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