(* Title: HOL/Bali/DeclConcepts.thy Author: Norbert Schirmer *) subsection‹Advanced concepts on Java declarations like overriding, inheritance, dynamic method lookup›
theory DeclConcepts imports TypeRel begin
subsubsection "access control (cf. 6.6), overriding and hiding (cf. 8.4.6.1)"
definition is_public :: "prog ==> qtname ==> bool"where "is_public G qn = (case class G qn of None ==> (case iface G qn of None ==> False | Some i ==> access i = Public) | Some c ==> access c = 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›
definition
is_acc_class :: "prog ==> pname ==> qtname ==> bool" where"is_acc_class G P C = (is_class G C ∧ G⊨(Class C) accessible_in P)"
definition
is_acc_iface :: "prog ==> pname ==> qtname ==> bool" where"is_acc_iface G P I = (is_iface G I ∧ G⊨(Iface I) accessible_in P)"
definition
is_acc_type :: "prog ==> pname ==> ty ==> bool" where"is_acc_type G P T = (is_type G T ∧ G⊨T accessible_in P)"
definition
is_acc_reftype :: "prog ==> pname ==> ref_ty ==> bool" where"is_acc_reftype G P T = (isrtype G T ∧ G⊨T accessible_in' P)"
lemma is_acc_classD: "is_acc_class G P C ==> is_class G C ∧ G⊨(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⊨(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⊨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⊨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 ‹accmodi›to select the access modifier out of various HOL types›
class has_accmodi = fixes accmodi:: "'a ==> acc_modi"
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‹ ‹G⊨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} › definition
inheritable_in :: "prog ==> (qtname × memberdecl) ==> pname ==> bool" (‹_ ⊨ _ inheritable'_in _› [61,61,61] 60) where "G⊨membr inheritable_in pack = (case (accmodi membr) of Private ==> False | Package ==> (pid (declclass membr)) = pack | Protected ==> True | Public ==> True)"
abbreviation
Method_inheritable_in_syntax:: "prog ==> (qtname × mdecl) ==> pname ==> bool"
(‹_ ⊨Method _ inheritable'_in _ › [61,61,61] 60) where"G⊨Method m inheritable_in p == G⊨methdMembr m inheritable_in p"
abbreviation
Methd_inheritable_in:: "prog ==> sig ==> (qtname × methd) ==> pname ==> bool"
(‹_ ⊨Methd _ _ inheritable'_in _ › [61,61,61,61] 60) where"G⊨Methd s m inheritable_in p == G⊨(method s m) inheritable_in p"
subsubsection "declared-in/undeclared-in"
definition
cdeclaredmethd :: "prog ==> qtname ==> (sig,methd) table"where "cdeclaredmethd G C = (case class G C of None ==> λ sig. None | Some c ==> table_of (methods c))"
definition
cdeclaredfield :: "prog ==> qtname ==> (vname,field) table"where "cdeclaredfield G C = (case class G C of None ==> λ sig. None | Some c ==> table_of (cfields c))"
definition
declared_in :: "prog ==> memberdecl ==> qtname ==> bool" (‹_⊨ _ declared'_in _› [61,61,61] 60) where "G⊨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)"
abbreviation
method_declared_in:: "prog ==> (qtname × mdecl) ==> qtname ==> bool"
(‹_⊨Method _ declared'_in _› [61,61,61] 60) where"G⊨Method m declared_in C == G⊨mdecl (mthd m) declared_in C"
abbreviation
methd_declared_in:: "prog ==> sig ==>(qtname × methd) ==> qtname ==> bool"
(‹_⊨Methd _ _ declared'_in _› [61,61,61,61] 60) where"G⊨Methd s m declared_in C == G⊨mdecl (s,mthd m) declared_in C"
lemma declared_in_classD: "G⊨m declared_in C ==> is_class G C" by (cases m)
(auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def)
definition
undeclared_in :: "prog ==> memberid ==> qtname ==> bool" (‹_⊨ _ undeclared'_in _›[61,61,61] 60) where "G⊨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 \<Rightarrow> qtname \<Rightarrow> memberdecl set because the class qtname changes to the superclass in the inductive definition below *)
inductive
members :: "prog ==> (qtname × memberdecl) ==> qtname ==> bool"
(‹_ ⊨ _ member'_of _› [61,61,61] 60) for G :: prog where
Immediate: "[G⊨mbr m declared_in C;declclass m = C]==> G⊨m member_of C"
| Inherited: "[G⊨m inheritable_in (pid C); G⊨memberid m undeclared_in C; G⊨C ≺🪙C1 S; G⊨(Class S) accessible_in (pid C);G⊨m member_of S ]==> G⊨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.›
abbreviation
method_member_of:: "prog ==> (qtname × mdecl) ==> qtname ==> bool"
(‹_ ⊨Method _ member'_of _› [61,61,61] 60) where"G⊨Method m member_of C == G⊨(methdMembr m) member_of C"
abbreviation
methd_member_of:: "prog ==> sig ==> (qtname × methd) ==> qtname ==> bool"
(‹_ ⊨Methd _ _ member'_of _› [61,61,61,61] 60) where"G⊨Methd s m member_of C == G⊨(method s m) member_of C"
abbreviation
fieldm_member_of:: "prog ==> vname ==> (qtname × field) ==> qtname ==> bool"
(‹_ ⊨Field _ _ member'_of _› [61,61,61] 60) where"G⊨Field n f member_of C == G⊨fieldm n f member_of C"
definition
inherits :: "prog ==> qtname ==> (qtname × memberdecl) ==> bool" (‹_ ⊨ _ inherits _› [61,61,61] 60) where "G⊨C inherits m = (G⊨m inheritable_in (pid C) ∧ G⊨memberid m undeclared_in C ∧ (∃S. G⊨C ≺🪙C1 S ∧ G⊨(Class S) accessible_in (pid C) ∧ G⊨m member_of S))"
lemma inherits_member: "G⊨C inherits m ==> G⊨m member_of C" by (auto simp add: inherits_def intro: members.Inherited)
definition
member_in :: "prog ==> (qtname × memberdecl) ==> qtname ==> bool" (‹_ ⊨ _ member'_in _› [61,61,61] 60) where"G⊨m member_in C = (∃ provC. G⊨ C ⪯🪙C provC ∧ G ⊨ 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.›
abbreviation
method_member_in:: "prog ==> (qtname × mdecl) ==> qtname ==> bool"
(‹_ ⊨Method _ member'_in _› [61,61,61] 60) where"G⊨Method m member_in C == G⊨(methdMembr m) member_in C"
abbreviation
methd_member_in:: "prog ==> sig ==> (qtname × methd) ==> qtname ==> bool"
(‹_ ⊨Methd _ _ member'_in _› [61,61,61,61] 60) where"G⊨Methd s m member_in C == G⊨(method s m) member_in C"
lemma member_inD: "G⊨m member_in C ==>∃ provC. G⊨ C ⪯🪙C provC ∧ G ⊨ m member_of provC" by (auto simp add: member_in_def)
lemma member_inI: "[G ⊨ m member_of provC;G⊨ C ⪯🪙C provC]==> G⊨m member_in C" by (auto simp add: member_in_def)
lemma member_of_to_member_in: "G ⊨ m member_of C ==> G ⊨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"
(‹_ ⊨ _ overrides🪙S _› [61,61,61] 60) for G :: prog where
Direct: "[¬ is_static new; msig new = msig old; G⊨Method new declared_in (declclass new); G⊨Method old declared_in (declclass old); G⊨Method old inheritable_in pid (declclass new); G⊨(declclass new) ≺🪙C1 superNew; G ⊨Method old member_of superNew ]==> G⊨new overrides🪙S old"
subsubsection "permits access" definition
permits_acc :: "prog ==> (qtname × memberdecl) ==> qtname ==> qtname ==> bool" (‹_ ⊨ _ in _ permits'_acc'_from _› [61,61,61,61] 60) where "G⊨membr in cls 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⊨accclass ≺🪙C declclass membr ∧ (G⊨cls ⪯🪙C accclass ∨ is_static membr)) | Public ==> True)" text‹ The subcondition of the 🍋‹Protected›case: 🍋‹G⊨accclass ≺🪙C declclass membr›could also be relaxed to: 🍋‹G⊨accclass ⪯🪙C declclass membr›since in case both classes are the same the other condition 🍋‹(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"
(‹_ ⊨ _ of _ accessible'_from _› [61,61,61,61] 60) and method_accessible_from :: "prog ==> (qtname × mdecl) ==> qtname ==> qtname ==>bool"
(‹_ ⊨Method _ of _ accessible'_from _› [61,61,61,61] 60) for G :: prog and accclass :: qtname where "G⊨membr of cls accessible_from accclass ≡ accessible_fromR G accclass membr cls"
| "G⊨Method m of cls accessible_from accclass ≡ accessible_fromR G accclass (methdMembr m) cls"
| Immediate: "!!membr class. [G⊨membr member_of class; G⊨(Class class) accessible_in (pid accclass); G⊨membr in class permits_acc_from accclass ]==> G⊨membr of class accessible_from accclass"
| Overriding: "!!membr class C new old supr. [G⊨membr member_of class; G⊨(Class class) accessible_in (pid accclass); membr=(C,mdecl new); G⊨(C,new) overrides🪙S old; G⊨class ≺🪙C supr; G⊨Method old of supr accessible_from accclass ]==> G⊨membr of class accessible_from accclass"
abbreviation
methd_accessible_from:: "prog ==> sig ==> (qtname × methd) ==> qtname ==> qtname ==> bool"
(‹_ ⊨Methd _ _ of _ accessible'_from _› [61,61,61,61,61] 60) where "G⊨Methd s m of cls accessible_from accclass == G⊨(method s m) of cls accessible_from accclass"
abbreviation
field_accessible_from:: "prog ==> vname ==> (qtname × field) ==> qtname ==> qtname ==> bool"
(‹_ ⊨Field _ _ of _ accessible'_from _› [61,61,61,61,61] 60) where "G⊨Field fn f of C accessible_from accclass == G⊨(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"
(‹_ ⊨ _ in _ dyn'_accessible'_from _› [61,61,61,61] 60) and method_dyn_accessible_from :: "prog ==> (qtname × mdecl) ==> qtname ==> qtname ==> bool"
(‹_ ⊨Method _ in _ dyn'_accessible'_from _› [61,61,61,61] 60) for G :: prog and accclass :: qtname where "G⊨membr in C dyn_accessible_from accC ≡ dyn_accessible_fromR G accC membr C"
| "G⊨Method m in C dyn_accessible_from accC ≡ dyn_accessible_fromR G accC (methdMembr m) C"
| Immediate: "!!class. [G⊨membr member_in class; G⊨membr in class permits_acc_from accclass ]==> G⊨membr in class dyn_accessible_from accclass"
| Overriding: "!!class. [G⊨membr member_in class; membr=(C,mdecl new); G⊨(C,new) overrides old; G⊨class ≺🪙C supr; G⊨Method old in supr dyn_accessible_from accclass ]==> G⊨membr in class dyn_accessible_from accclass"
abbreviation
methd_dyn_accessible_from:: "prog ==> sig ==> (qtname × methd) ==> qtname ==> qtname ==> bool"
(‹_ ⊨Methd _ _ in _ dyn'_accessible'_from _› [61,61,61,61,61] 60) where "G⊨Methd s m in C dyn_accessible_from accC == G⊨(method s m) in C dyn_accessible_from accC"
abbreviation
field_dyn_accessible_from:: "prog ==> vname ==> (qtname × field) ==> qtname ==> qtname ==> bool"
(‹_ ⊨Field _ _ in _ dyn'_accessible'_from _› [61,61,61,61,61] 60) where "G⊨Field fn f in dynC dyn_accessible_from accC == G⊨(fieldm fn f) in dynC dyn_accessible_from accC"
lemma accessible_from_commonD: "G⊨m of C accessible_from S ==> G⊨m member_of C ∧ G⊨(Class C) accessible_in (pid S)" by (auto elim: accessible_fromR.induct)
lemma unique_declaration: "[G⊨m declared_in C; G⊨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⊨m declared_in C ==>¬ G⊨ memberid m undeclared_in C" by (cases m) (auto simp add: declared_in_def undeclared_in_def)
lemma undeclared_not_declared: "G⊨ memberid m undeclared_in C ==>¬ G⊨ m declared_in C" by (cases m) (auto simp add: declared_in_def undeclared_in_def)
lemma not_undeclared_declared: "¬ G⊨ membr_id undeclared_in C ==> (∃ m. G⊨m declared_in C ∧ membr_id = memberid m)" proof - assume not_undecl:"¬ G⊨ 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⊨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⊨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⊨m declared_in C; G⊨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⊨n member_of C"and
m: "G⊨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⊨ mbr n declared_in C""declclass n = C" assume eqid: "memberid n = memberid m" assume"G ⊨ m member_of C" thenshow"n=m" proof (cases) case Immediate with eqid 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 with eqid member_n show ?thesis by (cases n) (auto dest: declared_not_undeclared) qed next case (Inherited n C S) assume undecl: "G⊨ memberid n undeclared_in C" assume super: "G⊨C≺🪙C1S" assume hyp: "[G ⊨ m member_of S; memberid n = memberid m]==> n = m" assume eqid: "memberid n = memberid m" assume"G ⊨ m member_of C" thenshow"n=m" proof (cases) case Immediate thenhave"G⊨ 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 ⊨ m member_of S" by (auto dest!: subcls1D) with eqid hyp show ?thesis by blast qed qed qed
lemma member_of_is_classD: "G⊨m member_of C ==> is_class G C" proof (induct set: members) case (Immediate m C) assume"G⊨ mbr m declared_in C" thenshow"is_class G C" by (cases "mbr m")
(auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def) next case (Inherited m C S) show"is_class G C"if"G⊨C≺🪙C1S"and"is_class G S" by (rule subcls_is_class2) (use that in auto) qed
lemma member_of_declC: "G⊨m member_of C ==> G⊨mbr m declared_in (declclass m)" by (induct set: members) auto
lemma member_of_member_of_declC: "G⊨m member_of C ==> G⊨m member_of (declclass m)" by (auto dest: member_of_declC intro: members.Immediate)
lemma member_of_class_relation: "G⊨m member_of C ==> G⊨C ⪯🪙C declclass m" proof (induct set: members) case (Immediate m C) thenshow"G⊨C ⪯🪙C declclass m"by simp next case (Inherited m C S) thenshow"G⊨C ⪯🪙C declclass m" by (auto dest: r_into_rtrancl intro: rtrancl_trans) qed
lemma member_in_class_relation: "G⊨m member_in C ==> G⊨C ⪯🪙C declclass m" by (auto dest: member_inD member_of_class_relation
intro: rtrancl_trans)
lemma stat_overrides_commonD: "[G⊨new overrides🪙S old]==> declclass new ≠ Object ∧¬ is_static new ∧ msig new = msig old ∧ G⊨(declclass new) ≺🪙C (declclass old) ∧ G⊨Method new declared_in (declclass new) ∧ G⊨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: assumes"G⊨m member_of C" and"accmodi m = Package" shows"pid (declclass m) = pid C" using assms proof induct case Immediate thenshow ?caseby simp next case Inherited thenshow ?caseby (auto simp add: inheritable_in_def) qed
lemma member_in_declC: "G⊨m member_in C==> G⊨m member_in (declclass m)" proof - assume member_in_C: "G⊨m member_in C" from member_in_C obtain provC where
subclseq_C_provC: "G⊨ C ⪯🪙C provC"and
member_of_provC: "G ⊨ m member_of provC" by (auto simp add: member_in_def) from member_of_provC have"G ⊨ m member_of declclass m" by (rule member_of_member_of_declC) moreover from member_in_C have"G⊨C ⪯🪙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⊨m in C dyn_accessible_from S ==> G⊨m member_in C" by (auto elim: dyn_accessible_fromR.induct)
lemma no_Private_stat_override: "[G⊨new overrides🪙S old]==> accmodi old ≠ Private" by (induct set: stat_overridesR) (auto simp add: inheritable_in_def)
lemma no_Private_override: "[G⊨new overrides old]==> accmodi old ≠ Private" by (induct set: overridesR) (auto simp add: inheritable_in_def)
lemma permits_acc_inheritance: "[G⊨m in statC permits_acc_from accC; G⊨dynC ⪯🪙C statC ]==> G⊨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⊨m in C permits_acc_from accC; G⊨m member_in C; is_static m ]==> G⊨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⊨m in C dyn_accessible_from accC"and
static: "is_static m" shows"G⊨m in (declclass m) dyn_accessible_from accC" proof - from acc_C static show"G⊨m in (declclass m) dyn_accessible_from accC" proof (induct) case (Immediate m C) thenshow ?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) thenhave"¬ is_static m" by (auto dest: overrides_commonD) moreover assume"is_static m" ultimatelyshow ?case by contradiction qed qed
lemma field_accessible_fromD: "[G⊨membr of C accessible_from accC;is_field membr] ==> G⊨membr member_of C ∧ G⊨(Class C) accessible_in (pid accC) ∧ G⊨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⊨membr of statC accessible_from accC; is_field membr; G ⊨ dynC ⪯🪙C statC] \ G⊨membr in dynC permits_acc_from accC" by (auto dest: field_accessible_fromD intro: permits_acc_inheritance)
(* lemma accessible_Package: "[G ⊨ m of C accessible_from S;accmodi m = Package; ∧ new old. G⊨new overrides🪙S old ==> accmodi old ≤ accmodi new] ==> pid S = pid C ∧ pid C = pid (declclass m)" proof - assume wf: "∧ new old. G⊨new overrides🪙S old ==> accmodi old ≤ accmodi new" assume "G ⊨ 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⊨m member_of C" "G ⊨ 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 ⊨ new member_of C" and acc_C: "G ⊨ Class C accessible_in pid S" and new: "new = (declC, mdecl newm)" and override: "G ⊨ (declC, newm) overrides🪙S old" and subcls_C_Sup: "G⊨C ≺🪙C Sup" and acc_old: "G ⊨ 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⊨membr of C accessible_from accC; is_field membr] ==> G⊨membr member_of C ∧ G⊨(Class C) accessible_in (pid accC) ∧ G⊨membr in C permits_acc_from accC" by (induct rule: accessible_fromR.induct) (auto dest: is_fieldD)
lemma member_of_Private: "[G⊨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⊨m member_of C ==> G⊨C ⪯🪙C declclass m" by (induct set: members) (auto dest: r_into_rtrancl intro: rtrancl_trans)
lemma member_of_inheritance: assumes m: "G⊨m member_of D"and
subclseq_D_C: "G⊨D ⪯🪙C C"and
subclseq_C_m: "G⊨C ⪯🪙C declclass m"and
ws: "ws_prog G" shows"G⊨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⊨D⪯🪙C C"and"G⊨C⪯🪙C declclass m" with ws have"D=C" by (auto intro: subclseq_acyclic) with Immediate show"G⊨m member_of C" by (auto intro: members.Immediate) next case (Inherited m D S) assume member_of_D_props: "G ⊨ m inheritable_in pid D" "G⊨ memberid m undeclared_in D" "G ⊨ Class S accessible_in pid D" "G ⊨ m member_of S" assume super: "G⊨D≺🪙C1S" assume hyp: "[G⊨S⪯🪙C C; G⊨C⪯🪙C declclass m]==> G ⊨ m member_of C" assume subclseq_C_m: "G⊨C⪯🪙C declclass m" assume"G⊨D⪯🪙C C" thenshow"G⊨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⊨D≺🪙C C" with super have"G⊨S⪯🪙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⊨old member_of C"and
new: "G⊨new member_of D"and
eqid: "memberid new = memberid old"and
subclseq_D_C: "G⊨D ⪯🪙C C"and
subcls_new_old: "G⊨declclass new ≺🪙C declclass old"and
ws: "ws_prog G" shows"G⊨D ≺🪙C C" proof - from old have subclseq_C_old: "G⊨C ⪯🪙C declclass old" by (auto dest: member_of_subclseq_declC) from new have subclseq_D_new: "G⊨D ⪯🪙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⊨(declclass new) ⪯🪙C C ∨ G⊨C ⪯🪙C (declclass new)" by (rule subcls_compareable) thenhave"G⊨(declclass new) ⪯🪙C C" proof assume"G⊨declclass new⪯🪙C C"thenshow ?thesis . next assume"G⊨C ⪯🪙C (declclass new)" with new subclseq_D_C ws have"G⊨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 thenshow ?thesis proof (cases rule: subclseq_cases) case Eq assume"declclass new = C" with new have"G⊨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⊨declclass new≺🪙C C" with subclseq_D_new show"G⊨D≺🪙C C" by (rule rtrancl_trancl_trancl) qed qed
corollary member_of_overrides_subcls: "[G⊨Methd sig old member_of C; G⊨Methd sig new member_of D;G⊨D ⪯🪙C C; G,sig⊨new overrides old; ws_prog G] ==> G⊨D ≺🪙C C" by (drule overrides_commonD) (auto intro: member_of_subcls)
corollary member_of_stat_overrides_subcls: "[G⊨Methd sig old member_of C; G⊨Methd sig new member_of D;G⊨D ⪯🪙C C; G,sig⊨new overrides🪙S old; ws_prog G] ==> G⊨D ≺🪙C C" by (drule stat_overrides_commonD) (auto intro: member_of_subcls)
lemma inherited_field_access: assumes stat_acc: "G⊨membr of statC accessible_from accC"and
is_field: "is_field membr"and
subclseq: "G ⊨ dynC ⪯🪙C statC" shows"G⊨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⊨m of statC accessible_from accC"and
subclseq: "G⊨dynC ⪯🪙C statC"and
member_dynC: "G⊨m member_of dynC"and
dynC_acc: "G⊨(Class dynC) accessible_in (pid accC)" shows"G⊨m of dynC accessible_from accC" proof - from stat_acc have member_statC: "G⊨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
definition
imethds :: "prog ==> qtname ==> (sig,qtname × mhead) tables"where "imethds G I = iface_rec G I (λI i ts. (Un_tables ts) ⊕⊕ (set_option ∘ table_of (map (λ(s,m). (s,I,m)) (imethods i))))" text‹methods of an interface, with overriding and inheritance, cf. 9.2›
definition
accimethds :: "prog ==> pname ==> qtname ==> (sig,qtname × mhead) tables"where "accimethds G pack I = (if G⊨Iface I accessible_in pack then imethds G I else (λ k. {}))" text‹only returns imethds if the interface is accessible›
definition
methd :: "prog ==> qtname ==> (sig,qtname × methd) table"where "methd G C = class_rec G C Map.empty (λC c subcls_mthds. filter_tab (λsig m. G⊨C inherits method sig m) subcls_mthds ++ table_of (map (λ(s,m). (s,C,m)) (methods c)))" text‹🍋‹methd G C›: methods of a class C (statically visible from C), with inheritance and hiding cf. 8.4.6; Overriding is captured by ‹dynmethd›. Every new method with the same signature coalesces the method of a superclass.›
definition
accmethd :: "prog ==> qtname ==> qtname ==> (sig,qtname × methd) table"where "accmethd G S C = filter_tab (λsig m. G⊨method sig m of C accessible_from S) (methd G C)" text‹🍋‹accmethd G S C›: only those methods of 🍋‹methd G C›, accessible from S›
text‹Note the class component in the accessibility filter. The class where method 🍋‹m›is declared (🍋‹declC›) isn't necessarily accessible from the current scope 🍋‹S›. The method can be made accessible through inheritance, too. So we must test accessibility of method 🍋‹m›of class 🍋‹C› (not 🍋‹declclass m›)›
definition
dynmethd :: "prog ==> qtname ==> qtname ==> (sig,qtname × methd) table"where "dynmethd G statC dynC = (λsig. (if G⊨dynC ⪯🪙C statC then (case methd G statC sig of None ==> None | Some statM ==> (class_rec G dynC Map.empty (λC c subcls_mthds. subcls_mthds ++ (filter_tab (λ _ dynM. G,sig⊨dynM overrides statM ∨ dynM=statM) (methd G C) )) ) sig ) else None))"
text‹🍋‹dynmethd G statC dynC›: dynamic method lookup of a reference with dynamic class 🍋‹dynC›and static class 🍋‹statC›\› text‹Note some kind of duality between 🍋‹methd›and 🍋‹dynmethd› in the 🍋‹class_rec›arguments. Whereas 🍋‹methd› filters the subclass methods (to get only the inherited ones), 🍋‹dynmethd› filters the new methods (to get only those methods which actually override the methods of the static class)›
definition
dynimethd :: "prog ==> qtname ==> qtname ==> (sig,qtname × methd) table"where "dynimethd G I dynC = (λsig. if imethds G I sig ≠ {} then methd G dynC sig else dynmethd G Object dynC sig)" text‹🍋‹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 (🍋‹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. ›
definition
dynlookup :: "prog ==> ref_ty ==> qtname ==> (sig,qtname × methd) table"where "dynlookup G statT dynC = (case statT of NullT ==> Map.empty | IfaceT I ==> dynimethd G I dynC | ClassT statC ==> dynmethd G statC dynC | ArrayT ty ==> dynmethd G Object dynC)" text‹🍋‹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›
definition
fields :: "prog ==> qtname ==> ((vname × qtname) × field) list"where "fields G C = class_rec G C [] (λC c ts. map (λ(n,t). ((n,C),t)) (cfields c) @ ts)" text‹🍋‹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›
definition
accfield :: "prog ==> qtname ==> qtname ==> (vname, qtname × field) table"where "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⊨ (declC,fdecl (n,f)) of C accessible_from S) field_tab)" text‹🍋‹accfield G C S›: fields of a class 🍋‹C› which are accessible from scope of class 🍋‹S›with inheritance and hiding, cf. 8.3› text‹note the class component in the accessibility filter (see also 🍋‹methd›). The class declaring field 🍋‹f›(🍋‹declC›) isn't necessarily accessible from scope 🍋‹S›. The field can be made visible through inheritance, too. So we must test accessibility of field 🍋‹f›of class 🍋‹C›(not 🍋‹declclass f›)›
definition
is_methd :: "prog ==> qtname ==> sig ==> bool" where"is_methd G = (λC sig. is_class G C ∧ methd G C sig ≠ None)"
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)) ⊕⊕ (set_option ∘ 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: assumes im: "im ∈ imethds G I sig" and ifI: "iface G I = Some i" and ws: "ws_prog G" obtains (NewMethod) "table_of (map (λ(s, mh). (s, I, mh)) (imethods i)) sig = Some im"
| (InheritedMethod) J where"J ∈ set (isuperIfs i)"and"im ∈ imethds G J sig" using assms by (auto simp add: imethds_rec)
subsection"accimethd"
lemma accimethds_simp [simp]: "G⊨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⊨Iface I accessible_in pack" by (auto simp add: accimethds_def)
lemma accimethdsI: "[im ∈ imethds G I sig;G⊨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 Map.empty else filter_tab (λsig m. G⊨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⊨C ⪯🪙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 ⊨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⊨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⊨Methd sig m declared_in (declclass m)" by (auto dest: methd_declC method_declared_inI)
lemma member_methd: assumes member_of: "G⊨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 ⊨Methd sig m member_of Object" thenhave"G⊨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 ⊨Methd sig m member_of super c ==> ?Methd (super c)"and
member_of: "G ⊨Methd sig m member_of C" from member_of show"?Methd C" proof (cases) case Immediate 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 S) with clsC have undecl: "G⊨mid sig undeclared_in C"and
super: "G ⊨Methd sig m member_of (super c)" by (auto dest: subcls1D) from 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 ⊨ 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 accmethd_SomeD: "accmethd G S C sig = Some m ==> methd G C sig = Some m ∧ G⊨method sig m of C accessible_from S" by (auto simp add: accmethd_def)
lemma accmethd_SomeI: "[methd G C sig = Some m; G⊨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⊨C ⪯🪙C (declclass m) ∧ methd G (declclass m) sig = Some m ∧ G⊨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⊨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,sig⊨ 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" thenshow"?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⊨Object ⪯🪙C statC") case False thenshow ?thesis by (simp add: dynmethd_def) next case True thenhave eq_statC_Obj: "statC = Object" .. show ?thesis proof (cases "methd G statC sig") case None thenshow ?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⊨dynC ⪯🪙C statC") case False thenshow ?thesis by (simp add: dynmethd_def) next case True note subclseq_dynC_statC = True show ?thesis proof (cases "methd G statC sig") case None thenshow ?thesis by (simp add: dynmethd_def) next case (Some statM) note statM = Some let ?filter = "λC. filter_tab (λ_ dynM. G,sig ⊨ dynM overrides statM ∨ dynM = statM) (methd G C)" let ?class_rec = "λC. class_rec G C Map.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⊨(super c) ⪯🪙C statC" by (blast dest: subclseq_superD) show ?thesis proof (cases "methd G dynC sig") case None thenhave"?filter dynC sig = None" by (rule filter_tab_None) thenhave"?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 ⊨ 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⊨dynC ⪯🪙C statC ∧ (∃ statM. methd G statC sig = Some statM)" by (auto simp add: dynmethd_rec)
lemma dynmethd_Some_cases: assumes dynM: "dynmethd G statC dynC sig = Some dynM" and is_cls_dynC: "is_class G dynC" and ws: "ws_prog G" obtains (Static) "methd G statC sig = Some dynM"
| (Overrides) statM where"methd G statC sig = Some statM" and"dynM ≠ statM" and"G,sig⊨dynM overrides statM" proof - from is_cls_dynC obtain dc where clsDynC: "class G dynC = Some dc"by blast from clsDynC ws dynM Static Overrides show ?thesis 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: assumes dynM: "dynmethd G statC dynC sig = Some dynM" and clsDynC: "class G dynC = Some c" and ws: "ws_prog G" obtains (Static) "methd G statC sig = Some dynM"
| (Override) statM where"methd G statC sig = Some statM" and"methd G dynC sig = Some dynM"and"statM ≠ dynM" and"G,sig⊨ dynM overrides statM"
| (Recursion) "dynC ≠ Object"and"dynmethd G statC (super c) sig = Some dynM" proof - from clsDynC have *: "is_class G dynC"by simp from ws clsDynC dynM Static Override Recursion show ?thesis by (auto simp add: dynmethd_rec dest: no_override_in_Object [OF dynM * ws]) 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⊨dynC ⪯🪙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⊨dynC ⪯🪙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⊨dynC ⪯🪙C statC"and
statM: "methd G statC sig = Some statM" by (blast dest: dynmethdSomeD) from clsDynC neq_dynC_Obj have subclseq_dynC_super: "G⊨dynC ⪯🪙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⊨dynC ⪯🪙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) thenobtain 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⊨super dc⪯🪙C statC ==> ?P (super dc)" assume subclseq': "G⊨dynC⪯🪙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⊨dynC≺🪙C statC" from this clsDynC' have"G⊨super dc⪯🪙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: assumes statM: "methd G statC sig = Some statM" and subclseq: "G⊨dynC ⪯🪙C statC" and is_cls_statC: "is_class G statC" and ws: "ws_prog G" obtains (Static) "dynmethd G statC dynC sig = Some statM"
| (Overrides) dynM where"dynmethd G statC dynC sig = Some dynM" and"dynM ≠ statM"and"G,sig⊨dynM overrides statM" proof - note hyp_static = Static and hyp_override = Overrides from subclseq is_cls_statC have is_cls_dynC: "is_class G dynC"by (rule subcls_is_class2) thenobtain 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⊨dynC ⪯🪙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⊨resTy dynM⪯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: assumes"dynlookup G statT dynC sig = x" obtains (NullT) "statT = NullT"and"Map.empty sig = x"
| (IfaceT) I where"statT = IfaceT I"and"dynimethd G I dynC sig = x"
| (ClassT) statC where"statT = ClassT statC"and"dynmethd G statC dynC sig = x"
| (ArrayT) ty where"statT = ArrayT ty"and"dynmethd G Object dynC sig = x" using assms 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⊨C ⪯🪙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 (case_prod (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 (case_prod (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⊨D ⪯🪙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 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⊨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
subsubsection "calculation of the superclasses of a class"
definition
superclasses :: "prog ==> qtname ==> qtname set"where "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: assumes clsrel: "G⊨C≺🪙C D" and 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" and x: "x∈superclasses G D" shows"x∈superclasses G C"using clsrel cls_C x proof (induct arbitrary: c rule: converse_trancl_induct) case (base C) with wf ws show ?case by (auto intro: no_subcls1_Object
simp add: superclasses_rec subcls1_def) next case (step C S) moreovernote wf ws moreoverfrom calculation have"x∈superclasses G S" by (force intro: no_subcls1_Object simp add: subcls1_def) moreoverfrom calculation have"super c = S" by (auto intro: no_subcls1_Object simp add: subcls1_def) ultimatelyshow ?case by (auto intro: no_subcls1_Object simp add: superclasses_rec) qed
lemma subclsEval: assumes clsrel: "G⊨C≺🪙C D" and 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" shows"D∈superclasses G C"using clsrel cls_C proof (induct arbitrary: c rule: converse_trancl_induct) case (base C) show ?case by (use ws wf base in‹auto intro: no_subcls1_Object simp add: superclasses_rec subcls1_def›) next case (step C S) show ?case by (rule superclasses_mono)
(use ws wf step in‹auto dest: no_subcls1_Object simp add: subcls1_def›) qed
end
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