Added conditional and (&&) and or (||).
(* Title: HOL/Bali/Eval.thy
ID: $Id$
Author: David von Oheimb
License: GPL (GNU GENERAL PUBLIC LICENSE)
*)
header {* Operational evaluation (big-step) semantics of Java expressions and
statements
*}
theory Eval = State + DeclConcepts:
text {*
improvements over Java Specification 1.0:
\begin{itemize}
\item dynamic method lookup does not need to consider the return type
(cf.15.11.4.4)
\item throw raises a NullPointer exception if a null reference is given, and
each throw of a standard exception yield a fresh exception object
(was not specified)
\item if there is not enough memory even to allocate an OutOfMemory exception,
evaluation/execution fails, i.e. simply stops (was not specified)
\item array assignment checks lhs (and may throw exceptions) before evaluating
rhs
\item fixed exact positions of class initializations
(immediate at first active use)
\end{itemize}
design issues:
\begin{itemize}
\item evaluation vs. (single-step) transition semantics
evaluation semantics chosen, because:
\begin{itemize}
\item[++] less verbose and therefore easier to read (and to handle in proofs)
\item[+] more abstract
\item[+] intermediate values (appearing in recursive rules) need not be
stored explicitly, e.g. no call body construct or stack of invocation
frames containing local variables and return addresses for method calls
needed
\item[+] convenient rule induction for subject reduction theorem
\item[-] no interleaving (for parallelism) can be described
\item[-] stating a property of infinite executions requires the meta-level
argument that this property holds for any finite prefixes of it
(e.g. stopped using a counter that is decremented to zero and then
throwing an exception)
\end{itemize}
\item unified evaluation for variables, expressions, expression lists,
statements
\item the value entry in statement rules is redundant
\item the value entry in rules is irrelevant in case of exceptions, but its full
inclusion helps to make the rule structure independent of exception occurence.
\item as irrelevant value entries are ignored, it does not matter if they are
unique.
For simplicity, (fixed) arbitrary values are preferred over "free" values.
\item the rule format is such that the start state may contain an exception.
\begin{itemize}
\item[++] faciliates exception handling
\item[+] symmetry
\end{itemize}
\item the rules are defined carefully in order to be applicable even in not
type-correct situations (yielding undefined values),
e.g. @{text "the_Addr (Val (Bool b)) = arbitrary"}.
\begin{itemize}
\item[++] fewer rules
\item[-] less readable because of auxiliary functions like @{text the_Addr}
\end{itemize}
Alternative: "defensive" evaluation throwing some InternalError exception
in case of (impossible, for correct programs) type mismatches
\item there is exactly one rule per syntactic construct
\begin{itemize}
\item[+] no redundancy in case distinctions
\end{itemize}
\item halloc fails iff there is no free heap address. When there is
only one free heap address left, it returns an OutOfMemory exception.
In this way it is guaranteed that when an OutOfMemory exception is thrown for
the first time, there is a free location on the heap to allocate it.
\item the allocation of objects that represent standard exceptions is deferred
until execution of any enclosing catch clause, which is transparent to
the program.
\begin{itemize}
\item[-] requires an auxiliary execution relation
\item[++] avoids copies of allocation code and awkward case distinctions
(whether there is enough memory to allocate the exception) in
evaluation rules
\end{itemize}
\item unfortunately @{text new_Addr} is not directly executable because of
Hilbert operator.
\end{itemize}
simplifications:
\begin{itemize}
\item local variables are initialized with default values
(no definite assignment)
\item garbage collection not considered, therefore also no finalizers
\item stack overflow and memory overflow during class initialization not
modelled
\item exceptions in initializations not replaced by ExceptionInInitializerError
\end{itemize}
*}
types vvar = "val \<times> (val \<Rightarrow> state \<Rightarrow> state)"
vals = "(val, vvar, val list) sum3"
translations
"vvar" <= (type) "val \<times> (val \<Rightarrow> state \<Rightarrow> state)"
"vals" <= (type)"(val, vvar, val list) sum3"
syntax (xsymbols)
dummy_res :: "vals" ("\<diamondsuit>")
translations
"\<diamondsuit>" == "In1 Unit"
constdefs
arbitrary3 :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> vals"
"arbitrary3 \<equiv> sum3_case (In1 \<circ> sum_case (\<lambda>x. arbitrary) (\<lambda>x. Unit))
(\<lambda>x. In2 arbitrary) (\<lambda>x. In3 arbitrary)"
lemma [simp]: "arbitrary3 (In1l x) = In1 arbitrary"
by (simp add: arbitrary3_def)
lemma [simp]: "arbitrary3 (In1r x) = \<diamondsuit>"
by (simp add: arbitrary3_def)
lemma [simp]: "arbitrary3 (In2 x) = In2 arbitrary"
by (simp add: arbitrary3_def)
lemma [simp]: "arbitrary3 (In3 x) = In3 arbitrary"
by (simp add: arbitrary3_def)
section "exception throwing and catching"
constdefs
throw :: "val \<Rightarrow> abopt \<Rightarrow> abopt"
"throw a' x \<equiv> abrupt_if True (Some (Xcpt (Loc (the_Addr a')))) (np a' x)"
lemma throw_def2:
"throw a' x = abrupt_if True (Some (Xcpt (Loc (the_Addr a')))) (np a' x)"
apply (unfold throw_def)
apply (simp (no_asm))
done
constdefs
fits :: "prog \<Rightarrow> st \<Rightarrow> val \<Rightarrow> ty \<Rightarrow> bool" ("_,_\<turnstile>_ fits _"[61,61,61,61]60)
"G,s\<turnstile>a' fits T \<equiv> (\<exists>rt. T=RefT rt) \<longrightarrow> a'=Null \<or> G\<turnstile>obj_ty(lookup_obj s a')\<preceq>T"
lemma fits_Null [simp]: "G,s\<turnstile>Null fits T"
by (simp add: fits_def)
lemma fits_Addr_RefT [simp]:
"G,s\<turnstile>Addr a fits RefT t = G\<turnstile>obj_ty (the (heap s a))\<preceq>RefT t"
by (simp add: fits_def)
lemma fitsD: "\<And>X. G,s\<turnstile>a' fits T \<Longrightarrow> (\<exists>pt. T = PrimT pt) \<or>
(\<exists>t. T = RefT t) \<and> a' = Null \<or>
(\<exists>t. T = RefT t) \<and> a' \<noteq> Null \<and> G\<turnstile>obj_ty (lookup_obj s a')\<preceq>T"
apply (unfold fits_def)
apply (case_tac "\<exists>pt. T = PrimT pt")
apply simp_all
apply (case_tac "T")
defer
apply (case_tac "a' = Null")
apply simp_all
done
constdefs
catch ::"prog \<Rightarrow> state \<Rightarrow> qtname \<Rightarrow> bool" ("_,_\<turnstile>catch _"[61,61,61]60)
"G,s\<turnstile>catch C\<equiv>\<exists>xc. abrupt s=Some (Xcpt xc) \<and>
G,store s\<turnstile>Addr (the_Loc xc) fits Class C"
lemma catch_Norm [simp]: "\<not>G,Norm s\<turnstile>catch tn"
apply (unfold catch_def)
apply (simp (no_asm))
done
lemma catch_XcptLoc [simp]:
"G,(Some (Xcpt (Loc a)),s)\<turnstile>catch C = G,s\<turnstile>Addr a fits Class C"
apply (unfold catch_def)
apply (simp (no_asm))
done
constdefs
new_xcpt_var :: "vname \<Rightarrow> state \<Rightarrow> state"
"new_xcpt_var vn \<equiv>
\<lambda>(x,s). Norm (lupd(VName vn\<mapsto>Addr (the_Loc (the_Xcpt (the x)))) s)"
lemma new_xcpt_var_def2 [simp]:
"new_xcpt_var vn (x,s) =
Norm (lupd(VName vn\<mapsto>Addr (the_Loc (the_Xcpt (the x)))) s)"
apply (unfold new_xcpt_var_def)
apply (simp (no_asm))
done
section "misc"
constdefs
assign :: "('a \<Rightarrow> state \<Rightarrow> state) \<Rightarrow> 'a \<Rightarrow> state \<Rightarrow> state"
"assign f v \<equiv> \<lambda>(x,s). let (x',s') = (if x = None then f v else id) (x,s)
in (x',if x' = None then s' else s)"
(*
lemma assign_Norm_Norm [simp]:
"f v \<lparr>abrupt=None,store=s\<rparr> = \<lparr>abrupt=None,store=s'\<rparr>
\<Longrightarrow> assign f v \<lparr>abrupt=None,store=s\<rparr> = \<lparr>abrupt=None,store=s'\<rparr>"
by (simp add: assign_def Let_def)
*)
lemma assign_Norm_Norm [simp]:
"f v (Norm s) = Norm s' \<Longrightarrow> assign f v (Norm s) = Norm s'"
by (simp add: assign_def Let_def)
(*
lemma assign_Norm_Some [simp]:
"\<lbrakk>abrupt (f v \<lparr>abrupt=None,store=s\<rparr>) = Some y\<rbrakk>
\<Longrightarrow> assign f v \<lparr>abrupt=None,store=s\<rparr> = \<lparr>abrupt=Some y,store =s\<rparr>"
by (simp add: assign_def Let_def split_beta)
*)
lemma assign_Norm_Some [simp]:
"\<lbrakk>abrupt (f v (Norm s)) = Some y\<rbrakk>
\<Longrightarrow> assign f v (Norm s) = (Some y,s)"
by (simp add: assign_def Let_def split_beta)
lemma assign_Some [simp]:
"assign f v (Some x,s) = (Some x,s)"
by (simp add: assign_def Let_def split_beta)
lemma assign_supd [simp]:
"assign (\<lambda>v. supd (f v)) v (x,s)
= (x, if x = None then f v s else s)"
apply auto
done
lemma assign_raise_if [simp]:
"assign (\<lambda>v (x,s). ((raise_if (b s v) xcpt) x, f v s)) v (x, s) =
(raise_if (b s v) xcpt x, if x=None \<and> \<not>b s v then f v s else s)"
apply (case_tac "x = None")
apply auto
done
(*
lemma assign_raise_if [simp]:
"assign (\<lambda>v s. \<lparr>abrupt=(raise_if (b (store s) v) xcpt) (abrupt s),
store = f v (store s)\<rparr>) v s =
\<lparr>abrupt=raise_if (b (store s) v) xcpt (abrupt s),
store= if (abrupt s)=None \<and> \<not>b (store s) v
then f v (store s) else (store s)\<rparr>"
apply (case_tac "abrupt s = None")
apply auto
done
*)
constdefs
init_comp_ty :: "ty \<Rightarrow> stmt"
"init_comp_ty T \<equiv> if (\<exists>C. T = Class C) then Init (the_Class T) else Skip"
lemma init_comp_ty_PrimT [simp]: "init_comp_ty (PrimT pt) = Skip"
apply (unfold init_comp_ty_def)
apply (simp (no_asm))
done
constdefs
(*
target :: "inv_mode \<Rightarrow> st \<Rightarrow> val \<Rightarrow> ref_ty \<Rightarrow> qtname"
"target m s a' t
\<equiv> if m = IntVir
then obj_class (lookup_obj s a')
else the_Class (RefT t)"
*)
invocation_class :: "inv_mode \<Rightarrow> st \<Rightarrow> val \<Rightarrow> ref_ty \<Rightarrow> qtname"
"invocation_class m s a' statT
\<equiv> (case m of
Static \<Rightarrow> if (\<exists> statC. statT = ClassT statC)
then the_Class (RefT statT)
else Object
| SuperM \<Rightarrow> the_Class (RefT statT)
| IntVir \<Rightarrow> obj_class (lookup_obj s a'))"
invocation_declclass::"prog \<Rightarrow> inv_mode \<Rightarrow> st \<Rightarrow> val \<Rightarrow> ref_ty \<Rightarrow> sig \<Rightarrow> qtname"
"invocation_declclass G m s a' statT sig
\<equiv> declclass (the (dynlookup G statT
(invocation_class m s a' statT)
sig))"
lemma invocation_class_IntVir [simp]:
"invocation_class IntVir s a' statT = obj_class (lookup_obj s a')"
by (simp add: invocation_class_def)
lemma dynclass_SuperM [simp]:
"invocation_class SuperM s a' statT = the_Class (RefT statT)"
by (simp add: invocation_class_def)
(*
lemma invocation_class_notIntVir [simp]:
"m \<noteq> IntVir \<Longrightarrow> invocation_class m s a' statT = the_Class (RefT statT)"
by (simp add: invocation_class_def)
*)
lemma invocation_class_Static [simp]:
"invocation_class Static s a' statT = (if (\<exists> statC. statT = ClassT statC)
then the_Class (RefT statT)
else Object)"
by (simp add: invocation_class_def)
constdefs
init_lvars :: "prog \<Rightarrow> qtname \<Rightarrow> sig \<Rightarrow> inv_mode \<Rightarrow> val \<Rightarrow> val list \<Rightarrow>
state \<Rightarrow> state"
"init_lvars G C sig mode a' pvs
\<equiv> \<lambda> (x,s).
let m = mthd (the (methd G C sig));
l = \<lambda> k.
(case k of
EName e
\<Rightarrow> (case e of
VNam v \<Rightarrow> (init_vals (table_of (lcls (mbody m)))
((pars m)[\<mapsto>]pvs)) v
| Res \<Rightarrow> Some (default_val (resTy m)))
| This
\<Rightarrow> (if mode=Static then None else Some a'))
in set_lvars l (if mode = Static then x else np a' x,s)"
lemma init_lvars_def2: "init_lvars G C sig mode a' pvs (x,s) =
set_lvars
(\<lambda> k.
(case k of
EName e
\<Rightarrow> (case e of
VNam v
\<Rightarrow> (init_vals
(table_of (lcls (mbody (mthd (the (methd G C sig))))))
((pars (mthd (the (methd G C sig))))[\<mapsto>]pvs)) v
| Res \<Rightarrow> Some (default_val (resTy (mthd (the (methd G C sig))))))
| This
\<Rightarrow> (if mode=Static then None else Some a')))
(if mode = Static then x else np a' x,s)"
apply (unfold init_lvars_def)
apply (simp (no_asm) add: Let_def)
done
constdefs
body :: "prog \<Rightarrow> qtname \<Rightarrow> sig \<Rightarrow> expr"
"body G C sig \<equiv> let m = the (methd G C sig)
in Body (declclass m) (stmt (mbody (mthd m)))"
lemma body_def2:
"body G C sig = Body (declclass (the (methd G C sig)))
(stmt (mbody (mthd (the (methd G C sig)))))"
apply (unfold body_def Let_def)
apply auto
done
section "variables"
constdefs
lvar :: "lname \<Rightarrow> st \<Rightarrow> vvar"
"lvar vn s \<equiv> (the (locals s vn), \<lambda>v. supd (lupd(vn\<mapsto>v)))"
fvar :: "qtname \<Rightarrow> bool \<Rightarrow> vname \<Rightarrow> val \<Rightarrow> state \<Rightarrow> vvar \<times> state"
"fvar C stat fn a' s
\<equiv> let (oref,xf) = if stat then (Stat C,id)
else (Heap (the_Addr a'),np a');
n = Inl (fn,C);
f = (\<lambda>v. supd (upd_gobj oref n v))
in ((the (values (the (globs (store s) oref)) n),f),abupd xf s)"
(*
"fvar C stat fn a' s
\<equiv> let (oref,xf) = if stat then (Stat C,id)
else (Heap (the_Addr a'),np a');
n = Inl (fn,C);
f = (\<lambda>v. supd (upd_gobj oref n v))
in ((the (values (the (globs (store s) oref)) n),f),abupd xf s)"
*)
avar :: "prog \<Rightarrow> val \<Rightarrow> val \<Rightarrow> state \<Rightarrow> vvar \<times> state"
"avar G i' a' s
\<equiv> let oref = Heap (the_Addr a');
i = the_Intg i';
n = Inr i;
(T,k,cs) = the_Arr (globs (store s) oref);
f = (\<lambda>v (x,s). (raise_if (\<not>G,s\<turnstile>v fits T)
ArrStore x
,upd_gobj oref n v s))
in ((the (cs n),f)
,abupd (raise_if (\<not>i in_bounds k) IndOutBound \<circ> np a') s)"
lemma fvar_def2: "fvar C stat fn a' s =
((the
(values
(the (globs (store s) (if stat then Stat C else Heap (the_Addr a'))))
(Inl (fn,C)))
,(\<lambda>v. supd (upd_gobj (if stat then Stat C else Heap (the_Addr a'))
(Inl (fn,C))
v)))
,abupd (if stat then id else np a') s)
"
apply (unfold fvar_def)
apply (simp (no_asm) add: Let_def split_beta)
done
lemma avar_def2: "avar G i' a' s =
((the ((snd(snd(the_Arr (globs (store s) (Heap (the_Addr a'))))))
(Inr (the_Intg i')))
,(\<lambda>v (x,s'). (raise_if (\<not>G,s'\<turnstile>v fits (fst(the_Arr (globs (store s)
(Heap (the_Addr a'))))))
ArrStore x
,upd_gobj (Heap (the_Addr a'))
(Inr (the_Intg i')) v s')))
,abupd (raise_if (\<not>(the_Intg i') in_bounds (fst(snd(the_Arr (globs (store s)
(Heap (the_Addr a'))))))) IndOutBound \<circ> np a')
s)"
apply (unfold avar_def)
apply (simp (no_asm) add: Let_def split_beta)
done
constdefs
check_field_access::
"prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> vname \<Rightarrow> bool \<Rightarrow> val \<Rightarrow> state \<Rightarrow> state"
"check_field_access G accC statDeclC fn stat a' s
\<equiv> let oref = if stat then Stat statDeclC
else Heap (the_Addr a');
dynC = case oref of
Heap a \<Rightarrow> obj_class (the (globs (store s) oref))
| Stat C \<Rightarrow> C;
f = (the (table_of (DeclConcepts.fields G dynC) (fn,statDeclC)))
in abupd
(error_if (\<not> G\<turnstile>Field fn (statDeclC,f) in dynC dyn_accessible_from accC)
AccessViolation)
s"
constdefs
check_method_access::
"prog \<Rightarrow> qtname \<Rightarrow> ref_ty \<Rightarrow> inv_mode \<Rightarrow> sig \<Rightarrow> val \<Rightarrow> state \<Rightarrow> state"
"check_method_access G accC statT mode sig a' s
\<equiv> let invC = invocation_class mode (store s) a' statT;
dynM = the (dynlookup G statT invC sig)
in abupd
(error_if (\<not> G\<turnstile>Methd sig dynM in invC dyn_accessible_from accC)
AccessViolation)
s"
section "evaluation judgments"
consts eval_unop :: "unop \<Rightarrow> val \<Rightarrow> val"
primrec
"eval_unop UPlus v = Intg (the_Intg v)"
"eval_unop UMinus v = Intg (- (the_Intg v))"
"eval_unop UBitNot v = Intg 42" -- "FIXME: Not yet implemented"
"eval_unop UNot v = Bool (\<not> the_Bool v)"
consts eval_binop :: "binop \<Rightarrow> val \<Rightarrow> val \<Rightarrow> val"
primrec
"eval_binop Mul v1 v2 = Intg ((the_Intg v1) * (the_Intg v2))"
"eval_binop Div v1 v2 = Intg ((the_Intg v1) div (the_Intg v2))"
"eval_binop Mod v1 v2 = Intg ((the_Intg v1) mod (the_Intg v2))"
"eval_binop Plus v1 v2 = Intg ((the_Intg v1) + (the_Intg v2))"
"eval_binop Minus v1 v2 = Intg ((the_Intg v1) - (the_Intg v2))"
-- "Be aware of the explicit coercion of the shift distance to nat"
"eval_binop LShift v1 v2 = Intg ((the_Intg v1) * (2^(nat (the_Intg v2))))"
"eval_binop RShift v1 v2 = Intg ((the_Intg v1) div (2^(nat (the_Intg v2))))"
"eval_binop RShiftU v1 v2 = Intg 42" --"FIXME: Not yet implemented"
"eval_binop Less v1 v2 = Bool ((the_Intg v1) < (the_Intg v2))"
"eval_binop Le v1 v2 = Bool ((the_Intg v1) \<le> (the_Intg v2))"
"eval_binop Greater v1 v2 = Bool ((the_Intg v2) < (the_Intg v1))"
"eval_binop Ge v1 v2 = Bool ((the_Intg v2) \<le> (the_Intg v1))"
"eval_binop Eq v1 v2 = Bool (v1=v2)"
"eval_binop Neq v1 v2 = Bool (v1\<noteq>v2)"
"eval_binop BitAnd v1 v2 = Intg 42" -- "FIXME: Not yet implemented"
"eval_binop And v1 v2 = Bool ((the_Bool v1) \<and> (the_Bool v2))"
"eval_binop BitXor v1 v2 = Intg 42" -- "FIXME: Not yet implemented"
"eval_binop Xor v1 v2 = Bool ((the_Bool v1) \<noteq> (the_Bool v2))"
"eval_binop BitOr v1 v2 = Intg 42" -- "FIXME: Not yet implemented"
"eval_binop Or v1 v2 = Bool ((the_Bool v1) \<or> (the_Bool v2))"
"eval_binop CondAnd v1 v2 = Bool ((the_Bool v1) \<and> (the_Bool v2))"
"eval_binop CondOr v1 v2 = Bool ((the_Bool v1) \<or> (the_Bool v2))"
constdefs need_second_arg :: "binop \<Rightarrow> val \<Rightarrow> bool"
"need_second_arg binop v1 \<equiv> \<not> ((binop=CondAnd \<and> \<not> the_Bool v1) \<or>
(binop=CondOr \<and> the_Bool v1))"
text {* @{term CondAnd} and @{term CondOr} only evalulate the second argument
if the value isn't already determined by the first argument*}
lemma need_second_arg_CondAnd [simp]: "need_second_arg CondAnd (Bool b) = b"
by (simp add: need_second_arg_def)
lemma need_second_arg_CondOr [simp]: "need_second_arg CondOr (Bool b) = (\<not> b)"
by (simp add: need_second_arg_def)
lemma need_second_arg_strict[simp]:
"\<lbrakk>binop\<noteq>CondAnd; binop\<noteq>CondOr\<rbrakk> \<Longrightarrow> need_second_arg binop b"
by (cases binop)
(simp_all add: need_second_arg_def)
consts
eval :: "prog \<Rightarrow> (state \<times> term \<times> vals \<times> state) set"
halloc:: "prog \<Rightarrow> (state \<times> obj_tag \<times> loc \<times> state) set"
sxalloc:: "prog \<Rightarrow> (state \<times> state) set"
syntax
eval ::"[prog,state,term,vals*state]=>bool"("_|-_ -_>-> _" [61,61,80, 61]60)
exec ::"[prog,state,stmt ,state]=>bool"("_|-_ -_-> _" [61,61,65, 61]60)
evar ::"[prog,state,var ,vvar,state]=>bool"("_|-_ -_=>_-> _"[61,61,90,61,61]60)
eval_::"[prog,state,expr ,val, state]=>bool"("_|-_ -_->_-> _"[61,61,80,61,61]60)
evals::"[prog,state,expr list ,
val list ,state]=>bool"("_|-_ -_#>_-> _"[61,61,61,61,61]60)
hallo::"[prog,state,obj_tag,
loc,state]=>bool"("_|-_ -halloc _>_-> _"[61,61,61,61,61]60)
sallo::"[prog,state ,state]=>bool"("_|-_ -sxalloc-> _"[61,61, 61]60)
syntax (xsymbols)
eval ::"[prog,state,term,vals\<times>state]\<Rightarrow>bool" ("_\<turnstile>_ \<midarrow>_\<succ>\<rightarrow> _" [61,61,80, 61]60)
exec ::"[prog,state,stmt ,state]\<Rightarrow>bool"("_\<turnstile>_ \<midarrow>_\<rightarrow> _" [61,61,65, 61]60)
evar ::"[prog,state,var ,vvar,state]\<Rightarrow>bool"("_\<turnstile>_ \<midarrow>_=\<succ>_\<rightarrow> _"[61,61,90,61,61]60)
eval_::"[prog,state,expr ,val ,state]\<Rightarrow>bool"("_\<turnstile>_ \<midarrow>_-\<succ>_\<rightarrow> _"[61,61,80,61,61]60)
evals::"[prog,state,expr list ,
val list ,state]\<Rightarrow>bool"("_\<turnstile>_ \<midarrow>_\<doteq>\<succ>_\<rightarrow> _"[61,61,61,61,61]60)
hallo::"[prog,state,obj_tag,
loc,state]\<Rightarrow>bool"("_\<turnstile>_ \<midarrow>halloc _\<succ>_\<rightarrow> _"[61,61,61,61,61]60)
sallo::"[prog,state, state]\<Rightarrow>bool"("_\<turnstile>_ \<midarrow>sxalloc\<rightarrow> _"[61,61, 61]60)
translations
"G\<turnstile>s \<midarrow>t \<succ>\<rightarrow> w___s' " == "(s,t,w___s') \<in> eval G"
"G\<turnstile>s \<midarrow>t \<succ>\<rightarrow> (w, s')" <= "(s,t,w, s') \<in> eval G"
"G\<turnstile>s \<midarrow>t \<succ>\<rightarrow> (w,x,s')" <= "(s,t,w,x,s') \<in> eval G"
"G\<turnstile>s \<midarrow>c \<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In1r c\<succ>\<rightarrow> (\<diamondsuit>,x,s')"
"G\<turnstile>s \<midarrow>c \<rightarrow> s' " == "G\<turnstile>s \<midarrow>In1r c\<succ>\<rightarrow> (\<diamondsuit> , s')"
"G\<turnstile>s \<midarrow>e-\<succ>v \<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In1l e\<succ>\<rightarrow> (In1 v ,x,s')"
"G\<turnstile>s \<midarrow>e-\<succ>v \<rightarrow> s' " == "G\<turnstile>s \<midarrow>In1l e\<succ>\<rightarrow> (In1 v , s')"
"G\<turnstile>s \<midarrow>e=\<succ>vf\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In2 e\<succ>\<rightarrow> (In2 vf,x,s')"
"G\<turnstile>s \<midarrow>e=\<succ>vf\<rightarrow> s' " == "G\<turnstile>s \<midarrow>In2 e\<succ>\<rightarrow> (In2 vf, s')"
"G\<turnstile>s \<midarrow>e\<doteq>\<succ>v \<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In3 e\<succ>\<rightarrow> (In3 v ,x,s')"
"G\<turnstile>s \<midarrow>e\<doteq>\<succ>v \<rightarrow> s' " == "G\<turnstile>s \<midarrow>In3 e\<succ>\<rightarrow> (In3 v , s')"
"G\<turnstile>s \<midarrow>halloc oi\<succ>a\<rightarrow> (x,s')" <= "(s,oi,a,x,s') \<in> halloc G"
"G\<turnstile>s \<midarrow>halloc oi\<succ>a\<rightarrow> s' " == "(s,oi,a, s') \<in> halloc G"
"G\<turnstile>s \<midarrow>sxalloc\<rightarrow> (x,s')" <= "(s ,x,s') \<in> sxalloc G"
"G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s' " == "(s , s') \<in> sxalloc G"
inductive "halloc G" intros (* allocating objects on the heap, cf. 12.5 *)
Abrupt:
"G\<turnstile>(Some x,s) \<midarrow>halloc oi\<succ>arbitrary\<rightarrow> (Some x,s)"
New: "\<lbrakk>new_Addr (heap s) = Some a;
(x,oi') = (if atleast_free (heap s) (Suc (Suc 0)) then (None,oi)
else (Some (Xcpt (Loc a)),CInst (SXcpt OutOfMemory)))\<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s \<midarrow>halloc oi\<succ>a\<rightarrow> (x,init_obj G oi' (Heap a) s)"
inductive "sxalloc G" intros (* allocating exception objects for
standard exceptions (other than OutOfMemory) *)
Norm: "G\<turnstile> Norm s \<midarrow>sxalloc\<rightarrow> Norm s"
XcptL: "G\<turnstile>(Some (Xcpt (Loc a) ),s) \<midarrow>sxalloc\<rightarrow> (Some (Xcpt (Loc a)),s)"
SXcpt: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>halloc (CInst (SXcpt xn))\<succ>a\<rightarrow> (x,s1)\<rbrakk> \<Longrightarrow>
G\<turnstile>(Some (Xcpt (Std xn)),s0) \<midarrow>sxalloc\<rightarrow> (Some (Xcpt (Loc a)),s1)"
text {*
\par
*} (* dummy text command to break paragraph for latex;
large paragraphs exhaust memory of debian pdflatex *)
inductive "eval G" intros
(* propagation of abrupt completion *)
(* cf. 14.1, 15.5 *)
Abrupt:
"G\<turnstile>(Some xc,s) \<midarrow>t\<succ>\<rightarrow> (arbitrary3 t,(Some xc,s))"
(* execution of statements *)
(* cf. 14.5 *)
Skip: "G\<turnstile>Norm s \<midarrow>Skip\<rightarrow> Norm s"
(* cf. 14.7 *)
Expr: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Expr e\<rightarrow> s1"
Lab: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c \<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>l\<bullet> c\<rightarrow> abupd (absorb l) s1"
(* cf. 14.2 *)
Comp: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1 \<rightarrow> s1;
G\<turnstile> s1 \<midarrow>c2 \<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>c1;; c2\<rightarrow> s2"
(* cf. 14.8.2 *)
If: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1;
G\<turnstile> s1\<midarrow>(if the_Bool b then c1 else c2)\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2 \<rightarrow> s2"
(* cf. 14.10, 14.10.1 *)
(* G\<turnstile>Norm s0 \<midarrow>If(e) (c;; While(e) c) Else Skip\<rightarrow> s3 *)
(* A "continue jump" from the while body c is handled by
this rule. If a continue jump with the proper label was invoked inside c
this label (Cont l) is deleted out of the abrupt component of the state
before the iterative evaluation of the while statement.
A "break jump" is handled by the Lab Statement (Lab l (while\<dots>).
*)
Loop: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1;
if normal s1 \<and> the_Bool b
then (G\<turnstile>s1 \<midarrow>c\<rightarrow> s2 \<and>
G\<turnstile>(abupd (absorb (Cont l)) s2) \<midarrow>l\<bullet> While(e) c\<rightarrow> s3)
else s3 = s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>l\<bullet> While(e) c\<rightarrow> s3"
Do: "G\<turnstile>Norm s \<midarrow>Do j\<rightarrow> (Some (Jump j), s)"
(* cf. 14.16 *)
Throw: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Throw e\<rightarrow> abupd (throw a') s1"
(* cf. 14.18.1 *)
Try: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> s1; G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2;
if G,s2\<turnstile>catch C then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<rightarrow> s3 else s3 = s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(C vn) c2\<rightarrow> s3"
(* cf. 14.18.2 *)
Fin: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> (x1,s1);
G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> s2;
s3=(if (\<exists> err. x1=Some (Error err))
then (x1,s1)
else abupd (abrupt_if (x1\<noteq>None) x1) s2) \<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<rightarrow> s3"
(* cf. 12.4.2, 8.5 *)
Init: "\<lbrakk>the (class G C) = c;
if inited C (globs s0) then s3 = Norm s0
else (G\<turnstile>Norm (init_class_obj G C s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<rightarrow> s1 \<and>
G\<turnstile>set_lvars empty s1 \<midarrow>init c\<rightarrow> s2 \<and> s3 = restore_lvars s1 s2)\<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Init C\<rightarrow> s3"
(* This class initialisation rule is a little bit inaccurate. Look at the
exact sequence:
1. The current class object (the static fields) are initialised
(init_class_obj)
2. Then the superclasses are initialised
3. The static initialiser of the current class is invoked
More precisely we should expect another ordering, namely 2 1 3.
But we can't just naively toggle 1 and 2. By calling init_class_obj
before initialising the superclasses we also implicitly record that
we have started to initialise the current class (by setting an
value for the class object). This becomes
crucial for the completeness proof of the axiomatic semantics
(AxCompl.thy). Static initialisation requires an induction on the number
of classes not yet initialised (or to be more precise, classes where the
initialisation has not yet begun).
So we could first assign a dummy value to the class before
superclass initialisation and afterwards set the correct values.
But as long as we don't take memory overflow into account
when allocating class objects, and don't model definite assignment in
the static initialisers, we can leave things as they are for convenience.
*)
(* evaluation of expressions *)
(* cf. 15.8.1, 12.4.1 *)
NewC: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init C\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<rightarrow> s2"
(* cf. 15.9.1, 12.4.1 *)
NewA: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e-\<succ>i'\<rightarrow> s2;
G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc (Arr T (the_Intg i'))\<succ>a\<rightarrow> s3\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<rightarrow> s3"
(* cf. 15.15 *)
Cast: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1;
s2 = abupd (raise_if (\<not>G,store s1\<turnstile>v fits T) ClassCast) s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<rightarrow> s2"
(* cf. 15.19.2 *)
Inst: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1;
b = (v\<noteq>Null \<and> G,store s1\<turnstile>v fits RefT T)\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<rightarrow> s1"
(* cf. 15.7.1 *)
Lit: "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<rightarrow> Norm s"
UnOp: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1\<rbrakk>
\<Longrightarrow> G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>(eval_unop unop v)\<rightarrow> s1"
BinOp: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<rightarrow> s1;
G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then (In1l e2) else (In1r Skip))
\<succ>\<rightarrow> (In1 v2,s2)
\<rbrakk>
\<Longrightarrow> G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>(eval_binop binop v1 v2)\<rightarrow> s2"
(* cf. 15.10.2 *)
Super: "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<rightarrow> Norm s"
(* cf. 15.2 *)
Acc: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v,f)\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<rightarrow> s1"
(* cf. 15.25.1 *)
Ass: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w,f)\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>e-\<succ>v \<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>va:=e-\<succ>v\<rightarrow> assign f v s2"
(* cf. 15.24 *)
Cond: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<rightarrow> s2"
(* cf. 15.11.4.1, 15.11.4.2, 15.11.4.4, 15.11.4.5 *)
Call:
"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1; G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2;
D = invocation_declclass G mode (store s2) a' statT \<lparr>name=mn,parTs=pTs\<rparr>;
s3=init_lvars G D \<lparr>name=mn,parTs=pTs\<rparr> mode a' vs s2;
s3' = check_method_access G accC statT mode \<lparr>name=mn,parTs=pTs\<rparr> a' s3;
G\<turnstile>s3' \<midarrow>Methd D \<lparr>name=mn,parTs=pTs\<rparr>-\<succ>v\<rightarrow> s4\<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e\<cdot>mn({pTs}args)-\<succ>v\<rightarrow> (restore_lvars s2 s4)"
(* The accessibility check is after init_lvars, to keep it simple. Init_lvars
already tests for the absence of a null-pointer reference in case of an
instance method invocation
*)
Methd: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<rightarrow> s1"
(* The local variables l are just a dummy here. The are only used by
the smallstep semantics *)
(* cf. 14.15, 12.4.1 *)
Body: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init D\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Body D c
-\<succ>the (locals (store s2) Result)\<rightarrow>abupd (absorb Ret) s2"
(* The local variables l are just a dummy here. The are only used by
the smallstep semantics *)
(* evaluation of variables *)
(* cf. 15.13.1, 15.7.2 *)
LVar: "G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<rightarrow> Norm s"
(* cf. 15.10.1, 12.4.1 *)
FVar: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2;
(v,s2') = fvar statDeclC stat fn a s2;
s3 = check_field_access G accC statDeclC fn stat a s2' \<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<rightarrow> s3"
(* The accessibility check is after fvar, to keep it simple. Fvar already
tests for the absence of a null-pointer reference in case of an instance
field
*)
(* cf. 15.12.1, 15.25.1 *)
AVar: "\<lbrakk>G\<turnstile> Norm s0 \<midarrow>e1-\<succ>a\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e2-\<succ>i\<rightarrow> s2;
(v,s2') = avar G i a s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<rightarrow> s2'"
(* evaluation of expression lists *)
(* cf. 15.11.4.2 *)
Nil:
"G\<turnstile>Norm s0 \<midarrow>[]\<doteq>\<succ>[]\<rightarrow> Norm s0"
(* cf. 15.6.4 *)
Cons: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e -\<succ> v \<rightarrow> s1;
G\<turnstile> s1 \<midarrow>es\<doteq>\<succ>vs\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e#es\<doteq>\<succ>v#vs\<rightarrow> s2"
(* Rearrangement of premisses:
[0,1(Abrupt),2(Skip),8(Do),4(Lab),30(Nil),31(Cons),27(LVar),17(Cast),18(Inst),
17(Lit),18(UnOp),19(BinOp),20(Super),21(Acc),3(Expr),5(Comp),25(Methd),26(Body),23(Cond),6(If),
7(Loop),11(Fin),9(Throw),13(NewC),14(NewA),12(Init),22(Ass),10(Try),28(FVar),
29(AVar),24(Call)]
*)
ML {*
bind_thm ("eval_induct_", rearrange_prems
[0,1,2,8,4,30,31,27,15,16,
17,18,19,20,21,3,5,25,26,23,6,
7,11,9,13,14,12,22,10,28,
29,24] (thm "eval.induct"))
*}
text {*
\par
*} (* dummy text command to break paragraph for latex;
large paragraphs exhaust memory of debian pdflatex *)
lemmas eval_induct = eval_induct_ [split_format and and and and and and and and
and and and and and and s1 (* Acc *) and and s2 (* Comp *) and and and and
and and
s2 (* Fin *) and and s2 (* NewC *)]
declare split_if [split del] split_if_asm [split del]
option.split [split del] option.split_asm [split del]
inductive_cases halloc_elim_cases:
"G\<turnstile>(Some xc,s) \<midarrow>halloc oi\<succ>a\<rightarrow> s'"
"G\<turnstile>(Norm s) \<midarrow>halloc oi\<succ>a\<rightarrow> s'"
inductive_cases sxalloc_elim_cases:
"G\<turnstile> Norm s \<midarrow>sxalloc\<rightarrow> s'"
"G\<turnstile>(Some (Xcpt (Loc a )),s) \<midarrow>sxalloc\<rightarrow> s'"
"G\<turnstile>(Some (Xcpt (Std xn)),s) \<midarrow>sxalloc\<rightarrow> s'"
inductive_cases sxalloc_cases: "G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'"
lemma sxalloc_elim_cases2: "\<lbrakk>G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s';
\<And>s. \<lbrakk>s' = Norm s\<rbrakk> \<Longrightarrow> P;
\<And>a s. \<lbrakk>s' = (Some (Xcpt (Loc a)),s)\<rbrakk> \<Longrightarrow> P
\<rbrakk> \<Longrightarrow> P"
apply cut_tac
apply (erule sxalloc_cases)
apply blast+
done
declare not_None_eq [simp del] (* IntDef.Zero_def [simp del] *)
declare split_paired_All [simp del] split_paired_Ex [simp del]
ML_setup {*
simpset_ref() := simpset() delloop "split_all_tac"
*}
inductive_cases eval_cases: "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> vs'"
inductive_cases eval_elim_cases:
"G\<turnstile>(Some xc,s) \<midarrow>t \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r Skip \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (Do j) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> c) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In3 ([]) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In3 (e#es) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Lit w) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (UnOp unop e) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (BinOp binop e1 e2) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In2 (LVar vn) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Cast T e) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (e InstOf T) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Super) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Acc va) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (Expr e) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (c1;; c2) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (Methd C sig) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (Body D c) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (e0 ? e1 : e2) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (If(e) c1 Else c2) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> While(e) c) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (c1 Finally c2) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (Throw e) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (NewC C) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (New T[e]) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Ass va e) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (Try c1 Catch(tn vn) c2) \<succ>\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In2 ({accC,statDeclC,stat}e..fn) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In2 (e1.[e2]) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l ({accC,statT,mode}e\<cdot>mn({pT}p)) \<succ>\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (Init C) \<succ>\<rightarrow> xs'"
declare not_None_eq [simp] (* IntDef.Zero_def [simp] *)
declare split_paired_All [simp] split_paired_Ex [simp]
ML_setup {*
simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
*}
declare split_if [split] split_if_asm [split]
option.split [split] option.split_asm [split]
lemma eval_Inj_elim:
"G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s')
\<Longrightarrow> case t of
In1 ec \<Rightarrow> (case ec of
Inl e \<Rightarrow> (\<exists>v. w = In1 v)
| Inr c \<Rightarrow> w = \<diamondsuit>)
| In2 e \<Rightarrow> (\<exists>v. w = In2 v)
| In3 e \<Rightarrow> (\<exists>v. w = In3 v)"
apply (erule eval_cases)
apply auto
apply (induct_tac "t")
apply (induct_tac "a")
apply auto
done
ML_setup {*
fun eval_fun nam inj rhs =
let
val name = "eval_" ^ nam ^ "_eq"
val lhs = "G\<turnstile>s \<midarrow>" ^ inj ^ " t\<succ>\<rightarrow> (w, s')"
val () = qed_goal name (the_context()) (lhs ^ " = (" ^ rhs ^ ")")
(K [Auto_tac, ALLGOALS (ftac (thm "eval_Inj_elim")) THEN Auto_tac])
fun is_Inj (Const (inj,_) $ _) = true
| is_Inj _ = false
fun pred (_ $ (Const ("Pair",_) $ _ $
(Const ("Pair", _) $ _ $ (Const ("Pair", _) $ x $ _ ))) $ _ ) = is_Inj x
in
make_simproc name lhs pred (thm name)
end
val eval_expr_proc =eval_fun "expr" "In1l" "\<exists>v. w=In1 v \<and> G\<turnstile>s \<midarrow>t-\<succ>v \<rightarrow> s'"
val eval_var_proc =eval_fun "var" "In2" "\<exists>vf. w=In2 vf \<and> G\<turnstile>s \<midarrow>t=\<succ>vf\<rightarrow> s'"
val eval_exprs_proc=eval_fun "exprs""In3" "\<exists>vs. w=In3 vs \<and> G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<rightarrow> s'"
val eval_stmt_proc =eval_fun "stmt" "In1r" " w=\<diamondsuit> \<and> G\<turnstile>s \<midarrow>t \<rightarrow> s'";
Addsimprocs [eval_expr_proc,eval_var_proc,eval_exprs_proc,eval_stmt_proc];
bind_thms ("AbruptIs", sum3_instantiate (thm "eval.Abrupt"))
*}
declare halloc.Abrupt [intro!] eval.Abrupt [intro!] AbruptIs [intro!]
text{* @{text Callee},@{text InsInitE}, @{text InsInitV}, @{text FinA} are only
used in smallstep semantics, not in the bigstep semantics. So their is no
valid evaluation of these terms
*}
lemma eval_Callee: "G\<turnstile>Norm s\<midarrow>Callee l e-\<succ>v\<rightarrow> s' = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In1l (Callee l e)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma eval_InsInitE: "G\<turnstile>Norm s\<midarrow>InsInitE c e-\<succ>v\<rightarrow> s' = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In1l (InsInitE c e)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma eval_InsInitV: "G\<turnstile>Norm s\<midarrow>InsInitV c w=\<succ>v\<rightarrow> s' = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In2 (InsInitV c w)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma eval_FinA: "G\<turnstile>Norm s\<midarrow>FinA a c\<rightarrow> s' = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In1r (FinA a c)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma eval_no_abrupt_lemma:
"\<And>s s'. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s') \<Longrightarrow> normal s' \<longrightarrow> normal s"
by (erule eval_cases, auto)
lemma eval_no_abrupt:
"G\<turnstile>(x,s) \<midarrow>t\<succ>\<rightarrow> (w,Norm s') =
(x = None \<and> G\<turnstile>Norm s \<midarrow>t\<succ>\<rightarrow> (w,Norm s'))"
apply auto
apply (frule eval_no_abrupt_lemma, auto)+
done
ML {*
local
fun is_None (Const ("Datatype.option.None",_)) = true
| is_None _ = false
fun pred (t as (_ $ (Const ("Pair",_) $
(Const ("Pair", _) $ x $ _) $ _ ) $ _)) = is_None x
in
val eval_no_abrupt_proc =
make_simproc "eval_no_abrupt" "G\<turnstile>(x,s) \<midarrow>e\<succ>\<rightarrow> (w,Norm s')" pred
(thm "eval_no_abrupt")
end;
Addsimprocs [eval_no_abrupt_proc]
*}
lemma eval_abrupt_lemma:
"G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (v,s') \<Longrightarrow> abrupt s=Some xc \<longrightarrow> s'= s \<and> v = arbitrary3 t"
by (erule eval_cases, auto)
lemma eval_abrupt:
" G\<turnstile>(Some xc,s) \<midarrow>t\<succ>\<rightarrow> (w,s') =
(s'=(Some xc,s) \<and> w=arbitrary3 t \<and>
G\<turnstile>(Some xc,s) \<midarrow>t\<succ>\<rightarrow> (arbitrary3 t,(Some xc,s)))"
apply auto
apply (frule eval_abrupt_lemma, auto)+
done
ML {*
local
fun is_Some (Const ("Pair",_) $ (Const ("Datatype.option.Some",_) $ _)$ _) =true
| is_Some _ = false
fun pred (_ $ (Const ("Pair",_) $
_ $ (Const ("Pair", _) $ _ $ (Const ("Pair", _) $ _ $
x))) $ _ ) = is_Some x
in
val eval_abrupt_proc =
make_simproc "eval_abrupt"
"G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<rightarrow> (w,s')" pred (thm "eval_abrupt")
end;
Addsimprocs [eval_abrupt_proc]
*}
lemma LitI: "G\<turnstile>s \<midarrow>Lit v-\<succ>(if normal s then v else arbitrary)\<rightarrow> s"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.Lit)
lemma SkipI [intro!]: "G\<turnstile>s \<midarrow>Skip\<rightarrow> s"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.Skip)
lemma ExprI: "G\<turnstile>s \<midarrow>e-\<succ>v\<rightarrow> s' \<Longrightarrow> G\<turnstile>s \<midarrow>Expr e\<rightarrow> s'"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.Expr)
lemma CompI: "\<lbrakk>G\<turnstile>s \<midarrow>c1\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c2\<rightarrow> s2\<rbrakk> \<Longrightarrow> G\<turnstile>s \<midarrow>c1;; c2\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.Comp)
lemma CondI:
"\<And>s1. \<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>b\<rightarrow> s1; G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>s \<midarrow>e ? e1 : e2-\<succ>(if normal s1 then v else arbitrary)\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.Cond)
lemma IfI: "\<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>v\<rightarrow> s1; G\<turnstile>s1 \<midarrow>(if the_Bool v then c1 else c2)\<rightarrow> s2\<rbrakk>
\<Longrightarrow> G\<turnstile>s \<midarrow>If(e) c1 Else c2\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.If)
lemma MethdI: "G\<turnstile>s \<midarrow>body G C sig-\<succ>v\<rightarrow> s'
\<Longrightarrow> G\<turnstile>s \<midarrow>Methd C sig-\<succ>v\<rightarrow> s'"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: eval.Methd)
lemma eval_Call:
"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1; G\<turnstile>s1 \<midarrow>ps\<doteq>\<succ>pvs\<rightarrow> s2;
D = invocation_declclass G mode (store s2) a' statT \<lparr>name=mn,parTs=pTs\<rparr>;
s3 = init_lvars G D \<lparr>name=mn,parTs=pTs\<rparr> mode a' pvs s2;
s3' = check_method_access G accC statT mode \<lparr>name=mn,parTs=pTs\<rparr> a' s3;
G\<turnstile>s3'\<midarrow>Methd D \<lparr>name=mn,parTs=pTs\<rparr>-\<succ> v\<rightarrow> s4;
s4' = restore_lvars s2 s4\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e\<cdot>mn({pTs}ps)-\<succ>v\<rightarrow> s4'"
apply (drule eval.Call, assumption)
apply (rule HOL.refl)
apply simp+
done
lemma eval_Init:
"\<lbrakk>if inited C (globs s0) then s3 = Norm s0
else G\<turnstile>Norm (init_class_obj G C s0)
\<midarrow>(if C = Object then Skip else Init (super (the (class G C))))\<rightarrow> s1 \<and>
G\<turnstile>set_lvars empty s1 \<midarrow>(init (the (class G C)))\<rightarrow> s2 \<and>
s3 = restore_lvars s1 s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Init C\<rightarrow> s3"
apply (rule eval.Init)
apply auto
done
lemma init_done: "initd C s \<Longrightarrow> G\<turnstile>s \<midarrow>Init C\<rightarrow> s"
apply (case_tac "s", simp)
apply (case_tac "a")
apply safe
apply (rule eval_Init)
apply auto
done
lemma eval_StatRef:
"G\<turnstile>s \<midarrow>StatRef rt-\<succ>(if abrupt s=None then Null else arbitrary)\<rightarrow> s"
apply (case_tac "s", simp)
apply (case_tac "a = None")
apply (auto del: eval.Abrupt intro!: eval.intros)
done
lemma SkipD [dest!]: "G\<turnstile>s \<midarrow>Skip\<rightarrow> s' \<Longrightarrow> s' = s"
apply (erule eval_cases)
by auto
lemma Skip_eq [simp]: "G\<turnstile>s \<midarrow>Skip\<rightarrow> s' = (s = s')"
by auto
(*unused*)
lemma init_retains_locals [rule_format (no_asm)]: "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s') \<Longrightarrow>
(\<forall>C. t=In1r (Init C) \<longrightarrow> locals (store s) = locals (store s'))"
apply (erule eval.induct)
apply (simp (no_asm_use) split del: split_if_asm option.split_asm)+
apply auto
done
lemma halloc_xcpt [dest!]:
"\<And>s'. G\<turnstile>(Some xc,s) \<midarrow>halloc oi\<succ>a\<rightarrow> s' \<Longrightarrow> s'=(Some xc,s)"
apply (erule_tac halloc_elim_cases)
by auto
(*
G\<turnstile>(x,(h,l)) \<midarrow>e\<succ>v\<rightarrow> (x',(h',l'))) \<Longrightarrow> l This = l' This"
G\<turnstile>(x,(h,l)) \<midarrow>s \<rightarrow> (x',(h',l'))) \<Longrightarrow> l This = l' This"
*)
lemma eval_Methd:
"G\<turnstile>s \<midarrow>In1l(body G C sig)\<succ>\<rightarrow> (w,s')
\<Longrightarrow> G\<turnstile>s \<midarrow>In1l(Methd C sig)\<succ>\<rightarrow> (w,s')"
apply (case_tac "s")
apply (case_tac "a")
apply clarsimp+
apply (erule eval.Methd)
apply (drule eval_abrupt_lemma)
apply force
done
lemma eval_binop_arg2_indep:
"\<not> need_second_arg binop v1 \<Longrightarrow> eval_binop binop v1 x = eval_binop binop v1 y"
by (cases binop)
(simp_all add: need_second_arg_def)
lemma eval_BinOp_arg2_indepI:
assumes eval_e1: "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<rightarrow> s1" and
no_need: "\<not> need_second_arg binop v1"
shows "G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>(eval_binop binop v1 v2)\<rightarrow> s1"
(is "?EvalBinOp v2")
proof -
from eval_e1
have "?EvalBinOp Unit"
by (rule eval.BinOp)
(simp add: no_need)
moreover
from no_need
have "eval_binop binop v1 Unit = eval_binop binop v1 v2"
by (simp add: eval_binop_arg2_indep)
ultimately
show ?thesis
by simp
qed
section "single valued"
lemma unique_halloc [rule_format (no_asm)]:
"\<And>s as as'. (s,oi,as)\<in>halloc G \<Longrightarrow> (s,oi,as')\<in>halloc G \<longrightarrow> as'=as"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule halloc.induct)
apply (auto elim!: halloc_elim_cases split del: split_if split_if_asm)
apply (drule trans [THEN sym], erule sym)
defer
apply (drule trans [THEN sym], erule sym)
apply auto
done
lemma single_valued_halloc:
"single_valued {((s,oi),(a,s')). G\<turnstile>s \<midarrow>halloc oi\<succ>a \<rightarrow> s'}"
apply (unfold single_valued_def)
by (clarsimp, drule (1) unique_halloc, auto)
lemma unique_sxalloc [rule_format (no_asm)]:
"\<And>s s'. G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s' \<Longrightarrow> G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'' \<longrightarrow> s'' = s'"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule sxalloc.induct)
apply (auto dest: unique_halloc elim!: sxalloc_elim_cases
split del: split_if split_if_asm)
done
lemma single_valued_sxalloc: "single_valued {(s,s'). G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'}"
apply (unfold single_valued_def)
apply (blast dest: unique_sxalloc)
done
lemma split_pairD: "(x,y) = p \<Longrightarrow> x = fst p & y = snd p"
by auto
lemma eval_Body: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init D\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c\<rightarrow> s2;
res=the (locals (store s2) Result);
s3=abupd (absorb Ret) s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>res\<rightarrow>s3"
by (auto elim: eval.Body)
lemma unique_eval [rule_format (no_asm)]:
"G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> ws \<Longrightarrow> (\<forall>ws'. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> ws' \<longrightarrow> ws' = ws)"
apply (case_tac "ws")
apply (simp only:)
apply (erule thin_rl)
apply (erule eval_induct)
apply (tactic {* ALLGOALS (EVERY'
[strip_tac, rotate_tac ~1, eresolve_tac (thms "eval_elim_cases")]) *})
(* 31 subgoals *)
prefer 28 (* Try *)
apply (simp (no_asm_use) only: split add: split_if_asm)
(* 34 subgoals *)
prefer 30 (* Init *)
apply (case_tac "inited C (globs s0)", (simp only: if_True if_False)+)
prefer 26 (* While *)
apply (simp (no_asm_use) only: split add: split_if_asm, blast)
apply (drule_tac x="(In1 bb, s1a)" in spec, drule (1) mp, simp)
apply (drule_tac x="(In1 bb, s1a)" in spec, drule (1) mp, simp)
apply blast
(* 33 subgoals *)
apply (blast dest: unique_sxalloc unique_halloc split_pairD)+
done
(* unused *)
lemma single_valued_eval:
"single_valued {((s,t),vs'). G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> vs'}"
apply (unfold single_valued_def)
by (clarify, drule (1) unique_eval, auto)
end