src/HOL/Bali/DeclConcepts.thy
changeset 12854 00d4a435777f
child 12859 f63315dfffd4
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Bali/DeclConcepts.thy	Mon Jan 28 17:00:19 2002 +0100
@@ -0,0 +1,2540 @@
+header {* Advanced concepts on Java declarations like overriding, inheritance,
+dynamic method lookup*}
+
+theory DeclConcepts = TypeRel:
+
+section "access control (cf. 6.6), overriding and hiding (cf. 8.4.6.1)"
+
+constdefs
+is_public :: "prog \<Rightarrow> qtname \<Rightarrow> bool"
+"is_public G qn \<equiv> (case class G qn of
+                     None       \<Rightarrow> (case iface G qn of
+                                      None       \<Rightarrow> False
+                                    | Some iface \<Rightarrow> access iface = Public)
+                   | Some class \<Rightarrow> access class = Public)"
+
+subsection "accessibility of types (cf. 6.6.1)"
+text {* 
+Primitive types are always accessible, interfaces and classes are accessible
+in their package or if they are defined public, an array type is accessible if
+its element type is accessible *}
+ 
+consts accessible_in   :: "prog \<Rightarrow> ty     \<Rightarrow> pname \<Rightarrow> bool"  
+                                      ("_ \<turnstile> _ accessible'_in _" [61,61,61] 60)
+       rt_accessible_in:: "prog \<Rightarrow> ref_ty \<Rightarrow> pname \<Rightarrow> bool" 
+                                      ("_ \<turnstile> _ accessible'_in' _" [61,61,61] 60) 
+primrec
+"G\<turnstile>(PrimT p)   accessible_in pack  = True"
+accessible_in_RefT_simp: 
+"G\<turnstile>(RefT  r)   accessible_in pack  = G\<turnstile>r accessible_in' pack"
+
+"G\<turnstile>(NullT)     accessible_in' pack = True"
+"G\<turnstile>(IfaceT I)  accessible_in' pack = ((pid I = pack) \<or> is_public G I)"
+"G\<turnstile>(ClassT C)  accessible_in' pack = ((pid C = pack) \<or> is_public G C)"
+"G\<turnstile>(ArrayT ty) accessible_in' pack = G\<turnstile>ty accessible_in pack"
+
+declare accessible_in_RefT_simp [simp del]
+
+constdefs
+  is_acc_class :: "prog \<Rightarrow> pname \<Rightarrow> qtname \<Rightarrow> bool"
+    "is_acc_class G P C \<equiv> is_class G C \<and> G\<turnstile>(Class C) accessible_in P"
+  is_acc_iface :: "prog \<Rightarrow> pname \<Rightarrow> qtname \<Rightarrow> bool"
+    "is_acc_iface G P I \<equiv> is_iface G I \<and> G\<turnstile>(Iface I) accessible_in P"
+  is_acc_type  :: "prog \<Rightarrow> pname \<Rightarrow> ty     \<Rightarrow> bool"
+    "is_acc_type  G P T \<equiv> is_type G T  \<and> G\<turnstile>T accessible_in P"
+  is_acc_reftype  :: "prog \<Rightarrow> pname \<Rightarrow> ref_ty \<Rightarrow> bool"
+  "is_acc_reftype  G P T \<equiv> isrtype G T  \<and> G\<turnstile>T accessible_in' P"
+
+lemma is_acc_classD:
+ "is_acc_class G P C \<Longrightarrow>  is_class G C \<and> G\<turnstile>(Class C) accessible_in P"
+by (simp add: is_acc_class_def)
+
+lemma is_acc_class_is_class: "is_acc_class G P C \<Longrightarrow> is_class G C"
+by (auto simp add: is_acc_class_def)
+
+lemma is_acc_ifaceD:
+  "is_acc_iface G P I \<Longrightarrow> is_iface G I \<and> G\<turnstile>(Iface I) accessible_in P"
+by (simp add: is_acc_iface_def)
+
+lemma is_acc_typeD:
+ "is_acc_type  G P T \<equiv> is_type G T  \<and> G\<turnstile>T accessible_in P"
+by (simp add: is_acc_type_def)
+
+lemma is_acc_reftypeD:
+"is_acc_reftype  G P T \<Longrightarrow> isrtype G T  \<and> G\<turnstile>T accessible_in' P"
+by (simp add: is_acc_reftype_def)
+
+subsection "accessibility of members"
+text {*
+The accessibility of members is more involved as the accessibility of types.
+We have to distinguish several cases to model the different effects of 
+accessibility during inheritance, overriding and ordinary member access 
+*}
+
+subsubsection {* Various technical conversion and selection functions *}
+
+text {* overloaded selector @{text accmodi} to select the access modifier 
+out of various HOL types *}
+
+axclass has_accmodi < "type"
+consts accmodi:: "'a::has_accmodi \<Rightarrow> acc_modi"
+
+instance acc_modi::has_accmodi
+by (intro_classes)
+
+defs (overloaded)
+acc_modi_accmodi_def: "accmodi (a::acc_modi) \<equiv> a"
+
+lemma acc_modi_accmodi_simp[simp]: "accmodi (a::acc_modi) = a"
+by (simp add: acc_modi_accmodi_def)
+
+instance access_field_type:: ("type","type") has_accmodi
+by (intro_classes)
+
+defs (overloaded)
+decl_acc_modi_def: "accmodi (d::('a:: type) decl_scheme) \<equiv> access d"
+
+
+lemma decl_acc_modi_simp[simp]: "accmodi (d::('a::type) decl_scheme) = access d"
+by (simp add: decl_acc_modi_def)
+
+instance * :: ("type",has_accmodi) has_accmodi
+by (intro_classes)
+
+defs (overloaded)
+pair_acc_modi_def: "accmodi p \<equiv> (accmodi (snd p))"
+
+lemma pair_acc_modi_simp[simp]: "accmodi (x,a) = (accmodi a)"
+by (simp add: pair_acc_modi_def)
+
+instance memberdecl :: has_accmodi
+by (intro_classes)
+
+defs (overloaded)
+memberdecl_acc_modi_def: "accmodi m \<equiv> (case m of
+                                          fdecl f \<Rightarrow> accmodi f
+                                        | mdecl m \<Rightarrow> accmodi m)"
+
+lemma memberdecl_fdecl_acc_modi_simp[simp]:
+ "accmodi (fdecl m) = accmodi m"
+by (simp add: memberdecl_acc_modi_def)
+
+lemma memberdecl_mdecl_acc_modi_simp[simp]:
+ "accmodi (mdecl m) = accmodi m"
+by (simp add: memberdecl_acc_modi_def)
+
+text {* overloaded selector @{text declclass} to select the declaring class 
+out of various HOL types *}
+
+axclass has_declclass < "type"
+consts declclass:: "'a::has_declclass \<Rightarrow> qtname"
+
+instance pid_field_type::("type","type") has_declclass
+by (intro_classes)
+
+defs (overloaded)
+qtname_declclass_def: "declclass (q::qtname) \<equiv> q"
+
+lemma qtname_declclass_simp[simp]: "declclass (q::qtname) = q"
+by (simp add: qtname_declclass_def)
+
+instance * :: ("has_declclass","type") has_declclass
+by (intro_classes)
+
+defs (overloaded)
+pair_declclass_def: "declclass p \<equiv> declclass (fst p)"
+
+lemma pair_declclass_simp[simp]: "declclass (c,x) = declclass c" 
+by (simp add: pair_declclass_def)
+
+text {* overloaded selector @{text is_static} to select the static modifier 
+out of various HOL types *}
+
+
+axclass has_static < "type"
+consts is_static :: "'a::has_static \<Rightarrow> bool"
+
+(*
+consts is_static :: "'a \<Rightarrow> bool"
+*)
+
+instance access_field_type :: ("type","has_static") has_static
+by (intro_classes) 
+
+defs (overloaded)
+decl_is_static_def: 
+ "is_static (m::('a::has_static) decl_scheme) \<equiv> is_static (Decl.decl.more m)" 
+
+instance static_field_type :: ("type","type") has_static
+by (intro_classes)
+
+defs (overloaded)
+static_field_type_is_static_def: 
+ "is_static (m::(bool,'b::type) static_field_type) \<equiv> static_val m"
+
+lemma member_is_static_simp: "is_static (m::'a member_scheme) = static m"
+apply (cases m)
+apply (simp add: static_field_type_is_static_def 
+                 decl_is_static_def Decl.member.dest_convs)
+done
+
+instance * :: ("type","has_static") has_static
+by (intro_classes)
+
+defs (overloaded)
+pair_is_static_def: "is_static p \<equiv> is_static (snd p)"
+
+lemma pair_is_static_simp [simp]: "is_static (x,s) = is_static s"
+by (simp add: pair_is_static_def)
+
+lemma pair_is_static_simp1: "is_static p = is_static (snd p)"
+by (simp add: pair_is_static_def)
+
+instance memberdecl:: has_static
+by (intro_classes)
+
+defs (overloaded)
+memberdecl_is_static_def: 
+ "is_static m \<equiv> (case m of
+                    fdecl f \<Rightarrow> is_static f
+                  | mdecl m \<Rightarrow> is_static m)"
+
+lemma memberdecl_is_static_fdecl_simp[simp]:
+ "is_static (fdecl f) = is_static f"
+by (simp add: memberdecl_is_static_def)
+
+lemma memberdecl_is_static_mdecl_simp[simp]:
+ "is_static (mdecl m) = is_static m"
+by (simp add: memberdecl_is_static_def)
+
+lemma mhead_static_simp [simp]: "is_static (mhead m) = is_static m"
+by (cases m) (simp add: mhead_def member_is_static_simp)
+
+constdefs  (* some mnemotic selectors for (qtname \<times> ('a::more) decl_scheme) 
+            * the first component is a class or interface name
+            * the second component is a method, field or method head *)
+(* "declclass":: "(qtname \<times> ('a::more) decl_scheme) \<Rightarrow> qtname"*)
+(* "declclass \<equiv> fst" *)          (* get the class component *)
+
+ "decliface":: "(qtname \<times> ('a::type) decl_scheme) \<Rightarrow> qtname"
+ "decliface \<equiv> fst"          (* get the interface component *)
+
+(*
+ "member"::   "(qtname \<times> ('a::type) decl_scheme) \<Rightarrow> ('a::type) decl_scheme"
+*)
+ "mbr"::   "(qtname \<times> memberdecl) \<Rightarrow> memberdecl"
+ "mbr \<equiv> snd"            (* get the memberdecl component *)
+
+ "mthd"::   "('b \<times> 'a) \<Rightarrow> 'a"
+                           (* also used for mdecl,mhead *)
+ "mthd \<equiv> snd"              (* get the method component *)
+
+ "fld"::   "('b \<times> ('a::type) decl_scheme) \<Rightarrow> ('a::type) decl_scheme"
+              (* also used for ((vname \<times> qtname)\<times> field) *)
+ "fld \<equiv> snd"               (* get the field component *)
+
+(* "accmodi" :: "('b \<times> ('a::type) decl_scheme) \<Rightarrow> acc_modi"*)
+                           (* also used for mdecl *)
+(* "accmodi \<equiv> access \<circ> snd"*)  (* get the access modifier *) 
+(*
+ "is_static" ::"('b \<times> ('a::type) member_scheme) \<Rightarrow> bool" *)
+                            (* also defined for emhead cf. WellType *)
+ (*"is_static \<equiv> static \<circ> snd"*) (* get the static modifier *)
+
+constdefs (* some mnemotic selectors for (vname \<times> qtname) *)
+ fname:: "(vname \<times> 'a) \<Rightarrow> vname" (* also used for fdecl *)
+ "fname \<equiv> fst"
+  
+  declclassf:: "(vname \<times> qtname) \<Rightarrow> qtname"
+ "declclassf \<equiv> snd"
+
+(*
+lemma declclass_simp[simp]: "declclass (C,m) = C"
+by (simp add: declclass_def)
+*)
+
+lemma decliface_simp[simp]: "decliface (I,m) = I"
+by (simp add: decliface_def) 
+
+lemma mbr_simp[simp]: "mbr (C,m) = m"
+by (simp add: mbr_def)
+
+lemma access_mbr_simp [simp]: "(accmodi (mbr m)) = accmodi m"
+by (cases m) (simp add:  mbr_def) 
+
+lemma mthd_simp[simp]: "mthd (C,m) = m"
+by (simp add: mthd_def)
+
+lemma fld_simp[simp]: "fld (C,f) = f"
+by (simp add: fld_def)
+
+lemma accmodi_simp[simp]: "accmodi (C,m) = access m"
+by (simp )
+
+lemma access_mthd_simp [simp]: "(access (mthd m)) = accmodi m"
+by (cases m) (simp add: mthd_def) 
+
+lemma access_fld_simp [simp]: "(access (fld f)) = accmodi f"
+by (cases f) (simp add:  fld_def) 
+
+(*
+lemma is_static_simp[simp]: "is_static (C,m) = static m"
+by (simp add: is_static_def)
+*)
+
+lemma static_mthd_simp[simp]: "static (mthd m) = is_static m"
+by (cases m) (simp add:  mthd_def member_is_static_simp)
+
+lemma mthd_is_static_simp [simp]: "is_static (mthd m) = is_static m"
+by (cases m) simp
+
+lemma static_fld_simp[simp]: "static (fld f) = is_static f"
+by (cases f) (simp add:  fld_def member_is_static_simp)
+
+lemma ext_field_simp [simp]: "(declclass f,fld f) = f"
+by (cases f) (simp add:  fld_def)
+
+lemma ext_method_simp [simp]: "(declclass m,mthd m) = m"
+by (cases m) (simp add:  mthd_def)
+
+lemma ext_mbr_simp [simp]: "(declclass m,mbr m) = m"
+by (cases m) (simp add: mbr_def)
+
+lemma fname_simp[simp]:"fname (n,c) = n"
+by (simp add: fname_def)
+
+lemma declclassf_simp[simp]:"declclassf (n,c) = c"
+by (simp add: declclassf_def)
+
+constdefs  (* some mnemotic selectors for (vname \<times> qtname) *)
+  "fldname"  :: "(vname \<times> qtname) \<Rightarrow> vname" 
+  "fldname \<equiv> fst"
+
+  "fldclass" :: "(vname \<times> qtname) \<Rightarrow> qtname"
+  "fldclass \<equiv> snd"
+
+lemma fldname_simp[simp]: "fldname (n,c) = n"
+by (simp add: fldname_def)
+
+lemma fldclass_simp[simp]: "fldclass (n,c) = c"
+by (simp add: fldclass_def)
+
+lemma ext_fieldname_simp[simp]: "(fldname f,fldclass f) = f"
+by (simp add: fldname_def fldclass_def)
+
+text {* Convert a qualified method declaration (qualified with its declaring 
+class) to a qualified member declaration:  @{text methdMembr}  *}
+
+constdefs
+methdMembr :: "(qtname \<times> mdecl) \<Rightarrow> (qtname \<times> memberdecl)"
+ "methdMembr m \<equiv> (fst m,mdecl (snd m))"
+
+lemma methdMembr_simp[simp]: "methdMembr (c,m) = (c,mdecl m)"
+by (simp add: methdMembr_def)
+
+lemma accmodi_methdMembr_simp[simp]: "accmodi (methdMembr m) = accmodi m"
+by (cases m) (simp add: methdMembr_def)
+
+lemma is_static_methdMembr_simp[simp]: "is_static (methdMembr m) = is_static m"
+by (cases m) (simp add: methdMembr_def)
+
+lemma declclass_methdMembr_simp[simp]: "declclass (methdMembr m) = declclass m"
+by (cases m) (simp add: methdMembr_def)
+
+text {* Convert a qualified method (qualified with its declaring 
+class) to a qualified member declaration:  @{text method}  *}
+
+constdefs
+method :: "sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> memberdecl)" 
+"method sig m \<equiv> (declclass m, mdecl (sig, mthd m))"
+
+lemma method_simp[simp]: "method sig (C,m) = (C,mdecl (sig,m))"
+by (simp add: method_def)
+
+lemma accmodi_method_simp[simp]: "accmodi (method sig m) = accmodi m"
+by (simp add: method_def)
+
+lemma declclass_method_simp[simp]: "declclass (method sig m) = declclass m"
+by (simp add: method_def)
+
+lemma is_static_method_simp[simp]: "is_static (method sig m) = is_static m"
+by (cases m) (simp add: method_def)
+
+lemma mbr_method_simp[simp]: "mbr (method sig m) = mdecl (sig,mthd m)"
+by (simp add: mbr_def method_def)
+
+lemma memberid_method_simp[simp]:  "memberid (method sig m) = mid sig"
+  by (simp add: method_def) 
+
+constdefs
+fieldm :: "vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> (qtname \<times> memberdecl)" 
+"fieldm n f \<equiv> (declclass f, fdecl (n, fld f))"
+
+lemma fieldm_simp[simp]: "fieldm n (C,f) = (C,fdecl (n,f))"
+by (simp add: fieldm_def)
+
+lemma accmodi_fieldm_simp[simp]: "accmodi (fieldm n f) = accmodi f"
+by (simp add: fieldm_def)
+
+lemma declclass_fieldm_simp[simp]: "declclass (fieldm n f) = declclass f"
+by (simp add: fieldm_def)
+
+lemma is_static_fieldm_simp[simp]: "is_static (fieldm n f) = is_static f"
+by (cases f) (simp add: fieldm_def)
+
+lemma mbr_fieldm_simp[simp]: "mbr (fieldm n f) = fdecl (n,fld f)"
+by (simp add: mbr_def fieldm_def)
+
+lemma memberid_fieldm_simp[simp]:  "memberid (fieldm n f) = fid n"
+by (simp add: fieldm_def) 
+
+text {* Select the signature out of a qualified method declaration:
+ @{text msig} *}
+
+constdefs msig:: "(qtname \<times> mdecl) \<Rightarrow> sig"
+"msig m \<equiv> fst (snd m)"
+
+lemma msig_simp[simp]: "msig (c,(s,m)) = s"
+by (simp add: msig_def)
+
+text {* Convert a qualified method (qualified with its declaring 
+class) to a qualified method declaration:  @{text qmdecl}  *}
+
+constdefs qmdecl :: "sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> mdecl)"
+"qmdecl sig m \<equiv> (declclass m, (sig,mthd m))"
+
+lemma qmdecl_simp[simp]: "qmdecl sig (C,m) = (C,(sig,m))"
+by (simp add: qmdecl_def)
+
+lemma declclass_qmdecl_simp[simp]: "declclass (qmdecl sig m) = declclass m"
+by (simp add: qmdecl_def)
+
+lemma accmodi_qmdecl_simp[simp]: "accmodi (qmdecl sig m) = accmodi m"
+by (simp add: qmdecl_def)
+
+lemma is_static_qmdecl_simp[simp]: "is_static (qmdecl sig m) = is_static m"
+by (cases m) (simp add: qmdecl_def)
+
+lemma msig_qmdecl_simp[simp]: "msig (qmdecl sig m) = sig"
+by (simp add: qmdecl_def)
+
+lemma mdecl_qmdecl_simp[simp]:  
+ "mdecl (mthd (qmdecl sig new)) = mdecl (sig, mthd new)" 
+by (simp add: qmdecl_def) 
+
+lemma methdMembr_qmdecl_simp [simp]: 
+ "methdMembr (qmdecl sig old) = method sig old"
+by (simp add: methdMembr_def qmdecl_def method_def)
+
+text {* overloaded selector @{text resTy} to select the result type 
+out of various HOL types *}
+
+axclass has_resTy < "type"
+consts resTy:: "'a::has_resTy \<Rightarrow> ty"
+
+instance access_field_type :: ("type","has_resTy") has_resTy
+by (intro_classes)
+
+defs (overloaded)
+decl_resTy_def: 
+ "resTy (m::('a::has_resTy) decl_scheme) \<equiv> resTy (Decl.decl.more m)" 
+
+instance static_field_type :: ("type","has_resTy") has_resTy
+by (intro_classes)
+
+defs (overloaded)
+static_field_type_resTy_def: 
+ "resTy (m::(bool,'b::has_resTy) static_field_type) 
+  \<equiv> resTy (static_more m)" 
+
+instance pars_field_type :: ("type","has_resTy") has_resTy
+by (intro_classes)
+
+defs (overloaded)
+pars_field_type_resTy_def: 
+ "resTy (m::(vname list,'b::has_resTy) pars_field_type) 
+  \<equiv> resTy (pars_more m)" 
+
+instance resT_field_type :: ("type","type") has_resTy
+by (intro_classes)
+
+defs (overloaded)
+resT_field_type_resTy_def: 
+ "resTy (m::(ty,'b::type) resT_field_type) 
+  \<equiv> resT_val m" 
+
+lemma mhead_resTy_simp: "resTy (m::'a mhead_scheme) = resT m"
+apply (cases m)
+apply (simp add: decl_resTy_def static_field_type_resTy_def 
+                 pars_field_type_resTy_def resT_field_type_resTy_def
+                 member.dest_convs mhead.dest_convs)
+done
+
+lemma resTy_mhead [simp]:"resTy (mhead m) = resTy m"
+by (simp add: mhead_def mhead_resTy_simp)
+
+instance * :: ("type","has_resTy") has_resTy
+by (intro_classes)
+
+defs (overloaded)
+pair_resTy_def: "resTy p \<equiv> resTy (snd p)"
+
+lemma pair_resTy_simp[simp]: "resTy (x,m) = resTy m"
+by (simp add: pair_resTy_def)
+
+lemma qmdecl_resTy_simp [simp]: "resTy (qmdecl sig m) = resTy m"
+by (cases m) (simp)
+
+lemma resTy_mthd [simp]:"resTy (mthd m) = resTy m"
+by (cases m) (simp add: mthd_def )
+
+subsubsection "inheritable-in"
+text {*
+@{text "G\<turnstile>m inheritable_in P"}: m can be inherited by
+classes in package P if:
+\begin{itemize} 
+\item the declaration class of m is accessible in P and
+\item the member m is declared with protected or public access or if it is
+      declared with default (package) access, the package of the declaration 
+      class of m is also P. If the member m is declared with private access
+      it is not accessible for inheritance at all.
+\end{itemize}
+*}
+constdefs
+inheritable_in:: 
+ "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> pname \<Rightarrow> bool"
+                  ("_ \<turnstile> _ inheritable'_in _" [61,61,61] 60)
+"G\<turnstile>membr inheritable_in pack 
+  \<equiv> (case (accmodi membr) of
+       Private   \<Rightarrow> False
+     | Package   \<Rightarrow> (pid (declclass membr)) = pack
+     | Protected \<Rightarrow> True
+     | Public    \<Rightarrow> True)"
+
+syntax
+Method_inheritable_in::
+ "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> pname \<Rightarrow> bool"
+                ("_ \<turnstile>Method _ inheritable'_in _ " [61,61,61] 60)
+
+translations
+"G\<turnstile>Method m inheritable_in p" == "G\<turnstile>methdMembr m inheritable_in p"
+
+syntax
+Methd_inheritable_in::
+ "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> pname \<Rightarrow> bool"
+                ("_ \<turnstile>Methd _ _ inheritable'_in _ " [61,61,61,61] 60)
+
+translations
+"G\<turnstile>Methd s m inheritable_in p" == "G\<turnstile>(method s m) inheritable_in p"
+
+subsubsection "declared-in/undeclared-in"
+
+constdefs cdeclaredmethd:: "prog \<Rightarrow> qtname \<Rightarrow> (sig,methd) table"
+"cdeclaredmethd G C 
+  \<equiv> (case class G C of
+       None \<Rightarrow> \<lambda> sig. None
+     | Some c \<Rightarrow> table_of (methods c)
+    )"
+
+constdefs
+cdeclaredfield:: "prog \<Rightarrow> qtname \<Rightarrow> (vname,field) table"
+"cdeclaredfield G C 
+  \<equiv> (case class G C of
+       None \<Rightarrow> \<lambda> sig. None
+     | Some c \<Rightarrow> table_of (cfields c)
+    )"
+
+
+constdefs
+declared_in:: "prog  \<Rightarrow> memberdecl \<Rightarrow> qtname \<Rightarrow> bool"
+                                 ("_\<turnstile> _ declared'_in _" [61,61,61] 60)
+"G\<turnstile>m declared_in C \<equiv> (case m of
+                        fdecl (fn,f ) \<Rightarrow> cdeclaredfield G C fn  = Some f
+                      | mdecl (sig,m) \<Rightarrow> cdeclaredmethd G C sig = Some m)"
+
+syntax
+method_declared_in:: "prog  \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> bool"
+                                 ("_\<turnstile>Method _ declared'_in _" [61,61,61] 60)
+translations
+"G\<turnstile>Method m declared_in C" == "G\<turnstile>mdecl (mthd m) declared_in C"
+
+syntax
+methd_declared_in:: "prog  \<Rightarrow> sig  \<Rightarrow>(qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> bool"
+                               ("_\<turnstile>Methd _  _ declared'_in _" [61,61,61,61] 60)
+translations
+"G\<turnstile>Methd s m declared_in C" == "G\<turnstile>mdecl (s,mthd m) declared_in C"
+
+lemma declared_in_classD:
+ "G\<turnstile>m declared_in C \<Longrightarrow> is_class G C"
+by (cases m) 
+   (auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def)
+
+constdefs
+undeclared_in:: "prog  \<Rightarrow> memberid \<Rightarrow> qtname \<Rightarrow> bool"
+                                 ("_\<turnstile> _ undeclared'_in _" [61,61,61] 60)
+
+"G\<turnstile>m undeclared_in C \<equiv> (case m of
+                            fid fn  \<Rightarrow> cdeclaredfield G C fn  = None
+                          | mid sig \<Rightarrow> cdeclaredmethd G C sig = None)"
+
+lemma method_declared_inI:
+  "\<lbrakk>class G C = Some c; table_of (methods c) sig = Some m\<rbrakk> 
+   \<Longrightarrow> G\<turnstile>mdecl (sig,m) declared_in C"
+by (auto simp add: declared_in_def cdeclaredmethd_def)
+
+
+subsubsection "members"
+
+consts
+members:: "prog \<Rightarrow> (qtname \<times> (qtname \<times> memberdecl)) set"
+(* Can't just take a function: prog \<Rightarrow> qtname \<Rightarrow> memberdecl set because
+   the class qtname changes to the superclass in the inductive definition
+   below
+*)
+
+syntax
+member_of:: "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile> _ member'_of _" [61,61,61] 60)
+
+translations
+ "G\<turnstile>m member_of C" \<rightleftharpoons> "(C,m) \<in> members G"
+
+inductive "members G"  intros
+
+Immediate: "\<lbrakk>G\<turnstile>mbr m declared_in C;declclass m = C\<rbrakk> \<Longrightarrow> G\<turnstile>m member_of C"
+Inherited: "\<lbrakk>G\<turnstile>m inheritable_in (pid C); G\<turnstile>memberid m undeclared_in C; 
+             G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S; G\<turnstile>(Class S) accessible_in (pid C);G\<turnstile>m member_of S 
+            \<rbrakk> \<Longrightarrow> G\<turnstile>m member_of C"
+text {* Note that in the case of an inherited member only the members of the
+direct superclass are concerned. If a member of a superclass of the direct
+superclass isn't inherited in the direct superclass (not member of the
+direct superclass) than it can't be a member of the class. E.g. If a
+member of a class A is defined with package access it isn't member of a 
+subclass S if S isn't in the same package as A. Any further subclasses 
+of S will not inherit the member, regardless if they are in the same
+package as A or not.*}
+
+syntax
+method_member_of:: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile>Method _ member'_of _" [61,61,61] 60)
+
+translations
+ "G\<turnstile>Method m member_of C" \<rightleftharpoons> "G\<turnstile>(methdMembr m) member_of C" 
+
+syntax
+methd_member_of:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile>Methd _ _ member'_of _" [61,61,61,61] 60)
+
+translations
+ "G\<turnstile>Methd s m member_of C" \<rightleftharpoons> "G\<turnstile>(method s m) member_of C" 
+
+syntax
+fieldm_member_of:: "prog \<Rightarrow> vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile>Field _  _ member'_of _" [61,61,61] 60)
+
+translations
+ "G\<turnstile>Field n f member_of C" \<rightleftharpoons> "G\<turnstile>fieldm n f member_of C" 
+
+constdefs
+inherits:: "prog \<Rightarrow> qtname \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> bool"
+                           ("_ \<turnstile> _ inherits _" [61,61,61] 60)
+"G\<turnstile>C inherits m 
+  \<equiv> G\<turnstile>m inheritable_in (pid C) \<and> G\<turnstile>memberid m undeclared_in C \<and> 
+    (\<exists> S. G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S \<and> G\<turnstile>(Class S) accessible_in (pid C) \<and> G\<turnstile>m member_of S)"
+
+lemma inherits_member: "G\<turnstile>C inherits m \<Longrightarrow> G\<turnstile>m member_of C"
+by (auto simp add: inherits_def intro: members.Inherited)
+
+
+constdefs member_in::"prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile> _ member'_in _" [61,61,61] 60)
+"G\<turnstile>m member_in C \<equiv> \<exists> provC. G\<turnstile> C \<preceq>\<^sub>C provC \<and> G \<turnstile> m member_of provC"
+text {* A member is in a class if it is member of the class or a superclass.
+If a member is in a class we can select this member. This additional notion
+is necessary since not all members are inherited to subclasses. So such
+members are not member-of the subclass but member-in the subclass.*}
+
+syntax
+method_member_in:: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile>Method _ member'_in _" [61,61,61] 60)
+
+translations
+ "G\<turnstile>Method m member_in C" \<rightleftharpoons> "G\<turnstile>(methdMembr m) member_in C" 
+
+syntax
+methd_member_in:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> bool"
+                           ("_ \<turnstile>Methd _ _ member'_in _" [61,61,61,61] 60)
+
+translations
+ "G\<turnstile>Methd s m member_in C" \<rightleftharpoons> "G\<turnstile>(method s m) member_in C" 
+
+consts stat_overridesR::
+  "prog  \<Rightarrow> ((qtname \<times> mdecl) \<times> (qtname \<times> mdecl)) set"
+
+lemma member_inD: "G\<turnstile>m member_in C 
+ \<Longrightarrow> \<exists> provC. G\<turnstile> C \<preceq>\<^sub>C provC \<and> G \<turnstile> m member_of provC"
+by (auto simp add: member_in_def)
+
+lemma member_inI: "\<lbrakk>G \<turnstile> m member_of provC;G\<turnstile> C \<preceq>\<^sub>C provC\<rbrakk> \<Longrightarrow>  G\<turnstile>m member_in C"
+by (auto simp add: member_in_def)
+
+lemma member_of_to_member_in: "G \<turnstile> m member_of C \<Longrightarrow> G \<turnstile>m member_in C"
+by (auto intro: member_inI)
+
+
+subsubsection "overriding"
+
+text {* Unfortunately the static notion of overriding (used during the
+typecheck of the compiler) and the dynamic notion of overriding (used during
+execution in the JVM) are not exactly the same. 
+*}
+
+text {* Static overriding (used during the typecheck of the compiler) *}
+syntax
+stat_overrides:: "prog  \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool" 
+                                  ("_ \<turnstile> _ overrides\<^sub>S _" [61,61,61] 60)
+translations
+ "G\<turnstile>new overrides\<^sub>S  old" == "(new,old) \<in> stat_overridesR G "
+
+inductive "stat_overridesR G" intros
+
+Direct: "\<lbrakk>\<not> is_static new; msig new = msig old; 
+         G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old);
+         G\<turnstile>Method new declared_in (declclass new);  
+         G\<turnstile>Method old declared_in (declclass old); 
+         G\<turnstile>Method old inheritable_in pid (declclass new);
+         G\<turnstile>(declclass new) \<prec>\<^sub>C\<^sub>1 superNew;
+         G \<turnstile>Method old member_of superNew
+         \<rbrakk> \<Longrightarrow> G\<turnstile>new overrides\<^sub>S old"
+
+Indirect: "\<lbrakk>G\<turnstile>new overrides\<^sub>S inter; G\<turnstile>inter overrides\<^sub>S old\<rbrakk>
+           \<Longrightarrow> G\<turnstile>new overrides\<^sub>S old"
+
+text {* Dynamic overriding (used during the typecheck of the compiler) *}
+consts overridesR::
+  "prog  \<Rightarrow> ((qtname \<times> mdecl) \<times> (qtname \<times> mdecl)) set"
+
+
+overrides:: "prog  \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool" 
+                                  ("_ \<turnstile> _ overrides _" [61,61,61] 60)
+translations
+ "G\<turnstile>new overrides old" == "(new,old) \<in> overridesR G "
+
+inductive "overridesR G" intros
+
+Direct: "\<lbrakk>\<not> is_static new; \<not> is_static old; accmodi new \<noteq> Private;
+         msig new = msig old; 
+         G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old);
+         G\<turnstile>Method new declared_in (declclass new);  
+         G\<turnstile>Method old declared_in (declclass old);    
+         G\<turnstile>Method old inheritable_in pid (declclass new);
+         G\<turnstile>resTy new \<preceq> resTy old
+         \<rbrakk> \<Longrightarrow> G\<turnstile>new overrides old"
+
+Indirect: "\<lbrakk>G\<turnstile>new overrides inter; G\<turnstile>inter overrides old\<rbrakk>
+           \<Longrightarrow> G\<turnstile>new overrides old"
+
+syntax
+sig_stat_overrides:: 
+ "prog  \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> methd) \<Rightarrow> bool" 
+                                  ("_,_\<turnstile> _ overrides\<^sub>S _" [61,61,61,61] 60)
+translations
+ "G,s\<turnstile>new overrides\<^sub>S old" \<rightharpoonup> "G\<turnstile>(qmdecl s new) overrides\<^sub>S (qmdecl s old)" 
+
+syntax
+sig_overrides:: "prog  \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> methd) \<Rightarrow> bool" 
+                                  ("_,_\<turnstile> _ overrides _" [61,61,61,61] 60)
+translations
+ "G,s\<turnstile>new overrides old" \<rightharpoonup> "G\<turnstile>(qmdecl s new) overrides (qmdecl s old)" 
+
+subsubsection "Hiding"
+
+constdefs hides::
+"prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool" 
+                                ("_\<turnstile> _ hides _" [61,61,61] 60)
+"G\<turnstile>new hides old
+  \<equiv> is_static new \<and> msig new = msig old \<and>
+    G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
+    G\<turnstile>Method new declared_in (declclass new) \<and>
+    G\<turnstile>Method old declared_in (declclass old) \<and> 
+    G\<turnstile>Method old inheritable_in pid (declclass new)"
+
+syntax
+sig_hides:: "prog  \<Rightarrow> sig \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool" 
+                                  ("_,_\<turnstile> _ hides _" [61,61,61,61] 60)
+translations
+ "G,s\<turnstile>new hides old" \<rightharpoonup> "G\<turnstile>(qmdecl s new) hides (qmdecl s old)" 
+
+lemma hidesI:
+"\<lbrakk>is_static new; msig new = msig old;
+  G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old);
+  G\<turnstile>Method new declared_in (declclass new);
+  G\<turnstile>Method old declared_in (declclass old);
+  G\<turnstile>Method old inheritable_in pid (declclass new)
+ \<rbrakk> \<Longrightarrow> G\<turnstile>new hides old"
+by (auto simp add: hides_def)
+
+lemma hidesD:
+"\<lbrakk>G\<turnstile>new hides old\<rbrakk> \<Longrightarrow>  
+  declclass new \<noteq> Object \<and> is_static new \<and> msig new = msig old \<and> 
+  G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
+  G\<turnstile>Method new declared_in (declclass new) \<and>   
+  G\<turnstile>Method old declared_in (declclass old)"
+by (auto simp add: hides_def)
+
+lemma overrides_commonD:
+"\<lbrakk>G\<turnstile>new overrides old\<rbrakk> \<Longrightarrow>  
+  declclass new \<noteq> Object \<and> \<not> is_static new \<and> \<not> is_static old \<and>
+  accmodi new \<noteq> Private \<and> 
+  msig new = msig old  \<and>
+  G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
+  G\<turnstile>Method new declared_in (declclass new) \<and>   
+  G\<turnstile>Method old declared_in (declclass old)"
+by (induct set: overridesR) (auto intro: trancl_trans)
+
+lemma ws_overrides_commonD:
+"\<lbrakk>G\<turnstile>new overrides old;ws_prog G\<rbrakk> \<Longrightarrow>  
+  declclass new \<noteq> Object \<and> \<not> is_static new \<and> \<not> is_static old \<and>
+  accmodi new \<noteq> Private \<and> G\<turnstile>resTy new \<preceq> resTy old \<and>
+  msig new = msig old  \<and>
+  G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
+  G\<turnstile>Method new declared_in (declclass new) \<and>   
+  G\<turnstile>Method old declared_in (declclass old)"
+by (induct set: overridesR) (auto intro: trancl_trans ws_widen_trans)
+
+lemma stat_overrides_commonD:
+"\<lbrakk>G\<turnstile>new overrides\<^sub>S old\<rbrakk> \<Longrightarrow>  
+  declclass new \<noteq> Object \<and> \<not> is_static new \<and> msig new = msig old \<and> 
+  G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
+  G\<turnstile>Method new declared_in (declclass new) \<and>   
+  G\<turnstile>Method old declared_in (declclass old)"
+by (induct set: stat_overridesR) (auto intro: trancl_trans)
+
+lemma overrides_eq_sigD: 
+ "\<lbrakk>G\<turnstile>new overrides old\<rbrakk> \<Longrightarrow> msig old=msig new"
+by (auto dest: overrides_commonD)
+
+lemma hides_eq_sigD: 
+ "\<lbrakk>G\<turnstile>new hides old\<rbrakk> \<Longrightarrow> msig old=msig new"
+by (auto simp add: hides_def)
+
+subsubsection "permits access" 
+constdefs 
+permits_acc:: 
+ "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                   ("_ \<turnstile> _ in _ permits'_acc'_to _" [61,61,61,61] 60)
+
+"G\<turnstile>membr in class permits_acc_to accclass 
+  \<equiv> (case (accmodi membr) of
+       Private   \<Rightarrow> (declclass membr = accclass)
+     | Package   \<Rightarrow> (pid (declclass membr) = pid accclass)
+     | Protected \<Rightarrow> (pid (declclass membr) = pid accclass)
+                    \<or>
+                    (G\<turnstile>accclass \<prec>\<^sub>C declclass membr \<and> G\<turnstile>class \<preceq>\<^sub>C accclass) 
+     | Public    \<Rightarrow> True)"
+text {*
+The subcondition of the @{term "Protected"} case: 
+@{term "G\<turnstile>accclass \<prec>\<^sub>C declclass membr"} could also be relaxed to:
+@{term "G\<turnstile>accclass \<preceq>\<^sub>C declclass membr"} since in case both classes are the
+same the other condition @{term "(pid (declclass membr) = pid accclass)"}
+holds anyway.
+*} 
+
+text {* Like in case of overriding, the static and dynamic accessibility 
+of members is not uniform.
+\begin{itemize}
+\item Statically the class/interface of the member must be accessible for the
+      member to be accessible. During runtime this is not necessary. For
+      Example, if a class is accessible and we are allowed to access a member
+      of this class (statically) we expect that we can access this member in 
+      an arbitrary subclass (during runtime). It's not intended to restrict
+      the access to accessible subclasses during runtime.
+\item Statically the member we want to access must be "member of" the class.
+      Dynamically it must only be "member in" the class.
+\end{itemize} 
+*} 
+
+
+consts
+accessible_fromR:: 
+ "prog \<Rightarrow> qtname \<Rightarrow> ((qtname \<times> memberdecl) \<times>  qtname) set"
+
+syntax 
+accessible_from:: 
+ "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                   ("_ \<turnstile> _ of _ accessible'_from _" [61,61,61,61] 60)
+
+translations
+"G\<turnstile>membr of cls accessible_from accclass"  
+ \<rightleftharpoons> "(membr,cls) \<in> accessible_fromR G accclass"
+
+syntax 
+method_accessible_from:: 
+ "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                   ("_ \<turnstile>Method _ of _ accessible'_from _" [61,61,61,61] 60)
+
+translations
+"G\<turnstile>Method m of cls accessible_from accclass"  
+ \<rightleftharpoons> "G\<turnstile>methdMembr m of cls accessible_from accclass"  
+
+syntax 
+methd_accessible_from:: 
+ "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                 ("_ \<turnstile>Methd _ _ of _ accessible'_from _" [61,61,61,61,61] 60)
+
+translations
+"G\<turnstile>Methd s m of cls accessible_from accclass"  
+ \<rightleftharpoons> "G\<turnstile>(method s m) of cls accessible_from accclass"  
+
+syntax 
+field_accessible_from:: 
+ "prog \<Rightarrow> vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                 ("_ \<turnstile>Field _  _ of _ accessible'_from _" [61,61,61,61,61] 60)
+
+translations
+"G\<turnstile>Field fn f of C accessible_from accclass"  
+ \<rightleftharpoons> "G\<turnstile>(fieldm fn f) of C accessible_from accclass" 
+
+inductive "accessible_fromR G accclass" intros
+immediate:  "\<lbrakk>G\<turnstile>membr member_of class;
+              G\<turnstile>(Class class) accessible_in (pid accclass);
+              G\<turnstile>membr in class permits_acc_to accclass 
+             \<rbrakk> \<Longrightarrow> G\<turnstile>membr of class accessible_from accclass"
+
+overriding: "\<lbrakk>G\<turnstile>membr member_of class;
+              G\<turnstile>(Class class) accessible_in (pid accclass);
+              membr=(C,mdecl new);
+              G\<turnstile>(C,new) overrides\<^sub>S old; 
+              G\<turnstile>class \<prec>\<^sub>C sup;
+              G\<turnstile>Method old of sup accessible_from accclass
+             \<rbrakk>\<Longrightarrow> G\<turnstile>membr of class accessible_from accclass"
+
+consts
+dyn_accessible_fromR:: 
+ "prog \<Rightarrow> qtname \<Rightarrow> ((qtname \<times> memberdecl) \<times>  qtname) set"
+
+syntax 
+dyn_accessible_from:: 
+ "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                   ("_ \<turnstile> _ in _ dyn'_accessible'_from _" [61,61,61,61] 60)
+
+translations
+"G\<turnstile>membr in C dyn_accessible_from accC"  
+ \<rightleftharpoons> "(membr,C) \<in> dyn_accessible_fromR G accC"
+
+syntax 
+method_dyn_accessible_from:: 
+ "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+                 ("_ \<turnstile>Method _ in _ dyn'_accessible'_from _" [61,61,61,61] 60)
+
+translations
+"G\<turnstile>Method m in C dyn_accessible_from accC"  
+ \<rightleftharpoons> "G\<turnstile>methdMembr m in C dyn_accessible_from accC"  
+
+syntax 
+methd_dyn_accessible_from:: 
+ "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+             ("_ \<turnstile>Methd _ _ in _ dyn'_accessible'_from _" [61,61,61,61,61] 60)
+
+translations
+"G\<turnstile>Methd s m in C dyn_accessible_from accC"  
+ \<rightleftharpoons> "G\<turnstile>(method s m) in C dyn_accessible_from accC"  
+
+syntax 
+field_dyn_accessible_from:: 
+ "prog \<Rightarrow> vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
+         ("_ \<turnstile>Field _ _ in _ dyn'_accessible'_from _" [61,61,61,61,61] 60)
+
+translations
+"G\<turnstile>Field fn f in dynC dyn_accessible_from accC"  
+ \<rightleftharpoons> "G\<turnstile>(fieldm fn f) in dynC dyn_accessible_from accC"
+  
+(* #### Testet JVM noch über den Bytecodeverifier hinaus ob der
+ statische Typ accessible ist bevor es den Zugriff erlaubt 
+ \<longrightarrow> Test mit Reflektion\<dots>
+*)
+inductive "dyn_accessible_fromR G accclass" intros
+immediate:  "\<lbrakk>G\<turnstile>membr member_in class;
+              G\<turnstile>membr in class permits_acc_to accclass 
+             \<rbrakk> \<Longrightarrow> G\<turnstile>membr in class dyn_accessible_from accclass"
+
+overriding: "\<lbrakk>G\<turnstile>membr member_in class;
+              membr=(C,mdecl new);
+              G\<turnstile>(C,new) overrides old; 
+              G\<turnstile>class \<prec>\<^sub>C sup;
+              G\<turnstile>Method old in sup dyn_accessible_from accclass
+             \<rbrakk>\<Longrightarrow> G\<turnstile>membr in class dyn_accessible_from accclass"
+
+
+lemma accessible_from_commonD: "G\<turnstile>m of C accessible_from S
+ \<Longrightarrow> G\<turnstile>m member_of C \<and> G\<turnstile>(Class C) accessible_in (pid S)"
+by (auto elim: accessible_fromR.induct)
+
+lemma declared_not_undeclared:
+  "G\<turnstile>m declared_in C \<Longrightarrow> \<not> G\<turnstile> memberid m undeclared_in C"
+by (cases m) (auto simp add: declared_in_def undeclared_in_def)
+
+lemma not_undeclared_declared:
+  "\<not> G\<turnstile> membr_id undeclared_in C \<Longrightarrow> (\<exists> m. G\<turnstile>m declared_in C \<and> 
+                                           membr_id = memberid m)"
+proof -
+  assume not_undecl:"\<not> G\<turnstile> membr_id undeclared_in C"
+  show ?thesis (is "?P membr_id")
+  proof (cases membr_id)
+    case (fid vname)
+    with not_undecl
+    obtain fld where
+      "G\<turnstile>fdecl (vname,fld) declared_in C" 
+      by (auto simp add: undeclared_in_def declared_in_def
+                         cdeclaredfield_def)
+    with fid show ?thesis 
+      by auto
+  next
+    case (mid sig)
+    with not_undecl
+    obtain mthd where
+      "G\<turnstile>mdecl (sig,mthd) declared_in C" 
+      by (auto simp add: undeclared_in_def declared_in_def
+                         cdeclaredmethd_def)
+    with mid show ?thesis 
+      by auto
+  qed
+qed
+
+lemma unique_declared_in:
+ "\<lbrakk>G\<turnstile>m declared_in C; G\<turnstile>n declared_in C; memberid m = memberid n\<rbrakk>
+ \<Longrightarrow> m = n"
+by (auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def
+            split: memberdecl.splits)
+
+lemma unique_member_of: 
+ (assumes n: "G\<turnstile>n member_of C" and  
+          m: "G\<turnstile>m member_of C" and
+       eqid: "memberid n = memberid m"
+ ) "n=m"
+proof -
+  from n m eqid  
+  show "n=m"
+  proof (induct)
+    case (Immediate C n)
+    assume member_n: "G\<turnstile> mbr n declared_in C" "declclass n = C"
+    assume eqid: "memberid n = memberid m"
+    assume "G \<turnstile> m member_of C"
+    then show "n=m"
+    proof (cases)
+      case (Immediate _ m')
+      with eqid 
+      have "m=m'"
+           "memberid n = memberid m" 
+           "G\<turnstile> mbr m declared_in C" 
+           "declclass m = C"
+	by auto
+      with member_n   
+      show ?thesis
+	by (cases n, cases m) 
+           (auto simp add: declared_in_def 
+                           cdeclaredmethd_def cdeclaredfield_def
+                    split: memberdecl.splits)
+    next
+      case (Inherited _ _ m')
+      then have "G\<turnstile> memberid m undeclared_in C"
+	by simp
+      with eqid member_n
+      show ?thesis
+	by (cases n) (auto dest: declared_not_undeclared)
+    qed
+  next
+    case (Inherited C S n)
+    assume undecl: "G\<turnstile> memberid n undeclared_in C"
+    assume  super: "G\<turnstile>C\<prec>\<^sub>C\<^sub>1S"
+    assume    hyp: "\<lbrakk>G \<turnstile> m member_of S; memberid n = memberid m\<rbrakk> \<Longrightarrow> n = m"
+    assume   eqid: "memberid n = memberid m"
+    assume "G \<turnstile> m member_of C"
+    then show "n=m"
+    proof (cases)
+      case Immediate
+      then have "G\<turnstile> mbr m declared_in C" by simp
+      with eqid undecl
+      show ?thesis
+	by (cases m) (auto dest: declared_not_undeclared)
+    next
+      case Inherited 
+      with super have "G \<turnstile> m member_of S"
+	by (auto dest!: subcls1D)
+      with eqid hyp
+      show ?thesis 
+	by blast
+    qed
+  qed
+qed
+
+lemma member_of_is_classD: "G\<turnstile>m member_of C \<Longrightarrow> is_class G C"
+proof (induct set: members)
+  case (Immediate C m)
+  assume "G\<turnstile> mbr m declared_in C"
+  then show "is_class G C"
+    by (cases "mbr m")
+       (auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def)
+next
+  case (Inherited C S m)  
+  assume "G\<turnstile>C\<prec>\<^sub>C\<^sub>1S" and "is_class G S"
+  then show "is_class G C"
+    by - (rule subcls_is_class2,auto)
+qed
+
+lemma member_of_declC: 
+ "G\<turnstile>m member_of C 
+  \<Longrightarrow> G\<turnstile>mbr m declared_in (declclass m)"
+by (induct set: members) auto
+
+lemma member_of_member_of_declC:
+ "G\<turnstile>m member_of C 
+  \<Longrightarrow> G\<turnstile>m member_of (declclass m)"
+by (auto dest: member_of_declC intro: members.Immediate)
+
+lemma member_of_class_relation:
+  "G\<turnstile>m member_of C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m"
+proof (induct set: members)
+  case (Immediate C m)
+  then show "G\<turnstile>C \<preceq>\<^sub>C declclass m" by simp
+next
+  case (Inherited C S m)
+  then show "G\<turnstile>C \<preceq>\<^sub>C declclass m" 
+    by (auto dest: r_into_rtrancl intro: rtrancl_trans)
+qed
+
+lemma member_in_class_relation:
+  "G\<turnstile>m member_in C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m"
+by (auto dest: member_inD member_of_class_relation
+        intro: rtrancl_trans)
+
+lemma member_of_Package: 
+ "\<lbrakk>G\<turnstile>m member_of C; accmodi m = Package\<rbrakk> 
+  \<Longrightarrow> pid (declclass m) = pid C" 
+proof -
+  assume   member: "G\<turnstile>m member_of C"
+  then show " accmodi m = Package \<Longrightarrow> ?thesis" (is "PROP ?P m C")
+  proof (induct rule: members.induct)
+    fix C m
+    assume     C: "declclass m = C"
+    then show "pid (declclass m) = pid C"
+      by simp
+  next
+    fix C S m  
+    assume inheritable: "G \<turnstile> m inheritable_in pid C"
+    assume         hyp: "PROP ?P m S" and
+           package_acc: "accmodi m = Package" 
+    with inheritable package_acc hyp
+    show "pid (declclass m) = pid C" 
+      by (auto simp add: inheritable_in_def)
+  qed
+qed
+
+lemma dyn_accessible_from_commonD: "G\<turnstile>m in C dyn_accessible_from S
+ \<Longrightarrow> G\<turnstile>m member_in C"
+by (auto elim: dyn_accessible_fromR.induct)
+
+(* ### Gilt nicht für wf_progs!dynmaisches Override, 
+  da die accmodi Bedingung nur für stat override gilt! *)
+(*
+lemma override_Package:
+ "\<lbrakk>G\<turnstile>new overrides old; 
+  \<And> new old. G\<turnstile>new overrides old \<Longrightarrow> accmodi old \<le> accmodi new;
+  accmodi old = Package; accmodi new = Package\<rbrakk>
+  \<Longrightarrow> pid (declclass old) = pid (declclass new)"
+proof - 
+  assume wf: "\<And> new old. G\<turnstile>new overrides old \<Longrightarrow> accmodi old \<le> accmodi new"
+  assume ovverride: "G\<turnstile>new overrides old"
+  then show "\<lbrakk>accmodi old = Package;accmodi new = Package\<rbrakk> \<Longrightarrow> ?thesis"
+    (is "?Pack old \<Longrightarrow> ?Pack new \<Longrightarrow> ?EqPid old new")
+  proof (induct rule: overridesR.induct)
+    case Direct
+    fix new old
+    assume "accmodi old = Package"
+           "G \<turnstile> methdMembr 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 wf have le_inter_new: "accmodi inter \<le> accmodi new"
+      by blast
+    assume "G \<turnstile> inter overrides old"
+    with wf have le_old_inter: "accmodi old \<le> accmodi inter"
+      by blast
+    from accmodi_old accmodi_new le_inter_new le_old_inter
+    have "accmodi inter = Package"
+      by(auto simp add: le_acc_def less_acc_def)
+    with Indirect accmodi_old accmodi_new
+    show "?EqPid old new"
+      by auto
+  qed
+qed
+
+lemma stat_override_Package:
+ "\<lbrakk>G\<turnstile>new overrides\<^sub>S old; 
+  \<And> new old. G\<turnstile>new overrides\<^sub>S old \<Longrightarrow> accmodi old \<le> accmodi new;
+  accmodi old = Package; accmodi new = Package\<rbrakk>
+  \<Longrightarrow> pid (declclass old) = pid (declclass new)"
+proof - 
+  assume wf: "\<And> new old. G\<turnstile>new overrides\<^sub>S old \<Longrightarrow> accmodi old \<le> accmodi new"
+  assume ovverride: "G\<turnstile>new overrides\<^sub>S old"
+  then show "\<lbrakk>accmodi old = Package;accmodi new = Package\<rbrakk> \<Longrightarrow> ?thesis"
+    (is "?Pack old \<Longrightarrow> ?Pack new \<Longrightarrow> ?EqPid old new")
+  proof (induct rule: stat_overridesR.induct)
+    case Direct
+    fix new old
+    assume "accmodi old = Package"
+           "G \<turnstile> methdMembr 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\<^sub>S inter"
+    with wf have le_inter_new: "accmodi inter \<le> accmodi new"
+      by blast
+    assume "G \<turnstile> inter overrides\<^sub>S old"
+    with wf have le_old_inter: "accmodi old \<le> accmodi inter"
+      by blast
+    from accmodi_old accmodi_new le_inter_new le_old_inter
+    have "accmodi inter = Package"
+      by(auto simp add: le_acc_def less_acc_def)
+    with Indirect accmodi_old accmodi_new
+    show "?EqPid old new"
+      by auto
+  qed
+qed
+
+*)
+lemma no_Private_stat_override: 
+ "\<lbrakk>G\<turnstile>new overrides\<^sub>S old\<rbrakk> \<Longrightarrow> accmodi old \<noteq> Private"
+by (induct set:  stat_overridesR) (auto simp add: inheritable_in_def)
+
+lemma no_Private_override: "\<lbrakk>G\<turnstile>new overrides old\<rbrakk> \<Longrightarrow> accmodi old \<noteq> Private"
+by (induct set: overridesR) (auto simp add: inheritable_in_def)
+
+lemma permits_acc_inheritance:
+ "\<lbrakk>G\<turnstile>m in statC permits_acc_to accC; G\<turnstile>dynC \<preceq>\<^sub>C statC
+  \<rbrakk> \<Longrightarrow> G\<turnstile>m in dynC permits_acc_to accC"
+by (cases "accmodi m")
+   (auto simp add: permits_acc_def
+            intro: subclseq_trans) 
+
+lemma field_accessible_fromD:
+ "\<lbrakk>G\<turnstile>membr of C accessible_from accC;is_field membr\<rbrakk> 
+  \<Longrightarrow> G\<turnstile>membr member_of C \<and>
+      G\<turnstile>(Class C) accessible_in (pid accC) \<and>
+      G\<turnstile>membr in C permits_acc_to accC"
+by (cases set: accessible_fromR)
+   (auto simp add: is_field_def split: memberdecl.splits) 
+
+lemma field_accessible_from_permits_acc_inheritance:
+"\<lbrakk>G\<turnstile>membr of statC accessible_from accC; is_field membr; G \<turnstile> dynC \<preceq>\<^sub>C statC\<rbrakk>
+\<Longrightarrow> G\<turnstile>membr in dynC permits_acc_to accC"
+by (auto dest: field_accessible_fromD intro: permits_acc_inheritance)
+
+
+(*
+lemma accessible_Package:
+ "\<lbrakk>G \<turnstile> m of C accessible_from S;accmodi m = Package;
+   \<And> new old. G\<turnstile>new overrides\<^sub>S old \<Longrightarrow> accmodi old \<le> accmodi new\<rbrakk>
+  \<Longrightarrow> pid S = pid C \<and> pid C = pid (declclass m)"
+proof -
+  assume wf: "\<And> new old. G\<turnstile>new overrides\<^sub>S old \<Longrightarrow> accmodi old \<le> accmodi new"
+  assume "G \<turnstile> m of C accessible_from S"
+  then show "accmodi m = Package \<Longrightarrow> pid S = pid C \<and> pid C = pid (declclass m)"
+    (is "?Pack m \<Longrightarrow> ?P C m")
+  proof (induct rule: accessible_fromR.induct)
+    fix C m
+    assume "G\<turnstile>m member_of C"
+           "G \<turnstile> m in C permits_acc_to S"
+           "accmodi m = Package"      
+    then show "?P C m"
+      by (auto dest: member_of_Package simp add: permits_acc_def)
+  next
+    fix declC C new newm old Sup
+    assume member_new: "G \<turnstile> new member_of C" and 
+                acc_C: "G \<turnstile> Class C accessible_in pid S" and
+                  new: "new = (declC, mdecl newm)" and
+             override: "G \<turnstile> (declC, newm) overrides\<^sub>S old" and
+         subcls_C_Sup: "G\<turnstile>C \<prec>\<^sub>C Sup" and
+              acc_old: "G \<turnstile> methdMembr old of Sup accessible_from S" and
+                  hyp: "?Pack (methdMembr old) \<Longrightarrow> ?P Sup (methdMembr old)" and
+          accmodi_new: "accmodi new = Package"
+    from override wf 
+    have accmodi_weaken: "accmodi old \<le> accmodi newm"
+      by (cases old,cases newm) auto
+    from override new
+    have "accmodi old \<noteq> Private"
+      by (simp add: no_Private_stat_override)
+    with accmodi_weaken accmodi_new new
+    have accmodi_old: "accmodi old = Package"
+      by (cases "accmodi old") (auto simp add: le_acc_def less_acc_def) 
+    with hyp 
+    have P_sup: "?P Sup (methdMembr old)"
+      by (simp)
+    from wf override new accmodi_old accmodi_new
+    have eq_pid_new_old: "pid (declclass new) = pid (declclass old)"
+      by (auto dest: stat_override_Package) 
+    from member_new accmodi_new
+    have "pid (declclass new) = pid C"
+      by (auto dest: member_of_Package)
+    with eq_pid_new_old P_sup show "?P C new"
+      by auto
+  qed
+qed
+*)
+lemma accessible_fieldD: 
+ "\<lbrakk>G\<turnstile>membr of C accessible_from accC; is_field membr\<rbrakk>
+ \<Longrightarrow> G\<turnstile>membr member_of C \<and>
+     G\<turnstile>(Class C) accessible_in (pid accC) \<and>
+     G\<turnstile>membr in C permits_acc_to accC"
+by (induct rule: accessible_fromR.induct) (auto dest: is_fieldD)
+      
+(* lemmata:
+ Wegen  G\<turnstile>Super accessible_in (pid C) folgt:
+  G\<turnstile>m declared_in C; G\<turnstile>m member_of D; accmodi m = Package (G\<turnstile>D \<preceq>\<^sub>C C)
+  \<Longrightarrow> pid C = pid D 
+
+  C package
+  m public in C
+  für alle anderen D: G\<turnstile>m undeclared_in C
+  m wird in alle subklassen vererbt, auch aus dem Package heraus!
+
+  G\<turnstile>m member_of C \<Longrightarrow> \<exists> D. G\<turnstile>C \<preceq>\<^sub>C D \<and> G\<turnstile>m declared_in D
+*)
+
+(* Begriff (C,m) overrides (D,m)
+    3 Fälle: Direkt,
+             Indirekt über eine Zwischenklasse (ohne weiteres override)
+             Indirekt über override
+*)
+   
+(*
+"G\<turnstile>m member_of C \<equiv> 
+constdefs declares_method:: "prog \<Rightarrow> sig \<Rightarrow> qtname \<Rightarrow> methd \<Rightarrow> bool"
+                                 ("_,_\<turnstile> _ declares'_method _" [61,61,61,61] 60)
+"G,sig\<turnstile>C declares_method m \<equiv> cdeclaredmethd G C sig = Some m" 
+
+constdefs is_declared:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> bool"
+"is_declared G sig em \<equiv> G,sig\<turnstile>declclass em declares_method mthd em"
+*)
+
+lemma member_of_Private:
+"\<lbrakk>G\<turnstile>m member_of C; accmodi m = Private\<rbrakk> \<Longrightarrow> declclass m = C"
+by (induct set: members) (auto simp add: inheritable_in_def)
+
+lemma member_of_subclseq_declC:
+  "G\<turnstile>m member_of C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m"
+by (induct set: members) (auto dest: r_into_rtrancl intro: rtrancl_trans)
+
+lemma member_of_inheritance:
+  (assumes    m: "G\<turnstile>m member_of D" and 
+     subclseq_D_C: "G\<turnstile>D \<preceq>\<^sub>C C" and
+     subclseq_C_m: "G\<turnstile>C \<preceq>\<^sub>C declclass m" and
+               ws: "ws_prog G" 
+  ) "G\<turnstile>m member_of C"
+proof -
+  from m subclseq_D_C subclseq_C_m
+  show ?thesis
+  proof (induct)
+    case (Immediate D m)
+    assume "declclass m = D" and
+           "G\<turnstile>D\<preceq>\<^sub>C C" and "G\<turnstile>C\<preceq>\<^sub>C declclass m"
+    with ws have "D=C" 
+      by (auto intro: subclseq_acyclic)
+    with Immediate 
+    show "G\<turnstile>m member_of C"
+      by (auto intro: members.Immediate)
+  next
+    case (Inherited D S m)
+    assume member_of_D_props: 
+            "G \<turnstile> m inheritable_in pid D" 
+            "G\<turnstile> memberid m undeclared_in D"  
+            "G \<turnstile> Class S accessible_in pid D" 
+            "G \<turnstile> m member_of S"
+    assume super: "G\<turnstile>D\<prec>\<^sub>C\<^sub>1S"
+    assume hyp: "\<lbrakk>G\<turnstile>S\<preceq>\<^sub>C C; G\<turnstile>C\<preceq>\<^sub>C declclass m\<rbrakk> \<Longrightarrow> G \<turnstile> m member_of C"
+    assume subclseq_C_m: "G\<turnstile>C\<preceq>\<^sub>C declclass m"
+    assume "G\<turnstile>D\<preceq>\<^sub>C C"
+    then show "G\<turnstile>m member_of C"
+    proof (cases rule: subclseq_cases)
+      case Eq
+      assume "D=C" 
+      with super member_of_D_props 
+      show ?thesis
+	by (auto intro: members.Inherited)
+    next
+      case Subcls
+      assume "G\<turnstile>D\<prec>\<^sub>C C"
+      with super 
+      have "G\<turnstile>S\<preceq>\<^sub>C C"
+	by (auto dest: subcls1D subcls_superD)
+      with subclseq_C_m hyp show ?thesis
+	by blast
+    qed
+  qed
+qed
+
+lemma member_of_subcls:
+  (assumes    old: "G\<turnstile>old member_of C" and 
+              new: "G\<turnstile>new member_of D" and
+             eqid: "memberid new = memberid old" and
+     subclseq_D_C: "G\<turnstile>D \<preceq>\<^sub>C C" and 
+   subcls_new_old: "G\<turnstile>declclass new \<prec>\<^sub>C declclass old" and
+               ws: "ws_prog G"
+  ) "G\<turnstile>D \<prec>\<^sub>C C"
+proof -
+  from old 
+  have subclseq_C_old: "G\<turnstile>C \<preceq>\<^sub>C declclass old"
+    by (auto dest: member_of_subclseq_declC)
+  from new 
+  have subclseq_D_new: "G\<turnstile>D \<preceq>\<^sub>C declclass new"
+    by (auto dest: member_of_subclseq_declC)
+  from subcls_new_old ws
+  have neq_new_old: "new\<noteq>old"
+    by (cases new,cases old) (auto dest: subcls_irrefl)
+  from subclseq_D_new subclseq_D_C
+  have "G\<turnstile>(declclass new) \<preceq>\<^sub>C C \<or> G\<turnstile>C \<preceq>\<^sub>C (declclass new)" 
+    by (rule subcls_compareable)
+  then have "G\<turnstile>(declclass new) \<preceq>\<^sub>C C"
+  proof
+    assume "G\<turnstile>declclass new\<preceq>\<^sub>C C" then show ?thesis .
+  next
+    assume "G\<turnstile>C \<preceq>\<^sub>C (declclass new)"
+    with new subclseq_D_C ws 
+    have "G\<turnstile>new member_of C"
+      by (blast intro: member_of_inheritance)
+    with eqid old 
+    have "new=old"
+      by (blast intro: unique_member_of)
+    with neq_new_old 
+    show ?thesis
+      by contradiction
+  qed
+  then show ?thesis
+  proof (cases rule: subclseq_cases)
+    case Eq
+    assume "declclass new = C"
+    with new have "G\<turnstile>new member_of C"
+      by (auto dest: member_of_member_of_declC)
+    with eqid old 
+    have "new=old"
+      by (blast intro: unique_member_of)
+    with neq_new_old 
+    show ?thesis
+      by contradiction
+  next
+    case Subcls
+    assume "G\<turnstile>declclass new\<prec>\<^sub>C C"
+    with subclseq_D_new
+    show "G\<turnstile>D\<prec>\<^sub>C C"
+      by (rule rtrancl_trancl_trancl)
+  qed
+qed
+
+corollary member_of_overrides_subcls:
+ "\<lbrakk>G\<turnstile>Methd sig old member_of C; G\<turnstile>Methd sig new member_of D;G\<turnstile>D \<preceq>\<^sub>C C;
+   G,sig\<turnstile>new overrides old; ws_prog G\<rbrakk>
+ \<Longrightarrow> G\<turnstile>D \<prec>\<^sub>C C"
+by (drule overrides_commonD) (auto intro: member_of_subcls)    
+
+corollary member_of_stat_overrides_subcls:
+ "\<lbrakk>G\<turnstile>Methd sig old member_of C; G\<turnstile>Methd sig new member_of D;G\<turnstile>D \<preceq>\<^sub>C C;
+   G,sig\<turnstile>new overrides\<^sub>S old; ws_prog G\<rbrakk>
+ \<Longrightarrow> G\<turnstile>D \<prec>\<^sub>C C"
+by (drule stat_overrides_commonD) (auto intro: member_of_subcls)    
+
+
+
+lemma inherited_field_access: 
+ (assumes stat_acc: "G\<turnstile>membr of statC accessible_from accC" and
+          is_field: "is_field membr" and 
+          subclseq: "G \<turnstile> dynC \<preceq>\<^sub>C statC"
+ ) "G\<turnstile>membr in dynC dyn_accessible_from accC"
+proof -
+  from stat_acc is_field subclseq 
+  show ?thesis
+    by (auto dest: accessible_fieldD 
+            intro: dyn_accessible_fromR.immediate 
+                   member_inI 
+                   permits_acc_inheritance)
+qed
+
+lemma accessible_inheritance:
+ (assumes stat_acc: "G\<turnstile>m of statC accessible_from accC" and
+          subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
+       member_dynC: "G\<turnstile>m member_of dynC" and
+          dynC_acc: "G\<turnstile>(Class dynC) accessible_in (pid accC)"
+ ) "G\<turnstile>m of dynC accessible_from accC"
+proof -
+  from stat_acc
+  have member_statC: "G\<turnstile>m member_of statC" 
+    by (auto dest: accessible_from_commonD)
+  from stat_acc
+  show ?thesis
+  proof (cases)
+    case immediate
+    with member_dynC member_statC subclseq dynC_acc
+    show ?thesis
+      by (auto intro: accessible_fromR.immediate permits_acc_inheritance)
+  next
+    case overriding
+    with member_dynC subclseq dynC_acc
+    show ?thesis
+      by (auto intro: accessible_fromR.overriding rtrancl_trancl_trancl)
+  qed
+qed
+
+section "fields and methods"
+
+
+types
+  fspec = "vname \<times> qtname"
+
+translations 
+  "fspec" <= (type) "vname \<times> qtname" 
+
+constdefs
+imethds:: "prog \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> mhead) tables"
+"imethds G I 
+  \<equiv> iface_rec (G,I)  
+              (\<lambda>I i ts. (Un_tables ts) \<oplus>\<oplus> 
+                        (o2s \<circ> table_of (map (\<lambda>(s,m). (s,I,m)) (imethods i))))"
+text {* methods of an interface, with overriding and inheritance, cf. 9.2 *}
+
+constdefs
+accimethds:: "prog \<Rightarrow> pname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> mhead) tables"
+"accimethds G pack I
+  \<equiv> if G\<turnstile>Iface I accessible_in pack 
+       then imethds G I
+       else \<lambda> k. {}"
+text {* only returns imethds if the interface is accessible *}
+
+constdefs
+methd:: "prog \<Rightarrow> qtname  \<Rightarrow> (sig,qtname \<times> methd) table"
+
+"methd G C 
+ \<equiv> class_rec (G,C) empty
+             (\<lambda>C c subcls_mthds. 
+               filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)
+                          subcls_mthds 
+               ++ 
+               table_of (map (\<lambda>(s,m). (s,C,m)) (methods c)))"
+text {* @{term "methd G C"}: methods of a class C (statically visible from C), 
+     with inheritance and hiding cf. 8.4.6;
+     Overriding is captured by @{text dynmethd}.
+     Every new method with the same signature coalesces the
+     method of a superclass. *}
+
+constdefs                      
+accmethd:: "prog \<Rightarrow> qtname \<Rightarrow> qtname  \<Rightarrow> (sig,qtname \<times> methd) table"
+"accmethd G S C 
+ \<equiv> filter_tab (\<lambda>sig m. G\<turnstile>method sig m of C accessible_from S) 
+              (methd G C)"
+text {* @{term "accmethd G S C"}: only those methods of @{term "methd G C"}, 
+        accessible from S *}
+
+text {* Note the class component in the accessibility filter. The class where
+    method @{term m} is declared (@{term declC}) isn't necessarily accessible 
+    from the current scope @{term S}. The method can be made accessible 
+    through inheritance, too.
+    So we must test accessibility of method @{term m} of class @{term C} 
+    (not @{term "declclass m"}) *}
+
+constdefs 
+dynmethd:: "prog  \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table"
+"dynmethd G statC dynC  
+  \<equiv> \<lambda> sig. 
+     (if G\<turnstile>dynC \<preceq>\<^sub>C statC
+        then (case methd G statC sig of
+                None \<Rightarrow> None
+              | Some statM 
+                  \<Rightarrow> (class_rec (G,dynC) empty
+                       (\<lambda>C c subcls_mthds. 
+                          subcls_mthds
+                          ++
+                          (filter_tab 
+                            (\<lambda> _ dynM. G,sig\<turnstile>dynM overrides statM \<or> dynM=statM)
+                            (methd G C) ))
+                      ) sig
+              )
+        else None)"
+(*
+"dynmethd G statC dynC  
+  \<equiv> \<lambda> sig. 
+     (if G\<turnstile>dynC \<preceq>\<^sub>C statC
+        then (case methd G statC sig of
+                None \<Rightarrow> None
+              | Some statM 
+                    \<Rightarrow> (class_rec (G,statC) empty
+                         (\<lambda>C c subcls_mthds. 
+                            subcls_mthds
+                            ++
+                            (filter_tab 
+                              (\<lambda> _ dynM. G,sig\<turnstile>dynM overrides statM)  
+                              (table_of (map (\<lambda>(s,m). (s,C,m)) (methods c)))))
+                        ) sig
+              )
+        else None)"*)
+text {* @{term "dynmethd G statC dynC"}: dynamic method lookup of a reference 
+        with dynamic class @{term dynC} and static class @{term statC} *}
+text {* Note some kind of duality between @{term methd} and @{term dynmethd} 
+        in the @{term class_rec} arguments. Whereas @{term methd} filters the 
+        subclass methods (to get only the inherited ones), @{term dynmethd} 
+        filters the new methods (to get only those methods which actually
+        override the methods of the static class) *}
+
+constdefs 
+dynimethd:: "prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table"
+"dynimethd G I dynC 
+  \<equiv> \<lambda> sig. if imethds G I sig \<noteq> {}
+               then methd G dynC sig
+               else dynmethd G Object dynC sig"
+text {* @{term "dynimethd G I dynC"}: dynamic method lookup of a reference with 
+        dynamic class dynC and static interface type I *}
+text {* 
+   When calling an interface method, we must distinguish if the method signature
+   was defined in the interface or if it must be an Object method in the other
+   case. If it was an interface method we search the class hierarchy
+   starting at the dynamic class of the object up to Object to find the 
+   first matching method (@{term methd}). Since all interface methods have 
+   public access the method can't be coalesced due to some odd visibility 
+   effects like in case of dynmethd. The method will be inherited or 
+   overridden in all classes from the first class implementing the interface 
+   down to the actual dynamic class.
+ *}
+
+constdefs
+dynlookup::"prog  \<Rightarrow> ref_ty \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table"
+"dynlookup G statT dynC
+  \<equiv> (case statT of
+       NullT        \<Rightarrow> empty
+     | IfaceT I     \<Rightarrow> dynimethd G I      dynC
+     | ClassT statC \<Rightarrow> dynmethd  G statC  dynC
+     | ArrayT ty    \<Rightarrow> dynmethd  G Object dynC)"
+text {* @{term "dynlookup G statT dynC"}: dynamic lookup of a method within the 
+    static reference type statT and the dynamic class dynC. 
+    In a wellformd context statT will not be NullT and in case
+    statT is an array type, dynC=Object *}
+
+constdefs
+fields:: "prog \<Rightarrow> qtname \<Rightarrow> ((vname \<times> qtname) \<times> field) list"
+"fields G C 
+  \<equiv> class_rec (G,C) [] (\<lambda>C c ts. map (\<lambda>(n,t). ((n,C),t)) (cfields c) @ ts)"
+text {* @{term "fields G C"} 
+     list of fields of a class, including all the fields of the superclasses
+     (private, inherited and hidden ones) not only the accessible ones
+     (an instance of a object allocates all these fields *}
+
+constdefs
+accfield:: "prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (vname, qtname  \<times>  field) table"
+"accfield G S C
+  \<equiv> let field_tab = table_of((map (\<lambda>((n,d),f).(n,(d,f)))) (fields G C))
+    in filter_tab (\<lambda>n (declC,f). G\<turnstile> (declC,fdecl (n,f)) of C accessible_from S)
+                  field_tab"
+text  {* @{term "accfield G C S"}: fields of a class @{term C} which are 
+         accessible from scope of class
+         @{term S} with inheritance and hiding, cf. 8.3 *}
+text {* note the class component in the accessibility filter (see also 
+        @{term methd}).
+   The class declaring field @{term f} (@{term declC}) isn't necessarily 
+   accessible from scope @{term S}. The field can be made visible through 
+   inheritance, too. So we must test accessibility of field @{term f} of class 
+   @{term C} (not @{term "declclass f"}) *} 
+
+constdefs
+
+  is_methd :: "prog \<Rightarrow> qtname  \<Rightarrow> sig \<Rightarrow> bool"
+ "is_methd G \<equiv> \<lambda>C sig. is_class G C \<and> methd G C sig \<noteq> None"
+
+constdefs efname:: "((vname \<times> qtname) \<times> field) \<Rightarrow> (vname \<times> qtname)"
+"efname \<equiv> fst"
+
+lemma efname_simp[simp]:"efname (n,f) = n"
+by (simp add: efname_def) 
+
+
+subsection "imethds"
+
+lemma imethds_rec: "\<lbrakk>iface G I = Some i; ws_prog G\<rbrakk> \<Longrightarrow>  
+  imethds G I = Un_tables ((\<lambda>J. imethds  G J)`set (isuperIfs i)) \<oplus>\<oplus>  
+                      (o2s \<circ> table_of (map (\<lambda>(s,mh). (s,I,mh)) (imethods i)))"
+apply (unfold imethds_def)
+apply (rule iface_rec [THEN trans])
+apply auto
+done
+
+
+(* local lemma *)
+lemma imethds_norec:
+  "\<lbrakk>iface G md = Some i; ws_prog G; table_of (imethods i) sig = Some mh\<rbrakk> \<Longrightarrow>  
+  (md, mh) \<in> imethds G md sig"
+apply (subst imethds_rec)
+apply assumption+
+apply (rule iffD2)
+apply (rule overrides_t_Some_iff)
+apply (rule disjI1)
+apply (auto elim: table_of_map_SomeI)
+done
+
+lemma imethds_declI: "\<lbrakk>m \<in> imethds G I sig; ws_prog G; is_iface G I\<rbrakk> \<Longrightarrow> 
+  (\<exists>i. iface G (decliface m) = Some i \<and> 
+  table_of (imethods i) sig = Some (mthd m)) \<and>  
+  (I,decliface m) \<in> (subint1 G)^* \<and> m \<in> imethds G (decliface m) sig"
+apply (erule make_imp)
+apply (rule ws_subint1_induct, assumption, assumption)
+apply (subst imethds_rec, erule conjunct1, assumption)
+apply (force elim: imethds_norec intro: rtrancl_into_rtrancl2)
+done
+
+lemma imethds_cases [consumes 3, case_names NewMethod InheritedMethod]:
+ (assumes im: "im \<in> imethds G I sig" and  
+         ifI: "iface G I = Some i" and
+          ws: "ws_prog G" and
+     hyp_new:  "table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)) sig 
+                = Some im \<Longrightarrow> P" and
+     hyp_inh: "\<And> J. \<lbrakk>J \<in> set (isuperIfs i); im \<in> imethds G J sig\<rbrakk> \<Longrightarrow> P"
+  ) "P"
+proof -
+  from ifI ws im hyp_new hyp_inh
+  show "P"
+    by (auto simp add: imethds_rec)
+qed
+
+subsection "accimethd"
+
+lemma accimethds_simp [simp]: 
+"G\<turnstile>Iface I accessible_in pack \<Longrightarrow> accimethds G pack I = imethds G I"
+by (simp add: accimethds_def)
+
+lemma accimethdsD:
+ "im \<in> accimethds G pack I sig 
+  \<Longrightarrow> im \<in> imethds G I sig \<and> G\<turnstile>Iface I accessible_in pack"
+by (auto simp add: accimethds_def)
+
+lemma accimethdsI: 
+"\<lbrakk>im \<in> imethds G I sig;G\<turnstile>Iface I accessible_in pack\<rbrakk>
+ \<Longrightarrow> im \<in> accimethds G pack I sig"
+by simp
+
+subsection "methd"
+
+lemma methd_rec: "\<lbrakk>class G C = Some c; ws_prog G\<rbrakk> \<Longrightarrow>  
+  methd G C 
+    = (if C = Object 
+          then empty 
+          else filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)
+                          (methd G (super c))) 
+      ++ table_of (map (\<lambda>(s,m). (s,C,m)) (methods c))"
+apply (unfold methd_def)
+apply (erule class_rec [THEN trans], assumption)
+apply (simp)
+done
+
+(* local lemma *)
+lemma methd_norec: 
+ "\<lbrakk>class G declC = Some c; ws_prog G;table_of (methods c) sig = Some m\<rbrakk> 
+  \<Longrightarrow> methd G declC sig = Some (declC, m)"
+apply (simp only: methd_rec)
+apply (rule disjI1 [THEN override_Some_iff [THEN iffD2]])
+apply (auto elim: table_of_map_SomeI)
+done
+
+
+lemma methd_declC: 
+"\<lbrakk>methd G C sig = Some m; ws_prog G;is_class G C\<rbrakk> \<Longrightarrow>
+ (\<exists>d. class G (declclass m)=Some d \<and> table_of (methods d) sig=Some (mthd m)) \<and> 
+ G\<turnstile>C \<preceq>\<^sub>C (declclass m) \<and> methd G (declclass m) sig = Some m"   
+apply (erule make_imp)
+apply (rule ws_subcls1_induct, assumption, assumption)
+apply (subst methd_rec, assumption)
+apply (case_tac "Ca=Object")
+apply   (force elim: methd_norec )
+
+apply   simp
+apply   (case_tac "table_of (map (\<lambda>(s, m). (s, Ca, m)) (methods c)) sig")
+apply     (force intro: rtrancl_into_rtrancl2)
+
+apply     (auto intro: methd_norec)
+done
+
+lemma methd_inheritedD:
+  "\<lbrakk>class G C = Some c; ws_prog G;methd G C sig = Some m\<rbrakk>
+  \<Longrightarrow>  (declclass m \<noteq> C \<longrightarrow> G \<turnstile>C inherits method sig m)"
+by (auto simp add: methd_rec)
+
+lemma methd_diff_cls:
+"\<lbrakk>ws_prog G; is_class G C; is_class G D;
+ methd G C sig = m; methd G D sig = n; m\<noteq>n
+\<rbrakk> \<Longrightarrow> C\<noteq>D"
+by (auto simp add: methd_rec)
+
+lemma method_declared_inI: 
+ "\<lbrakk>table_of (methods c) sig = Some m; class G C = Some c\<rbrakk>
+  \<Longrightarrow> G\<turnstile>mdecl (sig,m) declared_in C"
+by (auto simp add: cdeclaredmethd_def declared_in_def)
+
+lemma methd_declared_in_declclass: 
+ "\<lbrakk>methd G C sig = Some m; ws_prog G;is_class G C\<rbrakk> 
+ \<Longrightarrow> G\<turnstile>Methd sig m declared_in (declclass m)"
+by (auto dest: methd_declC method_declared_inI)
+
+lemma member_methd:
+ (assumes member_of: "G\<turnstile>Methd sig m member_of C" and
+                 ws: "ws_prog G"
+ ) "methd G C sig = Some m"
+proof -
+  from member_of 
+  have iscls_C: "is_class G C" 
+    by (rule member_of_is_classD)
+  from iscls_C ws member_of
+  show ?thesis (is "?Methd C")
+  proof (induct rule: ws_class_induct')
+    case (Object co)
+    assume "G \<turnstile>Methd sig m member_of Object"
+    then have "G\<turnstile>Methd sig m declared_in Object \<and> declclass m = Object"
+      by (cases set: members) (cases m, auto dest: subcls1D)
+    with ws Object 
+    show "?Methd Object"
+      by (cases m)
+         (auto simp add: declared_in_def cdeclaredmethd_def methd_rec
+                  intro:  table_of_mapconst_SomeI)
+  next
+    case (Subcls C c)
+    assume clsC: "class G C = Some c" and
+      neq_C_Obj: "C \<noteq> Object" and
+            hyp: "G \<turnstile>Methd sig m member_of super c \<Longrightarrow> ?Methd (super c)" and
+      member_of: "G \<turnstile>Methd sig m member_of C"
+    from member_of
+    show "?Methd C"
+    proof (cases)
+      case (Immediate Ca membr)
+      then have "Ca=C" "membr = method sig m" and 
+                "G\<turnstile>Methd sig m declared_in C" "declclass m = C"
+	by (cases m,auto)
+      with clsC 
+      have "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig = Some m"
+	by (cases m)
+	   (auto simp add: declared_in_def cdeclaredmethd_def
+	            intro: table_of_mapconst_SomeI)
+      with clsC neq_C_Obj ws 
+      show ?thesis
+	by (simp add: methd_rec)
+    next
+      case (Inherited Ca S membr)
+      with clsC
+      have eq_Ca_C: "Ca=C" and
+            undecl: "G\<turnstile>mid sig undeclared_in C" and
+             super: "G \<turnstile>Methd sig m member_of (super c)"
+	by (auto dest: subcls1D)
+      from eq_Ca_C clsC undecl 
+      have "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig = None"
+	by (auto simp add: undeclared_in_def cdeclaredmethd_def
+	            intro: table_of_mapconst_NoneI)
+      moreover
+      from Inherited have "G \<turnstile> C inherits (method sig m)"
+	by (auto simp add: inherits_def)
+      moreover
+      note clsC neq_C_Obj ws super hyp 
+      ultimately
+      show ?thesis
+	by (auto simp add: methd_rec intro: filter_tab_SomeI)
+    qed
+  qed
+qed
+
+(*unused*)
+lemma finite_methd:"ws_prog G \<Longrightarrow> finite {methd G C sig |sig C. is_class G C}"
+apply (rule finite_is_class [THEN finite_SetCompr2])
+apply (intro strip)
+apply (erule_tac ws_subcls1_induct, assumption)
+apply (subst methd_rec)
+apply (assumption)
+apply (auto intro!: finite_range_map_of finite_range_filter_tab finite_range_map_of_override)
+done
+
+lemma finite_dom_methd:
+ "\<lbrakk>ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> finite (dom (methd G C))"
+apply (erule_tac ws_subcls1_induct)
+apply assumption
+apply (subst methd_rec)
+apply (assumption)
+apply (auto intro!: finite_dom_map_of finite_dom_filter_tab)
+done
+
+
+subsection "accmethd"
+
+lemma accmethd_SomeD:
+"accmethd G S C sig = Some m
+ \<Longrightarrow> methd G C sig = Some m \<and> G\<turnstile>method sig m of C accessible_from S"
+by (auto simp add: accmethd_def dest: filter_tab_SomeD)
+
+lemma accmethd_SomeI:
+"\<lbrakk>methd G C sig = Some m; G\<turnstile>method sig m of C accessible_from S\<rbrakk> 
+ \<Longrightarrow> accmethd G S C sig = Some m"
+by (auto simp add: accmethd_def intro: filter_tab_SomeI)
+
+lemma accmethd_declC: 
+"\<lbrakk>accmethd G S C sig = Some m; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow>
+ (\<exists>d. class G (declclass m)=Some d \<and> 
+  table_of (methods d) sig=Some (mthd m)) \<and> 
+ G\<turnstile>C \<preceq>\<^sub>C (declclass m) \<and> methd G (declclass m) sig = Some m \<and> 
+ G\<turnstile>method sig m of C accessible_from S"
+by (auto dest: accmethd_SomeD methd_declC accmethd_SomeI)
+
+
+lemma finite_dom_accmethd:
+ "\<lbrakk>ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> finite (dom (accmethd G S C))"
+by (auto simp add: accmethd_def intro: finite_dom_filter_tab finite_dom_methd)
+
+
+subsection "dynmethd"
+
+lemma dynmethd_rec:
+"\<lbrakk>class G dynC = Some c; ws_prog G\<rbrakk> \<Longrightarrow>  
+ dynmethd G statC dynC sig
+   = (if G\<turnstile>dynC \<preceq>\<^sub>C statC
+        then (case methd G statC sig of
+                None \<Rightarrow> None
+              | Some statM 
+                  \<Rightarrow> (case methd G dynC sig of
+                        None \<Rightarrow> dynmethd G statC (super c) sig
+                      | Some dynM \<Rightarrow> 
+                          (if G,sig\<turnstile> dynM overrides statM \<or> dynM = statM 
+                              then Some dynM
+                              else (dynmethd G statC (super c) sig)
+                      )))
+         else None)" 
+   (is "_ \<Longrightarrow> _ \<Longrightarrow> ?Dynmethd_def dynC sig  = ?Dynmethd_rec dynC c sig") 
+proof -
+  assume clsDynC: "class G dynC = Some c" and 
+              ws: "ws_prog G"
+  then show "?Dynmethd_def dynC sig  = ?Dynmethd_rec dynC c sig" 
+  proof (induct rule: ws_class_induct'')
+    case (Object co)
+    show "?Dynmethd_def Object sig = ?Dynmethd_rec Object co sig"
+    proof (cases "G\<turnstile>Object \<preceq>\<^sub>C statC") 
+      case False
+      then show ?thesis by (simp add: dynmethd_def)
+    next
+      case True
+      then have eq_statC_Obj: "statC = Object" ..
+      show ?thesis 
+      proof (cases "methd G statC sig")
+	case None then show ?thesis by (simp add: dynmethd_def)
+      next
+	case Some
+	with True Object ws eq_statC_Obj 
+	show ?thesis
+	  by (auto simp add: dynmethd_def class_rec
+                      intro: filter_tab_SomeI)
+      qed
+    qed
+  next  
+    case (Subcls dynC c sc)
+    show "?Dynmethd_def dynC sig = ?Dynmethd_rec dynC c sig"
+    proof (cases "G\<turnstile>dynC \<preceq>\<^sub>C statC") 
+      case False
+      then show ?thesis by (simp add: dynmethd_def)
+    next
+      case True
+      note subclseq_dynC_statC = True
+      show ?thesis
+      proof (cases "methd G statC sig")
+	case None then show ?thesis by (simp add: dynmethd_def)
+      next
+	case (Some statM)
+	note statM = Some
+	let "?filter C" = 
+              "filter_tab
+                (\<lambda>_ dynM. G,sig \<turnstile> dynM overrides statM \<or> dynM = statM)
+                (methd G C)"
+        let "?class_rec C" =
+              "(class_rec (G, C) empty
+                           (\<lambda>C c subcls_mthds. subcls_mthds ++ (?filter C)))"
+	from statM Subcls ws subclseq_dynC_statC
+	have dynmethd_dynC_def:
+             "?Dynmethd_def dynC sig =
+                ((?class_rec (super c)) 
+                 ++
+                (?filter dynC)) sig"
+         by (simp (no_asm_simp) only: dynmethd_def class_rec)
+	    auto
+       show ?thesis
+       proof (cases "dynC = statC")
+	 case True
+	 with subclseq_dynC_statC statM dynmethd_dynC_def
+	 have "?Dynmethd_def dynC sig = Some statM"
+	   by (auto intro: override_find_right filter_tab_SomeI)
+	 with subclseq_dynC_statC True Some 
+	 show ?thesis
+	   by auto
+       next
+	 case False
+	 with subclseq_dynC_statC Subcls 
+	 have subclseq_super_statC: "G\<turnstile>(super c) \<preceq>\<^sub>C statC"
+	   by (blast dest: subclseq_superD)
+	 show ?thesis
+	 proof (cases "methd G dynC sig") 
+	   case None
+	   then have "?filter dynC sig = None"
+	     by (rule filter_tab_None)
+           then have "?Dynmethd_def dynC sig=?class_rec (super c) sig"
+	     by (simp add: dynmethd_dynC_def)
+	   with  subclseq_super_statC statM None
+	   have "?Dynmethd_def dynC sig = ?Dynmethd_def (super c) sig"
+	     by (auto simp add: empty_def dynmethd_def)
+           with None subclseq_dynC_statC statM
+	   show ?thesis 
+	     by simp
+	 next
+	   case (Some dynM)
+	   note dynM = Some
+	   let ?Termination = "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM \<or>
+                               dynM = statM"
+	   show ?thesis
+	   proof (cases "?filter dynC sig")
+	     case None
+	     with dynM 
+	     have no_termination: "\<not> ?Termination"
+	       by (simp add: filter_tab_def)
+	     from None 
+	     have "?Dynmethd_def dynC sig=?class_rec (super c) sig"
+	       by (simp add: dynmethd_dynC_def)
+	     with subclseq_super_statC statM dynM no_termination
+	     show ?thesis
+	       by (auto simp add: empty_def dynmethd_def)
+	   next
+	     case Some
+	     with dynM
+	     have termination: "?Termination"
+	       by (auto)
+	     with Some dynM
+	     have "?Dynmethd_def dynC sig=Some dynM"
+	      by (auto simp add: dynmethd_dynC_def)
+	     with subclseq_super_statC statM dynM termination
+	     show ?thesis
+	       by (auto simp add: dynmethd_def)
+	   qed
+	 qed
+       qed
+     qed
+   qed
+ qed
+qed
+	       
+lemma dynmethd_C_C:"\<lbrakk>is_class G C; ws_prog G\<rbrakk> 
+\<Longrightarrow> dynmethd G C C sig = methd G C sig"          
+apply (auto simp add: dynmethd_rec)
+done
+ 
+lemma dynmethdSomeD: 
+ "\<lbrakk>dynmethd G statC dynC sig = Some dynM; is_class G dynC; ws_prog G\<rbrakk> 
+  \<Longrightarrow> G\<turnstile>dynC \<preceq>\<^sub>C statC \<and> (\<exists> statM. methd G statC sig = Some statM)"
+apply clarify
+apply rotate_tac
+by (auto simp add: dynmethd_rec)
+ 
+lemma dynmethd_Some_cases [consumes 3, case_names Static Overrides]:
+  (assumes dynM: "dynmethd G statC dynC sig = Some dynM" and
+        is_cls_dynC: "is_class G dynC" and
+                 ws: "ws_prog G" and
+         hyp_static: "methd G statC sig = Some dynM \<Longrightarrow> P" and
+       hyp_override: "\<And> statM. \<lbrakk>methd G statC sig = Some statM;dynM\<noteq>statM;
+                       G,sig\<turnstile>dynM overrides statM\<rbrakk> \<Longrightarrow> P"
+   ) "P"
+proof -
+  from is_cls_dynC obtain dc where clsDynC: "class G dynC = Some dc" by blast
+  from clsDynC ws dynM hyp_static hyp_override
+  show "P"
+  proof (induct rule: ws_class_induct)
+    case (Object co)
+    with ws  have "statC = Object" 
+      by (auto simp add: dynmethd_rec)
+    with ws Object show ?thesis by (auto simp add: dynmethd_C_C)
+  next
+    case (Subcls C c)
+    with ws show ?thesis
+      by (auto simp add: dynmethd_rec)
+  qed
+qed
+
+lemma no_override_in_Object:
+  (assumes     dynM: "dynmethd G statC dynC sig = Some dynM" and
+            is_cls_dynC: "is_class G dynC" and
+                     ws: "ws_prog G" and
+                  statM: "methd G statC sig = Some statM" and
+         neq_dynM_statM: "dynM\<noteq>statM"
+   )
+   "dynC \<noteq> Object"
+proof -
+  from is_cls_dynC obtain dc where clsDynC: "class G dynC = Some dc" by blast
+  from clsDynC ws dynM statM neq_dynM_statM 
+  show ?thesis (is "?P dynC")
+  proof (induct rule: ws_class_induct)
+    case (Object co)
+    with ws  have "statC = Object" 
+      by (auto simp add: dynmethd_rec)
+    with ws Object show "?P Object" by (auto simp add: dynmethd_C_C)
+  next
+    case (Subcls dynC c)
+    with ws show "?P dynC"
+      by (auto simp add: dynmethd_rec)
+  qed
+qed
+
+
+lemma dynmethd_Some_rec_cases [consumes 3, 
+                               case_names Static Override  Recursion]:
+(assumes     dynM: "dynmethd G statC dynC sig = Some dynM" and
+                clsDynC: "class G dynC = Some c" and
+                     ws: "ws_prog G" and
+             hyp_static: "methd G statC sig = Some dynM \<Longrightarrow> P" and
+           hyp_override: "\<And> statM. \<lbrakk>methd G statC sig = Some statM;
+                                     methd G dynC sig = Some dynM; statM\<noteq>dynM;
+                                     G,sig\<turnstile> dynM overrides statM\<rbrakk> \<Longrightarrow> P" and
+
+          hyp_recursion: "\<lbrakk>dynC\<noteq>Object;
+                           dynmethd G statC (super c) sig = Some dynM\<rbrakk> \<Longrightarrow> P" 
+  ) "P"
+proof -
+  from clsDynC have "is_class G dynC" by simp
+  note no_override_in_Object' = no_override_in_Object [OF dynM this ws]
+  from ws clsDynC dynM hyp_static hyp_override hyp_recursion
+  show ?thesis
+    by (auto simp add: dynmethd_rec dest: no_override_in_Object')
+qed
+
+lemma dynmethd_declC: 
+"\<lbrakk>dynmethd G statC dynC sig = Some m;
+  is_class G statC;ws_prog G
+ \<rbrakk> \<Longrightarrow>
+  (\<exists>d. class G (declclass m)=Some d \<and> table_of (methods d) sig=Some (mthd m)) \<and>
+  G\<turnstile>dynC \<preceq>\<^sub>C (declclass m) \<and> methd G (declclass m) sig = Some m"
+proof - 
+  assume  is_cls_statC: "is_class G statC" 
+  assume            ws: "ws_prog G"  
+  assume             m: "dynmethd G statC dynC sig = Some m"
+  from m 
+  have "G\<turnstile>dynC \<preceq>\<^sub>C statC" by (auto simp add: dynmethd_def)
+  from this is_cls_statC 
+  have is_cls_dynC: "is_class G dynC" by (rule subcls_is_class2)
+  from is_cls_dynC ws m  
+  show ?thesis (is "?P dynC") 
+  proof (induct rule: ws_class_induct')
+    case (Object co)
+    with ws have "statC=Object" by (auto simp add: dynmethd_rec)
+    with ws Object  
+    show "?P Object" 
+      by (auto simp add: dynmethd_C_C dest: methd_declC)
+  next
+    case (Subcls dynC c)
+    assume   hyp: "dynmethd G statC (super c) sig = Some m \<Longrightarrow> ?P (super c)" and
+         clsDynC: "class G dynC = Some c"  and
+              m': "dynmethd G statC dynC sig = Some m" and
+    neq_dynC_Obj: "dynC \<noteq> Object"
+    from ws this obtain statM where
+      subclseq_dynC_statC: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and 
+                     statM: "methd G statC sig = Some statM"
+      by (blast dest: dynmethdSomeD)
+    from clsDynC neq_dynC_Obj 
+    have subclseq_dynC_super: "G\<turnstile>dynC \<preceq>\<^sub>C (super c)" 
+      by (auto intro: subcls1I) 
+    from m' clsDynC ws 
+    show "?P dynC"
+    proof (cases rule: dynmethd_Some_rec_cases) 
+      case Static
+      with is_cls_statC ws subclseq_dynC_statC 
+      show ?thesis 
+	by (auto intro: rtrancl_trans dest: methd_declC)
+    next
+      case Override
+      with clsDynC ws 
+      show ?thesis 
+	by (auto dest: methd_declC)
+    next
+      case Recursion
+      with hyp subclseq_dynC_super 
+      show ?thesis 
+	by (auto intro: rtrancl_trans) 
+    qed
+  qed
+qed
+
+lemma methd_Some_dynmethd_Some:
+  (assumes    statM: "methd G statC sig = Some statM" and 
+           subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
+       is_cls_statC: "is_class G statC" and
+                 ws: "ws_prog G"
+   ) "\<exists> dynM. dynmethd G statC dynC sig = Some dynM"
+   (is "?P dynC")
+proof -
+  from subclseq is_cls_statC 
+  have is_cls_dynC: "is_class G dynC" by (rule subcls_is_class2)
+  then obtain dc where
+    clsDynC: "class G dynC = Some dc" by blast
+  from clsDynC ws subclseq 
+  show "?thesis"
+  proof (induct rule: ws_class_induct)
+    case (Object co)
+    with ws  have "statC = Object" 
+      by (auto)
+    with ws Object statM
+    show "?P Object"  
+      by (auto simp add: dynmethd_C_C)
+  next
+    case (Subcls dynC dc)
+    assume clsDynC': "class G dynC = Some dc"
+    assume neq_dynC_Obj: "dynC \<noteq> Object"
+    assume hyp: "G\<turnstile>super dc\<preceq>\<^sub>C statC \<Longrightarrow> ?P (super dc)"
+    assume subclseq': "G\<turnstile>dynC\<preceq>\<^sub>C statC"
+    then
+    show "?P dynC"
+    proof (cases rule: subclseq_cases)
+      case Eq
+      with ws statM clsDynC' 
+      show ?thesis
+	by (auto simp add: dynmethd_rec)
+    next
+      case Subcls
+      assume "G\<turnstile>dynC\<prec>\<^sub>C statC"
+      from this clsDynC' 
+      have "G\<turnstile>super dc\<preceq>\<^sub>C statC" by (rule subcls_superD)
+      with hyp ws clsDynC' subclseq' statM
+      show ?thesis
+	by (auto simp add: dynmethd_rec)
+    qed
+  qed
+qed
+
+lemma dynmethd_cases [consumes 4, case_names Static Overrides]:
+  (assumes    statM: "methd G statC sig = Some statM" and 
+           subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
+       is_cls_statC: "is_class G statC" and
+                 ws: "ws_prog G" and
+         hyp_static: "dynmethd G statC dynC sig = Some statM \<Longrightarrow> P" and
+       hyp_override: "\<And> dynM. \<lbrakk>dynmethd G statC dynC sig = Some dynM;
+                                 dynM\<noteq>statM;
+                           G,sig\<turnstile>dynM overrides statM\<rbrakk> \<Longrightarrow> P"
+   ) "P"
+proof -
+  from subclseq is_cls_statC 
+  have is_cls_dynC: "is_class G dynC" by (rule subcls_is_class2)
+  then obtain dc where
+    clsDynC: "class G dynC = Some dc" by blast
+  from statM subclseq is_cls_statC ws 
+  obtain dynM
+    where dynM: "dynmethd G statC dynC sig = Some dynM"
+    by (blast dest: methd_Some_dynmethd_Some)
+  from dynM is_cls_dynC ws 
+  show ?thesis
+  proof (cases rule: dynmethd_Some_cases)
+    case Static
+    with hyp_static dynM statM show ?thesis by simp
+  next
+    case Overrides
+    with hyp_override dynM statM show ?thesis by simp
+  qed
+qed
+
+lemma ws_dynmethd:
+  (assumes statM: "methd G statC sig = Some statM" and
+        subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
+    is_cls_statC: "is_class G statC" and
+              ws: "ws_prog G"
+   )
+   "\<exists> dynM. dynmethd G statC dynC sig = Some dynM \<and>
+            is_static dynM = is_static statM \<and> G\<turnstile>resTy dynM\<preceq>resTy statM"
+proof - 
+  from statM subclseq is_cls_statC ws
+  show ?thesis
+  proof (cases rule: dynmethd_cases)
+    case Static
+    with statM 
+    show ?thesis
+      by simp
+  next
+    case Overrides
+    with ws
+    show ?thesis
+      by (auto dest: ws_overrides_commonD)
+  qed
+qed
+
+(*
+lemma dom_dynmethd: 
+  "dom (dynmethd G statC dynC) \<subseteq> dom (methd G statC) \<union> dom (methd G dynC)"
+by (auto simp add: dynmethd_def dom_def)
+
+lemma finite_dom_dynmethd:
+ "\<lbrakk>ws_prog G; is_class G statC; is_class G dynC\<rbrakk> 
+  \<Longrightarrow> finite (dom (dynmethd G statC dynC))"
+apply (rule_tac B="dom (methd G statC) \<union> dom (methd G dynC)" in finite_subset)
+apply (rule dom_dynmethd)
+apply (rule finite_UnI)
+apply (drule (2) finite_dom_methd)+
+done
+*)
+(*
+lemma dynmethd_SomeD: 
+"\<lbrakk>ws_prog G; is_class G statC; is_class G dynC;
+ methd G statC sig = Some sm; dynmethd G statC dynC sig = Some dm; sm \<noteq> dm
+ \<rbrakk> \<Longrightarrow> G\<turnstile>dynC \<prec>\<^sub>C statC \<and> 
+       (declclass dm \<noteq> dynC \<longrightarrow> G \<turnstile> dm accessible_through_inheritance_in dynC)"
+by (auto simp add: dynmethd_def 
+         dest: methd_inheritedD methd_diff_cls
+         intro: rtrancl_into_trancl3)
+*)
+
+subsection "dynlookup"
+
+lemma dynlookup_cases [consumes 1, case_names NullT IfaceT ClassT ArrayT]:
+"\<lbrakk>dynlookup G statT dynC sig = x;
+           \<lbrakk>statT = NullT       ; empty sig = x                  \<rbrakk> \<Longrightarrow> P;
+  \<And> I.    \<lbrakk>statT = IfaceT I    ; dynimethd G I      dynC sig = x\<rbrakk> \<Longrightarrow> P;
+  \<And> statC.\<lbrakk>statT = ClassT statC; dynmethd  G statC  dynC sig = x\<rbrakk> \<Longrightarrow> P;
+  \<And> ty.   \<lbrakk>statT = ArrayT ty   ; dynmethd  G Object dynC sig = x\<rbrakk> \<Longrightarrow> P
+ \<rbrakk> \<Longrightarrow> P"
+by (cases statT) (auto simp add: dynlookup_def)
+
+subsection "fields"
+
+lemma fields_rec: "\<lbrakk>class G C = Some c; ws_prog G\<rbrakk> \<Longrightarrow>  
+  fields G C = map (\<lambda>(fn,ft). ((fn,C),ft)) (cfields c) @  
+  (if C = Object then [] else fields G (super c))"
+apply (simp only: fields_def)
+apply (erule class_rec [THEN trans])
+apply assumption
+apply clarsimp
+done
+
+(* local lemma *)
+lemma fields_norec: 
+"\<lbrakk>class G fd = Some c; ws_prog G;  table_of (cfields c) fn = Some f\<rbrakk> 
+ \<Longrightarrow> table_of (fields G fd) (fn,fd) = Some f"
+apply (subst fields_rec)
+apply assumption+
+apply (subst map_of_override [symmetric])
+apply (rule disjI1 [THEN override_Some_iff [THEN iffD2]])
+apply (auto elim: table_of_map2_SomeI)
+done
+
+(* local lemma *)
+lemma table_of_fieldsD:
+"table_of (map (\<lambda>(fn,ft). ((fn,C),ft)) (cfields c)) efn = Some f
+\<Longrightarrow> (declclassf efn) = C \<and> table_of (cfields c) (fname efn) = Some f"
+apply (case_tac "efn")
+by auto
+
+lemma fields_declC: 
+ "\<lbrakk>table_of (fields G C) efn = Some f; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow>  
+  (\<exists>d. class G (declclassf efn) = Some d \<and> 
+                    table_of (cfields d) (fname efn)=Some f) \<and> 
+  G\<turnstile>C \<preceq>\<^sub>C (declclassf efn)  \<and> table_of (fields G (declclassf efn)) efn = Some f"
+apply (erule make_imp)
+apply (rule ws_subcls1_induct, assumption, assumption)
+apply (subst fields_rec, assumption)
+apply clarify
+apply (simp only: map_of_override [symmetric] del: map_of_override)
+apply (case_tac "table_of (map (split (\<lambda>fn. Pair (fn, Ca))) (cfields c)) efn") 
+apply   (force intro:rtrancl_into_rtrancl2 simp add: override_def)
+
+apply   (frule_tac fd="Ca" in fields_norec)
+apply     assumption
+apply     blast
+apply   (frule table_of_fieldsD)  
+apply   (frule_tac n="table_of (map (split (\<lambda>fn. Pair (fn, Ca))) (cfields c))"
+              and  m="table_of (if Ca = Object then [] else fields G (super c))"
+         in override_find_right)
+apply   (case_tac "efn")
+apply   (simp)
+done
+
+lemma fields_emptyI: "\<And>y. \<lbrakk>ws_prog G; class G C = Some c;cfields c = [];  
+  C \<noteq> Object \<longrightarrow> class G (super c) = Some y \<and> fields G (super c) = []\<rbrakk> \<Longrightarrow>  
+  fields G C = []"
+apply (subst fields_rec)
+apply assumption
+apply auto
+done
+
+(* easier than with table_of *)
+lemma fields_mono_lemma: 
+"\<lbrakk>x \<in> set (fields G C); G\<turnstile>D \<preceq>\<^sub>C C; ws_prog G\<rbrakk> 
+ \<Longrightarrow> x \<in> set (fields G D)"
+apply (erule make_imp)
+apply (erule converse_rtrancl_induct)
+apply  fast
+apply (drule subcls1D)
+apply clarsimp
+apply (subst fields_rec)
+apply   auto
+done
+
+
+lemma ws_unique_fields_lemma: 
+ "\<lbrakk>(efn,fd)  \<in> set (fields G (super c)); fc \<in> set (cfields c); ws_prog G;  
+   fname efn = fname fc; declclassf efn = C;
+   class G C = Some c; C \<noteq> Object; class G (super c) = Some d\<rbrakk> \<Longrightarrow> R"
+apply (frule_tac ws_prog_cdeclD [THEN conjunct2], assumption, assumption)
+apply (drule_tac weak_map_of_SomeI)
+apply (frule_tac subcls1I [THEN subcls1_irrefl], assumption, assumption)
+apply (auto dest: fields_declC [THEN conjunct2 [THEN conjunct1[THEN rtranclD]]])
+done
+
+lemma ws_unique_fields: "\<lbrakk>is_class G C; ws_prog G; 
+       \<And>C c. \<lbrakk>class G C = Some c\<rbrakk> \<Longrightarrow> unique (cfields c) \<rbrakk> \<Longrightarrow>
+      unique (fields G C)" 
+apply (rule ws_subcls1_induct, assumption, assumption)
+apply (subst fields_rec, assumption)            
+apply (auto intro!: unique_map_inj injI 
+            elim!: unique_append ws_unique_fields_lemma fields_norec
+            )
+done
+
+
+subsection "accfield"
+
+lemma accfield_fields: 
+ "accfield G S C fn = Some f 
+  \<Longrightarrow> table_of (fields G C) (fn, declclass f) = Some (fld f)"
+apply (simp only: accfield_def Let_def)
+apply (rule table_of_remap_SomeD)
+apply (auto dest: filter_tab_SomeD)
+done
+
+
+lemma accfield_declC_is_class: 
+ "\<lbrakk>is_class G C; accfield G S C en = Some (fd, f); ws_prog G\<rbrakk> \<Longrightarrow>  
+   is_class G fd"
+apply (drule accfield_fields)
+apply (drule fields_declC [THEN conjunct1], assumption)
+apply auto
+done
+
+lemma accfield_accessibleD: 
+  "accfield G S C fn = Some f \<Longrightarrow> G\<turnstile>Field fn f of C accessible_from S"
+by (auto simp add: accfield_def Let_def)
+
+subsection "is methd"
+
+lemma is_methdI: 
+"\<lbrakk>class G C = Some y; methd G C sig = Some b\<rbrakk> \<Longrightarrow> is_methd G C sig"
+apply (unfold is_methd_def)
+apply auto
+done
+
+lemma is_methdD: 
+"is_methd G C sig \<Longrightarrow> class G C \<noteq> None \<and> methd G C sig \<noteq> None"
+apply (unfold is_methd_def)
+apply auto
+done
+
+lemma finite_is_methd: 
+ "ws_prog G \<Longrightarrow> finite (Collect (split (is_methd G)))"
+apply (unfold is_methd_def)
+apply (subst SetCompr_Sigma_eq)
+apply (rule finite_is_class [THEN finite_SigmaI])
+apply (simp only: mem_Collect_eq)
+apply (fold dom_def)
+apply (erule finite_dom_methd)
+apply assumption
+done
+
+section "calculation of the superclasses of a class"
+
+constdefs 
+ superclasses:: "prog \<Rightarrow> qtname \<Rightarrow> qtname set"
+ "superclasses G C \<equiv> class_rec (G,C) {} 
+                       (\<lambda> C c superclss. (if C=Object 
+                                            then {} 
+                                            else insert (super c) superclss))"
+   
+lemma superclasses_rec: "\<lbrakk>class G C = Some c; ws_prog G\<rbrakk> \<Longrightarrow>  
+ superclasses G C 
+ = (if (C=Object) 
+       then {}
+       else insert (super c) (superclasses G (super c)))"
+apply (unfold superclasses_def)
+apply (erule class_rec [THEN trans], assumption)
+apply (simp)
+done
+
+lemma superclasses_mono:
+"\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D;ws_prog G; class G C = Some c;
+  \<And> C c. \<lbrakk>class G C = Some c;C\<noteq>Object\<rbrakk> \<Longrightarrow> \<exists> sc. class G (super c) = Some sc;
+  x\<in>superclasses G D 
+\<rbrakk> \<Longrightarrow> x\<in>superclasses G C" 
+proof -
+  
+  assume     ws: "ws_prog G"          and 
+          cls_C: "class G C = Some c" and
+             wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
+                  \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
+  assume clsrel: "G\<turnstile>C\<prec>\<^sub>C D"           
+  thus "\<And> c. \<lbrakk>class G C = Some c; x\<in>superclasses G D\<rbrakk>\<Longrightarrow>
+        x\<in>superclasses G C" (is "PROP ?P C"  
+                             is "\<And> c. ?CLS C c \<Longrightarrow> ?SUP D \<Longrightarrow> ?SUP C")
+  proof (induct ?P C  rule: converse_trancl_induct)
+    fix C c
+    assume "G\<turnstile>C\<prec>\<^sub>C\<^sub>1D" "class G C = Some c" "x \<in> superclasses G D"
+    with wf ws show "?SUP C"
+      by (auto    intro: no_subcls1_Object 
+               simp add: superclasses_rec subcls1_def)
+  next
+    fix C S c
+    assume clsrel': "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S" "G\<turnstile>S \<prec>\<^sub>C D"
+       and    hyp : "\<And> s. \<lbrakk>class G S = Some s; x \<in> superclasses G D\<rbrakk>
+                           \<Longrightarrow> x \<in> superclasses G S"
+       and  cls_C': "class G C = Some c"
+       and       x: "x \<in> superclasses G D"
+    moreover note wf ws
+    moreover from calculation 
+    have "?SUP S" 
+      by (force intro: no_subcls1_Object simp add: subcls1_def)
+    moreover from calculation 
+    have "super c = S" 
+      by (auto intro: no_subcls1_Object simp add: subcls1_def)
+    ultimately show "?SUP C" 
+      by (auto intro: no_subcls1_Object simp add: superclasses_rec)
+  qed
+qed
+
+lemma subclsEval:
+"\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D;ws_prog G; class G C = Some c;
+  \<And> C c. \<lbrakk>class G C = Some c;C\<noteq>Object\<rbrakk> \<Longrightarrow> \<exists> sc. class G (super c) = Some sc 
+ \<rbrakk> \<Longrightarrow> D\<in>superclasses G C" 
+proof -
+  note converse_trancl_induct 
+       = converse_trancl_induct [consumes 1,case_names Single Step]
+  assume 
+             ws: "ws_prog G"          and 
+          cls_C: "class G C = Some c" and
+             wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
+                  \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
+  assume clsrel: "G\<turnstile>C\<prec>\<^sub>C D"           
+  thus "\<And> c. class G C = Some c\<Longrightarrow> D\<in>superclasses G C" 
+    (is "PROP ?P C"  is "\<And> c. ?CLS C c  \<Longrightarrow> ?SUP C")
+  proof (induct ?P C  rule: converse_trancl_induct)
+    fix C c
+    assume "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D" "class G C = Some c"
+    with ws wf show "?SUP C"
+      by (auto intro: no_subcls1_Object simp add: superclasses_rec subcls1_def)
+  next
+    fix C S c
+    assume "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S" "G\<turnstile>S\<prec>\<^sub>C D" 
+           "\<And>s. class G S = Some s \<Longrightarrow> D \<in> superclasses G S"
+           "class G C = Some c" 
+    with ws wf show "?SUP C"
+      by - (rule superclasses_mono,
+            auto dest: no_subcls1_Object simp add: subcls1_def )
+  qed
+qed
+
+end
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