eq_sig: "msig old = msig new" and
   953             declC_new: "declclass new = C" 
   954           by (cases new_member) auto
   955         then have member_new: "G\<turnstile>Method new member_of C"
   956           by (cases new) (auto intro: members.Immediate)
   957         from declC_new super member_super
   958         have subcls_new_old: "G\<turnstile>declclass new \<prec>\<^sub>C declclass old"
   959           by (auto dest!: member_of_subclseq_declC
   960                     dest: r_into_trancl intro: trancl_rtrancl_trancl)
   961         show ?thesis
   962         proof (cases "is_static new")
   963           case False
   964           with eq_sig declC_new new old_declared inheritable
   965                super member_super subcls_new_old
   966           have "G\<turnstile>new overrides\<^sub>S old"
   967             by (auto intro!: stat_overridesR.Direct)
   968           with member_new show ?thesis
   969             by blast
   970         next
   971           case True
   972           with eq_sig declC_new subcls_new_old new old_declared inheritable
   973           have "G\<turnstile>new hides old"
   974             by (auto intro: hidesI)    
   975           with wf 
   976           have "is_static old"
   977             by (blast dest: wf_prog_hidesD)
   978           with instance_method
   979           show ?thesis
   980             by (contradiction)
   981         qed
   982       qed
   983     } note hyp_member_super = this
   984     from subclsC cls_C 
   985     have "G\<turnstile>(super c)\<preceq>\<^sub>C declclass old"
   986       by (rule subcls_superD)
   987     then
   988     show "?P C old"
   989     proof (cases rule: subclseq_cases) 
   990       case Eq
   991       assume "super c = declclass old"
   992       with old_declared 
   993       have "G\<turnstile>Method old member_of (super c)" 
   994         by (cases old) (auto intro: members.Immediate)
   995       with inheritable instance_method 
   996       show ?thesis
   997         by (blast dest: hyp_member_super)
   998     next
   999       case Subcls
  1000       assume "G\<turnstile>super c\<prec>\<^sub>C declclass old"
  1001       moreover
  1002       from inheritable accmodi_old
  1003       have "G \<turnstile>Method old inheritable_in pid (super c)"
  1004         by (cases "accmodi old") (auto simp add: inheritable_in_def)
  1005       ultimately
  1006       have "?P (super c) old"
  1007         by (blast dest: hyp)
  1008       then show ?thesis
  1009       proof
  1010         assume "G \<turnstile>Method old member_of super c"
  1011         with inheritable instance_method
  1012         show ?thesis
  1013           by (blast dest: hyp_member_super)
  1014       next
  1015         assume "\<exists>new. G \<turnstile> new overrides\<^sub>S old \<and> G \<turnstile>Method new member_of super c"
  1016         then obtain super_new where
  1017           super_new_override:  "G \<turnstile> super_new overrides\<^sub>S old" and
  1018             super_new_member:  "G \<turnstile>Method super_new member_of super c"
  1019           by blast
  1020         from super_new_override wf
  1021         have "accmodi old \<le> accmodi super_new"
  1022           by (auto dest: wf_prog_stat_overridesD)
  1023         with inheritable accmodi_old
  1024         have "G \<turnstile>Method super_new inheritable_in pid C"
  1025           by (auto simp add: inheritable_in_def 
  1026                       split: acc_modi.splits
  1027                        dest: acc_modi_le_Dests)
  1028         moreover
  1029         from super_new_override 
  1030         have "\<not> is_static super_new"
  1031           by (auto dest: stat_overrides_commonD)
  1032         moreover
  1033         note super_new_member
  1034         ultimately have "?P C super_new"
  1035           by (auto dest: hyp_member_super)
  1036         then show ?thesis
  1037         proof 
  1038           assume "G \<turnstile>Method super_new member_of C"
  1039           with super_new_override
  1040           show ?thesis
  1041             by blast
  1042         next
  1043           assume "\<exists>new. G \<turnstile> new overrides\<^sub>S super_new \<and> 
  1044                   G \<turnstile>Method new member_of C"
  1045           with super_new_override show ?thesis
  1046             by (blast intro: stat_overridesR.Indirect) 
  1047         qed
  1048       qed
  1049     qed
  1050   qed
  1051 qed
  1052 
  1053 lemma non_Package_instance_method_inheritance_cases [consumes 6,
  1054          case_names Inheritance Overriding]:
  1055   assumes old_inheritable: "G\<turnstile>Method old inheritable_in (pid C)" and
  1056               accmodi_old: "accmodi old \<noteq> Package" and 
  1057           instance_method: "\<not> is_static old" and
  1058                    subcls: "G\<turnstile>C \<prec>\<^sub>C declclass old" and
  1059              old_declared: "G\<turnstile>Method old declared_in (declclass old)" and
  1060                        wf: "wf_prog G" and
  1061               inheritance: "G\<turnstile>Method old member_of C \<Longrightarrow> P" and
  1062                overriding: "\<And> new.
  1063                            \<lbrakk>G\<turnstile> new overrides\<^sub>S old;G\<turnstile>Method new member_of C\<rbrakk>
  1064                            \<Longrightarrow> P"
  1065   shows P
  1066 proof -
  1067   from old_inheritable accmodi_old instance_method subcls old_declared wf 
  1068        inheritance overriding
  1069   show ?thesis
  1070     by (auto dest: non_Package_instance_method_inheritance)
  1071 qed
  1072 
  1073 lemma dynamic_to_static_overriding:
  1074   assumes dyn_override: "G\<turnstile> new overrides old" and
  1075            accmodi_old: "accmodi old \<noteq> Package" and
  1076                     wf: "wf_prog G"
  1077   shows "G\<turnstile> new overrides\<^sub>S old"  
  1078 proof - 
  1079   from dyn_override accmodi_old
  1080   show ?thesis (is "?Overrides new old")
  1081   proof (induct rule: overridesR.induct)
  1082     case (Direct new old)
  1083     assume   new_declared: "G\<turnstile>Method new declared_in declclass new"
  1084     assume eq_sig_new_old: "msig new = msig old"
  1085     assume subcls_new_old: "G\<turnstile>declclass new \<prec>\<^sub>C declclass old"
  1086     assume "G \<turnstile>Method old inheritable_in pid (declclass new)" and
  1087            "accmodi old \<noteq> Package" and
  1088            "\<not> is_static old" and
  1089            "G\<turnstile>declclass new\<prec>\<^sub>C declclass old" and
  1090            "G\<turnstile>Method old declared_in declclass old" 
  1091     from this wf
  1092     show "?Overrides new old"
  1093     proof (cases rule: non_Package_instance_method_inheritance_cases)
  1094       case Inheritance
  1095       assume "G \<turnstile>Method old member_of declclass new"
  1096       then have "G\<turnstile>mid (msig old) undeclared_in declclass new"
  1097       proof cases
  1098         case Immediate 
  1099         with subcls_new_old wf show ?thesis     
  1100           by (auto dest: subcls_irrefl)
  1101       next
  1102         case Inherited
  1103         then show ?thesis
  1104           by (cases old) auto
  1105       qed
  1106       with eq_sig_new_old new_declared
  1107       show ?thesis
  1108         by (cases old,cases new) (auto dest!: declared_not_undeclared)
  1109     next
  1110       case (Overriding new') 
  1111       assume stat_override_new': "G \<turnstile> new' overrides\<^sub>S old"
  1112       then have "msig new' = msig old"
  1113         by (auto dest: stat_overrides_commonD)
  1114       with eq_sig_new_old have eq_sig_new_new': "msig new=msig new'"
  1115         by simp
  1116       assume "G \<turnstile>Method new' member_of declclass new"
  1117       then show ?thesis
  1118       proof (cases)
  1119         case Immediate
  1120         then have declC_new: "declclass new' = declclass new" 
  1121           by auto
  1122         from Immediate 
  1123         have "G\<turnstile>Method new' declared_in declclass new"
  1124           by (cases new') auto
  1125         with new_declared eq_sig_new_new' declC_new 
  1126         have "new=new'"
  1127           by (cases new, cases new') (auto dest: unique_declared_in) 
  1128         with stat_override_new'
  1129         show ?thesis
  1130           by simp
  1131       next
  1132         case Inherited
  1133         then have "G\<turnstile>mid (msig new') undeclared_in declclass new"
  1134           by (cases new') (auto)
  1135         with eq_sig_new_new' new_declared
  1136         show ?thesis
  1137           by (cases new,cases new') (auto dest!: declared_not_undeclared)
  1138       qed
  1139     qed
  1140   next
  1141     case (Indirect new inter old)
  1142     assume accmodi_old: "accmodi old \<noteq> Package"
  1143     assume "accmodi old \<noteq> Package \<Longrightarrow> G \<turnstile> inter overrides\<^sub>S old"
  1144     with accmodi_old 
  1145     have stat_override_inter_old: "G \<turnstile> inter overrides\<^sub>S old"
  1146       by blast
  1147     moreover 
  1148     assume hyp_inter: "accmodi inter \<noteq> Package \<Longrightarrow> G \<turnstile> new overrides\<^sub>S inter"
  1149     moreover
  1150     have "accmodi inter \<noteq> Package"
  1151     proof -
  1152       from stat_override_inter_old wf 
  1153       have "accmodi old \<le> accmodi inter"
  1154         by (auto dest: wf_prog_stat_overridesD)
  1155       with stat_override_inter_old accmodi_old
  1156       show ?thesis
  1157         by (auto dest!: no_Private_stat_override
  1158                  split: acc_modi.splits 
  1159                  dest: acc_modi_le_Dests)
  1160     qed
  1161     ultimately show "?Overrides new old"
  1162       by (blast intro: stat_overridesR.Indirect)
  1163   qed
  1164 qed
  1165 
  1166 lemma wf_prog_dyn_override_prop:
  1167   assumes dyn_override: "G \<turnstile> new overrides old" and
  1168                     wf: "wf_prog G"
  1169   shows "accmodi old \<le> accmodi new"
  1170 proof (cases "accmodi old = Package")
  1171   case True
  1172   note old_Package = this
  1173   show ?thesis
  1174   proof (cases "accmodi old \<le> accmodi new")
  1175     case True then show ?thesis .
  1176   next
  1177     case False
  1178     with old_Package 
  1179     have "accmodi new = Private"
  1180       by (cases "accmodi new") (auto simp add: le_acc_def less_acc_def)
  1181     with dyn_override 
  1182     show ?thesis
  1183       by (auto dest: overrides_commonD)
  1184   qed    
  1185 next
  1186   case False
  1187   with dyn_override wf
  1188   have "G \<turnstile> new overrides\<^sub>S old"
  1189     by (blast intro: dynamic_to_static_overriding)
  1190   with wf 
  1191   show ?thesis
  1192    by (blast dest: wf_prog_stat_overridesD)
  1193 qed 
  1194 
  1195 lemma overrides_Package_old: 
  1196   assumes dyn_override: "G \<turnstile> new overrides old" and 
  1197            accmodi_new: "accmodi new = Package" and
  1198                     wf: "wf_prog G "
  1199   shows "accmodi old = Package"
  1200 proof (cases "accmodi old")
  1201   case Private
  1202   with dyn_override show ?thesis
  1203     by (simp add: no_Private_override)
  1204 next
  1205   case Package
  1206   then show ?thesis .
  1207 next
  1208   case Protected
  1209   with dyn_override wf
  1210   have "G \<turnstile> new overrides\<^sub>S old"
  1211     by (auto intro: dynamic_to_static_overriding)
  1212   with wf 
  1213   have "accmodi old \<le> accmodi new"
  1214     by (auto dest: wf_prog_stat_overridesD)
  1215   with Protected accmodi_new
  1216   show ?thesis
  1217     by (simp add: less_acc_def le_acc_def)
  1218 next
  1219   case Public
  1220   with dyn_override wf
  1221   have "G \<turnstile> new overrides\<^sub>S old"
  1222     by (auto intro: dynamic_to_static_overriding)
  1223   with wf 
  1224   have "accmodi old \<le> accmodi new"
  1225     by (auto dest: wf_prog_stat_overridesD)
  1226   with Public accmodi_new
  1227   show ?thesis
  1228     by (simp add: less_acc_def le_acc_def)
  1229 qed
  1230 
  1231 lemma dyn_override_Package:
  1232   assumes dyn_override: "G \<turnstile> new overrides old" and
  1233            accmodi_old: "accmodi old = Package" and 
  1234            accmodi_new: "accmodi new = Package" and
  1235                     wf: "wf_prog G"
  1236   shows "pid (declclass old) = pid (declclass new)"
  1237 proof - 
  1238   from dyn_override accmodi_old accmodi_new
  1239   show ?thesis (is "?EqPid old new")
  1240   proof (induct rule: overridesR.induct)
  1241     case (Direct new old)
  1242     assume "accmodi old = Package"
  1243            "G \<turnstile>Method old inheritable_in pid (declclass new)"
  1244     then show "pid (declclass old) =  pid (declclass new)"
  1245       by (auto simp add: inheritable_in_def)
  1246   next
  1247     case (Indirect new inter old)
  1248     assume accmodi_old: "accmodi old = Package" and
  1249            accmodi_new: "accmodi new = Package" 
  1250     assume "G \<turnstile> new overrides inter"
  1251     with accmodi_new wf
  1252     have "accmodi inter = Package"
  1253       by  (auto intro: overrides_Package_old)
  1254     with Indirect
  1255     show "pid (declclass old) =  pid (declclass new)"
  1256       by auto
  1257   qed
  1258 qed
  1259 
  1260 lemma dyn_override_Package_escape:
  1261   assumes dyn_override: "G \<turnstile> new overrides old" and
  1262            accmodi_old: "accmodi old = Package" and 
  1263           outside_pack: "pid (declclass old) \<noteq> pid (declclass new)" and
  1264                     wf: "wf_prog G"
  1265   shows "\<exists> inter. G \<turnstile> new overrides inter \<and> G \<turnstile> inter overrides old \<and>
  1266              pid (declclass old) = pid (declclass inter) \<and>
  1267              Protected \<le> accmodi inter"
  1268 proof -
  1269   from dyn_override accmodi_old outside_pack
  1270   show ?thesis (is "?P new old")
  1271   proof (induct rule: overridesR.induct)
  1272     case (Direct new old)
  1273     assume accmodi_old: "accmodi old = Package"
  1274     assume outside_pack: "pid (declclass old) \<noteq> pid (declclass new)"
  1275     assume "G \<turnstile>Method old inheritable_in pid (declclass new)"
  1276     with accmodi_old 
  1277     have "pid (declclass old) = pid (declclass new)"
  1278       by (simp add: inheritable_in_def)
  1279     with outside_pack 
  1280     show "?P new old"
  1281       by (contradiction)
  1282   next
  1283     case (Indirect new inter old)
  1284     assume accmodi_old: "accmodi old = Package"
  1285     assume outside_pack: "pid (declclass old) \<noteq> pid (declclass new)"
  1286     assume override_new_inter: "G \<turnstile> new overrides inter"
  1287     assume override_inter_old: "G \<turnstile> inter overrides old"
  1288     assume hyp_new_inter: "\<lbrakk>accmodi inter = Package; 
  1289                            pid (declclass inter) \<noteq> pid (declclass new)\<rbrakk>
  1290                            \<Longrightarrow> ?P new inter"
  1291     assume hyp_inter_old: "\<lbrakk>accmodi old = Package; 
  1292                            pid (declclass old) \<noteq> pid (declclass inter)\<rbrakk>
  1293                            \<Longrightarrow> ?P inter old"
  1294     show "?P new old"
  1295     proof (cases "pid (declclass old) = pid (declclass inter)")
  1296       case True
  1297       note same_pack_old_inter = this
  1298       show ?thesis
  1299       proof (cases "pid (declclass inter) = pid (declclass new)")
  1300         case True
  1301         with same_pack_old_inter outside_pack
  1302         show ?thesis
  1303           by auto
  1304       next
  1305         case False
  1306         note diff_pack_inter_new = this
  1307         show ?thesis
  1308         proof (cases "accmodi inter = Package")
  1309           case True
  1310           with diff_pack_inter_new hyp_new_inter  
  1311           obtain newinter where
  1312             over_new_newinter: "G \<turnstile> new overrides newinter" and
  1313             over_newinter_inter: "G \<turnstile> newinter overrides inter" and 
  1314             eq_pid: "pid (declclass inter) = pid (declclass newinter)" and
  1315             accmodi_newinter: "Protected \<le> accmodi newinter"
  1316             by auto
  1317           from over_newinter_inter override_inter_old
  1318           have "G\<turnstile>newinter overrides old"
  1319             by (rule overridesR.Indirect)
  1320           moreover
  1321           from eq_pid same_pack_old_inter 
  1322           have "pid (declclass old) = pid (declclass newinter)"
  1323             by simp
  1324           moreover
  1325           note over_new_newinter accmodi_newinter
  1326           ultimately show ?thesis
  1327             by blast
  1328         next
  1329           case False
  1330           with override_new_inter
  1331           have "Protected \<le> accmodi inter"
  1332             by (cases "accmodi inter") (auto dest: no_Private_override)
  1333           with override_new_inter override_inter_old same_pack_old_inter
  1334           show ?thesis
  1335             by blast
  1336         qed
  1337       qed
  1338     next
  1339       case False
  1340       with accmodi_old hyp_inter_old
  1341       obtain newinter where
  1342         over_inter_newinter: "G \<turnstile> inter overrides newinter" and
  1343           over_newinter_old: "G \<turnstile> newinter overrides old" and 
  1344                 eq_pid: "pid (declclass old) = pid (declclass newinter)" and
  1345         accmodi_newinter: "Protected \<le> accmodi newinter"
  1346         by auto
  1347       from override_new_inter over_inter_newinter 
  1348       have "G \<turnstile> new overrides newinter"
  1349         by (rule overridesR.Indirect)
  1350       with eq_pid over_newinter_old accmodi_newinter
  1351       show ?thesis
  1352         by blast
  1353     qed
  1354   qed
  1355 qed
  1356 
  1357 lemma declclass_widen[rule_format]: 
  1358  "wf_prog G 
  1359  \<longrightarrow> (\<forall>c m. class G C = Some c \<longrightarrow> methd G C sig = Some m 
  1360  \<longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m)" (is "?P G C")
  1361 proof (rule class_rec.induct,intro allI impI)
  1362   fix G C c m
  1363   assume Hyp: "\<forall>c. C \<noteq> Object \<and> ws_prog G \<and> class G C = Some c 
  1364                \<longrightarrow> ?P G (super c)"
  1365   assume wf: "wf_prog G" and cls_C: "class G C = Some c" and
  1366          m:  "methd G C sig = Some m"
  1367   show "G\<turnstile>C\<preceq>\<^sub>C declclass m" 
  1368   proof (cases "C=Object")
  1369     case True 
  1370     with wf m show ?thesis by (simp add: methd_Object_SomeD)
  1371   next
  1372     let ?filter="filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)"
  1373     let ?table = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
  1374     case False 
  1375     with cls_C wf m
  1376     have methd_C: "(?filter (methd G (super c)) ++ ?table) sig = Some m "
  1377       by (simp add: methd_rec)
  1378     show ?thesis
  1379     proof (cases "?table sig")
  1380       case None
  1381       from this methd_C have "?filter (methd G (super c)) sig = Some m"
  1382         by simp
  1383       moreover
  1384       from wf cls_C False obtain sup where "class G (super c) = Some sup"
  1385         by (blast dest: wf_prog_cdecl wf_cdecl_supD is_acc_class_is_class)
  1386       moreover note wf False cls_C  
  1387       ultimately have "G\<turnstile>super c \<preceq>\<^sub>C declclass m"  
  1388         by (auto intro: Hyp [rule_format])
  1389       moreover from cls_C False have  "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 super c" by (rule subcls1I)
  1390       ultimately show ?thesis by - (rule rtrancl_into_rtrancl2)
  1391     next
  1392       case Some
  1393       from this wf False cls_C methd_C show ?thesis by auto
  1394     qed
  1395   qed
  1396 qed
  1397 
  1398 lemma declclass_methd_Object: 
  1399  "\<lbrakk>wf_prog G; methd G Object sig = Some m\<rbrakk> \<Longrightarrow> declclass m = Object"
  1400 by auto
  1401 
  1402 lemma methd_declaredD: 
  1403  "\<lbrakk>wf_prog G; is_class G C;methd G C sig = Some m\<rbrakk> 
  1404   \<Longrightarrow> G\<turnstile>(mdecl (sig,mthd m)) declared_in (declclass m)"
  1405 proof -
  1406   assume    wf: "wf_prog G"
  1407   then have ws: "ws_prog G" ..
  1408   assume  clsC: "is_class G C"
  1409   from clsC ws 
  1410   show "methd G C sig = Some m 
  1411         \<Longrightarrow> G\<turnstile>(mdecl (sig,mthd m)) declared_in (declclass m)"
  1412     (is "PROP ?P C") 
  1413   proof (induct ?P C rule: ws_class_induct')
  1414     case Object
  1415     assume "methd G Object sig = Some m" 
  1416     with wf show ?thesis
  1417       by - (rule method_declared_inI, auto) 
  1418   next
  1419     case Subcls
  1420     fix C c
  1421     assume clsC: "class G C = Some c"
  1422     and       m: "methd G C sig = Some m"
  1423     and     hyp: "methd G (super c) sig = Some m \<Longrightarrow> ?thesis" 
  1424     let ?newMethods = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
  1425     show ?thesis
  1426     proof (cases "?newMethods sig")
  1427       case None
  1428       from None ws clsC m hyp 
  1429       show ?thesis by (auto intro: method_declared_inI simp add: methd_rec)
  1430     next
  1431       case Some
  1432       from Some ws clsC m 
  1433       show ?thesis by (auto intro: method_declared_inI simp add: methd_rec) 
  1434     qed
  1435   qed
  1436 qed
  1437 
  1438 lemma methd_rec_Some_cases [consumes 4, case_names NewMethod InheritedMethod]:
  1439   assumes methd_C: "methd G C sig = Some m" and
  1440                ws: "ws_prog G" and
  1441              clsC: "class G C = Some c" and
  1442         neq_C_Obj: "C\<noteq>Object"
  1443   shows
  1444 "\<lbrakk>table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig = Some m \<Longrightarrow> P;
  1445   \<lbrakk>G\<turnstile>C inherits (method sig m); methd G (super c) sig = Some m\<rbrakk> \<Longrightarrow> P 
  1446  \<rbrakk> \<Longrightarrow> P"
  1447 proof -
  1448   let ?inherited   = "filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m) 
  1449                               (methd G (super c))"
  1450   let ?new = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
  1451   from ws clsC neq_C_Obj methd_C 
  1452   have methd_unfold: "(?inherited ++ ?new) sig = Some m"
  1453     by (simp add: methd_rec)
  1454   assume NewMethod: "?new sig = Some m \<Longrightarrow> P"
  1455   assume InheritedMethod: "\<lbrakk>G\<turnstile>C inherits (method sig m); 
  1456                             methd G (super c) sig = Some m\<rbrakk> \<Longrightarrow> P"
  1457   show P
  1458   proof (cases "?new sig")
  1459     case None
  1460     with methd_unfold have "?inherited sig = Some m"
  1461       by (auto)
  1462     with InheritedMethod show P by blast
  1463   next
  1464     case Some
  1465     with methd_unfold have "?new sig = Some m"
  1466       by auto
  1467     with NewMethod show P by blast
  1468   qed
  1469 qed
  1470 
  1471   
  1472 lemma methd_member_of:
  1473   assumes wf: "wf_prog G"
  1474   shows
  1475     "\<lbrakk>is_class G C; methd G C sig = Some m\<rbrakk> \<Longrightarrow> G\<turnstile>Methd sig m member_of C" 
  1476   (is "?Class C \<Longrightarrow> ?Method C \<Longrightarrow> ?MemberOf C") 
  1477 proof -
  1478   from wf   have   ws: "ws_prog G" ..
  1479   assume defC: "is_class G C"
  1480   from defC ws 
  1481   show "?Class C \<Longrightarrow> ?Method C \<Longrightarrow> ?MemberOf C"
  1482   proof (induct rule: ws_class_induct')  
  1483     case Object
  1484     with wf have declC: "Object = declclass m"
  1485       by (simp add: declclass_methd_Object)
  1486     from Object wf have "G\<turnstile>Methd sig m declared_in Object"
  1487       by (auto intro: methd_declaredD simp add: declC)
  1488     with declC 
  1489     show "?MemberOf Object"
  1490       by (auto intro!: members.Immediate
  1491                   simp del: methd_Object)
  1492   next
  1493     case (Subcls C c)
  1494     assume  clsC: "class G C = Some c" and
  1495        neq_C_Obj: "C \<noteq> Object"  
  1496     assume methd: "?Method C"
  1497     from methd ws clsC neq_C_Obj
  1498     show "?MemberOf C"
  1499     proof (cases rule: methd_rec_Some_cases)
  1500       case NewMethod
  1501       with clsC show ?thesis
  1502         by (auto dest: method_declared_inI intro!: members.Immediate)
  1503     next
  1504       case InheritedMethod
  1505       then show "?thesis"
  1506         by (blast dest: inherits_member)
  1507     qed
  1508   qed
  1509 qed
  1510 
  1511 lemma current_methd: 
  1512       "\<lbrakk>table_of (methods c) sig = Some new;
  1513         ws_prog G; class G C = Some c; C \<noteq> Object; 
  1514         methd G (super c) sig = Some old\<rbrakk> 
  1515     \<Longrightarrow> methd G C sig = Some (C,new)"
  1516 by (auto simp add: methd_rec
  1517             intro: filter_tab_SomeI map_add_find_right table_of_map_SomeI)
  1518 
  1519 lemma wf_prog_staticD:
  1520   assumes     wf: "wf_prog G" and
  1521             clsC: "class G C = Some c" and
  1522        neq_C_Obj: "C \<noteq> Object" and 
  1523              old: "methd G (super c) sig = Some old" and 
  1524      accmodi_old: "Protected \<le> accmodi old" and
  1525              new: "table_of (methods c) sig = Some new"
  1526   shows "is_static new = is_static old"
  1527 proof -
  1528   from clsC wf 
  1529   have wf_cdecl: "wf_cdecl G (C,c)" by (rule wf_prog_cdecl)
  1530   from wf clsC neq_C_Obj
  1531   have is_cls_super: "is_class G (super c)" 
  1532     by (blast dest: wf_prog_acc_superD is_acc_classD)
  1533   from wf is_cls_super old 
  1534   have old_member_of: "G\<turnstile>Methd sig old member_of (super c)"  
  1535     by (rule methd_member_of)
  1536   from old wf is_cls_super 
  1537   have old_declared: "G\<turnstile>Methd sig old declared_in (declclass old)"
  1538     by (auto dest: methd_declared_in_declclass)
  1539   from new clsC 
  1540   have new_declared: "G\<turnstile>Methd sig (C,new) declared_in C"
  1541     by (auto intro: method_declared_inI)
  1542   note trancl_rtrancl_tranc = trancl_rtrancl_trancl [trans] (* ### in Basis *)
  1543   from clsC neq_C_Obj
  1544   have subcls1_C_super: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 super c"
  1545     by (rule subcls1I)
  1546   then have "G\<turnstile>C \<prec>\<^sub>C super c" ..
  1547   also from old wf is_cls_super
  1548   have "G\<turnstile>super c \<preceq>\<^sub>C (declclass old)" by (auto dest: methd_declC)
  1549   finally have subcls_C_old:  "G\<turnstile>C \<prec>\<^sub>C (declclass old)" .
  1550   from accmodi_old 
  1551   have inheritable: "G\<turnstile>Methd sig old inheritable_in pid C"
  1552     by (auto simp add: inheritable_in_def
  1553                  dest: acc_modi_le_Dests)
  1554   show ?thesis
  1555   proof (cases "is_static new")
  1556     case True
  1557     with subcls_C_old new_declared old_declared inheritable
  1558     have "G,sig\<turnstile>(C,new) hides old"
  1559       by (auto intro: hidesI)
  1560     with True wf_cdecl neq_C_Obj new 
  1561     show ?thesis
  1562       by (auto dest: wf_cdecl_hides_SomeD)
  1563   next
  1564     case False
  1565     with subcls_C_old new_declared old_declared inheritable subcls1_C_super
  1566          old_member_of
  1567     have "G,sig\<turnstile>(C,new) overrides\<^sub>S old"
  1568       by (auto intro: stat_overridesR.Direct)
  1569     with False wf_cdecl neq_C_Obj new 
  1570     show ?thesis
  1571       by (auto dest: wf_cdecl_overrides_SomeD)
  1572   qed
  1573 qed
  1574 
  1575 lemma inheritable_instance_methd: 
  1576   assumes subclseq_C_D: "G\<turnstile>C \<preceq>\<^sub>C D" and
  1577               is_cls_D: "is_class G D" and
  1578                     wf: "wf_prog G" and 
  1579                    old: "methd G D sig = Some old" and
  1580            accmodi_old: "Protected \<le> accmodi old" and  
  1581         not_static_old: "\<not> is_static old"
  1582   shows
  1583   "\<exists>new. methd G C sig = Some new \<and>
  1584          (new = old \<or> G,sig\<turnstile>new overrides\<^sub>S old)"
  1585  (is "(\<exists>new. (?Constraint C new old))")
  1586 proof -
  1587   from subclseq_C_D is_cls_D 
  1588   have is_cls_C: "is_class G C" by (rule subcls_is_class2) 
  1589   from wf 
  1590   have ws: "ws_prog G" ..
  1591   from is_cls_C ws subclseq_C_D 
  1592   show "\<exists>new. ?Constraint C new old"
  1593   proof (induct rule: ws_class_induct')
  1594     case (Object co)
  1595     then have eq_D_Obj: "D=Object" by auto
  1596     with old 
  1597     have "?Constraint Object old old"
  1598       by auto
  1599     with eq_D_Obj 
  1600     show "\<exists> new. ?Constraint Object new old" by auto
  1601   next
  1602     case (Subcls C c)
  1603     assume hyp: "G\<turnstile>super c\<preceq>\<^sub>C D \<Longrightarrow> \<exists>new. ?Constraint (super c) new old"
  1604     assume clsC: "class G C = Some c"
  1605     assume neq_C_Obj: "C\<noteq>Object"
  1606     from clsC wf 
  1607     have wf_cdecl: "wf_cdecl G (C,c)" 
  1608       by (rule wf_prog_cdecl)
  1609     from ws clsC neq_C_Obj
  1610     have is_cls_super: "is_class G (super c)"
  1611       by (auto dest: ws_prog_cdeclD)
  1612     from clsC wf neq_C_Obj 
  1613     have superAccessible: "G\<turnstile>(Class (super c)) accessible_in (pid C)" and
  1614          subcls1_C_super: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 super c"
  1615       by (auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_classD
  1616               intro: subcls1I)
  1617     show "\<exists>new. ?Constraint C new old"
  1618     proof (cases "G\<turnstile>super c\<preceq>\<^sub>C D")
  1619       case False
  1620       from False Subcls 
  1621       have eq_C_D: "C=D"
  1622         by (auto dest: subclseq_superD)
  1623       with old 
  1624       have "?Constraint C old old"
  1625         by auto
  1626       with eq_C_D 
  1627       show "\<exists> new. ?Constraint C new old" by auto
  1628     next
  1629       case True
  1630       with hyp obtain super_method
  1631         where super: "?Constraint (super c) super_method old" by blast
  1632       from super not_static_old
  1633       have not_static_super: "\<not> is_static super_method"
  1634         by (auto dest!: stat_overrides_commonD)
  1635       from super old wf accmodi_old
  1636       have accmodi_super_method: "Protected \<le> accmodi super_method"
  1637         by (auto dest!: wf_prog_stat_overridesD)
  1638       from super accmodi_old wf
  1639       have inheritable: "G\<turnstile>Methd sig super_method inheritable_in (pid C)"
  1640         by (auto dest!: wf_prog_stat_overridesD
  1641                         acc_modi_le_Dests
  1642               simp add: inheritable_in_def)                
  1643       from super wf is_cls_super
  1644       have member: "G\<turnstile>Methd sig super_method member_of (super c)"
  1645         by (auto intro: methd_member_of) 
  1646       from member
  1647       have decl_super_method:
  1648         "G\<turnstile>Methd sig super_method declared_in (declclass super_method)"
  1649         by (auto dest: member_of_declC)
  1650       from super subcls1_C_super ws is_cls_super 
  1651       have subcls_C_super: "G\<turnstile>C \<prec>\<^sub>C (declclass super_method)"
  1652         by (auto intro: rtrancl_into_trancl2 dest: methd_declC) 
  1653       show "\<exists> new. ?Constraint C new old"
  1654       proof (cases "methd G C sig")
  1655         case None
  1656         have "methd G (super c) sig = None"
  1657         proof -
  1658           from clsC ws None 
  1659           have no_new: "table_of (methods c) sig = None" 
  1660             by (auto simp add: methd_rec)
  1661           with clsC 
  1662           have undeclared: "G\<turnstile>mid sig undeclared_in C"
  1663             by (auto simp add: undeclared_in_def cdeclaredmethd_def)
  1664           with inheritable member superAccessible subcls1_C_super
  1665           have inherits: "G\<turnstile>C inherits (method sig super_method)"
  1666             by (auto simp add: inherits_def)
  1667           with clsC ws no_new super neq_C_Obj
  1668           have "methd G C sig = Some super_method"
  1669             by (auto simp add: methd_rec map_add_def intro: filter_tab_SomeI)
  1670           with None show ?thesis
  1671             by simp
  1672         qed
  1673         with super show ?thesis by auto
  1674       next
  1675         case (Some new)
  1676         from this ws clsC neq_C_Obj
  1677         show ?thesis
  1678         proof (cases rule: methd_rec_Some_cases)
  1679           case InheritedMethod
  1680           with super Some show ?thesis 
  1681             by auto
  1682         next
  1683           case NewMethod
  1684           assume new: "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig 
  1685                        = Some new"
  1686           from new 
  1687           have declcls_new: "declclass new = C" 
  1688             by auto
  1689           from wf clsC neq_C_Obj super new not_static_super accmodi_super_method
  1690           have not_static_new: "\<not> is_static new" 
  1691             by (auto dest: wf_prog_staticD) 
  1692           from clsC new
  1693           have decl_new: "G\<turnstile>Methd sig new declared_in C"
  1694             by (auto simp add: declared_in_def cdeclaredmethd_def)
  1695           from not_static_new decl_new decl_super_method
  1696                member subcls1_C_super inheritable declcls_new subcls_C_super 
  1697           have "G,sig\<turnstile> new overrides\<^sub>S super_method"
  1698             by (auto intro: stat_overridesR.Direct) 
  1699           with super Some
  1700           show ?thesis
  1701             by (auto intro: stat_overridesR.Indirect)
  1702         qed
  1703       qed
  1704     qed
  1705   qed
  1706 qed
  1707 
  1708 lemma inheritable_instance_methd_cases [consumes 6
  1709                                        , case_names Inheritance Overriding]: 
  1710   assumes subclseq_C_D: "G\<turnstile>C \<preceq>\<^sub>C D" and
  1711               is_cls_D: "is_class G D" and
  1712                     wf: "wf_prog G" and 
  1713                    old: "methd G D sig = Some old" and
  1714            accmodi_old: "Protected \<le> accmodi old" and  
  1715         not_static_old: "\<not> is_static old" and
  1716            inheritance:  "methd G C sig = Some old \<Longrightarrow> P" and
  1717             overriding:  "\<And> new. \<lbrakk>methd G C sig = Some new;
  1718                                    G,sig\<turnstile>new overrides\<^sub>S old\<rbrakk> \<Longrightarrow> P"
  1719         
  1720   shows P
  1721 proof -
  1722 from subclseq_C_D is_cls_D wf old accmodi_old not_static_old 
  1723 show ?thesis
  1724   by (auto dest: inheritable_instance_methd intro: inheritance overriding)
  1725 qed
  1726 
  1727 lemma inheritable_instance_methd_props: 
  1728   assumes subclseq_C_D: "G\<turnstile>C \<preceq>\<^sub>C D" and
  1729               is_cls_D: "is_class G D" and
  1730                     wf: "wf_prog G" and 
  1731                    old: "methd G D sig = Some old" and
  1732            accmodi_old: "Protected \<le> accmodi old" and  
  1733         not_static_old: "\<not> is_static old"
  1734   shows
  1735   "\<exists>new. methd G C sig = Some new \<and>
  1736           \<not> is_static new \<and> G\<turnstile>resTy new\<preceq>resTy old \<and> accmodi old \<le>accmodi new"
  1737  (is "(\<exists>new. (?Constraint C new old))")
  1738 proof -
  1739   from subclseq_C_D is_cls_D wf old accmodi_old not_static_old 
  1740   show ?thesis
  1741   proof (cases rule: inheritable_instance_methd_cases)
  1742     case Inheritance
  1743     with not_static_old accmodi_old show ?thesis by auto
  1744   next
  1745     case (Overriding new)
  1746     then have "\<not> is_static new" by (auto dest: stat_overrides_commonD)
  1747     with Overriding not_static_old accmodi_old wf 
  1748     show ?thesis 
  1749       by (auto dest!: wf_prog_stat_overridesD)
  1750   qed
  1751 qed
  1752               
  1753 (* local lemma *)
  1754 lemma bexI': "x \<in> A \<Longrightarrow> P x \<Longrightarrow> \<exists>x\<in>A. P x" by blast
  1755 lemma ballE': "\<forall>x\<in>A. P x \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> Q" by blast
  1756 
  1757 lemma subint_widen_imethds: 
  1758  "\<lbrakk>G\<turnstile>I\<preceq>I J; wf_prog G; is_iface G J; jm \<in> imethds G J sig\<rbrakk> \<Longrightarrow>   
  1759   \<exists> im \<in> imethds G I sig. is_static im = is_static jm \<and> 
  1760                           accmodi im = accmodi jm \<and>
  1761                           G\<turnstile>resTy im\<preceq>resTy jm"
  1762 proof -
  1763   assume irel: "G\<turnstile>I\<preceq>I J" and
  1764            wf: "wf_prog G" and
  1765      is_iface: "is_iface G J"
  1766   from irel show "jm \<in> imethds G J sig \<Longrightarrow> ?thesis" 
  1767                (is "PROP ?P I" is "PROP ?Prem J \<Longrightarrow> ?Concl I")
  1768   proof (induct ?P I rule: converse_rtrancl_induct) 
  1769     case Id
  1770     assume "jm \<in> imethds G J sig"
  1771     then show "?Concl J" by  (blast elim: bexI')
  1772   next
  1773     case Step
  1774     fix I SI
  1775     assume subint1_I_SI: "G\<turnstile>I \<prec>I1 SI" and 
  1776             subint_SI_J: "G\<turnstile>SI \<preceq>I J" and
  1777                     hyp: "PROP ?P SI" and
  1778                      jm: "jm \<in> imethds G J sig"
  1779     from subint1_I_SI 
  1780     obtain i where
  1781       ifI: "iface G I = Some i" and
  1782        SI: "SI \<in> set (isuperIfs i)"
  1783       by (blast dest: subint1D)
  1784 
  1785     let ?newMethods 
  1786           = "(Option.set \<circ> table_of (map (\<lambda>(sig, mh). (sig, I, mh)) (imethods i)))"
  1787     show "?Concl I"
  1788     proof (cases "?newMethods sig = {}")
  1789       case True
  1790       with ifI SI hyp wf jm 
  1791       show "?thesis" 
  1792         by (auto simp add: imethds_rec) 
  1793     next
  1794       case False
  1795       from ifI wf False
  1796       have imethds: "imethds G I sig = ?newMethods sig"
  1797         by (simp add: imethds_rec)
  1798       from False
  1799       obtain im where
  1800         imdef: "im \<in> ?newMethods sig" 
  1801         by (blast)
  1802       with imethds 
  1803       have im: "im \<in> imethds G I sig"
  1804         by (blast)
  1805       with im wf ifI 
  1806       obtain
  1807          imStatic: "\<not> is_static im" and
  1808          imPublic: "accmodi im = Public"
  1809         by (auto dest!: imethds_wf_mhead)       
  1810       from ifI wf 
  1811       have wf_I: "wf_idecl G (I,i)" 
  1812         by (rule wf_prog_idecl)
  1813       with SI wf  
  1814       obtain si where
  1815          ifSI: "iface G SI = Some si" and
  1816         wf_SI: "wf_idecl G (SI,si)" 
  1817         by (auto dest!: wf_idecl_supD is_acc_ifaceD
  1818                   dest: wf_prog_idecl)
  1819       from jm hyp 
  1820       obtain sim::"qtname \<times> mhead"  where
  1821                       sim: "sim \<in> imethds G SI sig" and
  1822          eq_static_sim_jm: "is_static sim = is_static jm" and 
  1823          eq_access_sim_jm: "accmodi sim = accmodi jm" and 
  1824         resTy_widen_sim_jm: "G\<turnstile>resTy sim\<preceq>resTy jm"
  1825         by blast
  1826       with wf_I SI imdef sim 
  1827       have "G\<turnstile>resTy im\<preceq>resTy sim"   
  1828         by (auto dest!: wf_idecl_hidings hidings_entailsD)
  1829       with wf resTy_widen_sim_jm 
  1830       have resTy_widen_im_jm: "G\<turnstile>resTy im\<preceq>resTy jm"
  1831         by (blast intro: widen_trans)
  1832       from sim wf ifSI  
  1833       obtain
  1834         simStatic: "\<not> is_static sim" and
  1835         simPublic: "accmodi sim = Public"
  1836         by (auto dest!: imethds_wf_mhead)
  1837       from im 
  1838            imStatic simStatic eq_static_sim_jm
  1839            imPublic simPublic eq_access_sim_jm
  1840            resTy_widen_im_jm
  1841       show ?thesis 
  1842         by auto 
  1843     qed
  1844   qed
  1845 qed
  1846      
  1847 (* Tactical version *)
  1848 (* 
  1849 lemma subint_widen_imethds: "\<lbrakk>G\<turnstile>I\<preceq>I J; wf_prog G; is_iface G J\<rbrakk> \<Longrightarrow>  
  1850   \<forall> jm \<in> imethds G J sig.  
  1851   \<exists> im \<in> imethds G I sig. static (mthd im)=static (mthd jm) \<and> 
  1852                           access (mthd im)= access (mthd jm) \<and>
  1853                           G\<turnstile>resTy (mthd im)\<preceq>resTy (mthd jm)"
  1854 apply (erule converse_rtrancl_induct)
  1855 apply  (clarsimp elim!: bexI')
  1856 apply (frule subint1D)
  1857 apply clarify
  1858 apply (erule ballE')
  1859 apply  fast
  1860 apply (erule_tac V = "?x \<in> imethds G J sig" in thin_rl)
  1861 apply clarsimp
  1862 apply (subst imethds_rec, assumption, erule wf_ws_prog)
  1863 apply (unfold overrides_t_def)
  1864 apply (drule (1) wf_prog_idecl)
  1865 apply (frule (3) imethds_wf_mhead [OF _ _ wf_idecl_supD [THEN conjunct1 
  1866                                        [THEN is_acc_ifaceD [THEN conjunct1]]]])
  1867 apply (case_tac "(Option.set \<circ> table_of (map (\<lambda>(s, mh). (s, y, mh)) (imethods i)))
  1868                   sig ={}")
  1869 apply   force
  1870 
  1871 apply   (simp only:)
  1872 apply   (simp)
  1873 apply   clarify
  1874 apply   (frule wf_idecl_hidings [THEN hidings_entailsD])
  1875 apply     blast
  1876 apply     blast
  1877 apply   (rule bexI')
  1878 apply     simp
  1879 apply     (drule table_of_map_SomeI [of _ "sig"])
  1880 apply     simp
  1881 
  1882 apply     (frule wf_idecl_mhead [of _ _ _ "sig"])
  1883 apply       (rule table_of_Some_in_set)
  1884 apply       assumption
  1885 apply     auto
  1886 done
  1887 *)
  1888     
  1889 
  1890 (* local lemma *)
  1891 lemma implmt1_methd: 
  1892  "\<And>sig. \<lbrakk>G\<turnstile>C\<leadsto>1I; wf_prog G; im \<in> imethds G I sig\<rbrakk> \<Longrightarrow>  
  1893   \<exists>cm \<in>methd G C sig: \<not> is_static cm \<and> \<not> is_static im \<and> 
  1894                        G\<turnstile>resTy cm\<preceq>resTy im \<and>
  1895                        accmodi im = Public \<and> accmodi cm = Public"
  1896 apply (drule implmt1D)
  1897 apply clarify
  1898 apply (drule (2) wf_prog_cdecl [THEN wf_cdecl_impD])
  1899 apply (frule (1) imethds_wf_mhead)
  1900 apply  (simp add: is_acc_iface_def)
  1901 apply (force)
  1902 done
  1903 
  1904 
  1905 (* local lemma *)
  1906 lemma implmt_methd [rule_format (no_asm)]: 
  1907 "\<lbrakk>wf_prog G; G\<turnstile>C\<leadsto>I\<rbrakk> \<Longrightarrow> is_iface G I \<longrightarrow>  
  1908  (\<forall> im    \<in>imethds G I   sig.  
  1909   \<exists> cm\<in>methd G C sig: \<not>is_static cm \<and> \<not> is_static im \<and> 
  1910                       G\<turnstile>resTy cm\<preceq>resTy im \<and>
  1911                       accmodi im = Public \<and> accmodi cm = Public)"
  1912 apply (frule implmt_is_class)
  1913 apply (erule implmt.induct)
  1914 apply   safe
  1915 apply   (drule (2) implmt1_methd)
  1916 apply   fast
  1917 apply  (drule (1) subint_widen_imethds)
  1918 apply   simp
  1919 apply   assumption
  1920 apply  clarify
  1921 apply  (drule (2) implmt1_methd)
  1922 apply  (force)
  1923 apply (frule subcls1D)
  1924 apply (drule (1) bspec)
  1925 apply clarify
  1926 apply (drule (3) r_into_rtrancl [THEN inheritable_instance_methd_props, 
  1927                                  OF _ implmt_is_class])
  1928 apply auto 
  1929 done
  1930 
  1931 lemma mheadsD [rule_format (no_asm)]: 
  1932 "emh \<in> mheads G S t sig \<longrightarrow> wf_prog G \<longrightarrow>
  1933  (\<exists>C D m. t = ClassT C \<and> declrefT emh = ClassT D \<and> 
  1934           accmethd G S C sig = Some m \<and>
  1935           (declclass m = D) \<and> mhead (mthd m) = (mhd emh)) \<or>
  1936  (\<exists>I. t = IfaceT I \<and> ((\<exists>im. im  \<in> accimethds G (pid S) I sig \<and> 
  1937           mthd im = mhd emh) \<or> 
  1938   (\<exists>m. G\<turnstile>Iface I accessible_in (pid S) \<and> accmethd G S Object sig = Some m \<and> 
  1939        accmodi m \<noteq> Private \<and> 
  1940        declrefT emh = ClassT Object \<and> mhead (mthd m) = mhd emh))) \<or>
  1941  (\<exists>T m. t = ArrayT T \<and> G\<turnstile>Array T accessible_in (pid S) \<and>
  1942         accmethd G S Object sig = Some m \<and> accmodi m \<noteq> Private \<and> 
  1943         declrefT emh = ClassT Object \<and> mhead (mthd m) = mhd emh)"
  1944 apply (rule_tac ref_ty1="t" in ref_ty_ty.induct [THEN conjunct1])
  1945 apply auto
  1946 apply (auto simp add: cmheads_def accObjectmheads_def Objectmheads_def)
  1947 apply (auto  dest!: accmethd_SomeD)
  1948 done
  1949 
  1950 lemma mheads_cases [consumes 2, case_names Class_methd 
  1951                     Iface_methd Iface_Object_methd Array_Object_methd]: 
  1952 "\<lbrakk>emh \<in> mheads G S t sig; wf_prog G;
  1953  \<And> C D m. \<lbrakk>t = ClassT C;declrefT emh = ClassT D; accmethd G S C sig = Some m;
  1954            (declclass m = D); mhead (mthd m) = (mhd emh)\<rbrakk> \<Longrightarrow> P emh; 
  1955  \<And> I im. \<lbrakk>t = IfaceT I; im  \<in> accimethds G (pid S) I sig; mthd im = mhd emh\<rbrakk>
  1956           \<Longrightarrow> P emh;
  1957  \<And> I m. \<lbrakk>t = IfaceT I; G\<turnstile>Iface I accessible_in (pid S);
  1958           accmethd G S Object sig = Some m; accmodi m \<noteq> Private;
  1959          declrefT emh = ClassT Object; mhead (mthd m) = mhd emh\<rbrakk> \<Longrightarrow> P emh;
  1960  \<And> T m. \<lbrakk>t = ArrayT T;G\<turnstile>Array T accessible_in (pid S);
  1961           accmethd G S Object sig = Some m; accmodi m \<noteq> Private; 
  1962           declrefT emh = ClassT Object; mhead (mthd m) = mhd emh\<rbrakk> \<Longrightarrow>  P emh 
  1963 \<rbrakk> \<Longrightarrow> P emh"
  1964 by (blast dest!: mheadsD)
  1965 
  1966 lemma declclassD[rule_format]:
  1967  "\<lbrakk>wf_prog G;class G C = Some c; methd G C sig = Some m; 
  1968    class G (declclass m) = Some d\<rbrakk>
  1969   \<Longrightarrow> table_of (methods d) sig  = Some (mthd m)"
  1970 proof -
  1971   assume    wf: "wf_prog G"
  1972   then have ws: "ws_prog G" ..
  1973   assume  clsC: "class G C = Some c"
  1974   from clsC ws 
  1975   show "\<And> m d. \<lbrakk>methd G C sig = Some m; class G (declclass m) = Some d\<rbrakk>
  1976         \<Longrightarrow> table_of (methods d) sig  = Some (mthd m)" 
  1977          (is "PROP ?P C") 
  1978   proof (induct ?P C rule: ws_class_induct)
  1979     case Object
  1980     fix m d
  1981     assume "methd G Object sig = Some m" 
  1982            "class G (declclass m) = Some d"
  1983     with wf show "?thesis m d" by auto
  1984   next
  1985     case Subcls
  1986     fix C c m d
  1987     assume hyp: "PROP ?P (super c)"
  1988     and      m: "methd G C sig = Some m"
  1989     and  declC: "class G (declclass m) = Some d"
  1990     and   clsC: "class G C = Some c"
  1991     and   nObj: "C \<noteq> Object"
  1992     let ?newMethods = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig"
  1993     show "?thesis m d" 
  1994     proof (cases "?newMethods")
  1995       case None
  1996       from None clsC nObj ws m declC
  1997       show "?thesis" by (auto simp add: methd_rec) (rule hyp)
  1998     next
  1999       case Some
  2000       from Some clsC nObj ws m declC
  2001       show "?thesis" 
  2002         by (auto simp add: methd_rec
  2003                      dest: wf_prog_cdecl wf_cdecl_supD is_acc_class_is_class)
  2004     qed
  2005   qed
  2006 qed
  2007 
  2008 (* Tactical version *)
  2009 (*
  2010 lemma declclassD[rule_format]:
  2011  "wf_prog G \<longrightarrow>  
  2012  (\<forall> c d m. class G C = Some c \<longrightarrow> methd G C sig = Some m \<longrightarrow> 
  2013   class G (declclass m) = Some d
  2014  \<longrightarrow> table_of (methods d) sig  = Some (mthd m))"
  2015 apply (rule class_rec.induct)
  2016 apply (rule impI)
  2017 apply (rule allI)+
  2018 apply (rule impI)
  2019 apply (case_tac "C=Object")
  2020 apply   (force simp add: methd_rec)
  2021 
  2022 apply   (subst methd_rec)
  2023 apply     (blast dest: wf_ws_prog)+
  2024 apply   (case_tac "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig")
  2025 apply     (auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_class_is_class)
  2026 done
  2027 *)
  2028 
  2029 lemma dynmethd_Object:
  2030   assumes statM: "methd G Object sig = Some statM" and
  2031         private: "accmodi statM = Private" and 
  2032        is_cls_C: "is_class G C" and
  2033              wf: "wf_prog G"
  2034   shows "dynmethd G Object C sig = Some statM"
  2035 proof -
  2036   from is_cls_C wf 
  2037   have subclseq: "G\<turnstile>C \<preceq>\<^sub>C Object" 
  2038     by (auto intro: subcls_ObjectI)
  2039   from wf have ws: "ws_prog G" 
  2040     by simp
  2041   from wf 
  2042   have is_cls_Obj: "is_class G Object" 
  2043     by simp
  2044   from statM subclseq is_cls_Obj ws private
  2045   show ?thesis
  2046   proof (cases rule: dynmethd_cases)
  2047     case Static then show ?thesis .
  2048   next
  2049     case Overrides 
  2050     with private show ?thesis 
  2051       by (auto dest: no_Private_override)
  2052   qed
  2053 qed
  2054 
  2055 lemma wf_imethds_hiding_objmethdsD: 
  2056   assumes     old: "methd G Object sig = Some old" and
  2057           is_if_I: "is_iface G I" and
  2058                wf: "wf_prog G" and    
  2059       not_private: "accmodi old \<noteq> Private" and
  2060               new: "new \<in> imethds G I sig" 
  2061   shows "G\<turnstile>resTy new\<preceq>resTy old \<and> is_static new = is_static old" (is "?P new")
  2062 proof -
  2063   from wf have ws: "ws_prog G" by simp
  2064   {
  2065     fix I i new
  2066     assume ifI: "iface G I = Some i"
  2067     assume new: "table_of (imethods i) sig = Some new" 
  2068     from ifI new not_private wf old  
  2069     have "?P (I,new)"
  2070       by (auto dest!: wf_prog_idecl wf_idecl_hiding cond_hiding_entailsD
  2071             simp del: methd_Object)
  2072   } note hyp_newmethod = this  
  2073   from is_if_I ws new 
  2074   show ?thesis
  2075   proof (induct rule: ws_interface_induct)
  2076     case (Step I i)
  2077     assume ifI: "iface G I = Some i" 
  2078     assume new: "new \<in> imethds G I sig" 
  2079     from Step
  2080     have hyp: "\<forall> J \<in> set (isuperIfs i). (new \<in> imethds G J sig \<longrightarrow> ?P new)"
  2081       by auto 
  2082     from new ifI ws
  2083     show "?P new"
  2084     proof (cases rule: imethds_cases)
  2085       case NewMethod
  2086       with ifI hyp_newmethod
  2087       show ?thesis
  2088         by auto
  2089     next
  2090       case (InheritedMethod J)
  2091       assume "J \<in> set (isuperIfs i)" 
  2092              "new \<in> imethds G J sig"
  2093       with hyp 
  2094       show "?thesis"
  2095         by auto
  2096     qed
  2097   qed
  2098 qed
  2099 
  2100 text {*
  2101 Which dynamic classes are valid to look up a member of a distinct static type?
  2102 We have to distinct class members (named static members in Java) 
  2103 from instance members. Class members are global to all Objects of a class,
  2104 instance members are local to a single Object instance. If a member is
  2105 equipped with the static modifier it is a class member, else it is an 
  2106 instance member.
  2107 The following table gives an overview of the current framework. We assume
  2108 to have a reference with static type statT and a dynamic class dynC. Between
  2109 both of these types the widening relation holds 
  2110 @{term "G\<turnstile>Class dynC\<preceq> statT"}. Unfortunately this ordinary widening relation 
  2111 isn't enough to describe the valid lookup classes, since we must cope the
  2112 special cases of arrays and interfaces,too. If we statically expect an array or
  2113 inteface we may lookup a field or a method in Object which isn't covered in 
  2114 the widening relation.
  2115 
  2116 statT      field         instance method       static (class) method
  2117 ------------------------------------------------------------------------
  2118  NullT      /                  /                   /
  2119  Iface      /                dynC                Object
  2120  Class    dynC               dynC                 dynC
  2121  Array      /                Object              Object
  2122 
  2123 In most cases we con lookup the member in the dynamic class. But as an
  2124 interface can't declare new static methods, nor an array can define new
  2125 methods at all, we have to lookup methods in the base class Object.
  2126 
  2127 The limitation to classes in the field column is artificial  and comes out
  2128 of the typing rule for the field access (see rule @{text "FVar"} in the 
  2129 welltyping relation @{term "wt"} in theory WellType). 
  2130 I stems out of the fact, that Object
  2131 indeed has no non private fields. So interfaces and arrays can actually
  2132 have no fields at all and a field access would be senseless. (In Java
  2133 interfaces are allowed to declare new fields but in current Bali not!).
  2134 So there is no principal reason why we should not allow Objects to declare
  2135 non private fields. Then we would get the following column:
  2136        
  2137  statT    field
  2138 ----------------- 
  2139  NullT      /  
  2140  Iface    Object 
  2141  Class    dynC 
  2142  Array    Object
  2143 *}
  2144 consts valid_lookup_cls:: "prog \<Rightarrow> ref_ty \<Rightarrow> qtname \<Rightarrow> bool \<Rightarrow> bool"
  2145                         ("_,_ \<turnstile> _ valid'_lookup'_cls'_for _" [61,61,61,61] 60)
  2146 primrec
  2147 "G,NullT    \<turnstile> dynC valid_lookup_cls_for static_membr = False"
  2148 "G,IfaceT I \<turnstile> dynC valid_lookup_cls_for static_membr 
  2149               = (if static_membr 
  2150                     then dynC=Object 
  2151                     else G\<turnstile>Class dynC\<preceq> Iface I)"
  2152 "G,ClassT C \<turnstile> dynC valid_lookup_cls_for static_membr = G\<turnstile>Class dynC\<preceq> Class C"
  2153 "G,ArrayT T \<turnstile> dynC valid_lookup_cls_for static_membr = (dynC=Object)"
  2154 
  2155 lemma valid_lookup_cls_is_class:
  2156   assumes dynC: "G,statT \<turnstile> dynC valid_lookup_cls_for static_membr" and
  2157       ty_statT: "isrtype G statT" and
  2158             wf: "wf_prog G"
  2159   shows "is_class G dynC"
  2160 proof (cases statT)
  2161   case NullT
  2162   with dynC ty_statT show ?thesis
  2163     by (auto dest: widen_NT2)
  2164 next
  2165   case (IfaceT I)
  2166   with dynC wf show ?thesis
  2167     by (auto dest: implmt_is_class)
  2168 next
  2169   case (ClassT C)
  2170   with dynC ty_statT show ?thesis
  2171     by (auto dest: subcls_is_class2)
  2172 next
  2173   case (ArrayT T)
  2174   with dynC wf show ?thesis
  2175     by (auto)
  2176 qed
  2177 
  2178 declare split_paired_All [simp del] split_paired_Ex [simp del]
  2179 declaration {* K (Simplifier.map_ss (fn ss => ss delloop "split_all_tac")) *}
  2180 declaration {* K (Classical.map_cs (fn cs => cs delSWrapper "split_all_tac")) *}
  2181 
  2182 lemma dynamic_mheadsD:   
  2183 "\<lbrakk>emh \<in> mheads G S statT sig;    
  2184   G,statT \<turnstile> dynC valid_lookup_cls_for (is_static emh);
  2185   isrtype G statT; wf_prog G
  2186  \<rbrakk> \<Longrightarrow> \<exists>m \<in> dynlookup G statT dynC sig: 
  2187           is_static m=is_static emh \<and> G\<turnstile>resTy m\<preceq>resTy emh"
  2188 proof - 
  2189   assume      emh: "emh \<in> mheads G S statT sig"
  2190   and          wf: "wf_prog G"
  2191   and   dynC_Prop: "G,statT \<turnstile> dynC valid_lookup_cls_for (is_static emh)"
  2192   and      istype: "isrtype G statT"
  2193   from dynC_Prop istype wf 
  2194   obtain y where
  2195     dynC: "class G dynC = Some y" 
  2196     by (auto dest: valid_lookup_cls_is_class)
  2197   from emh wf show ?thesis
  2198   proof (cases rule: mheads_cases)
  2199     case Class_methd
  2200     fix statC statDeclC sm
  2201     assume     statC: "statT = ClassT statC"
  2202     assume            "accmethd G S statC sig = Some sm"
  2203     then have     sm: "methd G statC sig = Some sm" 
  2204       by (blast dest: accmethd_SomeD)  
  2205     assume eq_mheads: "mhead (mthd sm) = mhd emh"
  2206     from statC 
  2207     have dynlookup: "dynlookup G statT dynC sig = dynmethd G statC dynC sig"
  2208       by (simp add: dynlookup_def)
  2209     from wf statC istype dynC_Prop sm 
  2210     obtain dm where
  2211       "dynmethd G statC dynC sig = Some dm"
  2212       "is_static dm = is_static sm" 
  2213       "G\<turnstile>resTy dm\<preceq>resTy sm"  
  2214       by (force dest!: ws_dynmethd accmethd_SomeD)
  2215     with dynlookup eq_mheads 
  2216     show ?thesis 
  2217       by (cases emh type: *) (auto)
  2218   next
  2219     case Iface_methd
  2220     fix I im
  2221     assume    statI: "statT = IfaceT I" and
  2222           eq_mheads: "mthd im = mhd emh" and
  2223                      "im \<in> accimethds G (pid S) I sig" 
  2224     then have im: "im \<in> imethds G I sig" 
  2225       by (blast dest: accimethdsD)
  2226     with istype statI eq_mheads wf 
  2227     have not_static_emh: "\<not> is_static emh"
  2228       by (cases emh) (auto dest: wf_prog_idecl imethds_wf_mhead)
  2229     from statI im
  2230     have dynlookup: "dynlookup G statT dynC sig = methd G dynC sig"
  2231       by (auto simp add: dynlookup_def dynimethd_def) 
  2232     from wf dynC_Prop statI istype im not_static_emh 
  2233     obtain dm where
  2234       "methd G dynC sig = Some dm"
  2235       "is_static dm = is_static im" 
  2236       "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mthd im)" 
  2237       by (force dest: implmt_methd)
  2238     with dynlookup eq_mheads
  2239     show ?thesis 
  2240       by (cases emh type: *) (auto)
  2241   next
  2242     case Iface_Object_methd
  2243     fix I sm
  2244     assume   statI: "statT = IfaceT I" and
  2245                 sm: "accmethd G S Object sig = Some sm" and 
  2246          eq_mheads: "mhead (mthd sm) = mhd emh" and
  2247              nPriv: "accmodi sm \<noteq> Private"
  2248      show ?thesis 
  2249      proof (cases "imethds G I sig = {}")
  2250        case True
  2251        with statI 
  2252        have dynlookup: "dynlookup G statT dynC sig = dynmethd G Object dynC sig"
  2253          by (simp add: dynlookup_def dynimethd_def)
  2254        from wf dynC 
  2255        have subclsObj: "G\<turnstile>dynC \<preceq>\<^sub>C Object"
  2256          by (auto intro: subcls_ObjectI)
  2257        from wf dynC dynC_Prop istype sm subclsObj 
  2258        obtain dm where
  2259          "dynmethd G Object dynC sig = Some dm"
  2260          "is_static dm = is_static sm" 
  2261          "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mthd sm)"  
  2262          by (auto dest!: ws_dynmethd accmethd_SomeD 
  2263                   intro: class_Object [OF wf] intro: that)
  2264        with dynlookup eq_mheads
  2265        show ?thesis 
  2266          by (cases emh type: *) (auto)
  2267      next
  2268        case False
  2269        with statI
  2270        have dynlookup: "dynlookup G statT dynC sig = methd G dynC sig"
  2271          by (simp add: dynlookup_def dynimethd_def)
  2272        from istype statI
  2273        have "is_iface G I"
  2274          by auto
  2275        with wf sm nPriv False 
  2276        obtain im where
  2277               im: "im \<in> imethds G I sig" and
  2278          eq_stat: "is_static im = is_static sm" and
  2279          resProp: "G\<turnstile>resTy (mthd im)\<preceq>resTy (mthd sm)"
  2280          by (auto dest: wf_imethds_hiding_objmethdsD accmethd_SomeD)
  2281        from im wf statI istype eq_stat eq_mheads
  2282        have not_static_sm: "\<not> is_static emh"
  2283          by (cases emh) (auto dest: wf_prog_idecl imethds_wf_mhead)
  2284        from im wf dynC_Prop dynC istype statI not_static_sm
  2285        obtain dm where
  2286          "methd G dynC sig = Some dm"
  2287          "is_static dm = is_static im" 
  2288          "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mthd im)" 
  2289          by (auto dest: implmt_methd)
  2290        with wf eq_stat resProp dynlookup eq_mheads
  2291        show ?thesis 
  2292          by (cases emh type: *) (auto intro: widen_trans)
  2293      qed
  2294   next
  2295     case Array_Object_methd
  2296     fix T sm
  2297     assume statArr: "statT = ArrayT T" and
  2298                 sm: "accmethd G S Object sig = Some sm" and 
  2299          eq_mheads: "mhead (mthd sm) = mhd emh" 
  2300     from statArr dynC_Prop wf
  2301     have dynlookup: "dynlookup G statT dynC sig = methd G Object sig"
  2302       by (auto simp add: dynlookup_def dynmethd_C_C)
  2303     with sm eq_mheads sm 
  2304     show ?thesis 
  2305       by (cases emh type: *) (auto dest: accmethd_SomeD)
  2306   qed
  2307 qed
  2308 declare split_paired_All [simp] split_paired_Ex [simp]
  2309 declaration {* K (Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))) *}
  2310 declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}
  2311 
  2312 (* Tactical version *)
  2313 (*
  2314 lemma dynamic_mheadsD: "  
  2315  \<lbrakk>emh \<in> mheads G S statT sig; wf_prog G; class G dynC = Some y;  
  2316    if (\<exists>T. statT=ArrayT T) then dynC=Object else G\<turnstile>Class dynC\<preceq>RefT statT; 
  2317    isrtype G statT\<rbrakk> \<Longrightarrow>  
  2318   \<exists>m \<in> dynlookup G statT dynC sig: 
  2319      static (mthd m)=static (mhd emh) \<and> G\<turnstile>resTy (mthd m)\<preceq>resTy (mhd emh)"
  2320 apply (drule mheadsD)
  2321 apply safe
  2322        -- reftype statT is a class  
  2323 apply  (case_tac "\<exists>T. ClassT C = ArrayT T")
  2324 apply    (simp)
  2325 
  2326 apply    (clarsimp simp add: dynlookup_def )
  2327 apply    (frule_tac statC="C" and dynC="dynC"  and sig="sig"  
  2328          in ws_dynmethd)
  2329 apply      assumption+
  2330 apply    (case_tac "emh")  
  2331 apply    (force dest: accmethd_SomeD)
  2332 
  2333        -- reftype statT is a interface, method defined in interface 
  2334 apply    (clarsimp simp add: dynlookup_def)
  2335 apply    (drule (1) implmt_methd)
  2336 apply      blast
  2337 apply      blast
  2338 apply    (clarify)  
  2339 apply    (unfold dynimethd_def)
  2340 apply    (rule_tac x="cm" in bexI)
  2341 apply      (case_tac "emh")
  2342 apply      force
  2343 
  2344 apply      force
  2345 
  2346         -- reftype statT is a interface, method defined in Object 
  2347 apply    (simp add: dynlookup_def)
  2348 apply    (simp only: dynimethd_def)
  2349 apply    (case_tac "imethds G I sig = {}")
  2350 apply       simp
  2351 apply       (frule_tac statC="Object" and dynC="dynC"  and sig="sig"  
  2352              in ws_dynmethd)
  2353 apply          (blast intro: subcls_ObjectI wf_ws_prog) 
  2354 apply          (blast dest: class_Object)
  2355 apply       (case_tac "emh") 
  2356 apply       (force dest: accmethd_SomeD)
  2357 
  2358 apply       simp
  2359 apply       (subgoal_tac "\<exists> im. im \<in> imethds G I sig") 
  2360 prefer 2      apply blast
  2361 apply       clarify
  2362 apply       (frule (1) implmt_methd)
  2363 apply         simp
  2364 apply         blast  
  2365 apply       (clarify dest!: accmethd_SomeD)
  2366 apply       (frule (4) iface_overrides_Object)
  2367 apply       clarify
  2368 apply       (case_tac emh)
  2369 apply       force
  2370 
  2371         -- reftype statT is a array
  2372 apply    (simp add: dynlookup_def)
  2373 apply    (case_tac emh)
  2374 apply    (force dest: accmethd_SomeD simp add: dynmethd_def)
  2375 done
  2376 *)
  2377 
  2378 (* FIXME occasionally convert to ws_class_induct*) 
  2379 lemma methd_declclass:
  2380 "\<lbrakk>class G C = Some c; wf_prog G; methd G C sig = Some m\<rbrakk> 
  2381  \<Longrightarrow> methd G (declclass m) sig = Some m"
  2382 proof -
  2383   assume asm: "class G C = Some c" "wf_prog G" "methd G C sig = Some m"
  2384   have "wf_prog G  \<longrightarrow> 
  2385            (\<forall> c m. class G C = Some c \<longrightarrow>  methd G C sig = Some m 
  2386                    \<longrightarrow>  methd G (declclass m) sig = Some m)"      (is "?P G C") 
  2387   proof (rule class_rec.induct,intro allI impI)
  2388     fix G C c m
  2389     assume hyp: "\<forall>c. C \<noteq> Object \<and> ws_prog G \<and> class G C = Some c \<longrightarrow>
  2390                      ?P G (super c)"
  2391     assume wf: "wf_prog G" and cls_C: "class G C = Some c" and
  2392             m: "methd G C sig = Some m"
  2393     show "methd G (declclass m) sig = Some m"
  2394     proof (cases "C=Object")
  2395       case True
  2396       with wf m show ?thesis by (auto intro: table_of_map_SomeI)
  2397     next
  2398       let ?filter="filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)"
  2399       let ?table = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
  2400       case False
  2401       with cls_C wf m
  2402       have methd_C: "(?filter (methd G (super c)) ++ ?table) sig = Some m "
  2403         by (simp add: methd_rec)
  2404       show ?thesis
  2405       proof (cases "?table sig")
  2406         case None
  2407         from this methd_C have "?filter (methd G (super c)) sig = Some m"
  2408           by simp
  2409         moreover
  2410         from wf cls_C False obtain sup where "class G (super c) = Some sup"
  2411           by (blast dest: wf_prog_cdecl wf_cdecl_supD is_acc_class_is_class)
  2412         moreover note wf False cls_C 
  2413         ultimately show ?thesis by (auto intro: hyp [rule_format])
  2414       next
  2415         case Some
  2416         from this methd_C m show ?thesis by auto 
  2417       qed
  2418     qed
  2419   qed   
  2420   with asm show ?thesis by auto
  2421 qed
  2422 
  2423 lemma dynmethd_declclass:
  2424  "\<lbrakk>dynmethd G statC dynC sig = Some m;
  2425    wf_prog G; is_class G statC
  2426   \<rbrakk> \<Longrightarrow> methd G (declclass m) sig = Some m"
  2427 by (auto dest: dynmethd_declC)
  2428 
  2429 lemma dynlookup_declC:
  2430  "\<lbrakk>dynlookup G statT dynC sig = Some m; wf_prog G;
  2431    is_class G dynC;isrtype G statT
  2432   \<rbrakk> \<Longrightarrow> G\<turnstile>dynC \<preceq>\<^sub>C (declclass m) \<and> is_class G (declclass m)"
  2433 by (cases "statT")
  2434    (auto simp add: dynlookup_def dynimethd_def 
  2435              dest: methd_declC dynmethd_declC)
  2436 
  2437 lemma dynlookup_Array_declclassD [simp]:
  2438 "\<lbrakk>dynlookup G (ArrayT T) Object sig = Some dm;wf_prog G\<rbrakk> 
  2439  \<Longrightarrow> declclass dm = Object"
  2440 proof -
  2441   assume dynL: "dynlookup G (ArrayT T) Object sig = Some dm"
  2442   assume wf: "wf_prog G"
  2443   from wf have ws: "ws_prog G" by auto
  2444   from wf have is_cls_Obj: "is_class G Object" by auto
  2445   from dynL wf
  2446   show ?thesis
  2447     by (auto simp add: dynlookup_def dynmethd_C_C [OF is_cls_Obj ws]
  2448                  dest: methd_Object_SomeD)
  2449 qed   
  2450   
  2451 
  2452 declare split_paired_All [simp del] split_paired_Ex [simp del]
  2453 declaration {* K (Simplifier.map_ss (fn ss => ss delloop "split_all_tac")) *}
  2454 declaration {* K (Classical.map_cs (fn cs => cs delSWrapper "split_all_tac")) *}
  2455 
  2456 lemma wt_is_type: "E,dt\<Turnstile>v\<Colon>T \<Longrightarrow>  wf_prog (prg E) \<longrightarrow> 
  2457   dt=empty_dt \<longrightarrow> (case T of 
  2458                      Inl T \<Rightarrow> is_type (prg E) T 
  2459                    | Inr Ts \<Rightarrow> Ball (set Ts) (is_type (prg E)))"
  2460 apply (unfold empty_dt_def)
  2461 apply (erule wt.induct)
  2462 apply (auto split del: split_if_asm simp del: snd_conv 
  2463             simp add: is_acc_class_def is_acc_type_def)
  2464 apply    (erule typeof_empty_is_type)
  2465 apply   (frule (1) wf_prog_cdecl [THEN wf_cdecl_supD], 
  2466         force simp del: snd_conv, clarsimp simp add: is_acc_class_def)
  2467 apply  (drule (1) max_spec2mheads [THEN conjunct1, THEN mheadsD])
  2468 apply  (drule_tac [2] accfield_fields) 
  2469 apply  (frule class_Object)
  2470 apply  (auto dest: accmethd_rT_is_type 
  2471                    imethds_wf_mhead [THEN conjunct1, THEN rT_is_acc_type]
  2472              dest!:accimethdsD
  2473              simp del: class_Object
  2474              simp add: is_acc_type_def
  2475     )
  2476 done
  2477 declare split_paired_All [simp] split_paired_Ex [simp]
  2478 declaration {* K (Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))) *}
  2479 declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}
  2480 
  2481 lemma ty_expr_is_type: 
  2482 "\<lbrakk>E\<turnstile>e\<Colon>-T; wf_prog (prg E)\<rbrakk> \<Longrightarrow> is_type (prg E) T"
  2483 by (auto dest!: wt_is_type)
  2484 lemma ty_var_is_type: 
  2485 "\<lbrakk>E\<turnstile>v\<Colon>=T; wf_prog (prg E)\<rbrakk> \<Longrightarrow> is_type (prg E) T"
  2486 by (auto dest!: wt_is_type)
  2487 lemma ty_exprs_is_type: 
  2488 "\<lbrakk>E\<turnstile>es\<Colon>\<doteq>Ts; wf_prog (prg E)\<rbrakk> \<Longrightarrow> Ball (set Ts) (is_type (prg E))"
  2489 by (auto dest!: wt_is_type)
  2490 
  2491 
  2492 lemma static_mheadsD: 
  2493  "\<lbrakk> emh \<in> mheads G S t sig; wf_prog G; E\<turnstile>e\<Colon>-RefT t; prg E=G ; 
  2494    invmode (mhd emh) e \<noteq> IntVir 
  2495   \<rbrakk> \<Longrightarrow> \<exists>m. (   (\<exists> C. t = ClassT C \<and> accmethd G S C sig = Some m)
  2496                \<or> (\<forall> C. t \<noteq> ClassT C \<and> accmethd G S Object sig = Some m )) \<and> 
  2497           declrefT emh = ClassT (declclass m) \<and>  mhead (mthd m) = (mhd emh)"
  2498 apply (subgoal_tac "is_static emh \<or> e = Super")
  2499 defer apply (force simp add: invmode_def)
  2500 apply (frule  ty_expr_is_type)
  2501 apply   simp
  2502 apply (case_tac "is_static emh")
  2503 apply  (frule (1) mheadsD)
  2504 apply  clarsimp
  2505 apply  safe
  2506 apply    blast
  2507 apply   (auto dest!: imethds_wf_mhead
  2508                      accmethd_SomeD 
  2509                      accimethdsD
  2510               simp add: accObjectmheads_def Objectmheads_def)
  2511 
  2512 apply  (erule wt_elim_cases)
  2513 apply  (force simp add: cmheads_def)
  2514 done
  2515 
  2516 lemma wt_MethdI:  
  2517 "\<lbrakk>methd G C sig = Some m; wf_prog G;  
  2518   class G C = Some c\<rbrakk> \<Longrightarrow>  
  2519  \<exists>T. \<lparr>prg=G,cls=(declclass m),
  2520       lcl=callee_lcl (declclass m) sig (mthd m)\<rparr>\<turnstile> Methd C sig\<Colon>-T \<and> G\<turnstile>T\<preceq>resTy m"
  2521 apply (frule (2) methd_wf_mdecl, clarify)
  2522 apply (force dest!: wf_mdecl_bodyD intro!: wt.Methd)
  2523 done
  2524 
  2525 subsection "accessibility concerns"
  2526 
  2527 lemma mheads_type_accessible:
  2528  "\<lbrakk>emh \<in> mheads G S T sig; wf_prog G\<rbrakk>
  2529  \<Longrightarrow> G\<turnstile>RefT T accessible_in (pid S)"
  2530 by (erule mheads_cases)
  2531    (auto dest: accmethd_SomeD accessible_from_commonD accimethdsD)
  2532 
  2533 lemma static_to_dynamic_accessible_from_aux:
  2534 "\<lbrakk>G\<turnstile>m of C accessible_from accC;wf_prog G\<rbrakk> 
  2535  \<Longrightarrow> G\<turnstile>m in C dyn_accessible_from accC"
  2536 proof (induct rule: accessible_fromR.induct)
  2537 qed (auto intro: dyn_accessible_fromR.intros 
  2538                  member_of_to_member_in
  2539                  static_to_dynamic_overriding)
  2540 
  2541 lemma static_to_dynamic_accessible_from:
  2542   assumes stat_acc: "G\<turnstile>m of statC accessible_from accC" and
  2543           subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
  2544                 wf: "wf_prog G"
  2545   shows "G\<turnstile>m in dynC dyn_accessible_from accC"
  2546 proof - 
  2547   from stat_acc subclseq 
  2548   show ?thesis (is "?Dyn_accessible m")
  2549   proof (induct rule: accessible_fromR.induct)
  2550     case (Immediate m statC)
  2551     then show "?Dyn_accessible m"
  2552       by (blast intro: dyn_accessible_fromR.Immediate
  2553                        member_inI
  2554                        permits_acc_inheritance)
  2555   next
  2556     case (Overriding m _ _)
  2557     with wf show "?Dyn_accessible m"
  2558       by (blast intro: dyn_accessible_fromR.Overriding
  2559                        member_inI
  2560                        static_to_dynamic_overriding  
  2561                        rtrancl_trancl_trancl 
  2562                        static_to_dynamic_accessible_from_aux)
  2563   qed
  2564 qed
  2565 
  2566 lemma static_to_dynamic_accessible_from_static:
  2567   assumes stat_acc: "G\<turnstile>m of statC accessible_from accC" and
  2568             static: "is_static m" and
  2569                 wf: "wf_prog G"
  2570   shows "G\<turnstile>m in (declclass m) dyn_accessible_from accC"
  2571 proof -
  2572   from stat_acc wf 
  2573   have "G\<turnstile>m in statC dyn_accessible_from accC"
  2574     by (auto intro: static_to_dynamic_accessible_from)
  2575   from this static
  2576   show ?thesis
  2577     by (rule dyn_accessible_from_static_declC)
  2578 qed
  2579 
  2580 lemma dynmethd_member_in:
  2581   assumes    m: "dynmethd G statC dynC sig = Some m" and
  2582    iscls_statC: "is_class G statC" and
  2583             wf: "wf_prog G"
  2584   shows "G\<turnstile>Methd sig m member_in dynC"
  2585 proof -
  2586   from m 
  2587   have subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC"
  2588     by (auto simp add: dynmethd_def)
  2589   from subclseq iscls_statC 
  2590   have iscls_dynC: "is_class G dynC"
  2591     by (rule subcls_is_class2)
  2592   from  iscls_dynC iscls_statC wf m
  2593   have "G\<turnstile>dynC \<preceq>\<^sub>C (declclass m) \<and> is_class G (declclass m) \<and>
  2594         methd G (declclass m) sig = Some m" 
  2595     by - (drule dynmethd_declC, auto)
  2596   with wf 
  2597   show ?thesis
  2598     by (auto intro: member_inI dest: methd_member_of)
  2599 qed
  2600 
  2601 lemma dynmethd_access_prop:
  2602   assumes statM: "methd G statC sig = Some statM" and
  2603        stat_acc: "G\<turnstile>Methd sig statM of statC accessible_from accC" and
  2604            dynM: "dynmethd G statC dynC sig = Some dynM" and
  2605              wf: "wf_prog G" 
  2606   shows "G\<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
  2607 proof -
  2608   from wf have ws: "ws_prog G" ..
  2609   from dynM 
  2610   have subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC"
  2611     by (auto simp add: dynmethd_def)
  2612   from stat_acc 
  2613   have is_cls_statC: "is_class G statC"
  2614     by (auto dest: accessible_from_commonD member_of_is_classD)
  2615   with subclseq 
  2616   have is_cls_dynC: "is_class G dynC"
  2617     by (rule subcls_is_class2)
  2618   from is_cls_statC statM wf 
  2619   have member_statC: "G\<turnstile>Methd sig statM member_of statC"
  2620     by (auto intro: methd_member_of)
  2621   from stat_acc 
  2622   have statC_acc: "G\<turnstile>Class statC accessible_in (pid accC)"
  2623     by (auto dest: accessible_from_commonD)
  2624   from statM subclseq is_cls_statC ws 
  2625   show ?thesis
  2626   proof (cases rule: dynmethd_cases)
  2627     case Static
  2628     assume dynmethd: "dynmethd G statC dynC sig = Some statM"
  2629     with dynM have eq_dynM_statM: "dynM=statM" 
  2630       by simp
  2631     with stat_acc subclseq wf 
  2632     show ?thesis
  2633       by (auto intro: static_to_dynamic_accessible_from)
  2634   next
  2635     case (Overrides newM)
  2636     assume dynmethd: "dynmethd G statC dynC sig = Some newM"
  2637     assume override: "G,sig\<turnstile>newM overrides statM"
  2638     assume      neq: "newM\<noteq>statM"
  2639     from dynmethd dynM 
  2640     have eq_dynM_newM: "dynM=newM" 
  2641       by simp
  2642     from dynmethd eq_dynM_newM wf is_cls_statC
  2643     have "G\<turnstile>Methd sig dynM member_in dynC"
  2644       by (auto intro: dynmethd_member_in)
  2645     moreover
  2646     from subclseq
  2647     have "G\<turnstile>dynC\<prec>\<^sub>C statC"
  2648     proof (cases rule: subclseq_cases)
  2649       case Eq
  2650       assume "dynC=statC"
  2651       moreover
  2652       from is_cls_statC obtain c
  2653         where "class G statC = Some c"
  2654         by auto
  2655       moreover 
  2656       note statM ws dynmethd 
  2657       ultimately
  2658       have "newM=statM" 
  2659         by (auto simp add: dynmethd_C_C)
  2660       with neq show ?thesis 
  2661         by (contradiction)
  2662     next
  2663       case Subcls then show ?thesis .
  2664     qed 
  2665     moreover
  2666     from stat_acc wf 
  2667     have "G\<turnstile>Methd sig statM in statC dyn_accessible_from accC"
  2668       by (blast intro: static_to_dynamic_accessible_from)
  2669     moreover
  2670     note override eq_dynM_newM
  2671     ultimately show ?thesis
  2672       by (cases dynM,cases statM) (auto intro: dyn_accessible_fromR.Overriding)
  2673   qed
  2674 qed
  2675 
  2676 lemma implmt_methd_access:
  2677   fixes accC::qtname
  2678   assumes iface_methd: "imethds G I sig \<noteq> {}" and
  2679            implements: "G\<turnstile>dynC\<leadsto>I"  and
  2680                isif_I: "is_iface G I" and
  2681                    wf: "wf_prog G" 
  2682   shows "\<exists> dynM. methd G dynC sig = Some dynM \<and> 
  2683             G\<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
  2684 proof -
  2685   from implements 
  2686   have iscls_dynC: "is_class G dynC" by (rule implmt_is_class)
  2687   from iface_methd
  2688   obtain im
  2689     where "im \<in> imethds G I sig"
  2690     by auto
  2691   with wf implements isif_I 
  2692   obtain dynM 
  2693     where dynM: "methd G dynC sig = Some dynM" and
  2694            pub: "accmodi dynM = Public"
  2695     by (blast dest: implmt_methd)
  2696   with iscls_dynC wf
  2697   have "G\<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
  2698     by (auto intro!: dyn_accessible_fromR.Immediate 
  2699               intro: methd_member_of member_of_to_member_in
  2700                      simp add: permits_acc_def)
  2701   with dynM    
  2702   show ?thesis
  2703     by blast
  2704 qed
  2705 
  2706 corollary implmt_dynimethd_access:
  2707   fixes accC::qtname
  2708   assumes iface_methd: "imethds G I sig \<noteq> {}" and
  2709            implements: "G\<turnstile>dynC\<leadsto>I"  and
  2710                isif_I: "is_iface G I" and
  2711                    wf: "wf_prog G" 
  2712   shows "\<exists> dynM. dynimethd G I dynC sig = Some dynM \<and> 
  2713             G\<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
  2714 proof -
  2715   from iface_methd
  2716   have "dynimethd G I dynC sig = methd G dynC sig"
  2717     by (simp add: dynimethd_def)
  2718   with iface_methd implements isif_I wf 
  2719   show ?thesis
  2720     by (simp only:)
  2721        (blast intro: implmt_methd_access)
  2722 qed
  2723 
  2724 lemma dynlookup_access_prop:
  2725   assumes emh: "emh \<in> mheads G accC statT sig" and
  2726          dynM: "dynlookup G statT dynC sig = Some dynM" and
  2727     dynC_prop: "G,statT \<turnstile> dynC valid_lookup_cls_for is_static emh" and
  2728     isT_statT: "isrtype G statT" and
  2729            wf: "wf_prog G"
  2730   shows "G \<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
  2731 proof -
  2732   from emh wf
  2733   have statT_acc: "G\<turnstile>RefT statT accessible_in (pid accC)"
  2734     by (rule mheads_type_accessible)
  2735   from dynC_prop isT_statT wf
  2736   have iscls_dynC: "is_class G dynC"
  2737     by (rule valid_lookup_cls_is_class)
  2738   from emh dynC_prop isT_statT wf dynM
  2739   have eq_static: "is_static emh = is_static dynM"
  2740     by (auto dest: dynamic_mheadsD)
  2741   from emh wf show ?thesis
  2742   proof (cases rule: mheads_cases)
  2743     case (Class_methd statC _ statM)
  2744     assume statT: "statT = ClassT statC"
  2745     assume "accmethd G accC statC sig = Some statM"
  2746     then have    statM: "methd G statC sig = Some statM" and
  2747               stat_acc: "G\<turnstile>Methd sig statM of statC accessible_from accC"
  2748       by (auto dest: accmethd_SomeD)
  2749     from dynM statT
  2750     have dynM': "dynmethd G statC dynC sig = Some dynM"
  2751       by (simp add: dynlookup_def) 
  2752     from statM stat_acc wf dynM'
  2753     show ?thesis
  2754       by (auto dest!: dynmethd_access_prop)
  2755   next
  2756     case (Iface_methd I im)
  2757     then have iface_methd: "imethds G I sig \<noteq> {}" and
  2758                  statT_acc: "G\<turnstile>RefT statT accessible_in (pid accC)" 
  2759       by (auto dest: accimethdsD)
  2760     assume   statT: "statT = IfaceT I"
  2761     assume      im: "im \<in>  accimethds G (pid accC) I sig"
  2762     assume eq_mhds: "mthd im = mhd emh"
  2763     from dynM statT
  2764     have dynM': "dynimethd G I dynC sig = Some dynM"
  2765       by (simp add: dynlookup_def)
  2766     from isT_statT statT 
  2767     have isif_I: "is_iface G I"
  2768       by simp
  2769     show ?thesis
  2770     proof (cases "is_static emh")
  2771       case False
  2772       with statT dynC_prop 
  2773       have widen_dynC: "G\<turnstile>Class dynC \<preceq> RefT statT"
  2774         by simp
  2775       from statT widen_dynC
  2776       have implmnt: "G\<turnstile>dynC\<leadsto>I"
  2777         by auto    
  2778       from eq_static False 
  2779       have not_static_dynM: "\<not> is_static dynM" 
  2780         by simp
  2781       from iface_methd implmnt isif_I wf dynM'
  2782       show ?thesis
  2783         by - (drule implmt_dynimethd_access, auto)
  2784     next
  2785       case True
  2786       assume "is_static emh"
  2787       moreover
  2788       from wf isT_statT statT im 
  2789       have "\<not> is_static im"
  2790         by (auto dest: accimethdsD wf_prog_idecl imethds_wf_mhead)
  2791       moreover note eq_mhds
  2792       ultimately show ?thesis
  2793         by (cases emh) auto
  2794     qed
  2795   next
  2796     case (Iface_Object_methd I statM)
  2797     assume statT: "statT = IfaceT I"
  2798     assume "accmethd G accC Object sig = Some statM"
  2799     then have    statM: "methd G Object sig = Some statM" and
  2800               stat_acc: "G\<turnstile>Methd sig statM of Object accessible_from accC"
  2801       by (auto dest: accmethd_SomeD)
  2802     assume not_Private_statM: "accmodi statM \<noteq> Private"
  2803     assume eq_mhds: "mhead (mthd statM) = mhd emh"
  2804     from iscls_dynC wf
  2805     have widen_dynC_Obj: "G\<turnstile>dynC \<preceq>\<^sub>C Object"
  2806       by (auto intro: subcls_ObjectI)
  2807     show ?thesis
  2808     proof (cases "imethds G I sig = {}")
  2809       case True
  2810       from dynM statT True
  2811       have dynM': "dynmethd G Object dynC sig = Some dynM"
  2812         by (simp add: dynlookup_def dynimethd_def)
  2813       from statT  
  2814       have "G\<turnstile>RefT statT \<preceq>Class Object"
  2815         by auto
  2816       with statM statT_acc stat_acc widen_dynC_Obj statT isT_statT 
  2817         wf dynM' eq_static dynC_prop  
  2818       show ?thesis
  2819         by - (drule dynmethd_access_prop,force+) 
  2820     next
  2821       case False
  2822       then obtain im where
  2823         im: "im \<in>  imethds G I sig"
  2824         by auto
  2825       have not_static_emh: "\<not> is_static emh"
  2826       proof -
  2827         from im statM statT isT_statT wf not_Private_statM 
  2828         have "is_static im = is_static statM"
  2829           by (fastsimp dest: wf_imethds_hiding_objmethdsD)
  2830         with wf isT_statT statT im 
  2831         have "\<not> is_static statM"
  2832           by (auto dest: wf_prog_idecl imethds_wf_mhead)
  2833         with eq_mhds
  2834         show ?thesis  
  2835           by (cases emh) auto
  2836       qed
  2837       with statT dynC_prop
  2838       have implmnt: "G\<turnstile>dynC\<leadsto>I"
  2839         by simp
  2840       with isT_statT statT
  2841       have isif_I: "is_iface G I"
  2842         by simp
  2843       from dynM statT
  2844       have dynM': "dynimethd G I dynC sig = Some dynM"
  2845         by (simp add: dynlookup_def) 
  2846       from False implmnt isif_I wf dynM'
  2847       show ?thesis
  2848         by - (drule implmt_dynimethd_access, auto)
  2849     qed
  2850   next
  2851     case (Array_Object_methd T statM)
  2852     assume statT: "statT = ArrayT T"
  2853     assume "accmethd G accC Object sig = Some statM"
  2854     then have    statM: "methd G Object sig = Some statM" and
  2855               stat_acc: "G\<turnstile>Methd sig statM of Object accessible_from accC"
  2856       by (auto dest: accmethd_SomeD)
  2857     from statT dynC_prop
  2858     have dynC_Obj: "dynC = Object" 
  2859       by simp
  2860     then
  2861     have widen_dynC_Obj: "G\<turnstile>Class dynC \<preceq> Class Object"
  2862       by simp
  2863     from dynM statT    
  2864     have dynM': "dynmethd G Object dynC sig = Some dynM"
  2865       by (simp add: dynlookup_def)
  2866     from statM statT_acc stat_acc dynM' wf widen_dynC_Obj  
  2867          statT isT_statT  
  2868     show ?thesis   
  2869       by - (drule dynmethd_access_prop, simp+) 
  2870   qed
  2871 qed
  2872 
  2873 lemma dynlookup_access:
  2874   assumes emh: "emh \<in> mheads G accC statT sig" and
  2875     dynC_prop: "G,statT \<turnstile> dynC valid_lookup_cls_for (is_static emh) " and
  2876     isT_statT: "isrtype G statT" and
  2877            wf: "wf_prog G"
  2878   shows "\<exists> dynM. dynlookup G statT dynC sig = Some dynM \<and> 
  2879             G\<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
  2880 proof - 
  2881   from dynC_prop isT_statT wf
  2882   have is_cls_dynC: "is_class G dynC"
  2883     by (auto dest: valid_lookup_cls_is_class)
  2884   with emh wf dynC_prop isT_statT
  2885   obtain dynM where 
  2886     "dynlookup G statT dynC sig = Some dynM"
  2887     by - (drule dynamic_mheadsD,auto)
  2888   with  emh dynC_prop isT_statT wf
  2889   show ?thesis 
  2890     by (blast intro: dynlookup_access_prop)
  2891 qed
  2892 
  2893 lemma stat_overrides_Package_old: 
  2894   assumes stat_override: "G \<turnstile> new overrides\<^sub>S old" and 
  2895           accmodi_new: "accmodi new = Package" and
  2896                    wf: "wf_prog G "
  2897   shows "accmodi old = Package"
  2898 proof -
  2899   from stat_override wf 
  2900   have "accmodi old \<le> accmodi new"
  2901     by (auto dest: wf_prog_stat_overridesD)
  2902   with stat_override accmodi_new show ?thesis
  2903     by (cases "accmodi old") (auto dest: no_Private_stat_override 
  2904                                    dest: acc_modi_le_Dests)
  2905 qed
  2906 
  2907 subsubsection {* Properties of dynamic accessibility *}
  2908 
  2909 lemma dyn_accessible_Private:
  2910  assumes dyn_acc: "G \<turnstile> m in C dyn_accessible_from accC" and
  2911             priv: "accmodi m = Private"
  2912    shows "accC = declclass m"
  2913 proof -
  2914   from dyn_acc priv
  2915   show ?thesis
  2916   proof (induct)
  2917     case (Immediate m C)
  2918     from `G \<turnstile> m in C permits_acc_from accC` and `accmodi m = Private`
  2919     show ?case
  2920       by (simp add: permits_acc_def)
  2921   next
  2922     case Overriding
  2923     then show ?case
  2924       by (auto dest!: overrides_commonD)
  2925   qed
  2926 qed
  2927 
  2928 text {* @{text dyn_accessible_Package} only works with the @{text wf_prog} assumption. 
  2929 Without it. it is easy to leaf the Package!
  2930 *}
  2931 lemma dyn_accessible_Package:
  2932  "\<lbrakk>G \<turnstile> m in C dyn_accessible_from accC; accmodi m = Package;
  2933    wf_prog G\<rbrakk>
  2934   \<Longrightarrow> pid accC = pid (declclass m)"
  2935 proof -
  2936   assume wf: "wf_prog G "
  2937   assume accessible: "G \<turnstile> m in C dyn_accessible_from accC"
  2938   then show "accmodi m = Package 
  2939             \<Longrightarrow> pid accC = pid (declclass m)"
  2940     (is "?Pack m \<Longrightarrow> ?P m")
  2941   proof (induct rule: dyn_accessible_fromR.induct)
  2942     case (Immediate m C)
  2943     assume "G\<turnstile>m member_in C"
  2944            "G \<turnstile> m in C permits_acc_from accC"
  2945            "accmodi m = Package"      
  2946     then show "?P m"
  2947       by (auto simp add: permits_acc_def)
  2948   next
  2949     case (Overriding new C declC newm old Sup)
  2950     assume member_new: "G \<turnstile> new member_in C" and
  2951                   new: "new = (declC, mdecl newm)" and
  2952              override: "G \<turnstile> (declC, newm) overrides old" and
  2953          subcls_C_Sup: "G\<turnstile>C \<prec>\<^sub>C Sup" and
  2954               acc_old: "G \<turnstile> methdMembr old in Sup dyn_accessible_from accC" and
  2955                   hyp: "?Pack (methdMembr old) \<Longrightarrow> ?P (methdMembr old)" and
  2956           accmodi_new: "accmodi new = Package"
  2957     from override accmodi_new new wf 
  2958     have accmodi_old: "accmodi old = Package"  
  2959       by (auto dest: overrides_Package_old)
  2960     with hyp 
  2961     have P_sup: "?P (methdMembr old)"
  2962       by (simp)
  2963     from wf override new accmodi_old accmodi_new
  2964     have eq_pid_new_old: "pid (declclass new) = pid (declclass old)"
  2965       by (auto dest: dyn_override_Package)
  2966     with eq_pid_new_old P_sup show "?P new"
  2967       by auto
  2968   qed
  2969 qed
  2970 
  2971 text {* For fields we don't need the wellformedness of the program, since
  2972 there is no overriding *}
  2973 lemma dyn_accessible_field_Package:
  2974  assumes dyn_acc: "G \<turnstile> f in C dyn_accessible_from accC" and
  2975             pack: "accmodi f = Package" and
  2976            field: "is_field f"
  2977    shows "pid accC = pid (declclass f)"
  2978 proof -
  2979   from dyn_acc pack field
  2980   show ?thesis
  2981   proof (induct)
  2982     case (Immediate f C)
  2983     from `G \<turnstile> f in C permits_acc_from accC` and `accmodi f = Package`
  2984     show ?case
  2985       by (simp add: permits_acc_def)
  2986   next
  2987     case Overriding
  2988     then show ?case by (simp add: is_field_def)
  2989   qed
  2990 qed
  2991 
  2992 text {* @{text dyn_accessible_instance_field_Protected} only works for fields
  2993 since methods can break the package bounds due to overriding
  2994 *}
  2995 lemma dyn_accessible_instance_field_Protected:
  2996   assumes dyn_acc: "G \<turnstile> f in C dyn_accessible_from accC" and
  2997              prot: "accmodi f = Protected" and
  2998             field: "is_field f" and
  2999    instance_field: "\<not> is_static f" and
  3000           outside: "pid (declclass f) \<noteq> pid accC"
  3001   shows "G\<turnstile> C \<preceq>\<^sub>C accC"
  3002 proof -
  3003   from dyn_acc prot field instance_field outside
  3004   show ?thesis
  3005   proof (induct)
  3006     case (Immediate f C)
  3007     note `G \<turnstile> f in C permits_acc_from accC`
  3008     moreover 
  3009     assume "accmodi f = Protected" and  "is_field f" and "\<not> is_static f" and
  3010            "pid (declclass f) \<noteq> pid accC"
  3011     ultimately 
  3012     show "G\<turnstile> C \<preceq>\<^sub>C accC"
  3013       by (auto simp add: permits_acc_def)
  3014   next
  3015     case Overriding
  3016     then show ?case by (simp add: is_field_def)
  3017   qed
  3018 qed
  3019    
  3020 lemma dyn_accessible_static_field_Protected:
  3021   assumes dyn_acc: "G \<turnstile> f in C dyn_accessible_from accC" and
  3022              prot: "accmodi f = Protected" and
  3023             field: "is_field f" and
  3024      static_field: "is_static f" and
  3025           outside: "pid (declclass f) \<noteq> pid accC"
  3026   shows "G\<turnstile> accC \<preceq>\<^sub>C declclass f  \<and> G\<turnstile>C \<preceq>\<^sub>C declclass f"
  3027 proof -
  3028   from dyn_acc prot field static_field outside
  3029   show ?thesis
  3030   proof (induct)
  3031     case (Immediate f C)
  3032     assume "accmodi f = Protected" and  "is_field f" and "is_static f" and
  3033            "pid (declclass f) \<noteq> pid accC"
  3034     moreover
  3035     note `G \<turnstile> f in C permits_acc_from accC`
  3036     ultimately
  3037     have "G\<turnstile> accC \<preceq>\<^sub>C declclass f"
  3038       by (auto simp add: permits_acc_def)
  3039     moreover
  3040     from `G \<turnstile> f member_in C`
  3041     have "G\<turnstile>C \<preceq>\<^sub>C declclass f"
  3042       by (rule member_in_class_relation)
  3043     ultimately show ?case
  3044       by blast
  3045   next
  3046     case Overriding
  3047     then show ?case by (simp add: is_field_def)
  3048   qed
  3049 qed
  3050 
  3051 end