tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/Bali/DeclConcepts.thy
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)"
definition is_public :: "prog \<Rightarrow> qtname \<Rightarrow> bool" where
"is_public G qn = (case class G qn of
None \<Rightarrow> (case iface G qn of
None \<Rightarrow> False
| Some i \<Rightarrow> access i = Public)
| Some c \<Rightarrow> 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 *}
primrec
accessible_in :: "prog \<Rightarrow> ty \<Rightarrow> pname \<Rightarrow> bool" ("_ \<turnstile> _ accessible'_in _" [61,61,61] 60) and
rt_accessible_in :: "prog \<Rightarrow> ref_ty \<Rightarrow> pname \<Rightarrow> bool" ("_ \<turnstile> _ accessible'_in' _" [61,61,61] 60)
where
"G\<turnstile>(PrimT p) accessible_in pack = True"
| accessible_in_RefT_simp:
"G\<turnstile>(RefT r) accessible_in pack = G\<turnstile>r accessible_in' pack"
| "G\<turnstile>(NullT) accessible_in' pack = True"
| "G\<turnstile>(IfaceT I) accessible_in' pack = ((pid I = pack) \<or> is_public G I)"
| "G\<turnstile>(ClassT C) accessible_in' pack = ((pid C = pack) \<or> is_public G C)"
| "G\<turnstile>(ArrayT ty) accessible_in' pack = G\<turnstile>ty accessible_in pack"
declare accessible_in_RefT_simp [simp del]
definition
is_acc_class :: "prog \<Rightarrow> pname \<Rightarrow> qtname \<Rightarrow> bool"
where "is_acc_class G P C = (is_class G C \<and> G\<turnstile>(Class C) accessible_in P)"
definition
is_acc_iface :: "prog \<Rightarrow> pname \<Rightarrow> qtname \<Rightarrow> bool"
where "is_acc_iface G P I = (is_iface G I \<and> G\<turnstile>(Iface I) accessible_in P)"
definition
is_acc_type :: "prog \<Rightarrow> pname \<Rightarrow> ty \<Rightarrow> bool"
where "is_acc_type G P T = (is_type G T \<and> G\<turnstile>T accessible_in P)"
definition
is_acc_reftype :: "prog \<Rightarrow> pname \<Rightarrow> ref_ty \<Rightarrow> bool"
where "is_acc_reftype G P T = (isrtype G T \<and> G\<turnstile>T accessible_in' P)"
lemma is_acc_classD:
"is_acc_class G P C \<Longrightarrow> is_class G C \<and> G\<turnstile>(Class C) accessible_in P"
by (simp add: is_acc_class_def)
lemma is_acc_class_is_class: "is_acc_class G P C \<Longrightarrow> is_class G C"
by (auto simp add: is_acc_class_def)
lemma is_acc_ifaceD:
"is_acc_iface G P I \<Longrightarrow> is_iface G I \<and> G\<turnstile>(Iface I) accessible_in P"
by (simp add: is_acc_iface_def)
lemma is_acc_typeD:
"is_acc_type G P T \<Longrightarrow> is_type G T \<and> G\<turnstile>T accessible_in P"
by (simp add: is_acc_type_def)
lemma is_acc_reftypeD:
"is_acc_reftype G P T \<Longrightarrow> isrtype G T \<and> G\<turnstile>T accessible_in' P"
by (simp add: is_acc_reftype_def)
subsection "accessibility of members"
text {*
The accessibility of members is more involved as the accessibility of types.
We have to distinguish several cases to model the different effects of
accessibility during inheritance, overriding and ordinary member access
*}
subsubsection {* Various technical conversion and selection functions *}
text {* overloaded selector @{text accmodi} to select the access modifier
out of various HOL types *}
class has_accmodi =
fixes accmodi:: "'a \<Rightarrow> acc_modi"
instantiation acc_modi :: has_accmodi
begin
definition
acc_modi_accmodi_def: "accmodi (a::acc_modi) = a"
instance ..
end
lemma acc_modi_accmodi_simp[simp]: "accmodi (a::acc_modi) = a"
by (simp add: acc_modi_accmodi_def)
instantiation decl_ext :: (type) has_accmodi
begin
definition
decl_acc_modi_def: "accmodi (d::('a:: type) decl_scheme) = access d"
instance ..
end
lemma decl_acc_modi_simp[simp]: "accmodi (d::('a::type) decl_scheme) = access d"
by (simp add: decl_acc_modi_def)
instantiation prod :: (type, has_accmodi) has_accmodi
begin
definition
pair_acc_modi_def: "accmodi p = accmodi (snd p)"
instance ..
end
lemma pair_acc_modi_simp[simp]: "accmodi (x,a) = (accmodi a)"
by (simp add: pair_acc_modi_def)
instantiation memberdecl :: has_accmodi
begin
definition
memberdecl_acc_modi_def: "accmodi m = (case m of
fdecl f \<Rightarrow> accmodi f
| mdecl m \<Rightarrow> accmodi m)"
instance ..
end
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 *}
class has_declclass =
fixes declclass:: "'a \<Rightarrow> qtname"
instantiation qtname_ext :: (type) has_declclass
begin
definition
"declclass q = \<lparr> pid = pid q, tid = tid q \<rparr>"
instance ..
end
lemma qtname_declclass_def:
"declclass q \<equiv> (q::qtname)"
by (induct q) (simp add: declclass_qtname_ext_def)
lemma qtname_declclass_simp[simp]: "declclass (q::qtname) = q"
by (simp add: qtname_declclass_def)
instantiation prod :: (has_declclass, type) has_declclass
begin
definition
pair_declclass_def: "declclass p = declclass (fst p)"
instance ..
end
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 *}
class has_static =
fixes is_static :: "'a \<Rightarrow> bool"
instantiation decl_ext :: (has_static) has_static
begin
instance ..
end
instantiation member_ext :: (type) has_static
begin
instance ..
end
axiomatization where
static_field_type_is_static_def: "is_static (m::('a member_scheme)) \<equiv> static m"
lemma member_is_static_simp: "is_static (m::'a member_scheme) = static m"
by (simp add: static_field_type_is_static_def)
instantiation prod :: (type, has_static) has_static
begin
definition
pair_is_static_def: "is_static p = is_static (snd p)"
instance ..
end
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)
instantiation memberdecl :: has_static
begin
definition
memberdecl_is_static_def:
"is_static m = (case m of
fdecl f \<Rightarrow> is_static f
| mdecl m \<Rightarrow> is_static m)"
instance ..
end
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)
-- {* some mnemotic selectors for various pairs *}
definition
decliface :: "qtname \<times> 'a decl_scheme \<Rightarrow> qtname" where
"decliface = fst" --{* get the interface component *}
definition
mbr :: "qtname \<times> memberdecl \<Rightarrow> memberdecl" where
"mbr = snd" --{* get the memberdecl component *}
definition
mthd :: "'b \<times> 'a \<Rightarrow> 'a" where
"mthd = snd" --{* get the method component *}
--{* also used for mdecl, mhead *}
definition
fld :: "'b \<times> 'a decl_scheme \<Rightarrow> 'a decl_scheme" where
"fld = snd" --{* get the field component *}
--{* also used for @{text "((vname \<times> qtname)\<times> field)"} *}
-- {* some mnemotic selectors for @{text "(vname \<times> qtname)"} *}
definition
fname:: "vname \<times> 'a \<Rightarrow> vname"
where "fname = fst"
--{* also used for fdecl *}
definition
declclassf:: "(vname \<times> qtname) \<Rightarrow> qtname"
where "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)
--{* some mnemotic selectors for @{text "(vname \<times> qtname)"} *}
definition
fldname :: "vname \<times> qtname \<Rightarrow> vname"
where "fldname = fst"
definition
fldclass :: "vname \<times> qtname \<Rightarrow> qtname"
where "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} *}
definition
methdMembr :: "qtname \<times> mdecl \<Rightarrow> qtname \<times> memberdecl"
where "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} *}
definition
method :: "sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> memberdecl)"
where "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)
definition
fieldm :: "vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> (qtname \<times> memberdecl)"
where "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} *}
definition
msig :: "(qtname \<times> mdecl) \<Rightarrow> sig"
where "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} *}
definition
qmdecl :: "sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> mdecl)"
where "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 *}
class has_resTy =
fixes resTy:: "'a \<Rightarrow> ty"
instantiation decl_ext :: (has_resTy) has_resTy
begin
instance ..
end
instantiation member_ext :: (has_resTy) has_resTy
begin
instance ..
end
instantiation mhead_ext :: (type) has_resTy
begin
instance ..
end
axiomatization where
mhead_ext_type_resTy_def: "resTy (m::('b mhead_scheme)) \<equiv> resT m"
lemma mhead_resTy_simp: "resTy (m::'a mhead_scheme) = resT m"
by (simp add: mhead_ext_type_resTy_def)
lemma resTy_mhead [simp]:"resTy (mhead m) = resTy m"
by (simp add: mhead_def mhead_resTy_simp)
instantiation prod :: (type, has_resTy) has_resTy
begin
definition
pair_resTy_def: "resTy p = resTy (snd p)"
instance ..
end
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}
*}
definition
inheritable_in :: "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> pname \<Rightarrow> bool" ("_ \<turnstile> _ inheritable'_in _" [61,61,61] 60)
where
"G\<turnstile>membr inheritable_in pack =
(case (accmodi membr) of
Private \<Rightarrow> False
| Package \<Rightarrow> (pid (declclass membr)) = pack
| Protected \<Rightarrow> True
| Public \<Rightarrow> True)"
abbreviation
Method_inheritable_in_syntax::
"prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> pname \<Rightarrow> bool"
("_ \<turnstile>Method _ inheritable'_in _ " [61,61,61] 60)
where "G\<turnstile>Method m inheritable_in p == G\<turnstile>methdMembr m inheritable_in p"
abbreviation
Methd_inheritable_in::
"prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> pname \<Rightarrow> bool"
("_ \<turnstile>Methd _ _ inheritable'_in _ " [61,61,61,61] 60)
where "G\<turnstile>Methd s m inheritable_in p == G\<turnstile>(method s m) inheritable_in p"
subsubsection "declared-in/undeclared-in"
definition
cdeclaredmethd :: "prog \<Rightarrow> qtname \<Rightarrow> (sig,methd) table" where
"cdeclaredmethd G C =
(case class G C of
None \<Rightarrow> \<lambda> sig. None
| Some c \<Rightarrow> table_of (methods c))"
definition
cdeclaredfield :: "prog \<Rightarrow> qtname \<Rightarrow> (vname,field) table" where
"cdeclaredfield G C =
(case class G C of
None \<Rightarrow> \<lambda> sig. None
| Some c \<Rightarrow> table_of (cfields c))"
definition
declared_in :: "prog \<Rightarrow> memberdecl \<Rightarrow> qtname \<Rightarrow> bool" ("_\<turnstile> _ declared'_in _" [61,61,61] 60)
where
"G\<turnstile>m declared_in C = (case m of
fdecl (fn,f ) \<Rightarrow> cdeclaredfield G C fn = Some f
| mdecl (sig,m) \<Rightarrow> cdeclaredmethd G C sig = Some m)"
abbreviation
method_declared_in:: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> bool"
("_\<turnstile>Method _ declared'_in _" [61,61,61] 60)
where "G\<turnstile>Method m declared_in C == G\<turnstile>mdecl (mthd m) declared_in C"
abbreviation
methd_declared_in:: "prog \<Rightarrow> sig \<Rightarrow>(qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> bool"
("_\<turnstile>Methd _ _ declared'_in _" [61,61,61,61] 60)
where "G\<turnstile>Methd s m declared_in C == G\<turnstile>mdecl (s,mthd m) declared_in C"
lemma declared_in_classD:
"G\<turnstile>m declared_in C \<Longrightarrow> is_class G C"
by (cases m)
(auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def)
definition
undeclared_in :: "prog \<Rightarrow> memberid \<Rightarrow> qtname \<Rightarrow> bool" ("_\<turnstile> _ undeclared'_in _" [61,61,61] 60)
where
"G\<turnstile>m undeclared_in C = (case m of
fid fn \<Rightarrow> cdeclaredfield G C fn = None
| mid sig \<Rightarrow> 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 \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile> _ member'_of _" [61,61,61] 60)
for G :: prog
where
Immediate: "\<lbrakk>G\<turnstile>mbr m declared_in C;declclass m = C\<rbrakk> \<Longrightarrow> G\<turnstile>m member_of C"
| Inherited: "\<lbrakk>G\<turnstile>m inheritable_in (pid C); G\<turnstile>memberid m undeclared_in C;
G\<turnstile>C \<prec>\<^sub>C1 S; G\<turnstile>(Class S) accessible_in (pid C);G\<turnstile>m member_of S
\<rbrakk> \<Longrightarrow> G\<turnstile>m member_of C"
text {* Note that in the case of an inherited member only the members of the
direct superclass are concerned. If a member of a superclass of the direct
superclass isn't inherited in the direct superclass (not member of the
direct superclass) than it can't be a member of the class. E.g. If a
member of a class A is defined with package access it isn't member of a
subclass S if S isn't in the same package as A. Any further subclasses
of S will not inherit the member, regardless if they are in the same
package as A or not.*}
abbreviation
method_member_of:: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Method _ member'_of _" [61,61,61] 60)
where "G\<turnstile>Method m member_of C == G\<turnstile>(methdMembr m) member_of C"
abbreviation
methd_member_of:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Methd _ _ member'_of _" [61,61,61,61] 60)
where "G\<turnstile>Methd s m member_of C == G\<turnstile>(method s m) member_of C"
abbreviation
fieldm_member_of:: "prog \<Rightarrow> vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Field _ _ member'_of _" [61,61,61] 60)
where "G\<turnstile>Field n f member_of C == G\<turnstile>fieldm n f member_of C"
definition
inherits :: "prog \<Rightarrow> qtname \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> bool" ("_ \<turnstile> _ inherits _" [61,61,61] 60)
where
"G\<turnstile>C inherits m =
(G\<turnstile>m inheritable_in (pid C) \<and> G\<turnstile>memberid m undeclared_in C \<and>
(\<exists>S. G\<turnstile>C \<prec>\<^sub>C1 S \<and> G\<turnstile>(Class S) accessible_in (pid C) \<and> G\<turnstile>m member_of S))"
lemma inherits_member: "G\<turnstile>C inherits m \<Longrightarrow> G\<turnstile>m member_of C"
by (auto simp add: inherits_def intro: members.Inherited)
definition
member_in :: "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> bool" ("_ \<turnstile> _ member'_in _" [61,61,61] 60)
where "G\<turnstile>m member_in C = (\<exists> provC. G\<turnstile> C \<preceq>\<^sub>C provC \<and> G \<turnstile> m member_of provC)"
text {* A member is in a class if it is member of the class or a superclass.
If a member is in a class we can select this member. This additional notion
is necessary since not all members are inherited to subclasses. So such
members are not member-of the subclass but member-in the subclass.*}
abbreviation
method_member_in:: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Method _ member'_in _" [61,61,61] 60)
where "G\<turnstile>Method m member_in C == G\<turnstile>(methdMembr m) member_in C"
abbreviation
methd_member_in:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Methd _ _ member'_in _" [61,61,61,61] 60)
where "G\<turnstile>Methd s m member_in C == G\<turnstile>(method s m) member_in C"
lemma member_inD: "G\<turnstile>m member_in C
\<Longrightarrow> \<exists> provC. G\<turnstile> C \<preceq>\<^sub>C provC \<and> G \<turnstile> m member_of provC"
by (auto simp add: member_in_def)
lemma member_inI: "\<lbrakk>G \<turnstile> m member_of provC;G\<turnstile> C \<preceq>\<^sub>C provC\<rbrakk> \<Longrightarrow> G\<turnstile>m member_in C"
by (auto simp add: member_in_def)
lemma member_of_to_member_in: "G \<turnstile> m member_of C \<Longrightarrow> G \<turnstile>m member_in C"
by (auto intro: member_inI)
subsubsection "overriding"
text {* Unfortunately the static notion of overriding (used during the
typecheck of the compiler) and the dynamic notion of overriding (used during
execution in the JVM) are not exactly the same.
*}
text {* Static overriding (used during the typecheck of the compiler) *}
inductive
stat_overridesR :: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool"
("_ \<turnstile> _ overrides\<^sub>S _" [61,61,61] 60)
for G :: prog
where
Direct: "\<lbrakk>\<not> 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>\<^sub>C1 superNew;
G \<turnstile>Method old member_of superNew
\<rbrakk> \<Longrightarrow> G\<turnstile>new overrides\<^sub>S old"
| Indirect: "\<lbrakk>G\<turnstile>new overrides\<^sub>S intr; G\<turnstile>intr overrides\<^sub>S old\<rbrakk>
\<Longrightarrow> G\<turnstile>new overrides\<^sub>S old"
text {* Dynamic overriding (used during the typecheck of the compiler) *}
inductive
overridesR :: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool"
("_ \<turnstile> _ overrides _" [61,61,61] 60)
for G :: prog
where
Direct: "\<lbrakk>\<not> is_static new; \<not> is_static old; accmodi new \<noteq> Private;
msig new = msig old;
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old);
G\<turnstile>Method new declared_in (declclass new);
G\<turnstile>Method old declared_in (declclass old);
G\<turnstile>Method old inheritable_in pid (declclass new);
G\<turnstile>resTy new \<preceq> resTy old
\<rbrakk> \<Longrightarrow> G\<turnstile>new overrides old"
| Indirect: "\<lbrakk>G\<turnstile>new overrides intr; G\<turnstile>intr overrides old\<rbrakk>
\<Longrightarrow> G\<turnstile>new overrides old"
abbreviation (input)
sig_stat_overrides::
"prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> methd) \<Rightarrow> bool"
("_,_\<turnstile> _ overrides\<^sub>S _" [61,61,61,61] 60)
where "G,s\<turnstile>new overrides\<^sub>S old == G\<turnstile>(qmdecl s new) overrides\<^sub>S (qmdecl s old)"
abbreviation (input)
sig_overrides:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> methd) \<Rightarrow> bool"
("_,_\<turnstile> _ overrides _" [61,61,61,61] 60)
where "G,s\<turnstile>new overrides old == G\<turnstile>(qmdecl s new) overrides (qmdecl s old)"
subsubsection "Hiding"
definition
hides :: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> bool" ("_\<turnstile> _ hides _" [61,61,61] 60)
where
"G\<turnstile>new hides old =
(is_static new \<and> msig new = msig old \<and>
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
G\<turnstile>Method new declared_in (declclass new) \<and>
G\<turnstile>Method old declared_in (declclass old) \<and>
G\<turnstile>Method old inheritable_in pid (declclass new))"
abbreviation
sig_hides:: "prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> (qtname \<times> methd) \<Rightarrow> bool"
("_,_\<turnstile> _ hides _" [61,61,61,61] 60)
where "G,s\<turnstile>new hides old == G\<turnstile>(qmdecl s new) hides (qmdecl s old)"
lemma hidesI:
"\<lbrakk>is_static new; msig new = msig old;
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old);
G\<turnstile>Method new declared_in (declclass new);
G\<turnstile>Method old declared_in (declclass old);
G\<turnstile>Method old inheritable_in pid (declclass new)
\<rbrakk> \<Longrightarrow> G\<turnstile>new hides old"
by (auto simp add: hides_def)
lemma hidesD:
"\<lbrakk>G\<turnstile>new hides old\<rbrakk> \<Longrightarrow>
declclass new \<noteq> Object \<and> is_static new \<and> msig new = msig old \<and>
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
G\<turnstile>Method new declared_in (declclass new) \<and>
G\<turnstile>Method old declared_in (declclass old)"
by (auto simp add: hides_def)
lemma overrides_commonD:
"\<lbrakk>G\<turnstile>new overrides old\<rbrakk> \<Longrightarrow>
declclass new \<noteq> Object \<and> \<not> is_static new \<and> \<not> is_static old \<and>
accmodi new \<noteq> Private \<and>
msig new = msig old \<and>
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
G\<turnstile>Method new declared_in (declclass new) \<and>
G\<turnstile>Method old declared_in (declclass old)"
by (induct set: overridesR) (auto intro: trancl_trans)
lemma ws_overrides_commonD:
"\<lbrakk>G\<turnstile>new overrides old;ws_prog G\<rbrakk> \<Longrightarrow>
declclass new \<noteq> Object \<and> \<not> is_static new \<and> \<not> is_static old \<and>
accmodi new \<noteq> Private \<and> G\<turnstile>resTy new \<preceq> resTy old \<and>
msig new = msig old \<and>
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
G\<turnstile>Method new declared_in (declclass new) \<and>
G\<turnstile>Method old declared_in (declclass old)"
by (induct set: overridesR) (auto intro: trancl_trans ws_widen_trans)
lemma overrides_eq_sigD:
"\<lbrakk>G\<turnstile>new overrides old\<rbrakk> \<Longrightarrow> msig old=msig new"
by (auto dest: overrides_commonD)
lemma hides_eq_sigD:
"\<lbrakk>G\<turnstile>new hides old\<rbrakk> \<Longrightarrow> msig old=msig new"
by (auto simp add: hides_def)
subsubsection "permits access"
definition
permits_acc :: "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool" ("_ \<turnstile> _ in _ permits'_acc'_from _" [61,61,61,61] 60)
where
"G\<turnstile>membr in cls permits_acc_from accclass =
(case (accmodi membr) of
Private \<Rightarrow> (declclass membr = accclass)
| Package \<Rightarrow> (pid (declclass membr) = pid accclass)
| Protected \<Rightarrow> (pid (declclass membr) = pid accclass)
\<or>
(G\<turnstile>accclass \<prec>\<^sub>C declclass membr
\<and> (G\<turnstile>cls \<preceq>\<^sub>C accclass \<or> is_static membr))
| Public \<Rightarrow> True)"
text {*
The subcondition of the @{term "Protected"} case:
@{term "G\<turnstile>accclass \<prec>\<^sub>C declclass membr"} could also be relaxed to:
@{term "G\<turnstile>accclass \<preceq>\<^sub>C declclass membr"} since in case both classes are the
same the other condition @{term "(pid (declclass membr) = pid accclass)"}
holds anyway.
*}
text {* Like in case of overriding, the static and dynamic accessibility
of members is not uniform.
\begin{itemize}
\item Statically the class/interface of the member must be accessible for the
member to be accessible. During runtime this is not necessary. For
Example, if a class is accessible and we are allowed to access a member
of this class (statically) we expect that we can access this member in
an arbitrary subclass (during runtime). It's not intended to restrict
the access to accessible subclasses during runtime.
\item Statically the member we want to access must be "member of" the class.
Dynamically it must only be "member in" the class.
\end{itemize}
*}
inductive
accessible_fromR :: "prog \<Rightarrow> qtname \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> bool"
and accessible_from :: "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile> _ of _ accessible'_from _" [61,61,61,61] 60)
and method_accessible_from :: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> 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 \<equiv> accessible_fromR G accclass membr cls"
| "G\<turnstile>Method m of cls accessible_from accclass \<equiv> accessible_fromR G accclass (methdMembr m) cls"
| Immediate: "!!membr class.
\<lbrakk>G\<turnstile>membr member_of class;
G\<turnstile>(Class class) accessible_in (pid accclass);
G\<turnstile>membr in class permits_acc_from accclass
\<rbrakk> \<Longrightarrow> G\<turnstile>membr of class accessible_from accclass"
| Overriding: "!!membr class C new old supr.
\<lbrakk>G\<turnstile>membr member_of class;
G\<turnstile>(Class class) accessible_in (pid accclass);
membr=(C,mdecl new);
G\<turnstile>(C,new) overrides\<^sub>S old;
G\<turnstile>class \<prec>\<^sub>C supr;
G\<turnstile>Method old of supr accessible_from accclass
\<rbrakk>\<Longrightarrow> G\<turnstile>membr of class accessible_from accclass"
abbreviation
methd_accessible_from::
"prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Methd _ _ of _ accessible'_from _" [61,61,61,61,61] 60)
where
"G\<turnstile>Methd s m of cls accessible_from accclass ==
G\<turnstile>(method s m) of cls accessible_from accclass"
abbreviation
field_accessible_from::
"prog \<Rightarrow> vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Field _ _ of _ accessible'_from _" [61,61,61,61,61] 60)
where
"G\<turnstile>Field fn f of C accessible_from accclass ==
G\<turnstile>(fieldm fn f) of C accessible_from accclass"
inductive
dyn_accessible_fromR :: "prog \<Rightarrow> qtname \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> bool"
and dyn_accessible_from' :: "prog \<Rightarrow> (qtname \<times> memberdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile> _ in _ dyn'_accessible'_from _" [61,61,61,61] 60)
and method_dyn_accessible_from :: "prog \<Rightarrow> (qtname \<times> mdecl) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Method _ in _ dyn'_accessible'_from _" [61,61,61,61] 60)
for G :: prog and accclass :: qtname
where
"G\<turnstile>membr in C dyn_accessible_from accC \<equiv> dyn_accessible_fromR G accC membr C"
| "G\<turnstile>Method m in C dyn_accessible_from accC \<equiv> dyn_accessible_fromR G accC (methdMembr m) C"
| Immediate: "!!class. \<lbrakk>G\<turnstile>membr member_in class;
G\<turnstile>membr in class permits_acc_from accclass
\<rbrakk> \<Longrightarrow> G\<turnstile>membr in class dyn_accessible_from accclass"
| Overriding: "!!class. \<lbrakk>G\<turnstile>membr member_in class;
membr=(C,mdecl new);
G\<turnstile>(C,new) overrides old;
G\<turnstile>class \<prec>\<^sub>C supr;
G\<turnstile>Method old in supr dyn_accessible_from accclass
\<rbrakk>\<Longrightarrow> G\<turnstile>membr in class dyn_accessible_from accclass"
abbreviation
methd_dyn_accessible_from::
"prog \<Rightarrow> sig \<Rightarrow> (qtname \<times> methd) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Methd _ _ in _ dyn'_accessible'_from _" [61,61,61,61,61] 60)
where
"G\<turnstile>Methd s m in C dyn_accessible_from accC ==
G\<turnstile>(method s m) in C dyn_accessible_from accC"
abbreviation
field_dyn_accessible_from::
"prog \<Rightarrow> vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool"
("_ \<turnstile>Field _ _ in _ dyn'_accessible'_from _" [61,61,61,61,61] 60)
where
"G\<turnstile>Field fn f in dynC dyn_accessible_from accC ==
G\<turnstile>(fieldm fn f) in dynC dyn_accessible_from accC"
lemma accessible_from_commonD: "G\<turnstile>m of C accessible_from S
\<Longrightarrow> G\<turnstile>m member_of C \<and> G\<turnstile>(Class C) accessible_in (pid S)"
by (auto elim: accessible_fromR.induct)
lemma unique_declaration:
"\<lbrakk>G\<turnstile>m declared_in C; G\<turnstile>n declared_in C; memberid m = memberid n \<rbrakk>
\<Longrightarrow> 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 \<Longrightarrow> \<not> 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 \<Longrightarrow> \<not> G\<turnstile> m declared_in C"
by (cases m) (auto simp add: declared_in_def undeclared_in_def)
lemma not_undeclared_declared:
"\<not> G\<turnstile> membr_id undeclared_in C \<Longrightarrow> (\<exists> m. G\<turnstile>m declared_in C \<and>
membr_id = memberid m)"
proof -
assume not_undecl:"\<not> G\<turnstile> membr_id undeclared_in C"
show ?thesis (is "?P membr_id")
proof (cases membr_id)
case (fid vname)
with not_undecl
obtain fld where
"G\<turnstile>fdecl (vname,fld) declared_in C"
by (auto simp add: undeclared_in_def declared_in_def
cdeclaredfield_def)
with fid show ?thesis
by auto
next
case (mid sig)
with not_undecl
obtain mthd where
"G\<turnstile>mdecl (sig,mthd) declared_in C"
by (auto simp add: undeclared_in_def declared_in_def
cdeclaredmethd_def)
with mid show ?thesis
by auto
qed
qed
lemma unique_declared_in:
"\<lbrakk>G\<turnstile>m declared_in C; G\<turnstile>n declared_in C; memberid m = memberid n\<rbrakk>
\<Longrightarrow> m = n"
by (auto simp add: declared_in_def cdeclaredmethd_def cdeclaredfield_def
split: memberdecl.splits)
lemma unique_member_of:
assumes n: "G\<turnstile>n member_of C" and
m: "G\<turnstile>m member_of C" and
eqid: "memberid n = memberid m"
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
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\<turnstile> memberid n undeclared_in C"
assume super: "G\<turnstile>C\<prec>\<^sub>C1S"
assume hyp: "\<lbrakk>G \<turnstile> m member_of S; memberid n = memberid m\<rbrakk> \<Longrightarrow> n = m"
assume eqid: "memberid n = memberid m"
assume "G \<turnstile> m member_of C"
then show "n=m"
proof (cases)
case Immediate
then have "G\<turnstile> mbr m declared_in C" by simp
with eqid undecl
show ?thesis
by (cases m) (auto dest: declared_not_undeclared)
next
case Inherited
with super have "G \<turnstile> m member_of S"
by (auto dest!: subcls1D)
with eqid hyp
show ?thesis
by blast
qed
qed
qed
lemma member_of_is_classD: "G\<turnstile>m member_of C \<Longrightarrow> is_class G C"
proof (induct set: members)
case (Immediate 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>\<^sub>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
\<Longrightarrow> G\<turnstile>mbr m declared_in (declclass m)"
by (induct set: members) auto
lemma member_of_member_of_declC:
"G\<turnstile>m member_of C
\<Longrightarrow> G\<turnstile>m member_of (declclass m)"
by (auto dest: member_of_declC intro: members.Immediate)
lemma member_of_class_relation:
"G\<turnstile>m member_of C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m"
proof (induct set: members)
case (Immediate m C)
then show "G\<turnstile>C \<preceq>\<^sub>C declclass m" by simp
next
case (Inherited m C S)
then show "G\<turnstile>C \<preceq>\<^sub>C declclass m"
by (auto dest: r_into_rtrancl intro: rtrancl_trans)
qed
lemma member_in_class_relation:
"G\<turnstile>m member_in C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m"
by (auto dest: member_inD member_of_class_relation
intro: rtrancl_trans)
lemma stat_override_declclasses_relation:
"\<lbrakk>G\<turnstile>(declclass new) \<prec>\<^sub>C1 superNew; G \<turnstile>Method old member_of superNew \<rbrakk>
\<Longrightarrow> G\<turnstile>(declclass new) \<prec>\<^sub>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:
"\<lbrakk>G\<turnstile>new overrides\<^sub>S old\<rbrakk> \<Longrightarrow>
declclass new \<noteq> Object \<and> \<not> is_static new \<and> msig new = msig old \<and>
G\<turnstile>(declclass new) \<prec>\<^sub>C (declclass old) \<and>
G\<turnstile>Method new declared_in (declclass new) \<and>
G\<turnstile>Method old declared_in (declclass old)"
apply (induct set: stat_overridesR)
apply (frule (1) stat_override_declclasses_relation)
apply (auto intro: trancl_trans)
done
lemma member_of_Package:
assumes "G\<turnstile>m member_of C"
and "accmodi m = Package"
shows "pid (declclass m) = pid C"
using assms
proof induct
case Immediate
then show ?case by simp
next
case Inherited
then show ?case by (auto simp add: inheritable_in_def)
qed
lemma member_in_declC: "G\<turnstile>m member_in C\<Longrightarrow> 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>\<^sub>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>\<^sub>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
\<Longrightarrow> G\<turnstile>m member_in C"
by (auto elim: dyn_accessible_fromR.induct)
lemma no_Private_stat_override:
"\<lbrakk>G\<turnstile>new overrides\<^sub>S old\<rbrakk> \<Longrightarrow> accmodi old \<noteq> Private"
by (induct set: stat_overridesR) (auto simp add: inheritable_in_def)
lemma no_Private_override: "\<lbrakk>G\<turnstile>new overrides old\<rbrakk> \<Longrightarrow> accmodi old \<noteq> Private"
by (induct set: overridesR) (auto simp add: inheritable_in_def)
lemma permits_acc_inheritance:
"\<lbrakk>G\<turnstile>m in statC permits_acc_from accC; G\<turnstile>dynC \<preceq>\<^sub>C statC
\<rbrakk> \<Longrightarrow> 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:
"\<lbrakk>G\<turnstile>m in C permits_acc_from accC; G\<turnstile>m member_in C; is_static m
\<rbrakk> \<Longrightarrow> 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 "\<not> is_static m"
by (auto dest: overrides_commonD)
moreover
assume "is_static m"
ultimately show ?case
by contradiction
qed
qed
lemma field_accessible_fromD:
"\<lbrakk>G\<turnstile>membr of C accessible_from accC;is_field membr\<rbrakk>
\<Longrightarrow> G\<turnstile>membr member_of C \<and>
G\<turnstile>(Class C) accessible_in (pid accC) \<and>
G\<turnstile>membr in C permits_acc_from accC"
by (cases set: accessible_fromR)
(auto simp add: is_field_def split: memberdecl.splits)
lemma field_accessible_from_permits_acc_inheritance:
"\<lbrakk>G\<turnstile>membr of statC accessible_from accC; is_field membr; G \<turnstile> dynC \<preceq>\<^sub>C statC\<rbrakk>
\<Longrightarrow> G\<turnstile>membr in dynC permits_acc_from accC"
by (auto dest: field_accessible_fromD intro: permits_acc_inheritance)
(*
lemma accessible_Package:
"\<lbrakk>G \<turnstile> m of C accessible_from S;accmodi m = Package;
\<And> new old. G\<turnstile>new overrides\<^sub>S old \<Longrightarrow> accmodi old \<le> accmodi new\<rbrakk>
\<Longrightarrow> pid S = pid C \<and> pid C = pid (declclass m)"
proof -
assume wf: "\<And> new old. G\<turnstile>new overrides\<^sub>S old \<Longrightarrow> accmodi old \<le> accmodi new"
assume "G \<turnstile> m of C accessible_from S"
then show "accmodi m = Package \<Longrightarrow> pid S = pid C \<and> pid C = pid (declclass m)"
(is "?Pack m \<Longrightarrow> ?P C m")
proof (induct rule: accessible_fromR.induct)
fix C m
assume "G\<turnstile>m member_of C"
"G \<turnstile> m in C permits_acc_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) overrides\<^sub>S old" and
subcls_C_Sup: "G\<turnstile>C \<prec>\<^sub>C Sup" and
acc_old: "G \<turnstile> methdMembr old of Sup accessible_from S" and
hyp: "?Pack (methdMembr old) \<Longrightarrow> ?P Sup (methdMembr old)" and
accmodi_new: "accmodi new = Package"
from override wf
have accmodi_weaken: "accmodi old \<le> accmodi newm"
by (cases old,cases newm) auto
from override new
have "accmodi old \<noteq> Private"
by (simp add: no_Private_stat_override)
with accmodi_weaken accmodi_new new
have accmodi_old: "accmodi old = Package"
by (cases "accmodi old") (auto simp add: le_acc_def less_acc_def)
with hyp
have P_sup: "?P Sup (methdMembr old)"
by (simp)
from wf override new accmodi_old accmodi_new
have eq_pid_new_old: "pid (declclass new) = pid (declclass old)"
by (auto dest: stat_override_Package)
from member_new accmodi_new
have "pid (declclass new) = pid C"
by (auto dest: member_of_Package)
with eq_pid_new_old P_sup show "?P C new"
by auto
qed
qed
*)
lemma accessible_fieldD:
"\<lbrakk>G\<turnstile>membr of C accessible_from accC; is_field membr\<rbrakk>
\<Longrightarrow> G\<turnstile>membr member_of C \<and>
G\<turnstile>(Class C) accessible_in (pid accC) \<and>
G\<turnstile>membr in C permits_acc_from accC"
by (induct rule: accessible_fromR.induct) (auto dest: is_fieldD)
lemma member_of_Private:
"\<lbrakk>G\<turnstile>m member_of C; accmodi m = Private\<rbrakk> \<Longrightarrow> declclass m = C"
by (induct set: members) (auto simp add: inheritable_in_def)
lemma member_of_subclseq_declC:
"G\<turnstile>m member_of C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m"
by (induct set: members) (auto dest: r_into_rtrancl intro: rtrancl_trans)
lemma member_of_inheritance:
assumes m: "G\<turnstile>m member_of D" and
subclseq_D_C: "G\<turnstile>D \<preceq>\<^sub>C C" and
subclseq_C_m: "G\<turnstile>C \<preceq>\<^sub>C declclass m" and
ws: "ws_prog G"
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>\<^sub>C C" and "G\<turnstile>C\<preceq>\<^sub>C declclass m"
with ws have "D=C"
by (auto intro: subclseq_acyclic)
with Immediate
show "G\<turnstile>m member_of C"
by (auto intro: members.Immediate)
next
case (Inherited 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>\<^sub>C1S"
assume hyp: "\<lbrakk>G\<turnstile>S\<preceq>\<^sub>C C; G\<turnstile>C\<preceq>\<^sub>C declclass m\<rbrakk> \<Longrightarrow> G \<turnstile> m member_of C"
assume subclseq_C_m: "G\<turnstile>C\<preceq>\<^sub>C declclass m"
assume "G\<turnstile>D\<preceq>\<^sub>C C"
then show "G\<turnstile>m member_of C"
proof (cases rule: subclseq_cases)
case Eq
assume "D=C"
with super member_of_D_props
show ?thesis
by (auto intro: members.Inherited)
next
case Subcls
assume "G\<turnstile>D\<prec>\<^sub>C C"
with super
have "G\<turnstile>S\<preceq>\<^sub>C C"
by (auto dest: subcls1D subcls_superD)
with subclseq_C_m hyp show ?thesis
by blast
qed
qed
qed
lemma member_of_subcls:
assumes old: "G\<turnstile>old member_of C" and
new: "G\<turnstile>new member_of D" and
eqid: "memberid new = memberid old" and
subclseq_D_C: "G\<turnstile>D \<preceq>\<^sub>C C" and
subcls_new_old: "G\<turnstile>declclass new \<prec>\<^sub>C declclass old" and
ws: "ws_prog G"
shows "G\<turnstile>D \<prec>\<^sub>C C"
proof -
from old
have subclseq_C_old: "G\<turnstile>C \<preceq>\<^sub>C declclass old"
by (auto dest: member_of_subclseq_declC)
from new
have subclseq_D_new: "G\<turnstile>D \<preceq>\<^sub>C declclass new"
by (auto dest: member_of_subclseq_declC)
from subcls_new_old ws
have neq_new_old: "new\<noteq>old"
by (cases new,cases old) (auto dest: subcls_irrefl)
from subclseq_D_new subclseq_D_C
have "G\<turnstile>(declclass new) \<preceq>\<^sub>C C \<or> G\<turnstile>C \<preceq>\<^sub>C (declclass new)"
by (rule subcls_compareable)
then have "G\<turnstile>(declclass new) \<preceq>\<^sub>C C"
proof
assume "G\<turnstile>declclass new\<preceq>\<^sub>C C" then show ?thesis .
next
assume "G\<turnstile>C \<preceq>\<^sub>C (declclass new)"
with new subclseq_D_C ws
have "G\<turnstile>new member_of C"
by (blast intro: member_of_inheritance)
with eqid old
have "new=old"
by (blast intro: unique_member_of)
with neq_new_old
show ?thesis
by contradiction
qed
then show ?thesis
proof (cases rule: subclseq_cases)
case Eq
assume "declclass new = C"
with new have "G\<turnstile>new member_of C"
by (auto dest: member_of_member_of_declC)
with eqid old
have "new=old"
by (blast intro: unique_member_of)
with neq_new_old
show ?thesis
by contradiction
next
case Subcls
assume "G\<turnstile>declclass new\<prec>\<^sub>C C"
with subclseq_D_new
show "G\<turnstile>D\<prec>\<^sub>C C"
by (rule rtrancl_trancl_trancl)
qed
qed
corollary member_of_overrides_subcls:
"\<lbrakk>G\<turnstile>Methd sig old member_of C; G\<turnstile>Methd sig new member_of D;G\<turnstile>D \<preceq>\<^sub>C C;
G,sig\<turnstile>new overrides old; ws_prog G\<rbrakk>
\<Longrightarrow> G\<turnstile>D \<prec>\<^sub>C C"
by (drule overrides_commonD) (auto intro: member_of_subcls)
corollary member_of_stat_overrides_subcls:
"\<lbrakk>G\<turnstile>Methd sig old member_of C; G\<turnstile>Methd sig new member_of D;G\<turnstile>D \<preceq>\<^sub>C C;
G,sig\<turnstile>new overrides\<^sub>S old; ws_prog G\<rbrakk>
\<Longrightarrow> G\<turnstile>D \<prec>\<^sub>C C"
by (drule stat_overrides_commonD) (auto intro: member_of_subcls)
lemma inherited_field_access:
assumes stat_acc: "G\<turnstile>membr of statC accessible_from accC" and
is_field: "is_field membr" and
subclseq: "G \<turnstile> dynC \<preceq>\<^sub>C statC"
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>\<^sub>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"
type_synonym
fspec = "vname \<times> qtname"
translations
(type) "fspec" <= (type) "vname \<times> qtname"
definition
imethds :: "prog \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> mhead) tables" where
"imethds G I =
iface_rec G I (\<lambda>I i ts. (Un_tables ts) \<oplus>\<oplus>
(Option.set \<circ> table_of (map (\<lambda>(s,m). (s,I,m)) (imethods i))))"
text {* methods of an interface, with overriding and inheritance, cf. 9.2 *}
definition
accimethds :: "prog \<Rightarrow> pname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> mhead) tables" where
"accimethds G pack I =
(if G\<turnstile>Iface I accessible_in pack
then imethds G I
else (\<lambda> k. {}))"
text {* only returns imethds if the interface is accessible *}
definition
methd :: "prog \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table" where
"methd G C =
class_rec G C empty
(\<lambda>C c subcls_mthds.
filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)
subcls_mthds
++
table_of (map (\<lambda>(s,m). (s,C,m)) (methods c)))"
text {* @{term "methd G C"}: methods of a class C (statically visible from C),
with inheritance and hiding cf. 8.4.6;
Overriding is captured by @{text dynmethd}.
Every new method with the same signature coalesces the
method of a superclass. *}
definition
accmethd :: "prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table" where
"accmethd G S C =
filter_tab (\<lambda>sig m. G\<turnstile>method sig m of C accessible_from S) (methd G C)"
text {* @{term "accmethd G S C"}: only those methods of @{term "methd G C"},
accessible from S *}
text {* Note the class component in the accessibility filter. The class where
method @{term m} is declared (@{term declC}) isn't necessarily accessible
from the current scope @{term S}. The method can be made accessible
through inheritance, too.
So we must test accessibility of method @{term m} of class @{term C}
(not @{term "declclass m"}) *}
definition
dynmethd :: "prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table" where
"dynmethd G statC dynC =
(\<lambda>sig.
(if G\<turnstile>dynC \<preceq>\<^sub>C statC
then (case methd G statC sig of
None \<Rightarrow> None
| Some statM
\<Rightarrow> (class_rec G dynC empty
(\<lambda>C c subcls_mthds.
subcls_mthds
++
(filter_tab
(\<lambda> _ dynM. G,sig\<turnstile>dynM overrides statM \<or> dynM=statM)
(methd G C) ))
) sig
)
else None))"
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) *}
definition
dynimethd :: "prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table" where
"dynimethd G I dynC =
(\<lambda>sig. if imethds G I sig \<noteq> {}
then methd G dynC sig
else dynmethd G Object dynC sig)"
text {* @{term "dynimethd G I dynC"}: dynamic method lookup of a reference with
dynamic class dynC and static interface type I *}
text {*
When calling an interface method, we must distinguish if the method signature
was defined in the interface or if it must be an Object method in the other
case. If it was an interface method we search the class hierarchy
starting at the dynamic class of the object up to Object to find the
first matching method (@{term methd}). Since all interface methods have
public access the method can't be coalesced due to some odd visibility
effects like in case of dynmethd. The method will be inherited or
overridden in all classes from the first class implementing the interface
down to the actual dynamic class.
*}
definition
dynlookup :: "prog \<Rightarrow> ref_ty \<Rightarrow> qtname \<Rightarrow> (sig,qtname \<times> methd) table" where
"dynlookup G statT dynC =
(case statT of
NullT \<Rightarrow> empty
| IfaceT I \<Rightarrow> dynimethd G I dynC
| ClassT statC \<Rightarrow> dynmethd G statC dynC
| ArrayT ty \<Rightarrow> dynmethd G Object dynC)"
text {* @{term "dynlookup G statT dynC"}: dynamic lookup of a method within the
static reference type statT and the dynamic class dynC.
In a wellformd context statT will not be NullT and in case
statT is an array type, dynC=Object *}
definition
fields :: "prog \<Rightarrow> qtname \<Rightarrow> ((vname \<times> qtname) \<times> field) list" where
"fields G C =
class_rec G C [] (\<lambda>C c ts. map (\<lambda>(n,t). ((n,C),t)) (cfields c) @ ts)"
text {* @{term "fields G C"}
list of fields of a class, including all the fields of the superclasses
(private, inherited and hidden ones) not only the accessible ones
(an instance of a object allocates all these fields *}
definition
accfield :: "prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> (vname, qtname \<times> field) table" where
"accfield G S C =
(let field_tab = table_of((map (\<lambda>((n,d),f).(n,(d,f)))) (fields G C))
in filter_tab (\<lambda>n (declC,f). G\<turnstile> (declC,fdecl (n,f)) of C accessible_from S)
field_tab)"
text {* @{term "accfield G C S"}: fields of a class @{term C} which are
accessible from scope of class
@{term S} with inheritance and hiding, cf. 8.3 *}
text {* note the class component in the accessibility filter (see also
@{term methd}).
The class declaring field @{term f} (@{term declC}) isn't necessarily
accessible from scope @{term S}. The field can be made visible through
inheritance, too. So we must test accessibility of field @{term f} of class
@{term C} (not @{term "declclass f"}) *}
definition
is_methd :: "prog \<Rightarrow> qtname \<Rightarrow> sig \<Rightarrow> bool"
where "is_methd G = (\<lambda>C sig. is_class G C \<and> methd G C sig \<noteq> None)"
definition
efname :: "((vname \<times> qtname) \<times> field) \<Rightarrow> (vname \<times> qtname)"
where "efname = fst"
lemma efname_simp[simp]:"efname (n,f) = n"
by (simp add: efname_def)
subsection "imethds"
lemma imethds_rec: "\<lbrakk>iface G I = Some i; ws_prog G\<rbrakk> \<Longrightarrow>
imethds G I = Un_tables ((\<lambda>J. imethds G J)`set (isuperIfs i)) \<oplus>\<oplus>
(Option.set \<circ> table_of (map (\<lambda>(s,mh). (s,I,mh)) (imethods i)))"
apply (unfold imethds_def)
apply (rule iface_rec [THEN trans])
apply auto
done
(* local lemma *)
lemma imethds_norec:
"\<lbrakk>iface G md = Some i; ws_prog G; table_of (imethods i) sig = Some mh\<rbrakk> \<Longrightarrow>
(md, mh) \<in> imethds G md sig"
apply (subst imethds_rec)
apply assumption+
apply (rule iffD2)
apply (rule overrides_t_Some_iff)
apply (rule disjI1)
apply (auto elim: table_of_map_SomeI)
done
lemma imethds_declI: "\<lbrakk>m \<in> imethds G I sig; ws_prog G; is_iface G I\<rbrakk> \<Longrightarrow>
(\<exists>i. iface G (decliface m) = Some i \<and>
table_of (imethods i) sig = Some (mthd m)) \<and>
(I,decliface m) \<in> (subint1 G)^* \<and> m \<in> imethds G (decliface m) sig"
apply (erule 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 \<in> imethds G I sig"
and ifI: "iface G I = Some i"
and ws: "ws_prog G"
obtains (NewMethod) "table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)) sig = Some im"
| (InheritedMethod) J where "J \<in> set (isuperIfs i)" and "im \<in> imethds G J sig"
using assms by (auto simp add: imethds_rec)
subsection "accimethd"
lemma accimethds_simp [simp]:
"G\<turnstile>Iface I accessible_in pack \<Longrightarrow> accimethds G pack I = imethds G I"
by (simp add: accimethds_def)
lemma accimethdsD:
"im \<in> accimethds G pack I sig
\<Longrightarrow> im \<in> imethds G I sig \<and> G\<turnstile>Iface I accessible_in pack"
by (auto simp add: accimethds_def)
lemma accimethdsI:
"\<lbrakk>im \<in> imethds G I sig;G\<turnstile>Iface I accessible_in pack\<rbrakk>
\<Longrightarrow> im \<in> accimethds G pack I sig"
by simp
subsection "methd"
lemma methd_rec: "\<lbrakk>class G C = Some c; ws_prog G\<rbrakk> \<Longrightarrow>
methd G C
= (if C = Object
then empty
else filter_tab (\<lambda>sig m. G\<turnstile>C inherits method sig m)
(methd G (super c)))
++ table_of (map (\<lambda>(s,m). (s,C,m)) (methods c))"
apply (unfold methd_def)
apply (erule class_rec [THEN trans], assumption)
apply (simp)
done
(* local lemma *)
lemma methd_norec:
"\<lbrakk>class G declC = Some c; ws_prog G;table_of (methods c) sig = Some m\<rbrakk>
\<Longrightarrow> methd G declC sig = Some (declC, m)"
apply (simp only: methd_rec)
apply (rule disjI1 [THEN map_add_Some_iff [THEN iffD2]])
apply (auto elim: table_of_map_SomeI)
done
lemma methd_declC:
"\<lbrakk>methd G C sig = Some m; ws_prog G;is_class G C\<rbrakk> \<Longrightarrow>
(\<exists>d. class G (declclass m)=Some d \<and> table_of (methods d) sig=Some (mthd m)) \<and>
G\<turnstile>C \<preceq>\<^sub>C (declclass m) \<and> methd G (declclass m) sig = Some m"
apply (erule 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 (\<lambda>(s, m). (s, Ca, m)) (methods c)) sig")
apply (force intro: rtrancl_into_rtrancl2)
apply (auto intro: methd_norec)
done
lemma methd_inheritedD:
"\<lbrakk>class G C = Some c; ws_prog G;methd G C sig = Some m\<rbrakk>
\<Longrightarrow> (declclass m \<noteq> C \<longrightarrow> G \<turnstile>C inherits method sig m)"
by (auto simp add: methd_rec)
lemma methd_diff_cls:
"\<lbrakk>ws_prog G; is_class G C; is_class G D;
methd G C sig = m; methd G D sig = n; m\<noteq>n
\<rbrakk> \<Longrightarrow> C\<noteq>D"
by (auto simp add: methd_rec)
lemma method_declared_inI:
"\<lbrakk>table_of (methods c) sig = Some m; class G C = Some c\<rbrakk>
\<Longrightarrow> G\<turnstile>mdecl (sig,m) declared_in C"
by (auto simp add: cdeclaredmethd_def declared_in_def)
lemma methd_declared_in_declclass:
"\<lbrakk>methd G C sig = Some m; ws_prog G;is_class G C\<rbrakk>
\<Longrightarrow> G\<turnstile>Methd sig m declared_in (declclass m)"
by (auto dest: methd_declC method_declared_inI)
lemma member_methd:
assumes member_of: "G\<turnstile>Methd sig m member_of C" and
ws: "ws_prog G"
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 \<and> declclass m = Object"
by (cases set: members) (cases m, auto dest: subcls1D)
with ws Object
show "?Methd Object"
by (cases m)
(auto simp add: declared_in_def cdeclaredmethd_def methd_rec
intro: table_of_mapconst_SomeI)
next
case (Subcls C c)
assume clsC: "class G C = Some c" and
neq_C_Obj: "C \<noteq> Object" and
hyp: "G \<turnstile>Methd sig m member_of super c \<Longrightarrow> ?Methd (super c)" and
member_of: "G \<turnstile>Methd sig m member_of C"
from member_of
show "?Methd C"
proof (cases)
case Immediate
with clsC
have "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig = Some m"
by (cases m)
(auto simp add: declared_in_def cdeclaredmethd_def
intro: table_of_mapconst_SomeI)
with clsC neq_C_Obj ws
show ?thesis
by (simp add: methd_rec)
next
case (Inherited S)
with clsC
have undecl: "G\<turnstile>mid sig undeclared_in C" and
super: "G \<turnstile>Methd sig m member_of (super c)"
by (auto dest: subcls1D)
from clsC undecl
have "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig = None"
by (auto simp add: undeclared_in_def cdeclaredmethd_def
intro: table_of_mapconst_NoneI)
moreover
from Inherited have "G \<turnstile> C inherits (method sig m)"
by (auto simp add: inherits_def)
moreover
note clsC neq_C_Obj ws super hyp
ultimately
show ?thesis
by (auto simp add: methd_rec intro: filter_tab_SomeI)
qed
qed
qed
(*unused*)
lemma finite_methd:"ws_prog G \<Longrightarrow> finite {methd G C sig |sig C. is_class G C}"
apply (rule finite_is_class [THEN finite_SetCompr2])
apply (intro strip)
apply (erule_tac ws_subcls1_induct, assumption)
apply (subst methd_rec)
apply (assumption)
apply (auto intro!: finite_range_map_of finite_range_filter_tab finite_range_map_of_map_add)
done
lemma finite_dom_methd:
"\<lbrakk>ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> finite (dom (methd G C))"
apply (erule_tac ws_subcls1_induct)
apply assumption
apply (subst methd_rec)
apply (assumption)
apply (auto intro!: finite_dom_map_of finite_dom_filter_tab)
done
subsection "accmethd"
lemma accmethd_SomeD:
"accmethd G S C sig = Some m
\<Longrightarrow> methd G C sig = Some m \<and> G\<turnstile>method sig m of C accessible_from S"
by (auto simp add: accmethd_def)
lemma accmethd_SomeI:
"\<lbrakk>methd G C sig = Some m; G\<turnstile>method sig m of C accessible_from S\<rbrakk>
\<Longrightarrow> accmethd G S C sig = Some m"
by (auto simp add: accmethd_def intro: filter_tab_SomeI)
lemma accmethd_declC:
"\<lbrakk>accmethd G S C sig = Some m; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow>
(\<exists>d. class G (declclass m)=Some d \<and>
table_of (methods d) sig=Some (mthd m)) \<and>
G\<turnstile>C \<preceq>\<^sub>C (declclass m) \<and> methd G (declclass m) sig = Some m \<and>
G\<turnstile>method sig m of C accessible_from S"
by (auto dest: accmethd_SomeD methd_declC accmethd_SomeI)
lemma finite_dom_accmethd:
"\<lbrakk>ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> finite (dom (accmethd G S C))"
by (auto simp add: accmethd_def intro: finite_dom_filter_tab finite_dom_methd)
subsection "dynmethd"
lemma dynmethd_rec:
"\<lbrakk>class G dynC = Some c; ws_prog G\<rbrakk> \<Longrightarrow>
dynmethd G statC dynC sig
= (if G\<turnstile>dynC \<preceq>\<^sub>C statC
then (case methd G statC sig of
None \<Rightarrow> None
| Some statM
\<Rightarrow> (case methd G dynC sig of
None \<Rightarrow> dynmethd G statC (super c) sig
| Some dynM \<Rightarrow>
(if G,sig\<turnstile> dynM overrides statM \<or> dynM = statM
then Some dynM
else (dynmethd G statC (super c) sig)
)))
else None)"
(is "_ \<Longrightarrow> _ \<Longrightarrow> ?Dynmethd_def dynC sig = ?Dynmethd_rec dynC c sig")
proof -
assume clsDynC: "class G dynC = Some c" and
ws: "ws_prog G"
then show "?Dynmethd_def dynC sig = ?Dynmethd_rec dynC c sig"
proof (induct rule: ws_class_induct'')
case (Object co)
show "?Dynmethd_def Object sig = ?Dynmethd_rec Object co sig"
proof (cases "G\<turnstile>Object \<preceq>\<^sub>C statC")
case False
then show ?thesis by (simp add: dynmethd_def)
next
case True
then have eq_statC_Obj: "statC = Object" ..
show ?thesis
proof (cases "methd G statC sig")
case None then show ?thesis by (simp add: dynmethd_def)
next
case Some
with True Object ws eq_statC_Obj
show ?thesis
by (auto simp add: dynmethd_def class_rec
intro: filter_tab_SomeI)
qed
qed
next
case (Subcls dynC c sc)
show "?Dynmethd_def dynC sig = ?Dynmethd_rec dynC c sig"
proof (cases "G\<turnstile>dynC \<preceq>\<^sub>C statC")
case False
then show ?thesis by (simp add: dynmethd_def)
next
case True
note subclseq_dynC_statC = True
show ?thesis
proof (cases "methd G statC sig")
case None then show ?thesis by (simp add: dynmethd_def)
next
case (Some statM)
note statM = Some
let ?filter =
"\<lambda>C. filter_tab
(\<lambda>_ dynM. G,sig \<turnstile> dynM overrides statM \<or> dynM = statM)
(methd G C)"
let ?class_rec =
"\<lambda>C. class_rec G C empty
(\<lambda>C c subcls_mthds. subcls_mthds ++ (?filter C))"
from statM Subcls ws subclseq_dynC_statC
have dynmethd_dynC_def:
"?Dynmethd_def dynC sig =
((?class_rec (super c))
++
(?filter dynC)) sig"
by (simp (no_asm_simp) only: dynmethd_def class_rec)
auto
show ?thesis
proof (cases "dynC = statC")
case True
with subclseq_dynC_statC statM dynmethd_dynC_def
have "?Dynmethd_def dynC sig = Some statM"
by (auto intro: 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>\<^sub>C statC"
by (blast dest: subclseq_superD)
show ?thesis
proof (cases "methd G dynC sig")
case None
then have "?filter dynC sig = None"
by (rule filter_tab_None)
then have "?Dynmethd_def dynC sig=?class_rec (super c) sig"
by (simp add: dynmethd_dynC_def)
with subclseq_super_statC statM None
have "?Dynmethd_def dynC sig = ?Dynmethd_def (super c) sig"
by (auto simp add: empty_def dynmethd_def)
with None subclseq_dynC_statC statM
show ?thesis
by simp
next
case (Some dynM)
note dynM = Some
let ?Termination = "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM \<or>
dynM = statM"
show ?thesis
proof (cases "?filter dynC sig")
case None
with dynM
have no_termination: "\<not> ?Termination"
by (simp add: filter_tab_def)
from None
have "?Dynmethd_def dynC sig=?class_rec (super c) sig"
by (simp add: dynmethd_dynC_def)
with subclseq_super_statC statM dynM no_termination
show ?thesis
by (auto simp add: empty_def dynmethd_def)
next
case Some
with dynM
have "termination": "?Termination"
by (auto)
with Some dynM
have "?Dynmethd_def dynC sig=Some dynM"
by (auto simp add: dynmethd_dynC_def)
with subclseq_super_statC statM dynM "termination"
show ?thesis
by (auto simp add: dynmethd_def)
qed
qed
qed
qed
qed
qed
qed
lemma dynmethd_C_C:"\<lbrakk>is_class G C; ws_prog G\<rbrakk>
\<Longrightarrow> dynmethd G C C sig = methd G C sig"
apply (auto simp add: dynmethd_rec)
done
lemma dynmethdSomeD:
"\<lbrakk>dynmethd G statC dynC sig = Some dynM; is_class G dynC; ws_prog G\<rbrakk>
\<Longrightarrow> G\<turnstile>dynC \<preceq>\<^sub>C statC \<and> (\<exists> statM. methd G statC sig = Some statM)"
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 \<noteq> statM"
and "G,sig\<turnstile>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\<noteq>statM"
shows "dynC \<noteq> Object"
proof -
from is_cls_dynC obtain dc where clsDynC: "class G dynC = Some dc" by blast
from clsDynC ws dynM statM neq_dynM_statM
show ?thesis (is "?P dynC")
proof (induct rule: ws_class_induct)
case (Object co)
with ws have "statC = Object"
by (auto simp add: dynmethd_rec)
with ws Object show "?P Object" by (auto simp add: dynmethd_C_C)
next
case (Subcls dynC c)
with ws show "?P dynC"
by (auto simp add: dynmethd_rec)
qed
qed
lemma dynmethd_Some_rec_cases:
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 \<noteq> dynM"
and "G,sig\<turnstile> dynM overrides statM"
| (Recursion) "dynC \<noteq> 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:
"\<lbrakk>dynmethd G statC dynC sig = Some m;
is_class G statC;ws_prog G
\<rbrakk> \<Longrightarrow>
(\<exists>d. class G (declclass m)=Some d \<and> table_of (methods d) sig=Some (mthd m)) \<and>
G\<turnstile>dynC \<preceq>\<^sub>C (declclass m) \<and> methd G (declclass m) sig = Some m"
proof -
assume is_cls_statC: "is_class G statC"
assume ws: "ws_prog G"
assume m: "dynmethd G statC dynC sig = Some m"
from m
have "G\<turnstile>dynC \<preceq>\<^sub>C statC" by (auto simp add: dynmethd_def)
from this is_cls_statC
have is_cls_dynC: "is_class G dynC" by (rule subcls_is_class2)
from is_cls_dynC ws m
show ?thesis (is "?P dynC")
proof (induct rule: ws_class_induct')
case (Object co)
with ws have "statC=Object" by (auto simp add: dynmethd_rec)
with ws Object
show "?P Object"
by (auto simp add: dynmethd_C_C dest: methd_declC)
next
case (Subcls dynC c)
assume hyp: "dynmethd G statC (super c) sig = Some m \<Longrightarrow> ?P (super c)" and
clsDynC: "class G dynC = Some c" and
m': "dynmethd G statC dynC sig = Some m" and
neq_dynC_Obj: "dynC \<noteq> Object"
from ws this obtain statM where
subclseq_dynC_statC: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
statM: "methd G statC sig = Some statM"
by (blast dest: dynmethdSomeD)
from clsDynC neq_dynC_Obj
have subclseq_dynC_super: "G\<turnstile>dynC \<preceq>\<^sub>C (super c)"
by (auto intro: subcls1I)
from m' clsDynC ws
show "?P dynC"
proof (cases rule: dynmethd_Some_rec_cases)
case Static
with is_cls_statC ws subclseq_dynC_statC
show ?thesis
by (auto intro: rtrancl_trans dest: methd_declC)
next
case Override
with clsDynC ws
show ?thesis
by (auto dest: methd_declC)
next
case Recursion
with hyp subclseq_dynC_super
show ?thesis
by (auto intro: rtrancl_trans)
qed
qed
qed
lemma methd_Some_dynmethd_Some:
assumes statM: "methd G statC sig = Some statM" and
subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
is_cls_statC: "is_class G statC" and
ws: "ws_prog G"
shows "\<exists> dynM. dynmethd G statC dynC sig = Some dynM"
(is "?P dynC")
proof -
from subclseq is_cls_statC
have is_cls_dynC: "is_class G dynC" by (rule subcls_is_class2)
then obtain dc where
clsDynC: "class G dynC = Some dc" by blast
from clsDynC ws subclseq
show "?thesis"
proof (induct rule: ws_class_induct)
case (Object co)
with ws have "statC = Object"
by (auto)
with ws Object statM
show "?P Object"
by (auto simp add: dynmethd_C_C)
next
case (Subcls dynC dc)
assume clsDynC': "class G dynC = Some dc"
assume neq_dynC_Obj: "dynC \<noteq> Object"
assume hyp: "G\<turnstile>super dc\<preceq>\<^sub>C statC \<Longrightarrow> ?P (super dc)"
assume subclseq': "G\<turnstile>dynC\<preceq>\<^sub>C statC"
then
show "?P dynC"
proof (cases rule: subclseq_cases)
case Eq
with ws statM clsDynC'
show ?thesis
by (auto simp add: dynmethd_rec)
next
case Subcls
assume "G\<turnstile>dynC\<prec>\<^sub>C statC"
from this clsDynC'
have "G\<turnstile>super dc\<preceq>\<^sub>C statC" by (rule subcls_superD)
with hyp ws clsDynC' subclseq' statM
show ?thesis
by (auto simp add: dynmethd_rec)
qed
qed
qed
lemma dynmethd_cases:
assumes statM: "methd G statC sig = Some statM"
and subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC"
and is_cls_statC: "is_class G statC"
and ws: "ws_prog G"
obtains (Static) "dynmethd G statC dynC sig = Some statM"
| (Overrides) dynM where "dynmethd G statC dynC sig = Some dynM"
and "dynM \<noteq> statM" and "G,sig\<turnstile>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)
then obtain dc where
clsDynC: "class G dynC = Some dc" by blast
from statM subclseq is_cls_statC ws
obtain dynM where dynM: "dynmethd G statC dynC sig = Some dynM"
by (blast dest: methd_Some_dynmethd_Some)
from dynM is_cls_dynC ws
show ?thesis
proof (cases rule: dynmethd_Some_cases)
case Static
with hyp_static dynM statM show ?thesis by simp
next
case Overrides
with hyp_override dynM statM show ?thesis by simp
qed
qed
lemma ws_dynmethd:
assumes statM: "methd G statC sig = Some statM" and
subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
is_cls_statC: "is_class G statC" and
ws: "ws_prog G"
shows
"\<exists> dynM. dynmethd G statC dynC sig = Some dynM \<and>
is_static dynM = is_static statM \<and> G\<turnstile>resTy dynM\<preceq>resTy statM"
proof -
from statM subclseq is_cls_statC ws
show ?thesis
proof (cases rule: dynmethd_cases)
case Static
with statM
show ?thesis
by simp
next
case Overrides
with ws
show ?thesis
by (auto dest: ws_overrides_commonD)
qed
qed
subsection "dynlookup"
lemma dynlookup_cases:
assumes "dynlookup G statT dynC sig = x"
obtains (NullT) "statT = NullT" and "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: "\<lbrakk>class G C = Some c; ws_prog G\<rbrakk> \<Longrightarrow>
fields G C = map (\<lambda>(fn,ft). ((fn,C),ft)) (cfields c) @
(if C = Object then [] else fields G (super c))"
apply (simp only: fields_def)
apply (erule class_rec [THEN trans])
apply assumption
apply clarsimp
done
(* local lemma *)
lemma fields_norec:
"\<lbrakk>class G fd = Some c; ws_prog G; table_of (cfields c) fn = Some f\<rbrakk>
\<Longrightarrow> table_of (fields G fd) (fn,fd) = Some f"
apply (subst fields_rec)
apply assumption+
apply (subst map_of_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 (\<lambda>(fn,ft). ((fn,C),ft)) (cfields c)) efn = Some f
\<Longrightarrow> (declclassf efn) = C \<and> table_of (cfields c) (fname efn) = Some f"
apply (case_tac "efn")
by auto
lemma fields_declC:
"\<lbrakk>table_of (fields G C) efn = Some f; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow>
(\<exists>d. class G (declclassf efn) = Some d \<and>
table_of (cfields d) (fname efn)=Some f) \<and>
G\<turnstile>C \<preceq>\<^sub>C (declclassf efn) \<and> table_of (fields G (declclassf efn)) efn = Some f"
apply (erule 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 (\<lambda>fn. 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 (\<lambda>fn. 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: "\<And>y. \<lbrakk>ws_prog G; class G C = Some c;cfields c = [];
C \<noteq> Object \<longrightarrow> class G (super c) = Some y \<and> fields G (super c) = []\<rbrakk> \<Longrightarrow>
fields G C = []"
apply (subst fields_rec)
apply assumption
apply auto
done
(* easier than with table_of *)
lemma fields_mono_lemma:
"\<lbrakk>x \<in> set (fields G C); G\<turnstile>D \<preceq>\<^sub>C C; ws_prog G\<rbrakk>
\<Longrightarrow> x \<in> set (fields G D)"
apply (erule 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:
"\<lbrakk>(efn,fd) \<in> set (fields G (super c)); fc \<in> set (cfields c); ws_prog G;
fname efn = fname fc; declclassf efn = C;
class G C = Some c; C \<noteq> Object; class G (super c) = Some d\<rbrakk> \<Longrightarrow> R"
apply (frule_tac ws_prog_cdeclD [THEN conjunct2], assumption, assumption)
apply (drule_tac weak_map_of_SomeI)
apply (frule_tac subcls1I [THEN subcls1_irrefl], assumption, assumption)
apply (auto dest: fields_declC [THEN conjunct2 [THEN conjunct1[THEN rtranclD]]])
done
lemma ws_unique_fields: "\<lbrakk>is_class G C; ws_prog G;
\<And>C c. \<lbrakk>class G C = Some c\<rbrakk> \<Longrightarrow> unique (cfields c) \<rbrakk> \<Longrightarrow>
unique (fields G C)"
apply (rule ws_subcls1_induct, assumption, assumption)
apply (subst fields_rec, assumption)
apply (auto intro!: unique_map_inj inj_onI
elim!: unique_append ws_unique_fields_lemma fields_norec)
done
subsection "accfield"
lemma accfield_fields:
"accfield G S C fn = Some f
\<Longrightarrow> table_of (fields G C) (fn, declclass f) = Some (fld f)"
apply (simp only: accfield_def Let_def)
apply (rule table_of_remap_SomeD)
apply auto
done
lemma accfield_declC_is_class:
"\<lbrakk>is_class G C; accfield G S C en = Some (fd, f); ws_prog G\<rbrakk> \<Longrightarrow>
is_class G fd"
apply (drule accfield_fields)
apply (drule fields_declC [THEN conjunct1], assumption)
apply auto
done
lemma accfield_accessibleD:
"accfield G S C fn = Some f \<Longrightarrow> G\<turnstile>Field fn f of C accessible_from S"
by (auto simp add: accfield_def Let_def)
subsection "is methd"
lemma is_methdI:
"\<lbrakk>class G C = Some y; methd G C sig = Some b\<rbrakk> \<Longrightarrow> is_methd G C sig"
apply (unfold is_methd_def)
apply auto
done
lemma is_methdD:
"is_methd G C sig \<Longrightarrow> class G C \<noteq> None \<and> methd G C sig \<noteq> None"
apply (unfold is_methd_def)
apply auto
done
lemma finite_is_methd:
"ws_prog G \<Longrightarrow> finite (Collect (split (is_methd G)))"
apply (unfold is_methd_def)
apply (subst SetCompr_Sigma_eq)
apply (rule finite_is_class [THEN finite_SigmaI])
apply (simp only: mem_Collect_eq)
apply (fold dom_def)
apply (erule finite_dom_methd)
apply assumption
done
section "calculation of the superclasses of a class"
definition
superclasses :: "prog \<Rightarrow> qtname \<Rightarrow> qtname set" where
"superclasses G C = class_rec G C {}
(\<lambda> C c superclss. (if C=Object
then {}
else insert (super c) superclss))"
lemma superclasses_rec: "\<lbrakk>class G C = Some c; ws_prog G\<rbrakk> \<Longrightarrow>
superclasses G C
= (if (C=Object)
then {}
else insert (super c) (superclasses G (super c)))"
apply (unfold superclasses_def)
apply (erule class_rec [THEN trans], assumption)
apply (simp)
done
lemma superclasses_mono:
assumes clsrel: "G\<turnstile>C\<prec>\<^sub>C D"
and ws: "ws_prog G"
and cls_C: "class G C = Some c"
and wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
\<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
and x: "x\<in>superclasses G D"
shows "x\<in>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)
moreover note wf ws
moreover from calculation
have "x\<in>superclasses G 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 ?case
by (auto intro: no_subcls1_Object simp add: superclasses_rec)
qed
lemma subclsEval:
assumes clsrel: "G\<turnstile>C\<prec>\<^sub>C D"
and ws: "ws_prog G"
and cls_C: "class G C = Some c"
and wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
\<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
shows "D\<in>superclasses G C" using clsrel cls_C
proof (induct arbitrary: c rule: converse_trancl_induct)
case (base C)
with ws wf show ?case
by (auto intro: no_subcls1_Object simp add: superclasses_rec subcls1_def)
next
case (step C S)
with ws wf show ?case
by - (rule superclasses_mono,
auto dest: no_subcls1_Object simp add: subcls1_def )
qed
end