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JVM_ImplemDomain.v
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(** * Bicolano: Semantic domains (interface implementation) *)
(* Hendra : trim the system to contain only Arithmetic *)
Require Export JVM_ImplemProgramWithMap.
Require Export JVM_Domain.
Ltac caseeq t := generalize (refl_equal t); pattern t at -1 in |- * ; case t.
(** All semantic domains and basic operation are encapsulated in a module signature *)
Module JVM_Dom <: JVM_SEMANTIC_DOMAIN.
(** We depend on the choices done for program data structures *)
Module JVM_Prog := JVM_ImplemProgramWithMap.JVM_Make.
Import JVM_Prog.
Open Scope Z_scope.
Module SByte <: Numeric.NUMSIZE. Definition power := 7%nat. End SByte.
Module SShort<: Numeric.NUMSIZE. Definition power := 15%nat. End SShort.
Module SInt <: Numeric.NUMSIZE. Definition power := 31%nat. End SInt.
Module Byte : Numeric.NUMERIC with Definition power := 7%nat := Numeric.Make SByte.
Module Short : Numeric.NUMERIC with Definition power := 15%nat := Numeric.Make SShort.
Module Int : Numeric.NUMERIC with Definition power := 31%nat := Numeric.Make SInt.
(** conversion *)
Definition b2i (b:Byte.t) : Int.t := Int.const (Byte.toZ b).
Definition s2i (s:Short.t) : Int.t := Int.const (Short.toZ s).
Definition i2b (i:Int.t) : Byte.t := Byte.const (Int.toZ i).
Definition i2s (i:Int.t) : Short.t := Short.const (Int.toZ i).
Definition i2bool (i:Int.t) : Byte.t := Byte.const (Int.toZ i mod 2).
Inductive JVM_num : Set :=
| I : Int.t -> JVM_num
| B : Byte.t -> JVM_num
| Sh : Short.t -> JVM_num.
(** Location is the domain of adresses in the heap *)
Definition JVM_Location : Set := N.
Definition JVM_Location_dec : forall loc1 loc2:JVM_Location,{loc1=loc2}+{~loc1=loc2} :=
(Aeq_Dec _ _ Neq_spec).
Inductive JVM_value : Set :=
| Num : JVM_num -> JVM_value
| Ref: JVM_Location -> JVM_value
| Null : JVM_value.
Definition init_value (t:JVM_type) : JVM_value :=
match t with
| JVM_ReferenceType _ => Null
| JVM_PrimitiveType _ => Num (I (Int.const 0))
end.
Definition init_field_value (f:JVM_Field) : JVM_value :=
match JVM_FIELD.initValue f with
| JVM_FIELD.Int z => Num (I (Int.const z))
| JVM_FIELD.NULL => Null
| JVM_FIELD.UNDEF => init_value (JVM_FIELDSIGNATURE.type (JVM_FIELD.signature f))
end.
(** Domain of local variables *)
Module Type JVM_LOCALVAR.
Parameter t : Type.
Parameter get : t-> JVM_Var -> option JVM_value.
Parameter update : t -> JVM_Var -> JVM_value -> t.
Parameter get_update_new : forall l x v, get (update l x v) x = Some v.
Parameter get_update_old : forall l x y v,
x<>y -> get (update l x v) y = get l y.
End JVM_LOCALVAR.
Module JVM_MapVar <: MAP with Definition key := JVM_Var := BinNatMap.
Module JVM_LocalVar <: JVM_LOCALVAR.
Definition t := JVM_MapVar.t JVM_value.
Definition get : t -> JVM_Var -> option JVM_value := @JVM_MapVar.get JVM_value.
Definition update : t -> JVM_Var -> JVM_value -> t := @JVM_MapVar.update JVM_value.
Lemma get_update_new : forall l x v, get (update l x v) x = Some v.
Proof. exact (JVM_MapVar.get_update1 JVM_value). Qed.
Lemma get_update_old : forall l x y v,
x<>y -> get (update l x v) y = get l y.
Proof.
intros;refine (JVM_MapVar.get_update2 JVM_value _ _ _ _ _).
intro;apply H;subst;trivial.
Qed.
Definition empty := JVM_MapVar.empty JVM_value.
End JVM_LocalVar.
(* Domain of operand stacks *)
Module Type JVM_OPERANDSTACK.
Definition t : Set := list JVM_value.
Definition empty : t := nil.
Definition push : JVM_value -> t -> t := fun v t => cons v t.
Definition size : t -> nat := fun t => length t .
Definition get_nth : t -> nat -> option JVM_value := fun s n => nth_error s n.
End JVM_OPERANDSTACK.
Module JVM_OperandStack <: JVM_OPERANDSTACK.
Definition t : Set := list JVM_value.
Definition empty : t := nil.
Definition push : JVM_value -> t -> t := fun v t => cons v t.
Definition size : t -> nat := fun t => length t .
Definition get_nth : t -> nat -> option JVM_value := fun s n => nth_error s n.
End JVM_OperandStack.
(** Transfert fonction between operand stack and local variables necessary for invoke instructions *)
Fixpoint stack2localvar_rec
(os:JVM_OperandStack.t) (n:nat) (l:JVM_LocalVar.t) {struct n}: JVM_LocalVar.t :=
match n with
| O => l
| S n =>
match os with
| nil => l
| v::os =>
stack2localvar_rec os n (JVM_LocalVar.update l (N_toVar n) v)
end
end.
Definition stack2localvar os n := stack2localvar_rec os n JVM_LocalVar.empty.
Lemma stack2locvar_rec_prop1 :
forall n l os x, (n <= Var_toN x)%nat ->
JVM_LocalVar.get (stack2localvar_rec os n l) x = JVM_LocalVar.get l x.
Proof.
induction n;simpl;intros;trivial.
destruct os;simpl;intros;trivial.
assert (n <= Var_toN x)%nat. omega.
rewrite IHn;trivial.
unfold JVM_LocalVar.get, JVM_LocalVar.update;rewrite JVM_MapVar.get_update2;trivial.
intro Heq;rewrite Heq in H.
rewrite Var_toN_bij2 in H;omega.
Qed.
Lemma stack2locvar_prop1 :
forall s n x, (n <= Var_toN x)%nat -> JVM_LocalVar.get (stack2localvar s n) x = None.
Proof.
unfold stack2localvar;intros.
rewrite stack2locvar_rec_prop1;trivial.
exact (JVM_MapVar.get_empty _ _).
Qed.
Lemma stack2locvar_rec_prop2 :
forall n os l x, (Var_toN x < n)%nat ->
(forall y, (Var_toN y < n)%nat -> JVM_LocalVar.get l y = None) ->
JVM_LocalVar.get (stack2localvar_rec os n l) x =
JVM_OperandStack.get_nth os (n-(Var_toN x)-1)%nat.
Proof.
induction n;simpl;intros.
elimtype False;omega.
destruct os.
rewrite H0;trivial.
destruct (Var_toN x);simpl;trivial.
destruct (n-0)%nat;trivial.
destruct (n-n0-1)%nat;trivial.
change (match Var_toN x with
| O => (S n)
| S l0 => (n - l0)
end - 1)%nat with (S n- Var_toN x -1)%nat.
assert (Var_toN x = n \/ Var_toN x < n)%nat. omega.
destruct H1.
pattern n at 2;rewrite <- H1;rewrite Var_toN_bij1.
rewrite stack2locvar_rec_prop1. 2:omega.
rewrite JVM_LocalVar.get_update_new.
replace (S n - Var_toN x - 1)%nat with O;try omega.
simpl;trivial.
rewrite IHn;trivial.
replace (S n - Var_toN x - 1)%nat with (S (n - Var_toN x - 1))%nat;try omega.
simpl;trivial.
intros;rewrite JVM_LocalVar.get_update_old.
apply H0;omega.
intro Heq;rewrite <-Heq in H2;rewrite Var_toN_bij2 in H2;omega.
Qed.
Lemma stack2locvar_prop2 :
forall s n x, (Var_toN x < n)%nat ->
JVM_LocalVar.get (stack2localvar s n) x = JVM_OperandStack.get_nth s (n-(Var_toN x)-1)%nat.
Proof.
unfold stack2localvar;intros;apply stack2locvar_rec_prop2;trivial.
intros;exact (JVM_MapVar.get_empty _ _).
Qed.
Fixpoint all_super_classes (p:JVM_Program) (c:JVM_Class) (n:nat) {struct n} : option (list JVM_Class) :=
match n with
| O => None
| S n =>
match JVM_CLASS.superClass c with
| None => Some nil
| Some super_name =>
match JVM_PROG.class p super_name with
| None => None
| Some super =>
match (all_super_classes p super n) with
| None => None
| Some l => Some (super::l)
end
end
end
end.
Ltac inv H := inversion H; subst; clear H.
Lemma clos_refl_trans_ind2 :
forall (A:Type) (R:A -> A -> Prop) (P:A -> A -> Prop),
(forall x, P x x) ->
(forall x y z:A, R x y -> clos_refl_trans A R y z -> P y z -> P x z) ->
forall x y, clos_refl_trans A R x y -> P x y.
Proof.
intros A R P H1 H2.
assert (forall x y, clos_refl_trans A R x y ->
forall z, clos_refl_trans A R y z -> P y z -> P x z).
induction 1; eauto; intros.
apply IHclos_refl_trans1.
constructor 3 with z; auto.
apply IHclos_refl_trans2; auto.
intros.
apply H with y; auto.
constructor 2.
Qed.
Lemma subclass_left : forall p c1 c2,
subclass p c1 c2 -> c1=c2 \/ (exists c, direct_subclass p c1 c /\ subclass p c c2).
Proof.
intros p; unfold subclass; apply clos_refl_trans_ind2; intros; auto.
destruct H1; subst; auto.
right; exists z; auto.
destruct H1 as [c [T1 T2]].
right; exists y; split; auto.
Qed.
Definition all_super_classes_correct : forall p n c l,
all_super_classes p c n = Some l ->
JVM_PROG.defined_Class p c ->
forall c', subclass p c c' -> In c' (c::l).
Proof.
induction n; simpl.
intros; discriminate.
intros c l; case_eq (JVM_CLASS.superClass c).
intros c' H'.
case_eq (JVM_PROG.class p c'); try (intros; discriminate).
intros c'' H''.
case_eq (all_super_classes p c'' n); try (intros; discriminate).
intros.
inv H0.
destruct (subclass_left _ _ _ H2); auto.
destruct H0 as [c0 [T1 T2]].
clear H2.
inv T1.
unfold JVM_PROG.defined_Class in *.
assert (c0 = c'') by congruence.
assert (c'=JVM_CLASS.name c'') by congruence.
subst.
clear H2 H3 H1.
right; apply (IHn _ _ H H'' _ T2).
intros.
destruct (subclass_left _ _ _ H2); auto.
destruct H3 as [c0 [T1 T2]].
inv T1.
congruence.
Qed.
Fixpoint all_super_interfaces (p:JVM_Program) (n:nat) {struct n} :
JVM_Interface -> option (list JVM_Interface) :=
match n with
| O => fun _ => None
| S n => fun c =>
List.fold_left
(fun o iname =>
match o with
| None => None
| Some l =>
match JVM_PROG.interface p iname with
| None => None
| Some itf =>
match all_super_interfaces p n itf with
| None => None
| Some l' => Some (itf::l++l')
end
end
end)
(JVM_INTERFACE.superInterfaces c)
(Some (c::nil))
end.
Lemma subinterface_left : forall p c1 c2,
subinterface p c1 c2 -> c1=c2 \/ (exists c, direct_subinterface p c1 c /\ subinterface p c c2).
Proof.
intros p; unfold subinterface; apply clos_refl_trans_ind2; intros; auto.
destruct H1; subst; auto.
right; exists z; auto.
destruct H1 as [c [T1 T2]].
right; exists y; split; auto.
Qed.
Lemma all_super_interfaces_aux : forall p n l3,
fold_left
(fun (o : option (list JVM_Interface)) (iname : JVM_InterfaceName) =>
match o with
| Some l6 =>
match JVM_PROG.interface p iname with
| Some itf =>
match all_super_interfaces p n itf with
| Some l' => Some (itf :: l6 ++ l')
| None => None (A:=list JVM_Interface)
end
| None => None (A:=list JVM_Interface)
end
| None => None (A:=list JVM_Interface)
end) l3 None=None.
Proof.
induction l3; simpl; auto.
Qed.
Lemma all_super_interfaces_aux' : forall p n l0 l1 l2,
fold_left
(fun (o : option (list JVM_Interface)) (iname : JVM_InterfaceName) =>
match o with
| Some l =>
match JVM_PROG.interface p iname with
| Some itf =>
match all_super_interfaces p n itf with
| Some l' => Some (itf :: l ++ l')
| None => None (A:=list JVM_Interface)
end
| None => None (A:=list JVM_Interface)
end
| None => None (A:=list JVM_Interface)
end) l0 (Some l1) = Some l2 -> incl l1 l2.
Proof.
induction l0; simpl.
intros.
inversion H; subst; intro; auto.
destruct (JVM_PROG.interface p a); try (intros; discriminate).
destruct (all_super_interfaces p n j); try (intros; discriminate).
intros.
assert (IH:=IHl0 _ _ H); clear H IHl0.
repeat intro; apply IH.
right; auto with datatypes.
rewrite all_super_interfaces_aux; intros; discriminate.
rewrite all_super_interfaces_aux; intros; discriminate.
Qed.
Definition all_super_interfaces_correct : forall p n c l,
all_super_interfaces p n c = Some l ->
JVM_PROG.defined_Interface p c ->
forall c', subinterface p c c' -> In c' (c::l).
Proof.
induction n; simpl.
intros; discriminate.
intros.
destruct (subinterface_left _ _ _ H1); clear H1; auto.
right; destruct H2 as [c0 [T1 T2]].
inv T1.
generalize dependent (Some (c::nil)).
generalize dependent (JVM_INTERFACE.superInterfaces c).
induction l0; simpl; intros.
elim H3.
destruct o; simpl in H; try discriminate.
case_eq (JVM_PROG.interface p a); intros.
rewrite H4 in H.
case_eq (all_super_interfaces p n j); intros.
rewrite H5 in H.
destruct H3; subst.
unfold JVM_PROG.defined_Interface in *.
assert (c0=j) by congruence; clear H4; subst.
generalize (IHn _ _ H5 H2 _ T2).
assert (T:=all_super_interfaces_aux' _ _ _ _ _ H).
simpl; intros.
apply T.
destruct H3.
left; auto.
right; auto with datatypes.
apply IHl0 with (Some (j :: l1 ++ l2)); auto.
rewrite H5 in H.
rewrite all_super_interfaces_aux in H; discriminate.
rewrite H4 in H.
rewrite all_super_interfaces_aux in H; discriminate.
rewrite all_super_interfaces_aux in H; discriminate.
Qed.
Definition all_interfaces (p:JVM_Program) (n:nat) (c:JVM_Class) : option (list JVM_Interface) :=
List.fold_left
(fun o iname =>
match o with
| None => None
| Some l =>
match JVM_PROG.interface p iname with
| None => None
| Some itf =>
match all_super_interfaces p n itf with
| None => None
| Some l' => Some (itf::l++l')
end
end
end)
(JVM_CLASS.superInterfaces c)
(Some nil).
Lemma all_interfaces_correct : forall p n c l,
all_interfaces p n c = Some l -> forall i I I',
In i (JVM_CLASS.superInterfaces c) ->
JVM_PROG.interface p i = Some I ->
subinterface p I I' ->
In I l.
Proof.
unfold all_interfaces.
intros p n c.
generalize (@nil JVM_Interface).
generalize (JVM_CLASS.superInterfaces c).
induction l; simpl.
intuition.
intros l0; case_eq (JVM_PROG.interface p a).
intros i Hi; case_eq (all_super_interfaces p n i); intros.
destruct H1; subst.
assert (I0=i) by congruence; subst; clear Hi.
apply (all_super_interfaces_aux' _ _ _ _ _ H0).
left; reflexivity.
eapply IHl ;eauto.
rewrite all_super_interfaces_aux in H0; discriminate.
intros.
rewrite all_super_interfaces_aux in H0; intros; discriminate.
Qed.
Set Implicit Arguments.
Module Type JVM_HEAP.
Parameter t : Type.
Inductive JVM_AdressingMode : Set :=
| StaticField : JVM_FieldSignature -> JVM_AdressingMode
| DynamicField : JVM_Location -> JVM_FieldSignature -> JVM_AdressingMode
| ArrayElement : JVM_Location -> Z -> JVM_AdressingMode.
Inductive JVM_LocationType : Type :=
| LocationObject : JVM_ClassName -> JVM_LocationType
| LocationArray : Int.t -> JVM_type -> JVM_Method*JVM_PC -> JVM_LocationType.
(** (LocationArray length element_type) *)
Parameter get : t -> JVM_AdressingMode -> option JVM_value.
Parameter update : t -> JVM_AdressingMode -> JVM_value -> t.
Parameter typeof : t -> JVM_Location -> option JVM_LocationType.
(** typeof h loc = None -> no object, no array allocated at location loc *)
Parameter new : t -> JVM_Program -> JVM_LocationType -> option (JVM_Location * t).
(** program is required to compute the size of the allocated element, i.e. to know
the Class associated with a ClassName *)
(** Compatibility between a heap and an adress *)
Inductive Compat (h:t) : JVM_AdressingMode -> Prop :=
| CompatStatic : forall f,
Compat h (StaticField f)
| CompatObject : forall cn loc f,
typeof h loc = Some (LocationObject cn) ->
Compat h (DynamicField loc f)
| CompatArray : forall length tp loc i a,
0 <= i < Int.toZ length ->
typeof h loc = Some (LocationArray length tp a) ->
Compat h (ArrayElement loc i).
Parameter get_update_same : forall h am v, Compat h am -> get (update h am v) am = Some v.
Parameter get_update_old : forall h am1 am2 v, am1<>am2 -> get (update h am1 v) am2 = get h am2.
Parameter get_uncompat : forall h am, ~ Compat h am -> get h am = None.
Parameter typeof_update_same : forall h loc am v,
typeof (update h am v) loc = typeof h loc.
Parameter new_fresh_location : forall (h:t) (p:JVM_Program) (lt:JVM_LocationType) (loc:JVM_Location) (h':t),
new h p lt = Some (loc,h') ->
typeof h loc = None.
Parameter new_typeof : forall (h:t) (p:JVM_Program) (lt:JVM_LocationType) (loc:JVM_Location) (h':t),
new h p lt = Some (loc,h') ->
typeof h' loc = Some lt.
Parameter new_typeof_old : forall (h:t) (p:JVM_Program) (lt:JVM_LocationType) (loc loc':JVM_Location) (h':t),
new h p lt = Some (loc,h') ->
loc <> loc' ->
typeof h' loc' = typeof h loc'.
Parameter new_defined_object_field : forall (h:t) (p:JVM_Program) (cn:JVM_ClassName) (fs:JVM_FieldSignature) (f:JVM_Field) (loc:JVM_Location) (h':t),
new h p (LocationObject cn) = Some (loc,h') ->
is_defined_field p cn fs f ->
get h' (DynamicField loc fs) = Some (init_field_value f).
Parameter new_undefined_object_field : forall (h:t) (p:JVM_Program) (cn:JVM_ClassName) (fs:JVM_FieldSignature) (loc:JVM_Location) (h':t),
new h p (LocationObject cn) = Some (loc,h') ->
~ defined_field p cn fs ->
get h' (DynamicField loc fs) = None.
Parameter new_object_no_change :
forall (h:t) (p:JVM_Program) (cn:JVM_ClassName) (loc:JVM_Location) (h':t) (am:JVM_AdressingMode),
new h p (LocationObject cn) = Some (loc,h') ->
(forall (fs:JVM_FieldSignature), am <> (DynamicField loc fs)) ->
get h' am = get h am.
Parameter new_valid_array_index : forall (h:t) (p:JVM_Program) (length:Int.t) (tp:JVM_type) a (i:Z) (loc:JVM_Location) (h':t),
new h p (LocationArray length tp a) = Some (loc,h') ->
0 <= i < Int.toZ length ->
get h' (ArrayElement loc i) = Some (init_value tp).
Parameter new_unvalid_array_index : forall (h:t) (p:JVM_Program) (length:Int.t) (tp:JVM_type) a (i:Z) (loc:JVM_Location) (h':t),
new h p (LocationArray length tp a) = Some (loc,h') ->
~ 0 <= i < Int.toZ length ->
get h' (ArrayElement loc i) = None.
Parameter new_array_no_change :
forall (h:t) (p:JVM_Program) (length:Int.t) (tp:JVM_type) a (loc:JVM_Location) (h':t) (am:JVM_AdressingMode),
new h p (LocationArray length tp a) = Some (loc,h') ->
(forall (i:Z), am <> (ArrayElement loc i)) ->
get h' am = get h am.
(* These properties should be useless
Parameter get_static_some : forall (h:t) (p:Program) (fs:FieldSignature),
isStatic p fs ->
exists v, get h (StaticField fs) = Some v.
Parameter get_static_some : forall (h:t) (p:Program) (fs:FieldSignature),
~ isStatic p fs ->
exists v, get h (StaticField fs) = None.
*)
End JVM_HEAP.
Module JVM_Heap <: JVM_HEAP.
Module JVM_Object.
Record t : Type := Obj { class : JVM_ClassName; fieldsval : JVM_MapFieldSignature.t JVM_value }.
Definition get (o:t) := JVM_MapFieldSignature.get JVM_value o.(fieldsval).
Definition op_get (o:option t) :=
match o with
| Some o => get o
| None => fun f => None
end.
Definition op_update (f:JVM_FieldSignature) (v:JVM_value) (o:option t) : option t :=
match o with
| None => None
| Some o => Some (Obj o.(class) (JVM_MapFieldSignature.update _ o.(fieldsval) f v))
end.
Definition init_fields : list (JVM_ClassName*JVM_Field) -> JVM_MapFieldSignature.t JVM_value :=
@fold_right (JVM_MapFieldSignature.t JVM_value) _
(fun f m => JVM_MapFieldSignature.update _ m (fst f,JVM_FIELD.signature (snd f)) (init_field_value (snd f)))
(JVM_MapFieldSignature.empty _).
Definition default (c:JVM_Class) : t :=
Obj
(JVM_CLASS.name c)
(init_fields (List.map (fun f => (JVM_CLASS.name c,f)) (JVM_CLASS.definedFields c))).
End JVM_Object.
Module JVM_Array.
Record t : Type := Arr {
length : Int.t;
element_type : JVM_type;
elements : BinNatMap.t JVM_value;
creation_point : JVM_Method * JVM_PC
}.
Definition Ncase (i:Z) : option N :=
match i with
| Z0 => Some N0
| Zpos p => Some (Npos p)
| _ => None
end.
Definition get (a:t) (i:Z) :=
match Ncase i with
| Some n =>
match Zcompare i (Int.toZ a.(length)) with
| Lt => BinNatMap.get JVM_value a.(elements) n
| _ => None
end
| None => None
end.
Definition op_get (a:option (t)) :=
match a with
| Some a => get a
| None => fun i => None
end.
Definition op_update (i:Z) (v:JVM_value) (a:option t) : option t :=
match a with
| None => None
| Some a =>
match Ncase i with
| Some n =>
match Zcompare i (Int.toZ a.(length)) with
| Lt => Some
(Arr a.(length) a.(element_type) (BinNatMap.update _ a.(elements) n v) a.(creation_point))
| _ => Some a
end
| _ => Some a
end
end.
Fixpoint init_array_rec (n:nat) (v:JVM_value) {struct n} : BinNatMap.t JVM_value :=
match n with
| O => BinNatMap.empty _
| S p => BinNatMap.update _ (init_array_rec p v) (N_of_nat p) v
end.
Definition default (lgth:Int.t) (n:nat) (tp:JVM_type) (a:JVM_Method*JVM_PC): t:=
Arr lgth tp (init_array_rec n (init_value tp)) a.
Definition N_to_nat (n:N) : nat :=
match n with
| Npos p => nat_of_P p
| N0 => O
end.
Lemma ZL4_inf : forall y:positive, {h : nat | nat_of_P y = S h}.
Proof.
intro y; induction y as [p H| p H1| ];
[ elim H; intros x H1; exists (S x + S x)%nat; unfold nat_of_P in |- *; simpl in |- *; rewrite ZL0; rewrite Pmult_nat_r_plus_morphism; unfold nat_of_P in H1; rewrite H1; auto with arith | elim H1; intros x H2; exists (x + S x)%nat; unfold nat_of_P in |- *; simpl in |- *; rewrite ZL0; rewrite Pmult_nat_r_plus_morphism; unfold nat_of_P in H2; rewrite H2; auto with arith | exists 0%nat; auto with arith ].
Qed.
Lemma N_of_nat_N_to_nat : forall n, N_of_nat (N_to_nat n) = n.
Proof.
unfold N_of_nat, N_to_nat; intros.
destruct n.
reflexivity.
destruct (ZL4_inf p).
rewrite e.
generalize (nat_of_P_o_P_of_succ_nat_eq_succ x).
rewrite <- e; intros.
assert (Zpos (P_of_succ_nat x) = Zpos p).
rewrite Zpos_eq_Z_of_nat_o_nat_of_P.
rewrite H.
rewrite Zpos_eq_Z_of_nat_o_nat_of_P.
auto.
congruence.
Qed.
Lemma N_to_nat_N_of_nat : forall n, N_to_nat (N_of_nat n) = n.
Proof.
unfold N_of_nat, N_to_nat; intros.
destruct n.
reflexivity.
rewrite nat_of_P_o_P_of_succ_nat_eq_succ.
reflexivity.
Qed.
Lemma get_init_array_rec: forall n v i,
(N_to_nat i<n)%nat ->
BinNatMap.get _ (init_array_rec n v) i = Some v.
Proof.
induction n; simpl; intros.
apply False_ind; omega.
assert (N_to_nat i < n \/ N_to_nat i = n)%nat by omega.
destruct H0; clear H.
rewrite BinNatMap.get_update2.
auto.
intro; subst.
rewrite N_to_nat_N_of_nat in H0; omega.
subst.
rewrite N_of_nat_N_to_nat.
rewrite BinNatMap.get_update1.
reflexivity.
Qed.
Lemma get_init_array_none_rec: forall n v i,
(N_to_nat i>=n)%nat ->
BinNatMap.get _ (init_array_rec n v) i = None.
Proof.
induction n; simpl; intros.
apply BinNatMap.get_empty.
rewrite BinNatMap.get_update2.
apply IHn.
omega.
intro; subst.
rewrite N_to_nat_N_of_nat in H; omega.
Qed.
End JVM_Array.
Module JVM_LocMap := BinNatMap.
Record t_ : Type := hp {
statics : JVM_MapFieldSignature.t JVM_value;
objects : JVM_LocMap.t JVM_Object.t;
arrays : JVM_LocMap.t JVM_Array.t;
next_loc : JVM_Location
}.
Definition t : Type := t_.
Inductive JVM_AdressingMode : Set :=
| StaticField : JVM_FieldSignature -> JVM_AdressingMode
| DynamicField : JVM_Location -> JVM_FieldSignature -> JVM_AdressingMode
| ArrayElement : JVM_Location -> Z -> JVM_AdressingMode.
Inductive JVM_LocationType : Type :=
| LocationObject : JVM_ClassName -> JVM_LocationType
| LocationArray : Int.t -> JVM_type -> JVM_Method*JVM_PC -> JVM_LocationType.
Definition typeof (h:t) (l:JVM_Location) : option (JVM_LocationType) :=
match JVM_LocMap.get _ h.(objects) l with
| Some o => Some (LocationObject o.(JVM_Object.class))
| None =>
match JVM_LocMap.get _ h.(arrays) l with
| Some a => Some (LocationArray a.(JVM_Array.length) a.(JVM_Array.element_type) a.(JVM_Array.creation_point))
| None => None
end
end.
(** Compatibility between a heap and an adress *)
Inductive Compat (h:t) : JVM_AdressingMode -> Prop :=
| CompatStatic : forall f,
Compat h (StaticField f)
| CompatObject : forall cn loc f,
typeof h loc = Some (LocationObject cn) ->
Compat h (DynamicField loc f)
| CompatArray : forall length tp loc i a,
0 <= i < Int.toZ length ->
typeof h loc = Some (LocationArray length tp a) ->
Compat h (ArrayElement loc i).
Definition check_compat (h:t) (am:JVM_AdressingMode) : bool :=
match am with
| StaticField f => true
| DynamicField loc f =>
match typeof h loc with
| Some (LocationObject _) => true
| _ => false
end
| ArrayElement loc i =>
match typeof h loc with
| Some (LocationArray length _ _ ) =>
match i ?= 0 with
| Lt => false
| _ =>
match i ?= Int.toZ length with
| Lt => true
| _ => false
end
end
| _ => false
end
end.
Lemma bounded_compare : forall i t:Z,
i < t -> i ?= t = Lt.
Proof.
intros.
generalize (Zlt_compare _ _ H); rewrite H; auto.
Qed.
Lemma case_compare : forall i t:Z,
(i ?= t = Lt /\ i<t) \/ (i ?=t = Gt) \/ (i ?=t = Eq).
Proof.
intros.
generalize (Zge_compare i t0).
destruct (i ?= t0); simpl; auto.
left; split; auto; omega.
Qed.
Lemma check_compat_correct : forall h am,
if check_compat h am then Compat h am else ~ Compat h am.
Proof.
unfold check_compat; intros.
destruct am.
constructor.
case_eq (typeof h j); intros.
destruct j1.
econstructor; eauto.
intro T; inversion T; subst.
rewrite H1 in H; discriminate.
intro T; inversion T; subst.
rewrite H1 in H; discriminate.
case_eq (typeof h j); intros.
destruct j0.
intro T; inversion T; subst.
rewrite H3 in H; discriminate.
generalize (Zge_compare z 0).
generalize (Zlt_compare z 0).
destruct (z ?= 0).
generalize (Zge_compare z (Int.toZ t0)).
generalize (Zlt_compare z (Int.toZ t0)).
destruct (z ?= (Int.toZ t0)).
repeat intro.
inversion H4; subst.
rewrite H8 in H; inversion H; subst; omega.
intros; econstructor; eauto.
omega.
repeat intro.
inversion H4; subst.
rewrite H8 in H; inversion H; subst; omega.
repeat intro.
inversion H2; subst.
rewrite H6 in H; inversion H; subst; omega.
generalize (Zge_compare z (Int.toZ t0)).
generalize (Zlt_compare z (Int.toZ t0)).
destruct (z ?= (Int.toZ t0)).
repeat intro.
inversion H4; subst.
rewrite H8 in H; inversion H; subst; omega.
intros; econstructor; eauto.
omega.
repeat intro.
inversion H4; subst.
rewrite H8 in H; inversion H; subst; omega.
intro.
inversion H0; subst.
rewrite H4 in H; inversion H.
Qed.
Definition get (h:t) (am:JVM_AdressingMode) : option JVM_value :=
if check_compat h am then
match am with
| StaticField f => JVM_MapFieldSignature.get _ h.(statics) f
| DynamicField l f => JVM_Object.op_get (JVM_LocMap.get _ h.(objects) l) f
| ArrayElement l i => JVM_Array.op_get (JVM_LocMap.get _ h.(arrays) l) i
end
else None.
Definition update (h:t) (am:JVM_AdressingMode) (v:JVM_value) : t :=
if check_compat h am then
match am with
| StaticField f =>
hp
(JVM_MapFieldSignature.update _ h.(statics) f v)
h.(objects)
h.(arrays)
h.(next_loc)
| DynamicField l f =>
hp
h.(statics)
(JVM_LocMap.modify _ h.(objects) l (JVM_Object.op_update f v))
h.(arrays)
h.(next_loc)
| ArrayElement l i =>
hp
h.(statics)
h.(objects)
(JVM_LocMap.modify _ h.(arrays) l (JVM_Array.op_update i v))
h.(next_loc)
end else h.
Definition Int_to_nat (i:Int.t) : option nat :=
match Int.toZ i with
| Zpos p => Some (nat_of_P p)
| Z0 => Some O
| Zneg _ => None
end.
Definition check_fresh (h:t) (l:JVM_Location) : bool :=
match JVM_LocMap.get _ h.(objects) l with
| Some _ => false
| _ =>
match JVM_LocMap.get _ h.(arrays) l with
| Some _ => false
| _ => true
end
end.
Lemma check_fresh_correct : forall h l,
check_fresh h l = true ->
(JVM_LocMap.get _ h.(objects) l = None
/\ JVM_LocMap.get _ h.(arrays) l = None).
Proof.
unfold check_fresh; intros.
destruct (JVM_LocMap.get _ (objects h) l); try discriminate.
destruct (JVM_LocMap.get _ (arrays h) l); try discriminate.
auto.
Qed.
Definition new (h:t) (p:JVM_Program) (ltp:JVM_LocationType) : option (JVM_Location * t) :=
if check_fresh h h.(next_loc) then
match ltp with
| LocationObject cn =>
match JVM_PROG.class p cn with
| None => None
| Some c =>
Some
(h.(next_loc),
hp
h.(statics)
(JVM_LocMap.update _ h.(objects) h.(next_loc) (JVM_Object.default c))
h.(arrays)
(Nsucc h.(next_loc)))
end
| LocationArray lgth tp a =>
match Int_to_nat lgth with
| None => None
| Some n =>
Some
(h.(next_loc),
hp
h.(statics)
h.(objects)
(JVM_LocMap.update _ h.(arrays) h.(next_loc) (JVM_Array.default lgth n tp a))
(Nsucc h.(next_loc)))
end
end
else None.
Lemma pos_Ncase : forall i:Z,
0<=i -> exists n, JVM_Array.Ncase i = Some n.
Proof.
destruct i; simpl; intros; eauto.
elim H; auto.
Qed.
Lemma case_Ncase : forall i:Z,
JVM_Array.Ncase i = None \/
exists n, JVM_Array.Ncase i = Some n /\ 0<=i.
Proof.
destruct i; simpl; eauto with zarith.
Qed.
Lemma Ncase_inj : forall i j,
JVM_Array.Ncase i = JVM_Array.Ncase j -> JVM_Array.Ncase i <> None -> i = j.
Proof.
destruct i; destruct j; intros H; inversion H; auto.
simpl; intuition.
Qed.
Lemma typeof_update_same : forall h loc am v,
typeof (update h am v) loc = typeof h loc.
Proof.
unfold typeof, update; intros.
generalize (check_compat_correct h am); destruct (check_compat h am); simpl; intros.
destruct H; simpl.
auto.
destruct (JVM_PC_eq_dec loc0 loc); subst.
unfold typeof in H.
rewrite JVM_LocMap.get_modify1; auto.
unfold JVM_LocMap.get in H.
unfold JVM_LocMap.get in *.
case_eq (BinNatMap_Base.get JVM_Object.t (objects h) loc); intros.
rewrite H0 in H.
reflexivity.
rewrite H0 in H.
reflexivity.
rewrite JVM_LocMap.get_modify2; auto.
destruct (JVM_PC_eq_dec loc0 loc); subst.
rewrite JVM_LocMap.get_modify1; auto.
unfold typeof in *.
destruct (JVM_LocMap.get JVM_Object.t (objects h) loc); try reflexivity.
unfold JVM_LocMap.get in *.
case_eq (BinNatMap_Base.get _ (arrays h) loc); simpl; intros.
rewrite H1 in H0.
inversion H0; clear H0; subst.
elim (@pos_Ncase i); [intros n H2|omega].
rewrite H2.
rewrite bounded_compare; try omega.
reflexivity.
reflexivity.
rewrite JVM_LocMap.get_modify2; auto.
reflexivity.
Qed.
Lemma Compat_update1 : forall h am am0 v0,
Compat h am -> Compat (update h am0 v0) am.
Proof.
destruct 1; econstructor; eauto.
rewrite typeof_update_same; eauto.
rewrite typeof_update_same; eauto.
Qed.
Lemma Compat_update2 : forall h am am0 v0,
Compat (update h am0 v0) am -> Compat h am.
Proof.
destruct 1; econstructor; eauto.
rewrite typeof_update_same in H; eauto.
rewrite typeof_update_same in H0; eauto.
Qed.
Lemma check_compat_update : forall h am v am',
check_compat (update h am v) am' = check_compat h am'.
Proof.
intros.
generalize (check_compat_correct (update h am v) am').
generalize (check_compat_correct h am').
destruct (check_compat (update h am v) am');
destruct (check_compat h am'); auto.
intros.
elim H; eapply Compat_update2; eauto.
intros.
elim H0; eapply Compat_update1; eauto.
Qed.
Lemma get_update_same : forall h am v, Compat h am -> get (update h am v) am = Some v.
Proof.
unfold get; intros.
rewrite check_compat_update.
unfold update.
generalize (check_compat_correct h am); destruct (check_compat h am).
clear H.
destruct 1; simpl.
rewrite JVM_MapFieldSignature.get_update1; auto.
rewrite JVM_LocMap.get_modify1; auto.
caseeq (BinNatMap_Base.get JVM_Object.t (objects h) loc); intros; simpl.
unfold JVM_Object.get; simpl.
rewrite JVM_MapFieldSignature.get_update1; auto.
unfold typeof in H.
replace JVM_LocMap.get with BinNatMap_Base.get in H; auto.
rewrite H0 in H.