src/HOL/Bali/WellForm.thy

"Definite Assignment Analysis" included, with proof of correctness. Large adjustments of type safety proof and soundness proof of the axiomatic semantics were necessary. Completeness proof of the loop rule of the axiomatic semantic was altered. So the additional polymorphic variants of some rules could be removed.

(*  Title:      HOL/Bali/WellForm.thy
    ID:         $Id$
    Author:     David von Oheimb and Norbert Schirmer
    License:    GPL (GNU GENERAL PUBLIC LICENSE)
*)

header {* Well-formedness of Java programs *}
theory WellForm = DefiniteAssignment:

text {*
For static checks on expressions and statements, see WellType.thy

improvements over Java Specification 1.0 (cf. 8.4.6.3, 8.4.6.4, 9.4.1):
\begin{itemize}
\item a method implementing or overwriting another method may have a result 
      type that widens to the result type of the other method 
      (instead of identical type)
\item if a method hides another method (both methods have to be static!)
  there are no restrictions to the result type 
  since the methods have to be static and there is no dynamic binding of 
  static methods
\item if an interface inherits more than one method with the same signature, the
  methods need not have identical return types
\end{itemize}
simplifications:
\begin{itemize}
\item Object and standard exceptions are assumed to be declared like normal 
      classes
\end{itemize}
*}

section "well-formed field declarations"
text  {* well-formed field declaration (common part for classes and interfaces),
        cf. 8.3 and (9.3) *}

constdefs
  wf_fdecl :: "prog \<Rightarrow> pname \<Rightarrow> fdecl \<Rightarrow> bool"
 "wf_fdecl G P \<equiv> \<lambda>(fn,f). is_acc_type G P (type f)"

lemma wf_fdecl_def2: "\<And>fd. wf_fdecl G P fd = is_acc_type G P (type (snd fd))"
apply (unfold wf_fdecl_def)
apply simp
done



section "well-formed method declarations"
  (*well-formed method declaration,cf. 8.4, 8.4.1, 8.4.3, 8.4.5, 14.3.2, (9.4)*)
  (* cf. 14.15, 15.7.2, for scope issues cf. 8.4.1 and 14.3.2 *)

text {*
A method head is wellformed if:
\begin{itemize}
\item the signature and the method head agree in the number of parameters
\item all types of the parameters are visible
\item the result type is visible
\item the parameter names are unique
\end{itemize} 
*}
constdefs
  wf_mhead :: "prog \<Rightarrow> pname \<Rightarrow> sig \<Rightarrow> mhead \<Rightarrow> bool"
 "wf_mhead G P \<equiv> \<lambda> sig mh. length (parTs sig) = length (pars mh) \<and>
			    \<spacespace> ( \<forall>T\<in>set (parTs sig). is_acc_type G P T) \<and> 
                            is_acc_type G P (resTy mh) \<and>
			    \<spacespace> distinct (pars mh)"


text {*
A method declaration is wellformed if:
\begin{itemize}
\item the method head is wellformed
\item the names of the local variables are unique
\item the types of the local variables must be accessible
\item the local variables don't shadow the parameters
\item the class of the method is defined
\item the body statement is welltyped with respect to the
      modified environment of local names, were the local variables, 
      the parameters the special result variable (Res) and This are assoziated
      with there types. 
\end{itemize}
*}

constdefs callee_lcl:: "qtname \<Rightarrow> sig \<Rightarrow> methd \<Rightarrow> lenv"
"callee_lcl C sig m 
 \<equiv> \<lambda> k. (case k of
            EName e 
            \<Rightarrow> (case e of 
                  VNam v 
                  \<Rightarrow>(table_of (lcls (mbody m))((pars m)[\<mapsto>](parTs sig))) v
                | Res \<Rightarrow> Some (resTy m))
	  | This \<Rightarrow> if is_static m then None else Some (Class C))"

constdefs parameters :: "methd \<Rightarrow> lname set"
"parameters m \<equiv>  set (map (EName \<circ> VNam) (pars m)) 
                  \<union> (if (static m) then {} else {This})"

constdefs
  wf_mdecl :: "prog \<Rightarrow> qtname \<Rightarrow> mdecl \<Rightarrow> bool"
 "wf_mdecl G C \<equiv> 
      \<lambda>(sig,m).
	  wf_mhead G (pid C) sig (mhead m) \<and> 
          unique (lcls (mbody m)) \<and> 
          (\<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T) \<and> 
	  (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None) \<and>
          jumpNestingOkS {Ret} (stmt (mbody m)) \<and> 
          is_class G C \<and>
          \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>(stmt (mbody m))\<Colon>\<surd> \<and>
          (\<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 
               \<and> Result \<in> nrm A)"

lemma callee_lcl_VNam_simp [simp]:
"callee_lcl C sig m (EName (VNam v)) 
  = (table_of (lcls (mbody m))((pars m)[\<mapsto>](parTs sig))) v"
by (simp add: callee_lcl_def)
 
lemma callee_lcl_Res_simp [simp]:
"callee_lcl C sig m (EName Res) = Some (resTy m)" 
by (simp add: callee_lcl_def)

lemma callee_lcl_This_simp [simp]:
"callee_lcl C sig m (This) = (if is_static m then None else Some (Class C))" 
by (simp add: callee_lcl_def)

lemma callee_lcl_This_static_simp:
"is_static m \<Longrightarrow> callee_lcl C sig m (This) = None"
by simp

lemma callee_lcl_This_not_static_simp:
"\<not> is_static m \<Longrightarrow> callee_lcl C sig m (This) = Some (Class C)"
by simp

lemma wf_mheadI: 
"\<lbrakk>length (parTs sig) = length (pars m); \<forall>T\<in>set (parTs sig). is_acc_type G P T;
  is_acc_type G P (resTy m); distinct (pars m)\<rbrakk> \<Longrightarrow>  
  wf_mhead G P sig m"
apply (unfold wf_mhead_def)
apply (simp (no_asm_simp))
done

lemma wf_mdeclI: "\<lbrakk>  
  wf_mhead G (pid C) sig (mhead m); unique (lcls (mbody m));  
  (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None); 
  \<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T;
  jumpNestingOkS {Ret} (stmt (mbody m));
  is_class G C;
  \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>(stmt (mbody m))\<Colon>\<surd>;
  (\<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
        \<and> Result \<in> nrm A)
  \<rbrakk> \<Longrightarrow>  
  wf_mdecl G C (sig,m)"
apply (unfold wf_mdecl_def)
apply simp
done

lemma wf_mdeclE [consumes 1]:  
  "\<lbrakk>wf_mdecl G C (sig,m); 
    \<lbrakk>wf_mhead G (pid C) sig (mhead m); unique (lcls (mbody m));  
     \<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None; 
     \<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T;
     jumpNestingOkS {Ret} (stmt (mbody m));
     is_class G C;
     \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>(stmt (mbody m))\<Colon>\<surd>;
   (\<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
        \<and> Result \<in> nrm A)
    \<rbrakk> \<Longrightarrow> P
  \<rbrakk> \<Longrightarrow> P"
by (unfold wf_mdecl_def) simp


lemma wf_mdeclD1: 
"wf_mdecl G C (sig,m) \<Longrightarrow>  
   wf_mhead G (pid C) sig (mhead m) \<and> unique (lcls (mbody m)) \<and>  
  (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None) \<and> 
  (\<forall>(vn,T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T)"
apply (unfold wf_mdecl_def)
apply simp
done

lemma wf_mdecl_bodyD: 
"wf_mdecl G C (sig,m) \<Longrightarrow>  
 (\<exists>T. \<lparr>prg=G,cls=C,lcl=callee_lcl C sig m\<rparr>\<turnstile>Body C (stmt (mbody m))\<Colon>-T \<and> 
      G\<turnstile>T\<preceq>(resTy m))"
apply (unfold wf_mdecl_def)
apply clarify
apply (rule_tac x="(resTy m)" in exI)
apply (unfold wf_mhead_def)
apply (auto simp add: wf_mhead_def is_acc_type_def intro: wt.Body )
done


(*
lemma static_Object_methodsE [elim!]: 
 "\<lbrakk>wf_mdecl G Object (sig, m);static m\<rbrakk> \<Longrightarrow> R"
apply (unfold wf_mdecl_def)
apply auto
done
*)

lemma rT_is_acc_type: 
  "wf_mhead G P sig m \<Longrightarrow> is_acc_type G P (resTy m)"
apply (unfold wf_mhead_def)
apply auto
done

section "well-formed interface declarations"
  (* well-formed interface declaration, cf. 9.1, 9.1.2.1, 9.1.3, 9.4 *)

text {*
A interface declaration is wellformed if:
\begin{itemize}
\item the interface hierarchy is wellstructured
\item there is no class with the same name
\item the method heads are wellformed and not static and have Public access
\item the methods are uniquely named
\item all superinterfaces are accessible
\item the result type of a method overriding a method of Object widens to the
      result type of the overridden method.
      Shadowing static methods is forbidden.
\item the result type of a method overriding a set of methods defined in the
      superinterfaces widens to each of the corresponding result types
\end{itemize}
*}
constdefs
  wf_idecl :: "prog  \<Rightarrow> idecl \<Rightarrow> bool"
 "wf_idecl G \<equiv> 
    \<lambda>(I,i). 
        ws_idecl G I (isuperIfs i) \<and> 
	\<not>is_class G I \<and>
	(\<forall>(sig,mh)\<in>set (imethods i). wf_mhead G (pid I) sig mh \<and> 
                                     \<not>is_static mh \<and>
                                      accmodi mh = Public) \<and>
	unique (imethods i) \<and>
        (\<forall> J\<in>set (isuperIfs i). is_acc_iface G (pid I) J) \<and>
        (table_of (imethods i)
          hiding (methd G Object)
          under  (\<lambda> new old. accmodi old \<noteq> Private)
          entails (\<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old \<and> 
                             is_static new = is_static old)) \<and> 
        (o2s \<circ> table_of (imethods i) 
               hidings Un_tables((\<lambda>J.(imethds G J))`set (isuperIfs i))
	       entails (\<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old))"

lemma wf_idecl_mhead: "\<lbrakk>wf_idecl G (I,i); (sig,mh)\<in>set (imethods i)\<rbrakk> \<Longrightarrow>  
  wf_mhead G (pid I) sig mh \<and> \<not>is_static mh \<and> accmodi mh = Public"
apply (unfold wf_idecl_def)
apply auto
done

lemma wf_idecl_hidings: 
"wf_idecl G (I, i) \<Longrightarrow> 
  (\<lambda>s. o2s (table_of (imethods i) s)) 
  hidings Un_tables ((\<lambda>J. imethds G J) ` set (isuperIfs i))  
  entails \<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old"
apply (unfold wf_idecl_def o_def)
apply simp
done

lemma wf_idecl_hiding:
"wf_idecl G (I, i) \<Longrightarrow> 
 (table_of (imethods i)
           hiding (methd G Object)
           under  (\<lambda> new old. accmodi old \<noteq> Private)
           entails (\<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old \<and> 
                              is_static new = is_static old))"
apply (unfold wf_idecl_def)
apply simp
done

lemma wf_idecl_supD: 
"\<lbrakk>wf_idecl G (I,i); J \<in> set (isuperIfs i)\<rbrakk> 
 \<Longrightarrow> is_acc_iface G (pid I) J \<and> (J, I) \<notin> (subint1 G)^+"
apply (unfold wf_idecl_def ws_idecl_def)
apply auto
done

section "well-formed class declarations"
  (* well-formed class declaration, cf. 8.1, 8.1.2.1, 8.1.2.2, 8.1.3, 8.1.4 and
   class method declaration, cf. 8.4.3.3, 8.4.6.1, 8.4.6.2, 8.4.6.3, 8.4.6.4 *)

text {*
A class declaration is wellformed if:
\begin{itemize}
\item there is no interface with the same name
\item all superinterfaces are accessible and for all methods implementing 
      an interface method the result type widens to the result type of 
      the interface method, the method is not static and offers at least 
      as much access 
      (this actually means that the method has Public access, since all 
      interface methods have public access)
\item all field declarations are wellformed and the field names are unique
\item all method declarations are wellformed and the method names are unique
\item the initialization statement is welltyped
\item the classhierarchy is wellstructured
\item Unless the class is Object:
      \begin{itemize}
      \item the superclass is accessible
      \item for all methods overriding another method (of a superclass )the
            result type widens to the result type of the overridden method,
            the access modifier of the new method provides at least as much
            access as the overwritten one.
      \item for all methods hiding a method (of a superclass) the hidden 
            method must be static and offer at least as much access rights.
            Remark: In contrast to the Java Language Specification we don't
            restrict the result types of the method
            (as in case of overriding), because there seems to be no reason,
            since there is no dynamic binding of static methods.
            (cf. 8.4.6.3 vs. 15.12.1).
            Stricly speaking the restrictions on the access rights aren't 
            necessary to, since the static type and the access rights 
            together determine which method is to be called statically. 
            But if a class gains more then one static method with the 
            same signature due to inheritance, it is confusing when the 
            method selection depends on the access rights only: 
            e.g.
              Class C declares static public method foo().
              Class D is subclass of C and declares static method foo()
              with default package access.
              D.foo() ? if this call is in the same package as D then
                        foo of class D is called, otherwise foo of class C.
      \end{itemize}

\end{itemize}
*}
(* to Table *)
constdefs entails:: "('a,'b) table \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"
                                 ("_ entails _" 20)
"t entails P \<equiv> \<forall>k. \<forall> x \<in> t k: P x"

lemma entailsD:
 "\<lbrakk>t entails P; t k = Some x\<rbrakk> \<Longrightarrow> P x"
by (simp add: entails_def)

lemma empty_entails[simp]: "empty entails P"
by (simp add: entails_def)

constdefs
 wf_cdecl :: "prog \<Rightarrow> cdecl \<Rightarrow> bool"
"wf_cdecl G \<equiv> 
   \<lambda>(C,c).
      \<not>is_iface G C \<and>
      (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and>
        (\<forall>s. \<forall> im \<in> imethds G I s.
      	    (\<exists> cm \<in> methd  G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and>
      	                             \<not> is_static cm \<and>
                                     accmodi im \<le> accmodi cm))) \<and>
      (\<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f) \<and> unique (cfields c) \<and> 
      (\<forall>m\<in>set (methods c). wf_mdecl G C m) \<and> unique (methods c) \<and>
      jumpNestingOkS {} (init c) \<and>
      (\<exists> A. \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<guillemotright> A) \<and>
      \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd> \<and> ws_cdecl G C (super c) \<and>
      (C \<noteq> Object \<longrightarrow> 
            (is_acc_class G (pid C) (super c) \<and>
            (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
             entails (\<lambda> new. \<forall> old sig. 
                       (G,sig\<turnstile>new overrides\<^sub>S old 
                        \<longrightarrow> (G\<turnstile>resTy new\<preceq>resTy old \<and>
                             accmodi old \<le> accmodi new \<and>
      	                     \<not>is_static old)) \<and>
                       (G,sig\<turnstile>new hides old 
                         \<longrightarrow> (accmodi old \<le> accmodi new \<and>
      	                      is_static old)))) 
            ))"

(*
constdefs
 wf_cdecl :: "prog \<Rightarrow> cdecl \<Rightarrow> bool"
"wf_cdecl G \<equiv> 
   \<lambda>(C,c).
      \<not>is_iface G C \<and>
      (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and>
        (\<forall>s. \<forall> im \<in> imethds G I s.
      	    (\<exists> cm \<in> methd  G C s: G\<turnstile>resTy (mthd cm)\<preceq>resTy (mthd im) \<and>
      	                             \<not> is_static cm \<and>
                                     accmodi im \<le> accmodi cm))) \<and>
      (\<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f) \<and> unique (cfields c) \<and> 
      (\<forall>m\<in>set (methods c). wf_mdecl G C m) \<and> unique (methods c) \<and> 
      \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd> \<and> ws_cdecl G C (super c) \<and>
      (C \<noteq> Object \<longrightarrow> 
            (is_acc_class G (pid C) (super c) \<and>
            (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
              hiding methd G (super c)
              under (\<lambda> new old. G\<turnstile>new overrides old)
              entails (\<lambda> new old. 
                           (G\<turnstile>resTy (mthd new)\<preceq>resTy (mthd old) \<and>
                            accmodi old \<le> accmodi new \<and>
      	                   \<not> is_static old)))  \<and>
            (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
              hiding methd G (super c)
              under (\<lambda> new old. G\<turnstile>new hides old)
              entails (\<lambda> new old. is_static old \<and> 
                                  accmodi old \<le> accmodi new))  \<and>
            (table_of (cfields c) hiding accfield G C (super c)
              entails (\<lambda> newF oldF. accmodi oldF \<le> access newF))))"
*)

lemma wf_cdeclE [consumes 1]: 
 "\<lbrakk>wf_cdecl G (C,c);
   \<lbrakk>\<not>is_iface G C;
    (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and>
        (\<forall>s. \<forall> im \<in> imethds G I s.
      	    (\<exists> cm \<in> methd  G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and>
      	                             \<not> is_static cm \<and>
                                     accmodi im \<le> accmodi cm))); 
      \<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f; unique (cfields c); 
      \<forall>m\<in>set (methods c). wf_mdecl G C m; unique (methods c);
      jumpNestingOkS {} (init c);
      \<exists> A. \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<guillemotright> A;
      \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd>; 
      ws_cdecl G C (super c); 
      (C \<noteq> Object \<longrightarrow> 
            (is_acc_class G (pid C) (super c) \<and>
            (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
             entails (\<lambda> new. \<forall> old sig. 
                       (G,sig\<turnstile>new overrides\<^sub>S old 
                        \<longrightarrow> (G\<turnstile>resTy new\<preceq>resTy old \<and>
                             accmodi old \<le> accmodi new \<and>
      	                     \<not>is_static old)) \<and>
                       (G,sig\<turnstile>new hides old 
                         \<longrightarrow> (accmodi old \<le> accmodi new \<and>
      	                      is_static old)))) 
            ))\<rbrakk> \<Longrightarrow> P
  \<rbrakk> \<Longrightarrow> P"
by (unfold wf_cdecl_def) simp

lemma wf_cdecl_unique: 
"wf_cdecl G (C,c) \<Longrightarrow> unique (cfields c) \<and> unique (methods c)"
apply (unfold wf_cdecl_def)
apply auto
done

lemma wf_cdecl_fdecl: 
"\<lbrakk>wf_cdecl G (C,c); f\<in>set (cfields c)\<rbrakk> \<Longrightarrow> wf_fdecl G (pid C) f"
apply (unfold wf_cdecl_def)
apply auto
done

lemma wf_cdecl_mdecl: 
"\<lbrakk>wf_cdecl G (C,c); m\<in>set (methods c)\<rbrakk> \<Longrightarrow> wf_mdecl G C m"
apply (unfold wf_cdecl_def)
apply auto
done

lemma wf_cdecl_impD: 
"\<lbrakk>wf_cdecl G (C,c); I\<in>set (superIfs c)\<rbrakk> 
\<Longrightarrow> is_acc_iface G (pid C) I \<and>  
    (\<forall>s. \<forall>im \<in> imethds G I s.  
        (\<exists>cm \<in> methd G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and> \<not>is_static cm \<and>
                                   accmodi im \<le> accmodi cm))"
apply (unfold wf_cdecl_def)
apply auto
done

lemma wf_cdecl_supD: 
"\<lbrakk>wf_cdecl G (C,c); C \<noteq> Object\<rbrakk> \<Longrightarrow>  
  is_acc_class G (pid C) (super c) \<and> (super c,C) \<notin> (subcls1 G)^+ \<and> 
   (table_of (map (\<lambda> (s,m). (s,C,m)) (methods c)) 
    entails (\<lambda> new. \<forall> old sig. 
                 (G,sig\<turnstile>new overrides\<^sub>S old 
                  \<longrightarrow> (G\<turnstile>resTy new\<preceq>resTy old \<and>
                       accmodi old \<le> accmodi new \<and>
                       \<not>is_static old)) \<and>
                 (G,sig\<turnstile>new hides old 
                   \<longrightarrow> (accmodi old \<le> accmodi new \<and>
                        is_static old))))"
apply (unfold wf_cdecl_def ws_cdecl_def)
apply auto
done


lemma wf_cdecl_overrides_SomeD:
"\<lbrakk>wf_cdecl G (C,c); C \<noteq> Object; table_of (methods c) sig = Some newM;
  G,sig\<turnstile>(C,newM) overrides\<^sub>S old
\<rbrakk> \<Longrightarrow>  G\<turnstile>resTy newM\<preceq>resTy old \<and>
       accmodi old \<le> accmodi newM \<and>
       \<not> is_static old" 
apply (drule (1) wf_cdecl_supD)
apply (clarify)
apply (drule entailsD)
apply   (blast intro: table_of_map_SomeI)
apply (drule_tac x="old" in spec)
apply (auto dest: overrides_eq_sigD simp add: msig_def)
done

lemma wf_cdecl_hides_SomeD:
"\<lbrakk>wf_cdecl G (C,c); C \<noteq> Object; table_of (methods c) sig = Some newM;
  G,sig\<turnstile>(C,newM) hides old
\<rbrakk> \<Longrightarrow>  accmodi old \<le> access newM \<and>
       is_static old" 
apply (drule (1) wf_cdecl_supD)
apply (clarify)
apply (drule entailsD)
apply   (blast intro: table_of_map_SomeI)
apply (drule_tac x="old" in spec)
apply (auto dest: hides_eq_sigD simp add: msig_def)
done

lemma wf_cdecl_wt_init: 
 "wf_cdecl G (C, c) \<Longrightarrow> \<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>init c\<Colon>\<surd>"
apply (unfold wf_cdecl_def)
apply auto
done


section "well-formed programs"
  (* well-formed program, cf. 8.1, 9.1 *)

text {*
A program declaration is wellformed if:
\begin{itemize}
\item the class ObjectC of Object is defined
\item every method of has an access modifier distinct from Package. This is
      necessary since every interface automatically inherits from Object.  
      We must know, that every time a Object method is "overriden" by an 
      interface method this is also overriden by the class implementing the
      the interface (see @{text "implement_dynmethd and class_mheadsD"})
\item all standard Exceptions are defined
\item all defined interfaces are wellformed
\item all defined classes are wellformed
\end{itemize}
*}
constdefs
  wf_prog  :: "prog \<Rightarrow> bool"
 "wf_prog G \<equiv> let is = ifaces G; cs = classes G in
	         ObjectC \<in> set cs \<and> 
                (\<forall> m\<in>set Object_mdecls. accmodi m \<noteq> Package) \<and>
                (\<forall>xn. SXcptC xn \<in> set cs) \<and>
		(\<forall>i\<in>set is. wf_idecl G i) \<and> unique is \<and>
		(\<forall>c\<in>set cs. wf_cdecl G c) \<and> unique cs"

lemma wf_prog_idecl: "\<lbrakk>iface G I = Some i; wf_prog G\<rbrakk> \<Longrightarrow> wf_idecl G (I,i)"
apply (unfold wf_prog_def Let_def)
apply simp
apply (fast dest: map_of_SomeD)
done

lemma wf_prog_cdecl: "\<lbrakk>class G C = Some c; wf_prog G\<rbrakk> \<Longrightarrow> wf_cdecl G (C,c)"
apply (unfold wf_prog_def Let_def)
apply simp
apply (fast dest: map_of_SomeD)
done

lemma wf_prog_Object_mdecls:
"wf_prog G \<Longrightarrow> (\<forall> m\<in>set Object_mdecls. accmodi m \<noteq> Package)"
apply (unfold wf_prog_def Let_def)
apply simp
done

lemma wf_prog_acc_superD:
 "\<lbrakk>wf_prog G; class G C = Some c; C \<noteq> Object \<rbrakk> 
  \<Longrightarrow> is_acc_class G (pid C) (super c)"
by (auto dest: wf_prog_cdecl wf_cdecl_supD)

lemma wf_ws_prog [elim!,simp]: "wf_prog G \<Longrightarrow> ws_prog G"
apply (unfold wf_prog_def Let_def)
apply (rule ws_progI)
apply  (simp_all (no_asm))
apply  (auto simp add: is_acc_class_def is_acc_iface_def 
             dest!: wf_idecl_supD wf_cdecl_supD )+
done

lemma class_Object [simp]: 
"wf_prog G \<Longrightarrow> 
  class G Object = Some \<lparr>access=Public,cfields=[],methods=Object_mdecls,
                                  init=Skip,super=arbitrary,superIfs=[]\<rparr>"
apply (unfold wf_prog_def Let_def ObjectC_def)
apply (fast dest!: map_of_SomeI)
done

lemma methd_Object[simp]: "wf_prog G \<Longrightarrow> methd G Object =  
  table_of (map (\<lambda>(s,m). (s, Object, m)) Object_mdecls)"
apply (subst methd_rec)
apply (auto simp add: Let_def)
done

lemma wf_prog_Object_methd:
"\<lbrakk>wf_prog G; methd G Object sig = Some m\<rbrakk> \<Longrightarrow> accmodi m \<noteq> Package"
by (auto dest!: wf_prog_Object_mdecls) (auto dest!: map_of_SomeD) 

lemma wf_prog_Object_is_public[intro]:
 "wf_prog G \<Longrightarrow> is_public G Object"
by (auto simp add: is_public_def dest: class_Object)

lemma class_SXcpt [simp]: 
"wf_prog G \<Longrightarrow> 
  class G (SXcpt xn) = Some \<lparr>access=Public,cfields=[],methods=SXcpt_mdecls,
                                   init=Skip,
                                   super=if xn = Throwable then Object 
                                                           else SXcpt Throwable,
                                   superIfs=[]\<rparr>"
apply (unfold wf_prog_def Let_def SXcptC_def)
apply (fast dest!: map_of_SomeI)
done

lemma wf_ObjectC [simp]: 
	"wf_cdecl G ObjectC = (\<not>is_iface G Object \<and> Ball (set Object_mdecls)
  (wf_mdecl G Object) \<and> unique Object_mdecls)"
apply (unfold wf_cdecl_def ws_cdecl_def ObjectC_def)
apply (auto intro: da.Skip)
done

lemma Object_is_class [simp,elim!]: "wf_prog G \<Longrightarrow> is_class G Object"
apply (simp (no_asm_simp))
done
 
lemma Object_is_acc_class [simp,elim!]: "wf_prog G \<Longrightarrow> is_acc_class G S Object"
apply (simp (no_asm_simp) add: is_acc_class_def is_public_def
                               accessible_in_RefT_simp)
done

lemma SXcpt_is_class [simp,elim!]: "wf_prog G \<Longrightarrow> is_class G (SXcpt xn)"
apply (simp (no_asm_simp))
done

lemma SXcpt_is_acc_class [simp,elim!]: 
"wf_prog G \<Longrightarrow> is_acc_class G S (SXcpt xn)"
apply (simp (no_asm_simp) add: is_acc_class_def is_public_def
                               accessible_in_RefT_simp)
done

lemma fields_Object [simp]: "wf_prog G \<Longrightarrow> DeclConcepts.fields G Object = []"
by (force intro: fields_emptyI)

lemma accfield_Object [simp]: 
 "wf_prog G \<Longrightarrow> accfield G S Object = empty"
apply (unfold accfield_def)
apply (simp (no_asm_simp) add: Let_def)
done

lemma fields_Throwable [simp]: 
 "wf_prog G \<Longrightarrow> DeclConcepts.fields G (SXcpt Throwable) = []"
by (force intro: fields_emptyI)

lemma fields_SXcpt [simp]: "wf_prog G \<Longrightarrow> DeclConcepts.fields G (SXcpt xn) = []"
apply (case_tac "xn = Throwable")
apply  (simp (no_asm_simp))
by (force intro: fields_emptyI)

lemmas widen_trans = ws_widen_trans [OF _ _ wf_ws_prog, elim]
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"
apply (erule (2) widen_trans)
done

lemma Xcpt_subcls_Throwable [simp]: 
"wf_prog G \<Longrightarrow> G\<turnstile>SXcpt xn\<preceq>\<^sub>C SXcpt Throwable"
apply (rule SXcpt_subcls_Throwable_lemma)
apply auto
done

lemma unique_fields: 
 "\<lbrakk>is_class G C; wf_prog G\<rbrakk> \<Longrightarrow> unique (DeclConcepts.fields G C)"
apply (erule ws_unique_fields)
apply  (erule wf_ws_prog)
apply (erule (1) wf_prog_cdecl [THEN wf_cdecl_unique [THEN conjunct1]])
done

lemma fields_mono: 
"\<lbrakk>table_of (DeclConcepts.fields G C) fn = Some f; G\<turnstile>D\<preceq>\<^sub>C C; 
  is_class G D; wf_prog G\<rbrakk> 
   \<Longrightarrow> table_of (DeclConcepts.fields G D) fn = Some f"
apply (rule map_of_SomeI)
apply  (erule (1) unique_fields)
apply (erule (1) map_of_SomeD [THEN fields_mono_lemma])
apply (erule wf_ws_prog)
done


lemma fields_is_type [elim]: 
"\<lbrakk>table_of (DeclConcepts.fields G C) m = Some f; wf_prog G; is_class G C\<rbrakk> \<Longrightarrow> 
      is_type G (type f)"
apply (frule wf_ws_prog)
apply (force dest: fields_declC [THEN conjunct1] 
                   wf_prog_cdecl [THEN wf_cdecl_fdecl]
             simp add: wf_fdecl_def2 is_acc_type_def)
done

lemma imethds_wf_mhead [rule_format (no_asm)]: 
"\<lbrakk>m \<in> imethds G I sig; wf_prog G; is_iface G I\<rbrakk> \<Longrightarrow>  
  wf_mhead G (pid (decliface m)) sig (mthd m) \<and> 
  \<not> is_static m \<and> accmodi m = Public"
apply (frule wf_ws_prog)
apply (drule (2) imethds_declI [THEN conjunct1])
apply clarify
apply (frule_tac I="(decliface m)" in wf_prog_idecl,assumption)
apply (drule wf_idecl_mhead)
apply (erule map_of_SomeD)
apply (cases m, simp)
done

lemma methd_wf_mdecl: 
 "\<lbrakk>methd G C sig = Some m; wf_prog G; class G C = Some y\<rbrakk> \<Longrightarrow>  
  G\<turnstile>C\<preceq>\<^sub>C (declclass m) \<and> is_class G (declclass m) \<and> 
  wf_mdecl G (declclass m) (sig,(mthd m))"
apply (frule wf_ws_prog)
apply (drule (1) methd_declC)
apply  fast
apply clarsimp
apply (frule (1) wf_prog_cdecl, erule wf_cdecl_mdecl, erule map_of_SomeD)
done

(*
This lemma doesn't hold!
lemma methd_rT_is_acc_type: 
"\<lbrakk>wf_prog G;methd G C C sig = Some (D,m);
    class G C = Some y\<rbrakk>
\<Longrightarrow> is_acc_type G (pid C) (resTy m)"
The result Type is only visible in the scope of defining class D 
"is_vis_type G (pid D) (resTy m)" but not necessarily in scope of class C!
(The same is true for the type of pramaters of a method)
*)


lemma methd_rT_is_type: 
"\<lbrakk>wf_prog G;methd G C sig = Some m;
    class G C = Some y\<rbrakk>
\<Longrightarrow> is_type G (resTy m)"
apply (drule (2) methd_wf_mdecl)
apply clarify
apply (drule wf_mdeclD1)
apply clarify
apply (drule rT_is_acc_type)
apply (cases m, simp add: is_acc_type_def)
done

lemma accmethd_rT_is_type:
"\<lbrakk>wf_prog G;accmethd G S C sig = Some m;
    class G C = Some y\<rbrakk>
\<Longrightarrow> is_type G (resTy m)"
by (auto simp add: accmethd_def  
         intro: methd_rT_is_type)

lemma methd_Object_SomeD:
"\<lbrakk>wf_prog G;methd G Object sig = Some m\<rbrakk> 
 \<Longrightarrow> declclass m = Object"
by (auto dest: class_Object simp add: methd_rec )

lemma wf_imethdsD: 
 "\<lbrakk>im \<in> imethds G I sig;wf_prog G; is_iface G I\<rbrakk> 
 \<Longrightarrow> \<not>is_static im \<and> accmodi im = Public"
proof -
  assume asm: "wf_prog G" "is_iface G I" "im \<in> imethds G I sig"
  have "wf_prog G \<longrightarrow> 
         (\<forall> i im. iface G I = Some i \<longrightarrow> im \<in> imethds G I sig
                  \<longrightarrow> \<not>is_static im \<and> accmodi im = Public)" (is "?P G I")
  proof (rule iface_rec.induct,intro allI impI)
    fix G I i im
    assume hyp: "\<forall> J i. J \<in> set (isuperIfs i) \<and> ws_prog G \<and> iface G I = Some i
                 \<longrightarrow> ?P G J"
    assume wf: "wf_prog G" and if_I: "iface G I = Some i" and 
           im: "im \<in> imethds G I sig" 
    show "\<not>is_static im \<and> accmodi im = Public" 
    proof -
      let ?inherited = "Un_tables (imethds G ` set (isuperIfs i))"
      let ?new = "(o2s \<circ> table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)))"
      from if_I wf im have imethds:"im \<in> (?inherited \<oplus>\<oplus> ?new) sig"
	by (simp add: imethds_rec)
      from wf if_I have 
	wf_supI: "\<forall> J. J \<in> set (isuperIfs i) \<longrightarrow> (\<exists> j. iface G J = Some j)"
	by (blast dest: wf_prog_idecl wf_idecl_supD is_acc_ifaceD)
      from wf if_I have
	"\<forall> im \<in> set (imethods i). \<not> is_static im \<and> accmodi im = Public"
	by (auto dest!: wf_prog_idecl wf_idecl_mhead)
      then have new_ok: "\<forall> im. table_of (imethods i) sig = Some im 
                         \<longrightarrow>  \<not> is_static im \<and> accmodi im = Public"
	by (auto dest!: table_of_Some_in_set)
      show ?thesis
	proof (cases "?new sig = {}")
	  case True
	  from True wf wf_supI if_I imethds hyp 
	  show ?thesis by (auto simp del:  split_paired_All)  
	next
	  case False
	  from False wf wf_supI if_I imethds new_ok hyp 
	  show ?thesis by (auto dest: wf_idecl_hidings hidings_entailsD)
	qed
      qed
    qed
  with asm show ?thesis by (auto simp del: split_paired_All)
qed

lemma wf_prog_hidesD:
  assumes hides: "G \<turnstile>new hides old" and wf: "wf_prog G"
  shows
   "accmodi old \<le> accmodi new \<and>
    is_static old"
proof -
  from hides 
  obtain c where 
    clsNew: "class G (declclass new) = Some c" and
    neqObj: "declclass new \<noteq> Object"
    by (auto dest: hidesD declared_in_classD)
  with hides obtain newM oldM where
    newM: "table_of (methods c) (msig new) = Some newM" and 
     new: "new = (declclass new,(msig new),newM)" and
     old: "old = (declclass old,(msig old),oldM)" and
          "msig new = msig old"
    by (cases new,cases old) 
       (auto dest: hidesD 
         simp add: cdeclaredmethd_def declared_in_def)
  with hides 
  have hides':
        "G,(msig new)\<turnstile>(declclass new,newM) hides (declclass old,oldM)"
    by auto
  from clsNew wf 
  have "wf_cdecl G (declclass new,c)" by (blast intro: wf_prog_cdecl)
  note wf_cdecl_hides_SomeD [OF this neqObj newM hides']
  with new old 
  show ?thesis
    by (cases new, cases old) auto
qed

text {* Compare this lemma about static  
overriding @{term "G \<turnstile>new overrides\<^sub>S old"} with the definition of 
dynamic overriding @{term "G \<turnstile>new overrides old"}. 
Conforming result types and restrictions on the access modifiers of the old 
and the new method are not part of the predicate for static overriding. But
they are enshured in a wellfromed program.  Dynamic overriding has 
no restrictions on the access modifiers but enforces confrom result types 
as precondition. But with some efford we can guarantee the access modifier
restriction for dynamic overriding, too. See lemma 
@{text wf_prog_dyn_override_prop}.
*}
lemma wf_prog_stat_overridesD:
  assumes stat_override: "G \<turnstile>new overrides\<^sub>S old" and wf: "wf_prog G"
  shows
   "G\<turnstile>resTy new\<preceq>resTy old \<and>
    accmodi old \<le> accmodi new \<and>
    \<not> is_static old"
proof -
  from stat_override 
  obtain c where 
    clsNew: "class G (declclass new) = Some c" and
    neqObj: "declclass new \<noteq> Object"
    by (auto dest: stat_overrides_commonD declared_in_classD)
  with stat_override obtain newM oldM where
    newM: "table_of (methods c) (msig new) = Some newM" and 
     new: "new = (declclass new,(msig new),newM)" and
     old: "old = (declclass old,(msig old),oldM)" and
          "msig new = msig old"
    by (cases new,cases old) 
       (auto dest: stat_overrides_commonD 
         simp add: cdeclaredmethd_def declared_in_def)
  with stat_override 
  have stat_override':
        "G,(msig new)\<turnstile>(declclass new,newM) overrides\<^sub>S (declclass old,oldM)"
    by auto
  from clsNew wf 
  have "wf_cdecl G (declclass new,c)" by (blast intro: wf_prog_cdecl)
  note wf_cdecl_overrides_SomeD [OF this neqObj newM stat_override']
  with new old 
  show ?thesis
    by (cases new, cases old) auto
qed
    
lemma static_to_dynamic_overriding: 
  assumes stat_override: "G\<turnstile>new overrides\<^sub>S old" and wf : "wf_prog G"
  shows "G\<turnstile>new overrides old"
proof -
  from stat_override 
  show ?thesis (is "?Overrides new old")
  proof (induct)
    case (Direct new old superNew)
    then have stat_override:"G\<turnstile>new overrides\<^sub>S old" 
      by (rule stat_overridesR.Direct)
    from stat_override wf
    have resTy_widen: "G\<turnstile>resTy new\<preceq>resTy old" and
      not_static_old: "\<not> is_static old" 
      by (auto dest: wf_prog_stat_overridesD)  
    have not_private_new: "accmodi new \<noteq> Private"
    proof -
      from stat_override 
      have "accmodi old \<noteq> Private"
	by (rule no_Private_stat_override)
      moreover
      from stat_override wf
      have "accmodi old \<le> accmodi new"
	by (auto dest: wf_prog_stat_overridesD)
      ultimately
      show ?thesis
	by (auto dest: acc_modi_bottom)
    qed
    with Direct resTy_widen not_static_old 
    show "?Overrides new old" 
      by (auto intro: overridesR.Direct stat_override_declclasses_relation) 
  next
    case (Indirect inter new old)
    then show "?Overrides new old" 
      by (blast intro: overridesR.Indirect) 
  qed
qed

lemma non_Package_instance_method_inheritance:
  assumes old_inheritable: "G\<turnstile>Method old inheritable_in (pid C)" and
              accmodi_old: "accmodi old \<noteq> Package" and 
          instance_method: "\<not> is_static old" and
                   subcls: "G\<turnstile>C \<prec>\<^sub>C declclass old" and
             old_declared: "G\<turnstile>Method old declared_in (declclass old)" and
                       wf: "wf_prog G"
  shows "G\<turnstile>Method old member_of C \<or>
   (\<exists> new. G\<turnstile> new overrides\<^sub>S old \<and> G\<turnstile>Method new member_of C)"
proof -
  from wf have ws: "ws_prog G" by auto
  from old_declared have iscls_declC_old: "is_class G (declclass old)"
    by (auto simp add: declared_in_def cdeclaredmethd_def)
  from subcls have  iscls_C: "is_class G C"
    by (blast dest:  subcls_is_class)
  from iscls_C ws old_inheritable subcls 
  show ?thesis (is "?P C old")
  proof (induct rule: ws_class_induct')
    case Object
    assume "G\<turnstile>Object\<prec>\<^sub>C declclass old"
    then show "?P Object old"
      by blast
  next
    case (Subcls C c)
    assume cls_C: "class G C = Some c" and 
       neq_C_Obj: "C \<noteq> Object" and
             hyp: "\<lbrakk>G \<turnstile>Method old inheritable_in pid (super c); 
                   G\<turnstile>super c\<prec>\<^sub>C declclass old\<rbrakk> \<Longrightarrow> ?P (super c) old" and
     inheritable: "G \<turnstile>Method old inheritable_in pid C" and
         subclsC: "G\<turnstile>C\<prec>\<^sub>C declclass old"
    from cls_C neq_C_Obj  
    have super: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 super c" 
      by (rule subcls1I)
    from wf cls_C neq_C_Obj
    have accessible_super: "G\<turnstile>(Class (super c)) accessible_in (pid C)" 
      by (auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_classD)
    {
      fix old
      assume    member_super: "G\<turnstile>Method old member_of (super c)"
      assume     inheritable: "G \<turnstile>Method old inheritable_in pid C"
      assume instance_method: "\<not> is_static old"
      from member_super
      have old_declared: "G\<turnstile>Method old declared_in (declclass old)"
       by (cases old) (auto dest: member_of_declC)
      have "?P C old"
      proof (cases "G\<turnstile>mid (msig old) undeclared_in C")
	case True
	with inheritable super accessible_super member_super
	have "G\<turnstile>Method old member_of C"
	  by (cases old) (auto intro: members.Inherited)
	then show ?thesis
	  by auto
      next
	case False
	then obtain new_member where
	     "G\<turnstile>new_member declared_in C" and
             "mid (msig old) = memberid new_member"
          by (auto dest: not_undeclared_declared)
	then obtain new where
	          new: "G\<turnstile>Method new declared_in C" and
               eq_sig: "msig old = msig new" and
	    declC_new: "declclass new = C" 
	  by (cases new_member) auto
	then have member_new: "G\<turnstile>Method new member_of C"
	  by (cases new) (auto intro: members.Immediate)
	from declC_new super member_super
	have subcls_new_old: "G\<turnstile>declclass new \<prec>\<^sub>C declclass old"
	  by (auto dest!: member_of_subclseq_declC
	            dest: r_into_trancl intro: trancl_rtrancl_trancl)
	show ?thesis
	proof (cases "is_static new")
	  case False
	  with eq_sig declC_new new old_declared inheritable
	       super member_super subcls_new_old
	  have "G\<turnstile>new overrides\<^sub>S old"
	    by (auto intro!: stat_overridesR.Direct)
	  with member_new show ?thesis
	    by blast
	next
	  case True
	  with eq_sig declC_new subcls_new_old new old_declared inheritable
	  have "G\<turnstile>new hides old"
	    by (auto intro: hidesI)    
	  with wf 
	  have "is_static old"
	    by (blast dest: wf_prog_hidesD)
	  with instance_method
	  show ?thesis
	    by (contradiction)
	qed
      qed
    } note hyp_member_super = this
    from subclsC cls_C 
    have "G\<turnstile>(super c)\<preceq>\<^sub>C declclass old"
      by (rule subcls_superD)
    then
    show "?P C old"
    proof (cases rule: subclseq_cases) 
      case Eq
      assume "super c = declclass old"
      with old_declared 
      have "G\<turnstile>Method old member_of (super c)" 
	by (cases old) (auto intro: members.Immediate)
      with inheritable instance_method 
      show ?thesis
	by (blast dest: hyp_member_super)
    next
      case Subcls
      assume "G\<turnstile>super c\<prec>\<^sub>C declclass old"
      moreover
      from inheritable accmodi_old
      have "G \<turnstile>Method old inheritable_in pid (super c)"
	by (cases "accmodi old") (auto simp add: inheritable_in_def)
      ultimately
      have "?P (super c) old"
	by (blast dest: hyp)
      then show ?thesis
      proof
	assume "G \<turnstile>Method old member_of super c"
	with inheritable instance_method
	show ?thesis
	  by (blast dest: hyp_member_super)
      next
	assume "\<exists>new. G \<turnstile> new overrides\<^sub>S old \<and> G \<turnstile>Method new member_of super c"
	then obtain super_new where
	  super_new_override:  "G \<turnstile> super_new overrides\<^sub>S old" and
            super_new_member:  "G \<turnstile>Method super_new member_of super c"
	  by blast
	from super_new_override wf
	have "accmodi old \<le> accmodi super_new"
	  by (auto dest: wf_prog_stat_overridesD)
	with inheritable accmodi_old
	have "G \<turnstile>Method super_new inheritable_in pid C"
	  by (auto simp add: inheritable_in_def 
	              split: acc_modi.splits
                       dest: acc_modi_le_Dests)
	moreover
	from super_new_override 
	have "\<not> is_static super_new"
	  by (auto dest: stat_overrides_commonD)
	moreover
	note super_new_member
	ultimately have "?P C super_new"
	  by (auto dest: hyp_member_super)
	then show ?thesis
	proof 
	  assume "G \<turnstile>Method super_new member_of C"
	  with super_new_override
	  show ?thesis
	    by blast
	next
	  assume "\<exists>new. G \<turnstile> new overrides\<^sub>S super_new \<and> 
                  G \<turnstile>Method new member_of C"
	  with super_new_override show ?thesis
	    by (blast intro: stat_overridesR.Indirect) 
	qed
      qed
    qed
  qed
qed

lemma non_Package_instance_method_inheritance_cases [consumes 6,
         case_names Inheritance Overriding]:
  assumes old_inheritable: "G\<turnstile>Method old inheritable_in (pid C)" and
              accmodi_old: "accmodi old \<noteq> Package" and 
          instance_method: "\<not> is_static old" and
                   subcls: "G\<turnstile>C \<prec>\<^sub>C declclass old" and
             old_declared: "G\<turnstile>Method old declared_in (declclass old)" and
                       wf: "wf_prog G" and
              inheritance: "G\<turnstile>Method old member_of C \<Longrightarrow> P" and
               overriding: "\<And> new.
                           \<lbrakk>G\<turnstile> new overrides\<^sub>S old;G\<turnstile>Method new member_of C\<rbrakk>
                           \<Longrightarrow> P"
  shows P
proof -
  from old_inheritable accmodi_old instance_method subcls old_declared wf 
       inheritance overriding
  show ?thesis
    by (auto dest: non_Package_instance_method_inheritance)
qed

lemma dynamic_to_static_overriding:
  assumes dyn_override: "G\<turnstile> new overrides old" and
           accmodi_old: "accmodi old \<noteq> Package" and
                    wf: "wf_prog G"
  shows "G\<turnstile> new overrides\<^sub>S old"  
proof - 
  from dyn_override accmodi_old
  show ?thesis (is "?Overrides new old")
  proof (induct rule: overridesR.induct)
    case (Direct new old)
    assume   new_declared: "G\<turnstile>Method new declared_in declclass new"
    assume eq_sig_new_old: "msig new = msig old"
    assume subcls_new_old: "G\<turnstile>declclass new \<prec>\<^sub>C declclass old"
    assume "G \<turnstile>Method old inheritable_in pid (declclass new)" and
           "accmodi old \<noteq> Package" and
           "\<not> is_static old" and
           "G\<turnstile>declclass new\<prec>\<^sub>C declclass old" and
           "G\<turnstile>Method old declared_in declclass old" 
    from this wf
    show "?Overrides new old"
    proof (cases rule: non_Package_instance_method_inheritance_cases)
      case Inheritance
      assume "G \<turnstile>Method old member_of declclass new"
      then have "G\<turnstile>mid (msig old) undeclared_in declclass new"
      proof cases
	case Immediate 
	with subcls_new_old wf show ?thesis 	
	  by (auto dest: subcls_irrefl)
      next
	case Inherited
	then show ?thesis
	  by (cases old) auto
      qed
      with eq_sig_new_old new_declared
      show ?thesis
	by (cases old,cases new) (auto dest!: declared_not_undeclared)
    next
      case (Overriding new') 
      assume stat_override_new': "G \<turnstile> new' overrides\<^sub>S old"
      then have "msig new' = msig old"
	by (auto dest: stat_overrides_commonD)
      with eq_sig_new_old have eq_sig_new_new': "msig new=msig new'"
	by simp
      assume "G \<turnstile>Method new' member_of declclass new"
      then show ?thesis
      proof (cases)
	case Immediate
	then have declC_new: "declclass new' = declclass new" 
	  by auto
	from Immediate 
	have "G\<turnstile>Method new' declared_in declclass new"
	  by (cases new') auto
	with new_declared eq_sig_new_new' declC_new 
	have "new=new'"
	  by (cases new, cases new') (auto dest: unique_declared_in) 
	with stat_override_new'
	show ?thesis
	  by simp
      next
	case Inherited
	then have "G\<turnstile>mid (msig new') undeclared_in declclass new"
	  by (cases new') (auto)
	with eq_sig_new_new' new_declared
	show ?thesis
	  by (cases new,cases new') (auto dest!: declared_not_undeclared)
      qed
    qed
  next
    case (Indirect inter new old)
    assume accmodi_old: "accmodi old \<noteq> Package"
    assume "accmodi old \<noteq> Package \<Longrightarrow> G \<turnstile> inter overrides\<^sub>S old"
    with accmodi_old 
    have stat_override_inter_old: "G \<turnstile> inter overrides\<^sub>S old"
      by blast
    moreover 
    assume hyp_inter: "accmodi inter \<noteq> Package \<Longrightarrow> G \<turnstile> new overrides\<^sub>S inter"
    moreover
    have "accmodi inter \<noteq> Package"
    proof -
      from stat_override_inter_old wf 
      have "accmodi old \<le> accmodi inter"
	by (auto dest: wf_prog_stat_overridesD)
      with stat_override_inter_old accmodi_old
      show ?thesis
	by (auto dest!: no_Private_stat_override
                 split: acc_modi.splits 
	         dest: acc_modi_le_Dests)
    qed
    ultimately show "?Overrides new old"
      by (blast intro: stat_overridesR.Indirect)
  qed
qed

lemma wf_prog_dyn_override_prop:
  assumes dyn_override: "G \<turnstile> new overrides old" and
                    wf: "wf_prog G"
  shows "accmodi old \<le> accmodi new"
proof (cases "accmodi old = Package")
  case True
  note old_Package = this
  show ?thesis
  proof (cases "accmodi old \<le> accmodi new")
    case True then show ?thesis .
  next
    case False
    with old_Package 
    have "accmodi new = Private"
      by (cases "accmodi new") (auto simp add: le_acc_def less_acc_def)
    with dyn_override 
    show ?thesis
      by (auto dest: overrides_commonD)
  qed    
next
  case False
  with dyn_override wf
  have "G \<turnstile> new overrides\<^sub>S old"
    by (blast intro: dynamic_to_static_overriding)
  with wf 
  show ?thesis
   by (blast dest: wf_prog_stat_overridesD)
qed 

lemma overrides_Package_old: 
  assumes dyn_override: "G \<turnstile> new overrides old" and 
           accmodi_new: "accmodi new = Package" and
                    wf: "wf_prog G "
  shows "accmodi old = Package"
proof (cases "accmodi old")
  case Private
  with dyn_override show ?thesis
    by (simp add: no_Private_override)
next
  case Package
  then show ?thesis .
next
  case Protected
  with dyn_override wf
  have "G \<turnstile> new overrides\<^sub>S old"
    by (auto intro: dynamic_to_static_overriding)
  with wf 
  have "accmodi old \<le> accmodi new"
    by (auto dest: wf_prog_stat_overridesD)
  with Protected accmodi_new
  show ?thesis
    by (simp add: less_acc_def le_acc_def)
next
  case Public
  with dyn_override wf
  have "G \<turnstile> new overrides\<^sub>S old"
    by (auto intro: dynamic_to_static_overriding)
  with wf 
  have "accmodi old \<le> accmodi new"
    by (auto dest: wf_prog_stat_overridesD)
  with Public accmodi_new
  show ?thesis
    by (simp add: less_acc_def le_acc_def)
qed

lemma dyn_override_Package:
  assumes dyn_override: "G \<turnstile> new overrides old" and
           accmodi_old: "accmodi old = Package" and 
           accmodi_new: "accmodi new = Package" and
                    wf: "wf_prog G"
  shows "pid (declclass old) = pid (declclass new)"
proof - 
  from dyn_override accmodi_old accmodi_new
  show ?thesis (is "?EqPid old new")
  proof (induct rule: overridesR.induct)
    case (Direct new old)
    assume "accmodi old = Package"
           "G \<turnstile>Method old inheritable_in pid (declclass new)"
    then show "pid (declclass old) =  pid (declclass new)"
      by (auto simp add: inheritable_in_def)
  next
    case (Indirect inter new old)
    assume accmodi_old: "accmodi old = Package" and
           accmodi_new: "accmodi new = Package" 
    assume "G \<turnstile> new overrides inter"
    with accmodi_new wf
    have "accmodi inter = Package"
      by  (auto intro: overrides_Package_old)
    with Indirect
    show "pid (declclass old) =  pid (declclass new)"
      by auto
  qed
qed

lemma dyn_override_Package_escape:
  assumes dyn_override: "G \<turnstile> new overrides old" and
           accmodi_old: "accmodi old = Package" and 
          outside_pack: "pid (declclass old) \<noteq> pid (declclass new)" and
                    wf: "wf_prog G"
  shows "\<exists> inter. G \<turnstile> new overrides inter \<and> G \<turnstile> inter overrides old \<and>
             pid (declclass old) = pid (declclass inter) \<and>
             Protected \<le> accmodi inter"
proof -
  from dyn_override accmodi_old outside_pack
  show ?thesis (is "?P new old")
  proof (induct rule: overridesR.induct)
    case (Direct new old)
    assume accmodi_old: "accmodi old = Package"
    assume outside_pack: "pid (declclass old) \<noteq> pid (declclass new)"
    assume "G \<turnstile>Method old inheritable_in pid (declclass new)"
    with accmodi_old 
    have "pid (declclass old) = pid (declclass new)"
      by (simp add: inheritable_in_def)
    with outside_pack 
    show "?P new old"
      by (contradiction)
  next
    case (Indirect inter new old)
    assume accmodi_old: "accmodi old = Package"
    assume outside_pack: "pid (declclass old) \<noteq> pid (declclass new)"
    assume override_new_inter: "G \<turnstile> new overrides inter"
    assume override_inter_old: "G \<turnstile> inter overrides old"
    assume hyp_new_inter: "\<lbrakk>accmodi inter = Package; 
                           pid (declclass inter) \<noteq> pid (declclass new)\<rbrakk>
                           \<Longrightarrow> ?P new inter"
    assume hyp_inter_old: "\<lbrakk>accmodi old = Package; 
                           pid (declclass old) \<noteq> pid (declclass inter)\<rbrakk>
                           \<Longrightarrow> ?P inter old"
    show "?P new old"
    proof (cases "pid (declclass old) = pid (declclass inter)")
      case True
      note same_pack_old_inter = this
      show ?thesis
      proof (cases "pid (declclass inter) = pid (declclass new)")
	case True
	with same_pack_old_inter outside_pack
	show ?thesis
	  by auto
      next
	case False
	note diff_pack_inter_new = this
	show ?thesis
	proof (cases "accmodi inter = Package")
	  case True
	  with diff_pack_inter_new hyp_new_inter  
	  obtain newinter where
	    over_new_newinter: "G \<turnstile> new overrides newinter" and
            over_newinter_inter: "G \<turnstile> newinter overrides inter" and 
            eq_pid: "pid (declclass inter) = pid (declclass newinter)" and
            accmodi_newinter: "Protected \<le> accmodi newinter"
	    by auto
	  from over_newinter_inter override_inter_old
	  have "G\<turnstile>newinter overrides old"
	    by (rule overridesR.Indirect)
	  moreover
	  from eq_pid same_pack_old_inter 
	  have "pid (declclass old) = pid (declclass newinter)"
	    by simp
	  moreover
	  note over_new_newinter accmodi_newinter
	  ultimately show ?thesis
	    by blast
	next
	  case False
	  with override_new_inter
	  have "Protected \<le> accmodi inter"
	    by (cases "accmodi inter") (auto dest: no_Private_override)
	  with override_new_inter override_inter_old same_pack_old_inter
	  show ?thesis
	    by blast
	qed
      qed
    next
      case False
      with accmodi_old hyp_inter_old
      obtain newinter where
	over_inter_newinter: "G \<turnstile> inter overrides newinter" and
          over_newinter_old: "G \<turnstile> newinter overrides old" and 
                eq_pid: "pid (declclass old) = pid (declclass newinter)" and
	accmodi_newinter: "Protected \<le> accmodi newinter"
	by auto
      from override_new_inter over_inter_newinter 
      have "G \<turnstile> new overrides newinter"
	by (rule overridesR.Indirect)
      with eq_pid over_newinter_old accmodi_newinter
      show ?thesis
	by blast
    qed
  qed
qed

lemma declclass_widen[rule_format]: 
 "wf_prog G 
 \<longrightarrow> (\<forall>c m. class G C = Some c \<longrightarrow> methd G C sig = Some m 
 \<longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m)" (is "?P G C")
proof (rule class_rec.induct,intro allI impI)
  fix G C c m
  assume Hyp: "\<forall>c. C \<noteq> Object \<and> ws_prog G \<and> class G C = Some c 
               \<longrightarrow> ?P G (super c)"
  assume wf: "wf_prog G" and cls_C: "class G C = Some c" and
         m:  "methd G C sig = Some m"
  show "G\<turnstile>C\<preceq>\<^sub>C declclass m" 
  proof (cases "C=Object")
    case True 
    with wf m show ?thesis by (simp add: methd_Object_SomeD)
  next
    let ?filter="filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)"
    let ?table = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
    case False 
    with cls_C wf m
    have methd_C: "(?filter (methd G (super c)) ++ ?table) sig = Some m "
      by (simp add: methd_rec)
    show ?thesis
    proof (cases "?table sig")
      case None
      from this methd_C have "?filter (methd G (super c)) sig = Some m"
	by simp
      moreover
      from wf cls_C False obtain sup where "class G (super c) = Some sup"
	by (blast dest: wf_prog_cdecl wf_cdecl_supD is_acc_class_is_class)
      moreover note wf False cls_C  
      ultimately have "G\<turnstile>super c \<preceq>\<^sub>C declclass m"  
	by (auto intro: Hyp [rule_format])
      moreover from cls_C False have  "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 super c" by (rule subcls1I)
      ultimately show ?thesis by - (rule rtrancl_into_rtrancl2)
    next
      case Some
      from this wf False cls_C methd_C show ?thesis by auto
    qed
  qed
qed

lemma declclass_methd_Object: 
 "\<lbrakk>wf_prog G; methd G Object sig = Some m\<rbrakk> \<Longrightarrow> declclass m = Object"
by auto

lemma methd_declaredD: 
 "\<lbrakk>wf_prog G; is_class G C;methd G C sig = Some m\<rbrakk> 
  \<Longrightarrow> G\<turnstile>(mdecl (sig,mthd m)) declared_in (declclass m)"
proof -
  assume    wf: "wf_prog G"
  then have ws: "ws_prog G" ..
  assume  clsC: "is_class G C"
  from clsC ws 
  show "methd G C sig = Some m 
        \<Longrightarrow> G\<turnstile>(mdecl (sig,mthd m)) declared_in (declclass m)"
    (is "PROP ?P C") 
  proof (induct ?P C rule: ws_class_induct')
    case Object
    assume "methd G Object sig = Some m" 
    with wf show ?thesis
      by - (rule method_declared_inI, auto) 
  next
    case Subcls
    fix C c
    assume clsC: "class G C = Some c"
    and       m: "methd G C sig = Some m"
    and     hyp: "methd G (super c) sig = Some m \<Longrightarrow> ?thesis" 
    let ?newMethods = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
    show ?thesis
    proof (cases "?newMethods sig")
      case None
      from None ws clsC m hyp 
      show ?thesis by (auto intro: method_declared_inI simp add: methd_rec)
    next
      case Some
      from Some ws clsC m 
      show ?thesis by (auto intro: method_declared_inI simp add: methd_rec) 
    qed
  qed
qed

lemma methd_rec_Some_cases [consumes 4, case_names NewMethod InheritedMethod]:
  assumes methd_C: "methd G C sig = Some m" and