mirror of
https://github.com/achlipala/frap.git
synced 2024-11-12 17:17:50 +00:00
EvaluationContexts: exceptions
This commit is contained in:
Adam Chlipala
parent
544e7fa500
commit
6866ca2f77
1 changed files with 345 additions and 0 deletions
|
|
@ -861,6 +861,351 @@ Module StlcSums.
|
|||
Qed.
|
||||
End StlcSums.
|
||||
|
||||
(** * Exceptions *)
|
||||
|
||||
(* Evaluation contexts are very helpful for concise modeling of control-flow
|
||||
* constructs like exceptions. Let's look at an example where exceptions are
|
||||
* just numbers, for simplicity. *)
|
||||
|
||||
Module StlcExceptions.
|
||||
Inductive exp : Set :=
|
||||
| Var (x : var)
|
||||
| Const (n : nat)
|
||||
| Plus (e1 e2 : exp)
|
||||
| Abs (x : var) (e1 : exp)
|
||||
| App (e1 e2 : exp)
|
||||
| Pair (e1 e2 : exp)
|
||||
| Fst (e1 : exp)
|
||||
| Snd (e2 : exp)
|
||||
| Inl (e1 : exp)
|
||||
| Inr (e2 : exp)
|
||||
| Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp)
|
||||
|
||||
| Throw (e1 : exp)
|
||||
| Catch (e1 : exp) (x : var) (e2 : exp).
|
||||
(* The last one roughly means "try e1 catch x => e2". *)
|
||||
|
||||
Inductive value : exp -> Prop :=
|
||||
| VConst : forall n, value (Const n)
|
||||
| VAbs : forall x e1, value (Abs x e1)
|
||||
| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
|
||||
| VInl : forall v, value v -> value (Inl v)
|
||||
| VInr : forall v, value v -> value (Inr v).
|
||||
|
||||
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
|
||||
match e2 with
|
||||
| Var y => if y ==v x then e1 else Var y
|
||||
| Const n => Const n
|
||||
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
|
||||
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
|
||||
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
|
||||
| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
|
||||
| Fst e2' => Fst (subst e1 x e2')
|
||||
| Snd e2' => Snd (subst e1 x e2')
|
||||
| Inl e2' => Inl (subst e1 x e2')
|
||||
| Inr e2' => Inr (subst e1 x e2')
|
||||
| Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
|
||||
x1 (if x1 ==v x then e21 else subst e1 x e21)
|
||||
x2 (if x2 ==v x then e22 else subst e1 x e22)
|
||||
|
||||
(* New cases: *)
|
||||
| Throw e2' => Throw (subst e1 x e2')
|
||||
| Catch e2' x1 e2'' => Catch (subst e1 x e2')
|
||||
x1 (if x1 ==v x then e2'' else subst e1 x e2'')
|
||||
end.
|
||||
|
||||
Inductive context : Set :=
|
||||
| Hole : context
|
||||
| Plus1 : context -> exp -> context
|
||||
| Plus2 : exp -> context -> context
|
||||
| App1 : context -> exp -> context
|
||||
| App2 : exp -> context -> context
|
||||
| Pair1 : context -> exp -> context
|
||||
| Pair2 : exp -> context -> context
|
||||
| Fst1 : context -> context
|
||||
| Snd1 : context -> context
|
||||
| Inl1 : context -> context
|
||||
| Inr1 : context -> context
|
||||
| Match1 : context -> var -> exp -> var -> exp -> context
|
||||
|
||||
(* New cases: *)
|
||||
| Throw1 : context -> context
|
||||
| Catch1 : context -> var -> exp -> context.
|
||||
|
||||
(* We modify [plug] with a new Boolean argument, to control whether [Catch1]
|
||||
* context kinds are allowed. *)
|
||||
Inductive plug : bool -> context -> exp -> exp -> Prop :=
|
||||
| PlugHole : forall ac e, plug ac Hole e e
|
||||
| PlugPlus1 : forall ac e e' C e2,
|
||||
plug ac C e e'
|
||||
-> plug ac (Plus1 C e2) e (Plus e' e2)
|
||||
| PlugPlus2 : forall ac e e' v1 C,
|
||||
value v1
|
||||
-> plug ac C e e'
|
||||
-> plug ac (Plus2 v1 C) e (Plus v1 e')
|
||||
| PlugApp1 : forall ac e e' C e2,
|
||||
plug ac C e e'
|
||||
-> plug ac (App1 C e2) e (App e' e2)
|
||||
| PlugApp2 : forall ac e e' v1 C,
|
||||
value v1
|
||||
-> plug ac C e e'
|
||||
-> plug ac (App2 v1 C) e (App v1 e')
|
||||
| PlugPair1 : forall ac e e' C e2,
|
||||
plug ac C e e'
|
||||
-> plug ac (Pair1 C e2) e (Pair e' e2)
|
||||
| PlugPair2 : forall ac e e' v1 C,
|
||||
value v1
|
||||
-> plug ac C e e'
|
||||
-> plug ac (Pair2 v1 C) e (Pair v1 e')
|
||||
| PlugFst1 : forall ac e e' C,
|
||||
plug ac C e e'
|
||||
-> plug ac (Fst1 C) e (Fst e')
|
||||
| PlugSnd1 : forall ac e e' C,
|
||||
plug ac C e e'
|
||||
-> plug ac (Snd1 C) e (Snd e')
|
||||
| PlugInl1 : forall ac e e' C,
|
||||
plug ac C e e'
|
||||
-> plug ac (Inl1 C) e (Inl e')
|
||||
| PlugInr1 : forall ac e e' C,
|
||||
plug ac C e e'
|
||||
-> plug ac (Inr1 C) e (Inr e')
|
||||
| PluMatch1 : forall ac e e' C x1 e1 x2 e2,
|
||||
plug ac C e e'
|
||||
-> plug ac (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2)
|
||||
| PlugThrow1 : forall ac e e' C,
|
||||
plug ac C e e'
|
||||
-> plug ac (Throw1 C) e (Throw e')
|
||||
| PlugCatch1 : forall e e' C x1 e1,
|
||||
plug true C e e'
|
||||
-> plug true (Catch1 C x1 e1) e (Catch e' x1 e1).
|
||||
|
||||
Inductive step0 : exp -> exp -> Prop :=
|
||||
| Beta : forall x e v,
|
||||
value v
|
||||
-> step0 (App (Abs x e) v) (subst v x e)
|
||||
| Add : forall n1 n2,
|
||||
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
|
||||
| FstPair : forall v1 v2,
|
||||
value v1
|
||||
-> value v2
|
||||
-> step0 (Fst (Pair v1 v2)) v1
|
||||
| SndPair : forall v1 v2,
|
||||
value v1
|
||||
-> value v2
|
||||
-> step0 (Snd (Pair v1 v2)) v2
|
||||
| MatchInl : forall v x1 e1 x2 e2,
|
||||
value v
|
||||
-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
|
||||
| MatchInr : forall v x1 e1 x2 e2,
|
||||
value v
|
||||
-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2)
|
||||
|
||||
| ThrowBubble : forall v C e,
|
||||
plug false C (Throw v) e
|
||||
-> value v
|
||||
-> C <> Hole
|
||||
-> step0 e (Throw v)
|
||||
| CatchValue : forall v x1 e1,
|
||||
value v
|
||||
-> step0 (Catch v x1 e1) v
|
||||
| CatchThrow : forall v x1 e1,
|
||||
value v
|
||||
-> step0 (Catch (Throw v) x1 e1) (subst v x1 e1).
|
||||
|
||||
Inductive step : exp -> exp -> Prop :=
|
||||
| StepRule : forall C e1 e2 e1' e2',
|
||||
plug true C e1 e1'
|
||||
-> plug true C e2 e2'
|
||||
-> step0 e1 e2
|
||||
-> step e1' e2'.
|
||||
|
||||
Definition trsys_of (e : exp) := {|
|
||||
Initial := {e};
|
||||
Step := step
|
||||
|}.
|
||||
|
||||
|
||||
Inductive type :=
|
||||
| Nat
|
||||
| Fun (dom ran : type)
|
||||
| Prod (t1 t2 : type)
|
||||
| Sum (t1 t2 : type).
|
||||
|
||||
Inductive hasty : fmap var type -> exp -> type -> Prop :=
|
||||
| HtVar : forall G x t,
|
||||
G $? x = Some t
|
||||
-> hasty G (Var x) t
|
||||
| HtConst : forall G n,
|
||||
hasty G (Const n) Nat
|
||||
| HtPlus : forall G e1 e2,
|
||||
hasty G e1 Nat
|
||||
-> hasty G e2 Nat
|
||||
-> hasty G (Plus e1 e2) Nat
|
||||
| HtAbs : forall G x e1 t1 t2,
|
||||
hasty (G $+ (x, t1)) e1 t2
|
||||
-> hasty G (Abs x e1) (Fun t1 t2)
|
||||
| HtApp : forall G e1 e2 t1 t2,
|
||||
hasty G e1 (Fun t1 t2)
|
||||
-> hasty G e2 t1
|
||||
-> hasty G (App e1 e2) t2
|
||||
| HtPair : forall G e1 e2 t1 t2,
|
||||
hasty G e1 t1
|
||||
-> hasty G e2 t2
|
||||
-> hasty G (Pair e1 e2) (Prod t1 t2)
|
||||
| HtFst : forall G e1 t1 t2,
|
||||
hasty G e1 (Prod t1 t2)
|
||||
-> hasty G (Fst e1) t1
|
||||
| HtSnd : forall G e1 t1 t2,
|
||||
hasty G e1 (Prod t1 t2)
|
||||
-> hasty G (Snd e1) t2
|
||||
| HtInl : forall G e1 t1 t2,
|
||||
hasty G e1 t1
|
||||
-> hasty G (Inl e1) (Sum t1 t2)
|
||||
| HtInr : forall G e1 t1 t2,
|
||||
hasty G e1 t2
|
||||
-> hasty G (Inr e1) (Sum t1 t2)
|
||||
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
|
||||
hasty G e (Sum t1 t2)
|
||||
-> hasty (G $+ (x1, t1)) e1 t
|
||||
-> hasty (G $+ (x2, t2)) e2 t
|
||||
-> hasty G (Match e x1 e1 x2 e2) t
|
||||
|
||||
(* New cases: *)
|
||||
| HtThrow : forall G e1 t,
|
||||
hasty G e1 Nat
|
||||
-> hasty G (Throw e1) t
|
||||
| HtCatch : forall G e x1 e1 t,
|
||||
hasty G e t
|
||||
-> hasty (G $+ (x1, Nat)) e1 t
|
||||
-> hasty G (Catch e x1 e1) t.
|
||||
|
||||
Local Hint Constructors value plug step0 step hasty : core.
|
||||
|
||||
Ltac t0 := match goal with
|
||||
| [ H : ex _ |- _ ] => invert H
|
||||
| [ H : _ /\ _ |- _ ] => invert H
|
||||
| [ |- context[?x ==v ?y] ] => cases (x ==v y)
|
||||
| [ H : Some _ = Some _ |- _ ] => invert H
|
||||
|
||||
| [ H : step _ _ |- _ ] => invert H
|
||||
| [ H : step0 _ _ |- _ ] => invert1 H
|
||||
| [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; []
|
||||
|
||||
| [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; [|]
|
||||
|
||||
| [ H : hasty _ _ _ |- _ ] => invert1 H
|
||||
| [ H : plug _ _ _ _ |- _ ] => invert1 H (* added an underscore *)
|
||||
end; subst.
|
||||
|
||||
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
|
||||
|
||||
Local Hint Extern 1 (_ <> Hole) => equality : core.
|
||||
|
||||
Lemma progress : forall e t,
|
||||
hasty $0 e t
|
||||
-> value e
|
||||
\/ (exists n : nat, e = Throw (Const n))
|
||||
\/ (exists e' : exp, step e e').
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
||||
Lemma weakening_override : forall (G G' : fmap var type) x t,
|
||||
(forall x' t', G $? x' = Some t' -> G' $? x' = Some t')
|
||||
-> (forall x' t', G $+ (x, t) $? x' = Some t'
|
||||
-> G' $+ (x, t) $? x' = Some t').
|
||||
Proof.
|
||||
simplify.
|
||||
cases (x ==v x'); simplify; eauto.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve weakening_override : core.
|
||||
|
||||
Lemma weakening : forall G e t,
|
||||
hasty G e t
|
||||
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
|
||||
-> hasty G' e t.
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve weakening : core.
|
||||
|
||||
(* Replacing a typing context with an equal one has no effect (useful to guide
|
||||
* proof search as a hint). *)
|
||||
Lemma hasty_change : forall G e t,
|
||||
hasty G e t
|
||||
-> forall G', G' = G
|
||||
-> hasty G' e t.
|
||||
Proof.
|
||||
t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve hasty_change : core.
|
||||
|
||||
Lemma substitution : forall G x t' e t e',
|
||||
hasty (G $+ (x, t')) e t
|
||||
-> hasty $0 e' t'
|
||||
-> hasty G (subst e' x e) t.
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve substitution : core.
|
||||
|
||||
Lemma throw_well_typed : forall C v e,
|
||||
plug false C (Throw v) e
|
||||
-> forall t, hasty $0 e t
|
||||
-> hasty $0 v Nat.
|
||||
Proof.
|
||||
induct 1; invert 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve throw_well_typed : core.
|
||||
|
||||
Lemma preservation0 : forall e1 e2,
|
||||
step0 e1 e2
|
||||
-> forall t, hasty $0 e1 t
|
||||
-> hasty $0 e2 t.
|
||||
Proof.
|
||||
invert 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve preservation0 : core.
|
||||
|
||||
Lemma preservation' : forall C e1 e1',
|
||||
plug true C e1 e1'
|
||||
-> forall e2 e2' t, plug true C e2 e2'
|
||||
-> step0 e1 e2
|
||||
-> hasty $0 e1' t
|
||||
-> hasty $0 e2' t.
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve preservation' : core.
|
||||
|
||||
Lemma preservation : forall e1 e2,
|
||||
step e1 e2
|
||||
-> forall t, hasty $0 e1 t
|
||||
-> hasty $0 e2 t.
|
||||
Proof.
|
||||
invert 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve progress preservation : core.
|
||||
|
||||
Theorem safety : forall e t, hasty $0 e t
|
||||
-> invariantFor (trsys_of e)
|
||||
(fun e' => value e'
|
||||
\/ (exists n, e' = Throw (Const n))
|
||||
\/ exists e'', step e' e'').
|
||||
Proof.
|
||||
simplify.
|
||||
apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t); eauto.
|
||||
apply invariant_induction; simplify; eauto; equality.
|
||||
Qed.
|
||||
End StlcExceptions.
|
||||
|
||||
(** * Mutable Variables *)
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue