(* Title: HOL/Bali/DeclConcepts.thy ID: $Id: DeclConcepts.thy,v 1.20 2007/10/22 13:24:58 wenzelm Exp $ Author: Norbert Schirmer *) header {* Advanced concepts on Java declarations like overriding, inheritance, dynamic method lookup*} theory DeclConcepts imports TypeRel begin section "access control (cf. 6.6), overriding and hiding (cf. 8.4.6.1)" constdefs is_public :: "prog => qtname => bool" "is_public G qn ≡ (case class G qn of None => (case iface G qn of None => False | Some iface => access iface = Public) | Some class => 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 => ty => pname => bool" ("_ \<turnstile> _ accessible'_in _" [61,61,61] 60) rt_accessible_in:: "prog => ref_ty => pname => 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) ∨ is_public G I)" "G\<turnstile>(ClassT C) accessible_in' pack = ((pid C = pack) ∨ 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 => pname => qtname => bool" "is_acc_class G P C ≡ is_class G C ∧ G\<turnstile>(Class C) accessible_in P" is_acc_iface :: "prog => pname => qtname => bool" "is_acc_iface G P I ≡ is_iface G I ∧ G\<turnstile>(Iface I) accessible_in P" is_acc_type :: "prog => pname => ty => bool" "is_acc_type G P T ≡ is_type G T ∧ G\<turnstile>T accessible_in P" is_acc_reftype :: "prog => pname => ref_ty => bool" "is_acc_reftype G P T ≡ isrtype G T ∧ G\<turnstile>T accessible_in' P" lemma is_acc_classD: "is_acc_class G P C ==> is_class G C ∧ 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 ==> is_class G C" by (auto simp add: is_acc_class_def) lemma is_acc_ifaceD: "is_acc_iface G P I ==> is_iface G I ∧ G\<turnstile>(Iface I) accessible_in P" by (simp add: is_acc_iface_def) lemma is_acc_typeD: "is_acc_type G P T ==> is_type G T ∧ G\<turnstile>T accessible_in P" by (simp add: is_acc_type_def) lemma is_acc_reftypeD: "is_acc_reftype G P T ==> isrtype G T ∧ 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 => acc_modi" instance acc_modi::has_accmodi .. defs (overloaded) acc_modi_accmodi_def: "accmodi (a::acc_modi) ≡ a" lemma acc_modi_accmodi_simp[simp]: "accmodi (a::acc_modi) = a" by (simp add: acc_modi_accmodi_def) instance decl_ext_type:: ("type") has_accmodi .. defs (overloaded) decl_acc_modi_def: "accmodi (d::('a:: type) decl_scheme) ≡ 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 .. defs (overloaded) pair_acc_modi_def: "accmodi p ≡ (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 .. defs (overloaded) memberdecl_acc_modi_def: "accmodi m ≡ (case m of fdecl f => accmodi f | mdecl m => 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 => qtname" instance qtname_ext_type::("type") has_declclass .. defs (overloaded) qtname_declclass_def: "declclass (q::qtname) ≡ q" lemma qtname_declclass_simp[simp]: "declclass (q::qtname) = q" by (simp add: qtname_declclass_def) instance * :: ("has_declclass","type") has_declclass .. defs (overloaded) pair_declclass_def: "declclass p ≡ 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 => bool" instance decl_ext_type :: ("has_static") has_static .. defs (overloaded) decl_is_static_def: "is_static (m::('a::has_static) decl_scheme) ≡ is_static (Decl.decl.more m)" instance member_ext_type :: ("type") has_static .. defs (overloaded) static_field_type_is_static_def: "is_static (m::('b::type) member_ext_type) ≡ static_sel 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 .. defs (overloaded) pair_is_static_def: "is_static p ≡ 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 .. defs (overloaded) memberdecl_is_static_def: "is_static m ≡ (case m of fdecl f => is_static f | mdecl m => 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 various pairs *} "decliface":: "(qtname × ('a::type) decl_scheme) => qtname" "decliface ≡ fst" --{* get the interface component *} "mbr":: "(qtname × memberdecl) => memberdecl" "mbr ≡ snd" --{* get the memberdecl component *} "mthd":: "('b × 'a) => 'a" --{* also used for mdecl, mhead *} "mthd ≡ snd" --{* get the method component *} "fld":: "('b × ('a::type) decl_scheme) => ('a::type) decl_scheme" --{* also used for @{text "((vname × qtname)× field)"} *} "fld ≡ snd" --{* get the field component *} constdefs --{* some mnemotic selectors for @{text "(vname × qtname)"} *} fname:: "(vname × 'a) => vname" --{* also used for fdecl *} "fname ≡ fst" declclassf:: "(vname × qtname) => qtname" "declclassf ≡ snd" 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 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 @{text "(vname × qtname)"} *} "fldname" :: "(vname × qtname) => vname" "fldname ≡ fst" "fldclass" :: "(vname × qtname) => qtname" "fldclass ≡ 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 × mdecl) => (qtname × memberdecl)" "methdMembr m ≡ (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 => (qtname × methd) => (qtname × memberdecl)" "method sig m ≡ (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 => (qtname × field) => (qtname × memberdecl)" "fieldm n f ≡ (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 × mdecl) => sig" "msig m ≡ 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 => (qtname × methd) => (qtname × mdecl)" "qmdecl sig m ≡ (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 => ty" instance decl_ext_type :: ("has_resTy") has_resTy .. defs (overloaded) decl_resTy_def: "resTy (m::('a::has_resTy) decl_scheme) ≡ resTy (Decl.decl.more m)" instance member_ext_type :: ("has_resTy") has_resTy .. defs (overloaded) member_ext_type_resTy_def: "resTy (m::('b::has_resTy) member_ext_type) ≡ resTy (member.more_sel m)" instance mhead_ext_type :: ("type") has_resTy .. defs (overloaded) mhead_ext_type_resTy_def: "resTy (m::('b mhead_ext_type)) ≡ resT_sel m" lemma mhead_resTy_simp: "resTy (m::'a mhead_scheme) = resT m" apply (cases m) apply (simp add: decl_resTy_def member_ext_type_resTy_def mhead_ext_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 .. defs (overloaded) pair_resTy_def: "resTy p ≡ 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 => (qtname × memberdecl) => pname => bool" ("_ \<turnstile> _ inheritable'_in _" [61,61,61] 60) "G\<turnstile>membr inheritable_in pack ≡ (case (accmodi membr) of Private => False | Package => (pid (declclass membr)) = pack | Protected => True | Public => True)" syntax Method_inheritable_in:: "prog => (qtname × mdecl) => pname => 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 => sig => (qtname × methd) => pname => 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 => qtname => (sig,methd) table" "cdeclaredmethd G C ≡ (case class G C of None => λ sig. None | Some c => table_of (methods c) )" constdefs cdeclaredfield:: "prog => qtname => (vname,field) table" "cdeclaredfield G C ≡ (case class G C of None => λ sig. None | Some c => table_of (cfields c) )" constdefs declared_in:: "prog => memberdecl => qtname => bool" ("_\<turnstile> _ declared'_in _" [61,61,61] 60) "G\<turnstile>m declared_in C ≡ (case m of fdecl (fn,f ) => cdeclaredfield G C fn = Some f | mdecl (sig,m) => cdeclaredmethd G C sig = Some m)" syntax method_declared_in:: "prog => (qtname × mdecl) => qtname => 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 => sig =>(qtname × methd) => qtname => 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 ==> is_class G C" by (cases m) (auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def) constdefs undeclared_in:: "prog => memberid => qtname => bool" ("_\<turnstile> _ undeclared'_in _" [61,61,61] 60) "G\<turnstile>m undeclared_in C ≡ (case m of fid fn => cdeclaredfield G C fn = None | mid sig => cdeclaredmethd G C sig = None)" subsubsection "members" (* Can't just take a function: prog => qtname => memberdecl set because the class qtname changes to the superclass in the inductive definition below *) inductive members :: "prog => (qtname × memberdecl) => qtname => bool" ("_ \<turnstile> _ member'_of _" [61,61,61] 60) for G :: prog where Immediate: "[|G\<turnstile>mbr m declared_in C;declclass m = C|] ==> G\<turnstile>m member_of C" | Inherited: "[|G\<turnstile>m inheritable_in (pid C); G\<turnstile>memberid m undeclared_in C; G\<turnstile>C \<prec>C1 S; G\<turnstile>(Class S) accessible_in (pid C);G\<turnstile>m member_of S |] ==> 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 => (qtname × mdecl) => qtname => 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 => sig => (qtname × methd) => qtname => 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 => vname => (qtname × field) => qtname => 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 => qtname => (qtname × memberdecl) => bool" ("_ \<turnstile> _ inherits _" [61,61,61] 60) "G\<turnstile>C inherits m ≡ G\<turnstile>m inheritable_in (pid C) ∧ G\<turnstile>memberid m undeclared_in C ∧ (∃ S. G\<turnstile>C \<prec>C1 S ∧ G\<turnstile>(Class S) accessible_in (pid C) ∧ G\<turnstile>m member_of S)" lemma inherits_member: "G\<turnstile>C inherits m ==> G\<turnstile>m member_of C" by (auto simp add: inherits_def intro: members.Inherited) constdefs member_in::"prog => (qtname × memberdecl) => qtname => bool" ("_ \<turnstile> _ member'_in _" [61,61,61] 60) "G\<turnstile>m member_in C ≡ ∃ provC. G\<turnstile> C \<preceq>C provC ∧ 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 => (qtname × mdecl) => qtname => 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 => sig => (qtname × methd) => qtname => 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" lemma member_inD: "G\<turnstile>m member_in C ==> ∃ provC. G\<turnstile> C \<preceq>C provC ∧ G \<turnstile> m member_of provC" by (auto simp add: member_in_def) lemma member_inI: "[|G \<turnstile> m member_of provC;G\<turnstile> C \<preceq>C provC|] ==> 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 ==> 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) *} inductive stat_overridesR :: "prog => (qtname × mdecl) => (qtname × mdecl) => bool" ("_ \<turnstile> _ overridesS _" [61,61,61] 60) for G :: prog where Direct: "[|¬ is_static new; msig new = msig 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>C1 superNew; G \<turnstile>Method old member_of superNew |] ==> G\<turnstile>new overridesS old" | Indirect: "[|G\<turnstile>new overridesS inter; G\<turnstile>inter overridesS old|] ==> G\<turnstile>new overridesS old" text {* Dynamic overriding (used during the typecheck of the compiler) *} inductive overridesR :: "prog => (qtname × mdecl) => (qtname × mdecl) => bool" ("_ \<turnstile> _ overrides _" [61,61,61] 60) for G :: prog where Direct: "[|¬ is_static new; ¬ is_static old; accmodi new ≠ Private; msig new = msig old; G\<turnstile>(declclass new) \<prec>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 |] ==> G\<turnstile>new overrides old" | Indirect: "[|G\<turnstile>new overrides inter; G\<turnstile>inter overrides old|] ==> G\<turnstile>new overrides old" syntax sig_stat_overrides:: "prog => sig => (qtname × methd) => (qtname × methd) => bool" ("_,_\<turnstile> _ overridesS _" [61,61,61,61] 60) translations "G,s\<turnstile>new overridesS old" \<rightharpoonup> "G\<turnstile>(qmdecl s new) overridesS (qmdecl s old)" syntax sig_overrides:: "prog => sig => (qtname × methd) => (qtname × methd) => 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 => (qtname × mdecl) => (qtname × mdecl) => bool" ("_\<turnstile> _ hides _" [61,61,61] 60) "G\<turnstile>new hides old ≡ is_static new ∧ msig new = msig old ∧ G\<turnstile>(declclass new) \<prec>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)" syntax sig_hides:: "prog => sig => (qtname × mdecl) => (qtname × mdecl) => 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: "[|is_static new; msig new = msig old; G\<turnstile>(declclass new) \<prec>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>new hides old" by (auto simp add: hides_def) lemma hidesD: "[|G\<turnstile>new hides old|] ==> declclass new ≠ Object ∧ is_static new ∧ msig new = msig old ∧ G\<turnstile>(declclass new) \<prec>C (declclass old) ∧ G\<turnstile>Method new declared_in (declclass new) ∧ G\<turnstile>Method old declared_in (declclass old)" by (auto simp add: hides_def) lemma overrides_commonD: "[|G\<turnstile>new overrides old|] ==> declclass new ≠ Object ∧ ¬ is_static new ∧ ¬ is_static old ∧ accmodi new ≠ Private ∧ msig new = msig old ∧ G\<turnstile>(declclass new) \<prec>C (declclass old) ∧ G\<turnstile>Method new declared_in (declclass new) ∧ G\<turnstile>Method old declared_in (declclass old)" by (induct set: overridesR) (auto intro: trancl_trans) lemma ws_overrides_commonD: "[|G\<turnstile>new overrides old;ws_prog G|] ==> declclass new ≠ Object ∧ ¬ is_static new ∧ ¬ is_static old ∧ accmodi new ≠ Private ∧ G\<turnstile>resTy new \<preceq> resTy old ∧ msig new = msig old ∧ G\<turnstile>(declclass new) \<prec>C (declclass old) ∧ G\<turnstile>Method new declared_in (declclass new) ∧ G\<turnstile>Method old declared_in (declclass old)" by (induct set: overridesR) (auto intro: trancl_trans ws_widen_trans) lemma overrides_eq_sigD: "[|G\<turnstile>new overrides old|] ==> msig old=msig new" by (auto dest: overrides_commonD) lemma hides_eq_sigD: "[|G\<turnstile>new hides old|] ==> msig old=msig new" by (auto simp add: hides_def) subsubsection "permits access" constdefs permits_acc:: "prog => (qtname × memberdecl) => qtname => qtname => bool" ("_ \<turnstile> _ in _ permits'_acc'_from _" [61,61,61,61] 60) "G\<turnstile>membr in class permits_acc_from accclass ≡ (case (accmodi membr) of Private => (declclass membr = accclass) | Package => (pid (declclass membr) = pid accclass) | Protected => (pid (declclass membr) = pid accclass) ∨ (G\<turnstile>accclass \<prec>C declclass membr ∧ (G\<turnstile>class \<preceq>C accclass ∨ is_static membr)) | Public => True)" text {* The subcondition of the @{term "Protected"} case: @{term "G\<turnstile>accclass \<prec>C declclass membr"} could also be relaxed to: @{term "G\<turnstile>accclass \<preceq>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} *} inductive accessible_fromR :: "prog => qtname => (qtname × memberdecl) => qtname => bool" and accessible_from :: "prog => (qtname × memberdecl) => qtname => qtname => bool" ("_ \<turnstile> _ of _ accessible'_from _" [61,61,61,61] 60) and method_accessible_from :: "prog => (qtname × mdecl) => qtname => qtname => bool" ("_ \<turnstile>Method _ of _ accessible'_from _" [61,61,61,61] 60) for G :: prog and accclass :: qtname where "G\<turnstile>membr of cls accessible_from accclass ≡ accessible_fromR G accclass membr cls" | "G\<turnstile>Method m of cls accessible_from accclass ≡ accessible_fromR G accclass (methdMembr m) cls" | Immediate: "[|G\<turnstile>membr member_of class; G\<turnstile>(Class class) accessible_in (pid accclass); G\<turnstile>membr in class permits_acc_from accclass |] ==> G\<turnstile>membr of class accessible_from accclass" | Overriding: "[|G\<turnstile>membr member_of class; G\<turnstile>(Class class) accessible_in (pid accclass); membr=(C,mdecl new); G\<turnstile>(C,new) overridesS old; G\<turnstile>class \<prec>C supr; G\<turnstile>Method old of supr accessible_from accclass |]==> G\<turnstile>membr of class accessible_from accclass" syntax methd_accessible_from:: "prog => sig => (qtname × methd) => qtname => qtname => 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 => vname => (qtname × field) => qtname => qtname => 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 dyn_accessible_fromR :: "prog => qtname => (qtname × memberdecl) => qtname => bool" and dyn_accessible_from' :: "prog => (qtname × memberdecl) => qtname => qtname => bool" ("_ \<turnstile> _ in _ dyn'_accessible'_from _" [61,61,61,61] 60) and method_dyn_accessible_from :: "prog => (qtname × mdecl) => qtname => qtname => bool" ("_ \<turnstile>Method _ in _ dyn'_accessible'_from _" [61,61,61,61] 60) for G :: prog and accclass :: qtname where "G\<turnstile>membr in C dyn_accessible_from accC ≡ dyn_accessible_fromR G accC membr C" | "G\<turnstile>Method m in C dyn_accessible_from accC ≡ dyn_accessible_fromR G accC (methdMembr m) C" | Immediate: "[|G\<turnstile>membr member_in class; G\<turnstile>membr in class permits_acc_from accclass |] ==> G\<turnstile>membr in class dyn_accessible_from accclass" | Overriding: "[|G\<turnstile>membr member_in class; membr=(C,mdecl new); G\<turnstile>(C,new) overrides old; G\<turnstile>class \<prec>C supr; G\<turnstile>Method old in supr dyn_accessible_from accclass |]==> G\<turnstile>membr in class dyn_accessible_from accclass" syntax methd_dyn_accessible_from:: "prog => sig => (qtname × methd) => qtname => qtname => 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 => vname => (qtname × field) => qtname => qtname => 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" lemma accessible_from_commonD: "G\<turnstile>m of C accessible_from S ==> G\<turnstile>m member_of C ∧ G\<turnstile>(Class C) accessible_in (pid S)" by (auto elim: accessible_fromR.induct) lemma unique_declaration: "[|G\<turnstile>m declared_in C; G\<turnstile>n declared_in C; memberid m = memberid n |] ==> m = n" apply (cases m) apply (cases n, auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def)+ done lemma declared_not_undeclared: "G\<turnstile>m declared_in C ==> ¬ G\<turnstile> memberid m undeclared_in C" by (cases m) (auto simp add: declared_in_def undeclared_in_def) lemma undeclared_not_declared: "G\<turnstile> memberid m undeclared_in C ==> ¬ G\<turnstile> m declared_in C" by (cases m) (auto simp add: declared_in_def undeclared_in_def) lemma not_undeclared_declared: "¬ G\<turnstile> membr_id undeclared_in C ==> (∃ m. G\<turnstile>m declared_in C ∧ membr_id = memberid m)" proof - assume not_undecl:"¬ 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: "[|G\<turnstile>m declared_in C; G\<turnstile>n declared_in C; memberid m = memberid n|] ==> 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" shows "n=m" proof - from n m eqid show "n=m" proof (induct) case (Immediate n C) 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 n C S) assume undecl: "G\<turnstile> memberid n undeclared_in C" assume super: "G\<turnstile>C\<prec>C1S" assume hyp: "[|G \<turnstile> m member_of S; memberid n = memberid m|] ==> 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 ==> is_class G C" proof (induct set: members) case (Immediate m C) 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 m C S) assume "G\<turnstile>C\<prec>C1S" 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 ==> 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 ==> 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 ==> G\<turnstile>C \<preceq>C declclass m" proof (induct set: members) case (Immediate m C) then show "G\<turnstile>C \<preceq>C declclass m" by simp next case (Inherited m C S) then show "G\<turnstile>C \<preceq>C declclass m" by (auto dest: r_into_rtrancl intro: rtrancl_trans) qed lemma member_in_class_relation: "G\<turnstile>m member_in C ==> G\<turnstile>C \<preceq>C declclass m" by (auto dest: member_inD member_of_class_relation intro: rtrancl_trans) lemma stat_override_declclasses_relation: "[|G\<turnstile>(declclass new) \<prec>C1 superNew; G \<turnstile>Method old member_of superNew |] ==> G\<turnstile>(declclass new) \<prec>C (declclass old)" apply (rule trancl_rtrancl_trancl) apply (erule r_into_trancl) apply (cases old) apply (auto dest: member_of_class_relation) done lemma stat_overrides_commonD: "[|G\<turnstile>new overridesS old|] ==> declclass new ≠ Object ∧ ¬ is_static new ∧ msig new = msig old ∧ G\<turnstile>(declclass new) \<prec>C (declclass old) ∧ G\<turnstile>Method new declared_in (declclass new) ∧ G\<turnstile>Method old declared_in (declclass old)" apply (induct set: stat_overridesR) apply (frule (1) stat_override_declclasses_relation) apply (auto intro: trancl_trans) done lemma member_of_Package: "[|G\<turnstile>m member_of C; accmodi m = Package|] ==> pid (declclass m) = pid C" proof - assume member: "G\<turnstile>m member_of C" then show " accmodi m = Package ==> ?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 member_in_declC: "G\<turnstile>m member_in C==> G\<turnstile>m member_in (declclass m)" proof - assume member_in_C: "G\<turnstile>m member_in C" from member_in_C obtain provC where subclseq_C_provC: "G\<turnstile> C \<preceq>C provC" and member_of_provC: "G \<turnstile> m member_of provC" by (auto simp add: member_in_def) from member_of_provC have "G \<turnstile> m member_of declclass m" by (rule member_of_member_of_declC) moreover from member_in_C have "G\<turnstile>C \<preceq>C declclass m" by (rule member_in_class_relation) ultimately show ?thesis by (auto simp add: member_in_def) qed lemma dyn_accessible_from_commonD: "G\<turnstile>m in C dyn_accessible_from S ==> G\<turnstile>m member_in C" by (auto elim: dyn_accessible_fromR.induct) lemma no_Private_stat_override: "[|G\<turnstile>new overridesS old|] ==> accmodi old ≠ Private" by (induct set: stat_overridesR) (auto simp add: inheritable_in_def) lemma no_Private_override: "[|G\<turnstile>new overrides old|] ==> accmodi old ≠ Private" by (induct set: overridesR) (auto simp add: inheritable_in_def) lemma permits_acc_inheritance: "[|G\<turnstile>m in statC permits_acc_from accC; G\<turnstile>dynC \<preceq>C statC |] ==> G\<turnstile>m in dynC permits_acc_from accC" by (cases "accmodi m") (auto simp add: permits_acc_def intro: subclseq_trans) lemma permits_acc_static_declC: "[|G\<turnstile>m in C permits_acc_from accC; G\<turnstile>m member_in C; is_static m |] ==> G\<turnstile>m in (declclass m) permits_acc_from accC" by (cases "accmodi m") (auto simp add: permits_acc_def) lemma dyn_accessible_from_static_declC: assumes acc_C: "G\<turnstile>m in C dyn_accessible_from accC" and static: "is_static m" shows "G\<turnstile>m in (declclass m) dyn_accessible_from accC" proof - from acc_C static show "G\<turnstile>m in (declclass m) dyn_accessible_from accC" proof (induct) case (Immediate m C) then show ?case by (auto intro!: dyn_accessible_fromR.Immediate dest: member_in_declC permits_acc_static_declC) next case (Overriding m C declCNew new old sup) then have "¬ is_static m" by (auto dest: overrides_commonD) moreover assume "is_static m" ultimately show ?case by contradiction qed qed lemma field_accessible_fromD: "[|G\<turnstile>membr of C accessible_from accC;is_field membr|] ==> G\<turnstile>membr member_of C ∧ G\<turnstile>(Class C) accessible_in (pid accC) ∧ G\<turnstile>membr in C permits_acc_from accC" by (cases set: accessible_fromR) (auto simp add: is_field_def split: memberdecl.splits) lemma field_accessible_from_permits_acc_inheritance: "[|G\<turnstile>membr of statC accessible_from accC; is_field membr; G \<turnstile> dynC \<preceq>C statC|] ==> G\<turnstile>membr in dynC permits_acc_from accC" by (auto dest: field_accessible_fromD intro: permits_acc_inheritance) (* lemma accessible_Package: "[|G \<turnstile> m of C accessible_from S;accmodi m = Package; !! new old. G\<turnstile>new overridesS old ==> accmodi old ≤ accmodi new|] ==> pid S = pid C ∧ pid C = pid (declclass m)" proof - assume wf: "!! new old. G\<turnstile>new overridesS old ==> accmodi old ≤ accmodi new" assume "G \<turnstile> m of C accessible_from S" then show "accmodi m = Package ==> pid S = pid C ∧ pid C = pid (declclass m)" (is "?Pack m ==> ?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_from 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) overridesS old" and subcls_C_Sup: "G\<turnstile>C \<prec>C Sup" and acc_old: "G \<turnstile> methdMembr old of Sup accessible_from S" and hyp: "?Pack (methdMembr old) ==> ?P Sup (methdMembr old)" and accmodi_new: "accmodi new = Package" from override wf have accmodi_weaken: "accmodi old ≤ accmodi newm" by (cases old,cases newm) auto from override new have "accmodi old ≠ 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: "[|G\<turnstile>membr of C accessible_from accC; is_field membr|] ==> G\<turnstile>membr member_of C ∧ G\<turnstile>(Class C) accessible_in (pid accC) ∧ G\<turnstile>membr in C permits_acc_from accC" by (induct rule: accessible_fromR.induct) (auto dest: is_fieldD) lemma member_of_Private: "[|G\<turnstile>m member_of C; accmodi m = Private|] ==> declclass m = C" by (induct set: members) (auto simp add: inheritable_in_def) lemma member_of_subclseq_declC: "G\<turnstile>m member_of C ==> G\<turnstile>C \<preceq>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>C C" and subclseq_C_m: "G\<turnstile>C \<preceq>C declclass m" and ws: "ws_prog G" shows "G\<turnstile>m member_of C" proof - from m subclseq_D_C subclseq_C_m show ?thesis proof (induct) case (Immediate m D) assume "declclass m = D" and "G\<turnstile>D\<preceq>C C" and "G\<turnstile>C\<preceq>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 m D S) 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>C1S" assume hyp: "[|G\<turnstile>S\<preceq>C C; G\<turnstile>C\<preceq>C declclass m|] ==> G \<turnstile> m member_of C" assume subclseq_C_m: "G\<turnstile>C\<preceq>C declclass m" assume "G\<turnstile>D\<preceq>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>C C" with super have "G\<turnstile>S\<preceq>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>C C" and subcls_new_old: "G\<turnstile>declclass new \<prec>C declclass old" and ws: "ws_prog G" shows "G\<turnstile>D \<prec>C C" proof - from old have subclseq_C_old: "G\<turnstile>C \<preceq>C declclass old" by (auto dest: member_of_subclseq_declC) from new have subclseq_D_new: "G\<turnstile>D \<preceq>C declclass new" by (auto dest: member_of_subclseq_declC) from subcls_new_old ws have neq_new_old: "new≠old" by (cases new,cases old) (auto dest: subcls_irrefl) from subclseq_D_new subclseq_D_C have "G\<turnstile>(declclass new) \<preceq>C C ∨ G\<turnstile>C \<preceq>C (declclass new)" by (rule subcls_compareable) then have "G\<turnstile>(declclass new) \<preceq>C C" proof assume "G\<turnstile>declclass new\<preceq>C C" then show ?thesis . next assume "G\<turnstile>C \<preceq>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>C C" with subclseq_D_new show "G\<turnstile>D\<prec>C C" by (rule rtrancl_trancl_trancl) qed qed corollary member_of_overrides_subcls: "[|G\<turnstile>Methd sig old member_of C; G\<turnstile>Methd sig new member_of D;G\<turnstile>D \<preceq>C C; G,sig\<turnstile>new overrides old; ws_prog G|] ==> G\<turnstile>D \<prec>C C" by (drule overrides_commonD) (auto intro: member_of_subcls) corollary member_of_stat_overrides_subcls: "[|G\<turnstile>Methd sig old member_of C; G\<turnstile>Methd sig new member_of D;G\<turnstile>D \<preceq>C C; G,sig\<turnstile>new overridesS old; ws_prog G|] ==> G\<turnstile>D \<prec>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>C statC" shows "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>C statC" and member_dynC: "G\<turnstile>m member_of dynC" and dynC_acc: "G\<turnstile>(Class dynC) accessible_in (pid accC)" shows "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 × qtname" translations "fspec" <= (type) "vname × qtname" constdefs imethds:: "prog => qtname => (sig,qtname × mhead) tables" "imethds G I ≡ iface_rec (G,I) (λI i ts. (Un_tables ts) ⊕⊕ (o2s o table_of (map (λ(s,m). (s,I,m)) (imethods i))))" text {* methods of an interface, with overriding and inheritance, cf. 9.2 *} constdefs accimethds:: "prog => pname => qtname => (sig,qtname × mhead) tables" "accimethds G pack I ≡ if G\<turnstile>Iface I accessible_in pack then imethds G I else λ k. {}" text {* only returns imethds if the interface is accessible *} constdefs methd:: "prog => qtname => (sig,qtname × methd) table" "methd G C ≡ class_rec (G,C) empty (λC c subcls_mthds. filter_tab (λsig m. G\<turnstile>C inherits method sig m) subcls_mthds ++ table_of (map (λ(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 => qtname => qtname => (sig,qtname × methd) table" "accmethd G S C ≡ filter_tab (λsig 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 => qtname => qtname => (sig,qtname × methd) table" "dynmethd G statC dynC ≡ λ sig. (if G\<turnstile>dynC \<preceq>C statC then (case methd G statC sig of None => None | Some statM => (class_rec (G,dynC) empty (λC c subcls_mthds. subcls_mthds ++ (filter_tab (λ _ dynM. G,sig\<turnstile>dynM overrides statM ∨ dynM=statM) (methd G 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 => qtname => qtname => (sig,qtname × methd) table" "dynimethd G I dynC ≡ λ sig. if imethds G I sig ≠ {} 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 => ref_ty => qtname => (sig,qtname × methd) table" "dynlookup G statT dynC ≡ (case statT of NullT => empty | IfaceT I => dynimethd G I dynC | ClassT statC => dynmethd G statC dynC | ArrayT ty => 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 => qtname => ((vname × qtname) × field) list" "fields G C ≡ class_rec (G,C) [] (λC c ts. map (λ(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 => qtname => qtname => (vname, qtname × field) table" "accfield G S C ≡ let field_tab = table_of((map (λ((n,d),f).(n,(d,f)))) (fields G C)) in filter_tab (λn (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 => qtname => sig => bool" "is_methd G ≡ λC sig. is_class G C ∧ methd G C sig ≠ None" constdefs efname:: "((vname × qtname) × field) => (vname × qtname)" "efname ≡ fst" lemma efname_simp[simp]:"efname (n,f) = n" by (simp add: efname_def) subsection "imethds" lemma imethds_rec: "[|iface G I = Some i; ws_prog G|] ==> imethds G I = Un_tables ((λJ. imethds G J)`set (isuperIfs i)) ⊕⊕ (o2s o table_of (map (λ(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: "[|iface G md = Some i; ws_prog G; table_of (imethods i) sig = Some mh|] ==> (md, mh) ∈ 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: "[|m ∈ imethds G I sig; ws_prog G; is_iface G I|] ==> (∃i. iface G (decliface m) = Some i ∧ table_of (imethods i) sig = Some (mthd m)) ∧ (I,decliface m) ∈ (subint1 G)^* ∧ m ∈ imethds G (decliface m) sig" apply (erule rev_mp) 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 ∈ imethds G I sig" and ifI: "iface G I = Some i" and ws: "ws_prog G" and hyp_new: "table_of (map (λ(s, mh). (s, I, mh)) (imethods i)) sig = Some im ==> P" and hyp_inh: "!! J. [|J ∈ set (isuperIfs i); im ∈ imethds G J sig|] ==> P" shows 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 ==> accimethds G pack I = imethds G I" by (simp add: accimethds_def) lemma accimethdsD: "im ∈ accimethds G pack I sig ==> im ∈ imethds G I sig ∧ G\<turnstile>Iface I accessible_in pack" by (auto simp add: accimethds_def) lemma accimethdsI: "[|im ∈ imethds G I sig;G\<turnstile>Iface I accessible_in pack|] ==> im ∈ accimethds G pack I sig" by simp subsection "methd" lemma methd_rec: "[|class G C = Some c; ws_prog G|] ==> methd G C = (if C = Object then empty else filter_tab (λsig m. G\<turnstile>C inherits method sig m) (methd G (super c))) ++ table_of (map (λ(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: "[|class G declC = Some c; ws_prog G;table_of (methods c) sig = Some m|] ==> methd G declC sig = Some (declC, m)" apply (simp only: methd_rec) apply (rule disjI1 [THEN map_add_Some_iff [THEN iffD2]]) apply (auto elim: table_of_map_SomeI) done lemma methd_declC: "[|methd G C sig = Some m; ws_prog G;is_class G C|] ==> (∃d. class G (declclass m)=Some d ∧ table_of (methods d) sig=Some (mthd m)) ∧ G\<turnstile>C \<preceq>C (declclass m) ∧ methd G (declclass m) sig = Some m" apply (erule rev_mp) 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 (λ(s, m). (s, Ca, m)) (methods c)) sig") apply (force intro: rtrancl_into_rtrancl2) apply (auto intro: methd_norec) done lemma methd_inheritedD: "[|class G C = Some c; ws_prog G;methd G C sig = Some m|] ==> (declclass m ≠ C --> G \<turnstile>C inherits method sig m)" by (auto simp add: methd_rec) lemma methd_diff_cls: "[|ws_prog G; is_class G C; is_class G D; methd G C sig = m; methd G D sig = n; m≠n |] ==> C≠D" by (auto simp add: methd_rec) lemma method_declared_inI: "[|table_of (methods c) sig = Some m; class G C = Some c|] ==> G\<turnstile>mdecl (sig,m) declared_in C" by (auto simp add: cdeclaredmethd_def declared_in_def) lemma methd_declared_in_declclass: "[|methd G C sig = Some m; ws_prog G;is_class G C|] ==> 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" shows "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 ∧ 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 ≠ Object" and hyp: "G \<turnstile>Methd sig m member_of super c ==> ?Methd (super c)" and member_of: "G \<turnstile>Methd sig m member_of C" from member_of show "?Methd C" proof (cases) case (Immediate membr Ca) 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 (λ(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 membr Ca S) 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 (λ(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 ==> 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_map_add) done lemma finite_dom_methd: "[|ws_prog G; is_class G C|] ==> 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 ==> methd G C sig = Some m ∧ G\<turnstile>method sig m of C accessible_from S" by (auto simp add: accmethd_def dest: filter_tab_SomeD) lemma accmethd_SomeI: "[|methd G C sig = Some m; G\<turnstile>method sig m of C accessible_from S|] ==> accmethd G S C sig = Some m" by (auto simp add: accmethd_def intro: filter_tab_SomeI) lemma accmethd_declC: "[|accmethd G S C sig = Some m; ws_prog G; is_class G C|] ==> (∃d. class G (declclass m)=Some d ∧ table_of (methods d) sig=Some (mthd m)) ∧ G\<turnstile>C \<preceq>C (declclass m) ∧ methd G (declclass m) sig = Some m ∧ G\<turnstile>method sig m of C accessible_from S" by (auto dest: accmethd_SomeD methd_declC accmethd_SomeI) lemma finite_dom_accmethd: "[|ws_prog G; is_class G C|] ==> 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: "[|class G dynC = Some c; ws_prog G|] ==> dynmethd G statC dynC sig = (if G\<turnstile>dynC \<preceq>C statC then (case methd G statC sig of None => None | Some statM => (case methd G dynC sig of None => dynmethd G statC (super c) sig | Some dynM => (if G,sig\<turnstile> dynM overrides statM ∨ dynM = statM then Some dynM else (dynmethd G statC (super c) sig) ))) else None)" (is "_ ==> _ ==> ?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>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>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 (λ_ dynM. G,sig \<turnstile> dynM overrides statM ∨ dynM = statM) (methd G C)" let "?class_rec C" = "(class_rec (G, C) empty (λC 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: map_add_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>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 ∨ dynM = statM" show ?thesis proof (cases "?filter dynC sig") case None with dynM have no_termination: "¬ ?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:"[|is_class G C; ws_prog G|] ==> dynmethd G C C sig = methd G C sig" apply (auto simp add: dynmethd_rec) done lemma dynmethdSomeD: "[|dynmethd G statC dynC sig = Some dynM; is_class G dynC; ws_prog G|] ==> G\<turnstile>dynC \<preceq>C statC ∧ (∃ statM. methd G statC sig = Some statM)" 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 ==> P" and hyp_override: "!! statM. [|methd G statC sig = Some statM;dynM≠statM; G,sig\<turnstile>dynM overrides statM|] ==> P" shows 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≠statM" shows "dynC ≠ 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 ==> P" and hyp_override: "!! statM. [|methd G statC sig = Some statM; methd G dynC sig = Some dynM; statM≠dynM; G,sig\<turnstile> dynM overrides statM|] ==> P" and hyp_recursion: "[|dynC≠Object; dynmethd G statC (super c) sig = Some dynM|] ==> P" shows 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: "[|dynmethd G statC dynC sig = Some m; is_class G statC;ws_prog G |] ==> (∃d. class G (declclass m)=Some d ∧ table_of (methods d) sig=Some (mthd m)) ∧ G\<turnstile>dynC \<preceq>C (declclass m) ∧ 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>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 ==> ?P (super c)" and clsDynC: "class G dynC = Some c" and m': "dynmethd G statC dynC sig = Some m" and neq_dynC_Obj: "dynC ≠ Object" from ws this obtain statM where subclseq_dynC_statC: "G\<turnstile>dynC \<preceq>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>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>C statC" and is_cls_statC: "is_class G statC" and ws: "ws_prog G" shows "∃ 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 ≠ Object" assume hyp: "G\<turnstile>super dc\<preceq>C statC ==> ?P (super dc)" assume subclseq': "G\<turnstile>dynC\<preceq>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>C statC" from this clsDynC' have "G\<turnstile>super dc\<preceq>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>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 ==> P" and hyp_override: "!! dynM. [|dynmethd G statC dynC sig = Some dynM; dynM≠statM; G,sig\<turnstile>dynM overrides statM|] ==> P" shows 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>C statC" and is_cls_statC: "is_class G statC" and ws: "ws_prog G" shows "∃ dynM. dynmethd G statC dynC sig = Some dynM ∧ is_static dynM = is_static statM ∧ 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 subsection "dynlookup" lemma dynlookup_cases [consumes 1, case_names NullT IfaceT ClassT ArrayT]: "[|dynlookup G statT dynC sig = x; [|statT = NullT ; empty sig = x |] ==> P; !! I. [|statT = IfaceT I ; dynimethd G I dynC sig = x|] ==> P; !! statC.[|statT = ClassT statC; dynmethd G statC dynC sig = x|] ==> P; !! ty. [|statT = ArrayT ty ; dynmethd G Object dynC sig = x|] ==> P |] ==> P" by (cases statT) (auto simp add: dynlookup_def) subsection "fields" lemma fields_rec: "[|class G C = Some c; ws_prog G|] ==> fields G C = map (λ(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: "[|class G fd = Some c; ws_prog G; table_of (cfields c) fn = Some f|] ==> table_of (fields G fd) (fn,fd) = Some f" apply (subst fields_rec) apply assumption+ apply (subst map_of_append) apply (rule disjI1 [THEN map_add_Some_iff [THEN iffD2]]) apply (auto elim: table_of_map2_SomeI) done (* local lemma *) lemma table_of_fieldsD: "table_of (map (λ(fn,ft). ((fn,C),ft)) (cfields c)) efn = Some f ==> (declclassf efn) = C ∧ table_of (cfields c) (fname efn) = Some f" apply (case_tac "efn") by auto lemma fields_declC: "[|table_of (fields G C) efn = Some f; ws_prog G; is_class G C|] ==> (∃d. class G (declclassf efn) = Some d ∧ table_of (cfields d) (fname efn)=Some f) ∧ G\<turnstile>C \<preceq>C (declclassf efn) ∧ table_of (fields G (declclassf efn)) efn = Some f" apply (erule rev_mp) apply (rule ws_subcls1_induct, assumption, assumption) apply (subst fields_rec, assumption) apply clarify apply (simp only: map_of_append) apply (case_tac "table_of (map (split (λfn. Pair (fn, Ca))) (cfields c)) efn") apply (force intro:rtrancl_into_rtrancl2 simp add: map_add_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 (λfn. Pair (fn, Ca))) (cfields c))" and m="table_of (if Ca = Object then [] else fields G (super c))" in map_add_find_right) apply (case_tac "efn") apply (simp) done lemma fields_emptyI: "!!y. [|ws_prog G; class G C = Some c;cfields c = []; C ≠ Object --> class G (super c) = Some y ∧ fields G (super c) = []|] ==> fields G C = []" apply (subst fields_rec) apply assumption apply auto done (* easier than with table_of *) lemma fields_mono_lemma: "[|x ∈ set (fields G C); G\<turnstile>D \<preceq>C C; ws_prog G|] ==> x ∈ set (fields G D)" apply (erule rev_mp) apply (erule converse_rtrancl_induct) apply fast apply (drule subcls1D) apply clarsimp apply (subst fields_rec) apply auto done lemma ws_unique_fields_lemma: "[|(efn,fd) ∈ set (fields G (super c)); fc ∈ set (cfields c); ws_prog G; fname efn = fname fc; declclassf efn = C; class G C = Some c; C ≠ Object; class G (super c) = Some d|] ==> 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: "[|is_class G C; ws_prog G; !!C c. [|class G C = Some c|] ==> unique (cfields c) |] ==> unique (fields G C)" apply (rule ws_subcls1_induct, assumption, assumption) apply (subst fields_rec, assumption) apply (auto intro!: unique_map_inj inj_onI elim!: unique_append ws_unique_fields_lemma fields_norec) done subsection "accfield" lemma accfield_fields: "accfield G S C fn = Some f ==> 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: "[|is_class G C; accfield G S C en = Some (fd, f); ws_prog G|] ==> 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 ==> G\<turnstile>Field fn f of C accessible_from S" by (auto simp add: accfield_def Let_def) subsection "is methd" lemma is_methdI: "[|class G C = Some y; methd G C sig = Some b|] ==> is_methd G C sig" apply (unfold is_methd_def) apply auto done lemma is_methdD: "is_methd G C sig ==> class G C ≠ None ∧ methd G C sig ≠ None" apply (unfold is_methd_def) apply auto done lemma finite_is_methd: "ws_prog G ==> 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 => qtname => qtname set" "superclasses G C ≡ class_rec (G,C) {} (λ C c superclss. (if C=Object then {} else insert (super c) superclss))" lemma superclasses_rec: "[|class G C = Some c; ws_prog G|] ==> 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: "[|G\<turnstile>C \<prec>C D;ws_prog G; class G C = Some c; !! C c. [|class G C = Some c;C≠Object|] ==> ∃ sc. class G (super c) = Some sc; x∈superclasses G D |] ==> x∈superclasses G C" proof - assume ws: "ws_prog G" and cls_C: "class G C = Some c" and wf: "!!C c. [|class G C = Some c; C ≠ Object|] ==> ∃sc. class G (super c) = Some sc" assume clsrel: "G\<turnstile>C\<prec>C D" thus "!! c. [|class G C = Some c; x∈superclasses G D|]==> x∈superclasses G C" (is "PROP ?P C" is "!! c. ?CLS C c ==> ?SUP D ==> ?SUP C") proof (induct ?P C rule: converse_trancl_induct) fix C c assume "G\<turnstile>C\<prec>C1D" "class G C = Some c" "x ∈ 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>C1 S" "G\<turnstile>S \<prec>C D" and hyp : "!! s. [|class G S = Some s; x ∈ superclasses G D|] ==> x ∈ superclasses G S" and cls_C': "class G C = Some c" and x: "x ∈ 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: "[|G\<turnstile>C \<prec>C D;ws_prog G; class G C = Some c; !! C c. [|class G C = Some c;C≠Object|] ==> ∃ sc. class G (super c) = Some sc |] ==> D∈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: "!!C c. [|class G C = Some c; C ≠ Object|] ==> ∃sc. class G (super c) = Some sc" assume clsrel: "G\<turnstile>C\<prec>C D" thus "!! c. class G C = Some c==> D∈superclasses G C" (is "PROP ?P C" is "!! c. ?CLS C c ==> ?SUP C") proof (induct ?P C rule: converse_trancl_induct) fix C c assume "G\<turnstile>C \<prec>C1 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>C1 S" "G\<turnstile>S\<prec>C D" "!!s. class G S = Some s ==> D ∈ 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
lemma is_acc_classD:
is_acc_class G P C ==> is_class G C ∧ G \<turnstile> Class C accessible_in P
lemma is_acc_class_is_class:
is_acc_class G P C ==> is_class G C
lemma is_acc_ifaceD:
is_acc_iface G P I ==> is_iface G I ∧ G \<turnstile> Iface I accessible_in P
lemma is_acc_typeD:
is_acc_type G P T ==> is_type G T ∧ G \<turnstile> T accessible_in P
lemma is_acc_reftypeD:
is_acc_reftype G P T ==> isrtype G T ∧ G \<turnstile> T accessible_in' P
lemma acc_modi_accmodi_simp:
accmodi a = a
lemma decl_acc_modi_simp:
accmodi d = access d
lemma pair_acc_modi_simp:
accmodi (x, a) = accmodi a
lemma memberdecl_fdecl_acc_modi_simp:
accmodi (fdecl m) = accmodi m
lemma memberdecl_mdecl_acc_modi_simp:
accmodi (mdecl m) = accmodi m
lemma qtname_declclass_simp:
declclass q = q
lemma pair_declclass_simp:
declclass (c, x) = declclass c
lemma member_is_static_simp:
is_static m = static m
lemma pair_is_static_simp:
is_static (x, s) = is_static s
lemma pair_is_static_simp1:
is_static p = is_static (snd p)
lemma memberdecl_is_static_fdecl_simp:
is_static (fdecl f) = is_static f
lemma memberdecl_is_static_mdecl_simp:
is_static (mdecl m) = is_static m
lemma mhead_static_simp:
is_static (mhead m) = is_static m
lemma decliface_simp:
decliface (I, m) = I
lemma mbr_simp:
mbr (C, m) = m
lemma access_mbr_simp:
accmodi (mbr m) = accmodi m
lemma mthd_simp:
mthd (C, m) = m
lemma fld_simp:
fld (C, f) = f
lemma accmodi_simp:
accmodi (C, m) = access m
lemma access_mthd_simp:
access (mthd m) = accmodi m
lemma access_fld_simp:
access (fld f) = accmodi f
lemma static_mthd_simp:
static (mthd m) = is_static m
lemma mthd_is_static_simp:
is_static (mthd m) = is_static m
lemma static_fld_simp:
static (fld f) = is_static f
lemma ext_field_simp:
(declclass f, fld f) = f
lemma ext_method_simp:
(declclass m, mthd m) = m
lemma ext_mbr_simp:
(declclass m, mbr m) = m
lemma fname_simp:
fname (n, c) = n
lemma declclassf_simp:
declclassf (n, c) = c
lemma fldname_simp:
fldname (n, c) = n
lemma fldclass_simp:
fldclass (n, c) = c
lemma ext_fieldname_simp:
(fldname f, fldclass f) = f
lemma methdMembr_simp:
methdMembr (c, m) = (c, mdecl m)
lemma accmodi_methdMembr_simp:
accmodi (methdMembr m) = accmodi m
lemma is_static_methdMembr_simp:
is_static (methdMembr m) = is_static m
lemma declclass_methdMembr_simp:
declclass (methdMembr m) = declclass m
lemma method_simp:
method sig (C, m) = (C, mdecl (sig, m))
lemma accmodi_method_simp:
accmodi (method sig m) = accmodi m
lemma declclass_method_simp:
declclass (method sig m) = declclass m
lemma is_static_method_simp:
is_static (method sig m) = is_static m
lemma mbr_method_simp:
mbr (method sig m) = mdecl (sig, mthd m)
lemma memberid_method_simp:
memberid (method sig m) = mid sig
lemma fieldm_simp:
fieldm n (C, f) = (C, fdecl (n, f))
lemma accmodi_fieldm_simp:
accmodi (fieldm n f) = accmodi f
lemma declclass_fieldm_simp:
declclass (fieldm n f) = declclass f
lemma is_static_fieldm_simp:
is_static (fieldm n f) = is_static f
lemma mbr_fieldm_simp:
mbr (fieldm n f) = fdecl (n, fld f)
lemma memberid_fieldm_simp:
memberid (fieldm n f) = fid n
lemma msig_simp:
msig (c, s, m) = s
lemma qmdecl_simp:
qmdecl sig (C, m) = (C, sig, m)
lemma declclass_qmdecl_simp:
declclass (qmdecl sig m) = declclass m
lemma accmodi_qmdecl_simp:
accmodi (qmdecl sig m) = accmodi m
lemma is_static_qmdecl_simp:
is_static (qmdecl sig m) = is_static m
lemma msig_qmdecl_simp:
msig (qmdecl sig m) = sig
lemma mdecl_qmdecl_simp:
mdecl (mthd (qmdecl sig new)) = mdecl (sig, mthd new)
lemma methdMembr_qmdecl_simp:
methdMembr (qmdecl sig old) = method sig old
lemma mhead_resTy_simp:
resTy m = resT m
lemma resTy_mhead:
resTy (mhead m) = resTy m
lemma pair_resTy_simp:
resTy (x, m) = resTy m
lemma qmdecl_resTy_simp:
resTy (qmdecl sig m) = resTy m
lemma resTy_mthd:
resTy (mthd m) = resTy m
lemma declared_in_classD:
G\<turnstile> m declared_in C ==> is_class G C
lemma inherits_member:
G \<turnstile> C inherits m ==> G \<turnstile> m member_of C
lemma member_inD:
G \<turnstile> m member_in C
==> ∃provC. G|-C<=:C provC ∧ G \<turnstile> m member_of provC
lemma member_inI:
[| G \<turnstile> m member_of provC; G|-C<=:C provC |]
==> G \<turnstile> m member_in C
lemma member_of_to_member_in:
G \<turnstile> m member_of C ==> G \<turnstile> m member_in C
lemma hidesI:
[| is_static new; msig new = msig old; G|-declclass new<: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> new hides old
lemma hidesD:
G\<turnstile> new hides old
==> declclass new ≠ Object ∧
is_static new ∧
msig new = msig old ∧
G|-declclass new<:C declclass old ∧
G\<turnstile>Method new declared_in declclass new ∧
G\<turnstile>Method old declared_in declclass old
lemma overrides_commonD:
G \<turnstile> new overrides old
==> declclass new ≠ Object ∧
¬ is_static new ∧
¬ is_static old ∧
accmodi new ≠ Private ∧
msig new = msig old ∧
G|-declclass new<:C declclass old ∧
G\<turnstile>Method new declared_in declclass new ∧
G\<turnstile>Method old declared_in declclass old
lemma ws_overrides_commonD:
[| G \<turnstile> new overrides old; ws_prog G |]
==> declclass new ≠ Object ∧
¬ is_static new ∧
¬ is_static old ∧
accmodi new ≠ Private ∧
G\<turnstile>resTy new\<preceq>resTy old ∧
msig new = msig old ∧
G|-declclass new<:C declclass old ∧
G\<turnstile>Method new declared_in declclass new ∧
G\<turnstile>Method old declared_in declclass old
lemma overrides_eq_sigD:
G \<turnstile> new overrides old ==> msig old = msig new
lemma hides_eq_sigD:
G\<turnstile> new hides old ==> msig old = msig new
lemma accessible_from_commonD:
G \<turnstile> m of C accessible_from S
==> G \<turnstile> m member_of C ∧ G \<turnstile> Class C accessible_in pid S
lemma unique_declaration:
[| G\<turnstile> m declared_in C; G\<turnstile> n declared_in C;
memberid m = memberid n |]
==> m = n
lemma declared_not_undeclared:
G\<turnstile> m declared_in C ==> ¬ G\<turnstile> memberid m undeclared_in C
lemma undeclared_not_declared:
G\<turnstile> memberid m undeclared_in C ==> ¬ G\<turnstile> m declared_in C
lemma not_undeclared_declared:
¬ G\<turnstile> membr_id undeclared_in C
==> ∃m. G\<turnstile> m declared_in C ∧ membr_id = memberid m
lemma unique_declared_in:
[| G\<turnstile> m declared_in C; G\<turnstile> n declared_in C;
memberid m = memberid n |]
==> m = n
lemma unique_member_of:
[| G \<turnstile> n member_of C; G \<turnstile> m member_of C;
memberid n = memberid m |]
==> n = m
lemma member_of_is_classD:
G \<turnstile> m member_of C ==> is_class G C
lemma member_of_declC:
G \<turnstile> m member_of C ==> G\<turnstile> mbr m declared_in declclass m
lemma member_of_member_of_declC:
G \<turnstile> m member_of C ==> G \<turnstile> m member_of declclass m
lemma member_of_class_relation:
G \<turnstile> m member_of C ==> G|-C<=:C declclass m
lemma member_in_class_relation:
G \<turnstile> m member_in C ==> G|-C<=:C declclass m
lemma stat_override_declclasses_relation:
[| G|-declclass new<:C1superNew; G \<turnstile>Method old member_of superNew |]
==> G|-declclass new<:C declclass old
lemma stat_overrides_commonD:
G \<turnstile> new overridesS old
==> declclass new ≠ Object ∧
¬ is_static new ∧
msig new = msig old ∧
G|-declclass new<:C declclass old ∧
G\<turnstile>Method new declared_in declclass new ∧
G\<turnstile>Method old declared_in declclass old
lemma member_of_Package:
[| G \<turnstile> m member_of C; accmodi m = Package |]
==> pid (declclass m) = pid C
lemma member_in_declC:
G \<turnstile> m member_in C ==> G \<turnstile> m member_in declclass m
lemma dyn_accessible_from_commonD:
G \<turnstile> m in C dyn_accessible_from S ==> G \<turnstile> m member_in C
lemma no_Private_stat_override:
G \<turnstile> new overridesS old ==> accmodi old ≠ Private
lemma no_Private_override:
G \<turnstile> new overrides old ==> accmodi old ≠ Private
lemma permits_acc_inheritance:
[| G \<turnstile> m in statC permits_acc_from accC; G|-dynC<=:C statC |]
==> G \<turnstile> m in dynC permits_acc_from accC
lemma permits_acc_static_declC:
[| G \<turnstile> m in C permits_acc_from accC; G \<turnstile> m member_in C;
is_static m |]
==> G \<turnstile> m in declclass m permits_acc_from accC
lemma dyn_accessible_from_static_declC:
[| G \<turnstile> m in C dyn_accessible_from accC; is_static m |]
==> G \<turnstile> m in declclass m dyn_accessible_from accC
lemma field_accessible_fromD:
[| G \<turnstile> membr of C accessible_from accC; is_field membr |]
==> G \<turnstile> membr member_of C ∧
G \<turnstile> Class C accessible_in pid accC ∧
G \<turnstile> membr in C permits_acc_from accC
lemma field_accessible_from_permits_acc_inheritance:
[| G \<turnstile> membr of statC accessible_from accC; is_field membr;
G|-dynC<=:C statC |]
==> G \<turnstile> membr in dynC permits_acc_from accC
lemma accessible_fieldD:
[| G \<turnstile> membr of C accessible_from accC; is_field membr |]
==> G \<turnstile> membr member_of C ∧
G \<turnstile> Class C accessible_in pid accC ∧
G \<turnstile> membr in C permits_acc_from accC
lemma member_of_Private:
[| G \<turnstile> m member_of C; accmodi m = Private |] ==> declclass m = C
lemma member_of_subclseq_declC:
G \<turnstile> m member_of C ==> G|-C<=:C declclass m
lemma member_of_inheritance:
[| G \<turnstile> m member_of D; G|-D<=:C C; G|-C<=:C declclass m; ws_prog G |]
==> G \<turnstile> m member_of C
lemma member_of_subcls:
[| G \<turnstile> old member_of C; G \<turnstile> new member_of D;
memberid new = memberid old; G|-D<=:C C; G|-declclass new<:C declclass old;
ws_prog G |]
==> G|-D<:C C
corollary member_of_overrides_subcls:
[| G \<turnstile>Methd sig old member_of C;
G \<turnstile>Methd sig new member_of D; G|-D<=:C C;
G \<turnstile> qmdecl sig new overrides qmdecl sig old; ws_prog G |]
==> G|-D<:C C
corollary member_of_stat_overrides_subcls:
[| G \<turnstile>Methd sig old member_of C;
G \<turnstile>Methd sig new member_of D; G|-D<=:C C;
G \<turnstile> qmdecl sig new overridesS qmdecl sig old; ws_prog G |]
==> G|-D<:C C
lemma inherited_field_access:
[| G \<turnstile> membr of statC accessible_from accC; is_field membr;
G|-dynC<=:C statC |]
==> G \<turnstile> membr in dynC dyn_accessible_from accC
lemma accessible_inheritance:
[| G \<turnstile> m of statC accessible_from accC; G|-dynC<=:C statC;
G \<turnstile> m member_of dynC;
G \<turnstile> Class dynC accessible_in pid accC |]
==> G \<turnstile> m of dynC accessible_from accC
lemma efname_simp:
efname (n, f) = n
lemma imethds_rec:
[| iface G I = Some i; ws_prog G |]
==> DeclConcepts.imethds G I =
Un_tables (DeclConcepts.imethds G ` set (isuperIfs i)) ⊕⊕
(o2s o table_of (map (λ(s, mh). (s, I, mh)) (imethods i)))
lemma imethds_norec:
[| iface G md = Some i; ws_prog G; table_of (imethods i) sig = Some mh |]
==> (md, mh) ∈ DeclConcepts.imethds G md sig
lemma imethds_declI:
[| m ∈ DeclConcepts.imethds G I sig; ws_prog G; is_iface G I |]
==> (∃i. iface G (decliface m) = Some i ∧
table_of (imethods i) sig = Some (mthd m)) ∧
G|-I<=:I decliface m ∧ m ∈ DeclConcepts.imethds G (decliface m) sig
lemma imethds_cases:
[| im ∈ DeclConcepts.imethds G I sig; iface G I = Some i; ws_prog G;
table_of (map (λ(s, mh). (s, I, mh)) (imethods i)) sig = Some im ==> P;
!!J. [| J ∈ set (isuperIfs i); im ∈ DeclConcepts.imethds G J sig |] ==> P |]
==> P
lemma accimethds_simp:
G \<turnstile> Iface I accessible_in pack
==> accimethds G pack I = DeclConcepts.imethds G I
lemma accimethdsD:
im ∈ accimethds G pack I sig
==> im ∈ DeclConcepts.imethds G I sig ∧
G \<turnstile> Iface I accessible_in pack
lemma accimethdsI:
[| im ∈ DeclConcepts.imethds G I sig;
G \<turnstile> Iface I accessible_in pack |]
==> im ∈ accimethds G pack I sig
lemma methd_rec:
[| class G C = Some c; ws_prog G |]
==> methd G C =
(if C = Object then empty
else filter_tab (λsig m. G \<turnstile> C inherits method sig m)
(methd G (super c))) ++
table_of (map (λ(s, m). (s, C, m)) (methods c))
lemma methd_norec:
[| class G declC = Some c; ws_prog G; table_of (methods c) sig = Some m |]
==> methd G declC sig = Some (declC, m)
lemma methd_declC:
[| methd G C sig = Some m; ws_prog G; is_class G C |]
==> (∃d. class G (declclass m) = Some d ∧
table_of (methods d) sig = Some (mthd m)) ∧
G|-C<=:C declclass m ∧ methd G (declclass m) sig = Some m
lemma methd_inheritedD:
[| class G C = Some c; ws_prog G; methd G C sig = Some m |]
==> declclass m ≠ C --> G \<turnstile> C inherits method sig m
lemma methd_diff_cls:
[| ws_prog G; is_class G C; is_class G D; methd G C sig = m; methd G D sig = n;
m ≠ n |]
==> C ≠ D
lemma method_declared_inI:
[| table_of (methods c) sig = Some m; class G C = Some c |]
==> G\<turnstile> mdecl (sig, m) declared_in C
lemma methd_declared_in_declclass:
[| methd G C sig = Some m; ws_prog G; is_class G C |]
==> G\<turnstile>Methd sig m declared_in declclass m
lemma member_methd:
[| G \<turnstile>Methd sig m member_of C; ws_prog G |]
==> methd G C sig = Some m
lemma finite_methd:
ws_prog G ==> finite {methd G C sig |sig C. is_class G C}
lemma finite_dom_methd:
[| ws_prog G; is_class G C |] ==> finite (dom (methd G C))
lemma accmethd_SomeD:
accmethd G S C sig = Some m
==> methd G C sig = Some m ∧ G \<turnstile>Methd sig m of C accessible_from S
lemma accmethd_SomeI:
[| methd G C sig = Some m; G \<turnstile>Methd sig m of C accessible_from S |]
==> accmethd G S C sig = Some m
lemma accmethd_declC:
[| accmethd G S C sig = Some m; ws_prog G; is_class G C |]
==> (∃d. class G (declclass m) = Some d ∧
table_of (methods d) sig = Some (mthd m)) ∧
G|-C<=:C declclass m ∧
methd G (declclass m) sig = Some m ∧
G \<turnstile>Methd sig m of C accessible_from S
lemma finite_dom_accmethd:
[| ws_prog G; is_class G C |] ==> finite (dom (accmethd G S C))
lemma dynmethd_rec:
[| class G dynC = Some c; ws_prog G |]
==> dynmethd G statC dynC sig =
(if G|-dynC<=:C statC
then case methd G statC sig of None => None
| Some statM =>
case methd G dynC sig of None => dynmethd G statC (super c) sig
| Some dynM =>
if G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM ∨
dynM = statM
then Some dynM else dynmethd G statC (super c) sig
else None)
lemma dynmethd_C_C:
[| is_class G C; ws_prog G |] ==> dynmethd G C C sig = methd G C sig
lemma dynmethdSomeD:
[| dynmethd G statC dynC sig = Some dynM; is_class G dynC; ws_prog G |]
==> G|-dynC<=:C statC ∧ (∃statM. methd G statC sig = Some statM)
lemma dynmethd_Some_cases:
[| dynmethd G statC dynC sig = Some dynM; is_class G dynC; ws_prog G;
methd G statC sig = Some dynM ==> P;
!!statM.
[| methd G statC sig = Some statM; dynM ≠ statM;
G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM |]
==> P |]
==> P
lemma no_override_in_Object:
[| dynmethd G statC dynC sig = Some dynM; is_class G dynC; ws_prog G;
methd G statC sig = Some statM; dynM ≠ statM |]
==> dynC ≠ Object
lemma dynmethd_Some_rec_cases:
[| dynmethd G statC dynC sig = Some dynM; class G dynC = Some c; ws_prog G;
methd G statC sig = Some dynM ==> P;
!!statM.
[| methd G statC sig = Some statM; methd G dynC sig = Some dynM;
statM ≠ dynM;
G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM |]
==> P;
[| dynC ≠ Object; dynmethd G statC (super c) sig = Some dynM |] ==> P |]
==> P
lemma dynmethd_declC:
[| dynmethd G statC dynC sig = Some m; is_class G statC; ws_prog G |]
==> (∃d. class G (declclass m) = Some d ∧
table_of (methods d) sig = Some (mthd m)) ∧
G|-dynC<=:C declclass m ∧ methd G (declclass m) sig = Some m
lemma methd_Some_dynmethd_Some:
[| methd G statC sig = Some statM; G|-dynC<=:C statC; is_class G statC;
ws_prog G |]
==> ∃dynM. dynmethd G statC dynC sig = Some dynM
lemma dynmethd_cases:
[| methd G statC sig = Some statM; G|-dynC<=:C statC; is_class G statC;
ws_prog G; dynmethd G statC dynC sig = Some statM ==> P;
!!dynM.
[| dynmethd G statC dynC sig = Some dynM; dynM ≠ statM;
G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM |]
==> P |]
==> P
lemma ws_dynmethd:
[| methd G statC sig = Some statM; G|-dynC<=:C statC; is_class G statC;
ws_prog G |]
==> ∃dynM. dynmethd G statC dynC sig = Some dynM ∧
is_static dynM = is_static statM ∧
G\<turnstile>resTy dynM\<preceq>resTy statM
lemma dynlookup_cases:
[| dynlookup G statT dynC sig = x; [| statT = NullT; None = x |] ==> P;
!!I. [| statT = IfaceT I; dynimethd G I dynC sig = x |] ==> P;
!!statC. [| statT = ClassT statC; dynmethd G statC dynC sig = x |] ==> P;
!!ty. [| statT = ArrayT ty; dynmethd G Object dynC sig = x |] ==> P |]
==> P
lemma fields_rec:
[| class G C = Some c; ws_prog G |]
==> DeclConcepts.fields G C =
map (λ(fn, ft). ((fn, C), ft)) (cfields c) @
(if C = Object then [] else DeclConcepts.fields G (super c))
lemma fields_norec:
[| class G fd = Some c; ws_prog G; table_of (cfields c) fn = Some f |]
==> table_of (DeclConcepts.fields G fd) (fn, fd) = Some f
lemma table_of_fieldsD:
table_of (map (λ(fn, ft). ((fn, C), ft)) (cfields c)) efn = Some f
==> declclassf efn = C ∧ table_of (cfields c) (fname efn) = Some f
lemma fields_declC:
[| table_of (DeclConcepts.fields G C) efn = Some f; ws_prog G; is_class G C |]
==> (∃d. class G (declclassf efn) = Some d ∧
table_of (cfields d) (fname efn) = Some f) ∧
G|-C<=:C declclassf efn ∧
table_of (DeclConcepts.fields G (declclassf efn)) efn = Some f
lemma fields_emptyI:
[| ws_prog G; class G C = Some c; cfields c = [];
C ≠ Object -->
class G (super c) = Some y ∧ DeclConcepts.fields G (super c) = [] |]
==> DeclConcepts.fields G C = []
lemma fields_mono_lemma:
[| x ∈ set (DeclConcepts.fields G C); G|-D<=:C C; ws_prog G |]
==> x ∈ set (DeclConcepts.fields G D)
lemma ws_unique_fields_lemma:
[| (efn, fd) ∈ set (DeclConcepts.fields G (super c)); fc ∈ set (cfields c);
ws_prog G; fname efn = fname fc; declclassf efn = C; class G C = Some c;
C ≠ Object; class G (super c) = Some d |]
==> R
lemma ws_unique_fields:
[| is_class G C; ws_prog G; !!C c. class G C = Some c ==> unique (cfields c) |]
==> unique (DeclConcepts.fields G C)
lemma accfield_fields:
accfield G S C fn = Some f
==> table_of (DeclConcepts.fields G C) (fn, declclass f) = Some (fld f)
lemma accfield_declC_is_class:
[| is_class G C; accfield G S C en = Some (fd, f); ws_prog G |]
==> is_class G fd
lemma accfield_accessibleD:
accfield G S C fn = Some f ==> G \<turnstile>Field fn f of C accessible_from S
lemma is_methdI:
[| class G C = Some y; methd G C sig = Some b |] ==> is_methd G C sig
lemma is_methdD:
is_methd G C sig ==> is_class G C ∧ methd G C sig ≠ None
lemma finite_is_methd:
ws_prog G ==> finite {(x, y). is_methd G x y}
lemma superclasses_rec:
[| class G C = Some c; ws_prog G |]
==> superclasses G C =
(if C = Object then {} else insert (super c) (superclasses G (super c)))
lemma superclasses_mono:
[| G|-C<:C D; ws_prog G; class G C = Some c;
!!C c. [| class G C = Some c; C ≠ Object |]
==> ∃sc. class G (super c) = Some sc;
x ∈ superclasses G D |]
==> x ∈ superclasses G C
lemma subclsEval:
[| G|-C<:C D; ws_prog G; class G C = Some c;
!!C c. [| class G C = Some c; C ≠ Object |]
==> ∃sc. class G (super c) = Some sc |]
==> D ∈ superclasses G C