EvaluationContexts: mutable variables

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Adam Chlipala 2021-03-28 14:51:12 -04:00
parent af135d6853
commit 95a28b26f6

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@ -111,13 +111,6 @@ Module Stlc.
Local Hint Constructors value plug step0 step hasty : core. Local Hint Constructors value plug step0 step hasty : core.
Infix "-->" := Fun (at level 60, right associativity).
Coercion Const : nat >-> exp.
Infix "^+^" := Plus (at level 50).
Coercion Var : var >-> exp.
Notation "\ x , e" := (Abs x e) (at level 51).
Infix "@" := App (at level 49, left associativity).
(** * Now we adapt the automated proof of type soundness. *) (** * Now we adapt the automated proof of type soundness. *)
Ltac t0 := match goal with Ltac t0 := match goal with
@ -451,13 +444,6 @@ Module StlcPairs.
Local Hint Constructors value plug step0 step hasty : core. Local Hint Constructors value plug step0 step hasty : core.
Infix "-->" := Fun (at level 60, right associativity).
Coercion Const : nat >-> exp.
Infix "^+^" := Plus (at level 50).
Coercion Var : var >-> exp.
Notation "\ x , e" := (Abs x e) (at level 51).
Infix "@" := App (at level 49, left associativity).
Ltac t0 := match goal with Ltac t0 := match goal with
| [ H : ex _ |- _ ] => invert H | [ H : ex _ |- _ ] => invert H
| [ H : _ /\ _ |- _ ] => invert H | [ H : _ /\ _ |- _ ] => invert H
@ -760,13 +746,6 @@ Module StlcSums.
Local Hint Constructors value plug step0 step hasty : core. Local Hint Constructors value plug step0 step hasty : core.
Infix "-->" := Fun (at level 60, right associativity).
Coercion Const : nat >-> exp.
Infix "^+^" := Plus (at level 50).
Coercion Var : var >-> exp.
Notation "\ x , e" := (Abs x e) (at level 51).
Infix "@" := App (at level 49, left associativity).
Ltac t0 := match goal with Ltac t0 := match goal with
| [ H : ex _ |- _ ] => invert H | [ H : ex _ |- _ ] => invert H
| [ H : _ /\ _ |- _ ] => invert H | [ H : _ /\ _ |- _ ] => invert H
@ -881,3 +860,415 @@ Module StlcSums.
apply invariant_induction; simplify; eauto; equality. apply invariant_induction; simplify; eauto; equality.
Qed. Qed.
End StlcSums. End StlcSums.
(** * Mutable Variables *)
(* Let's show how to add in another classic language feature, showing off how
* abstracting with the idea of evaluation contexts saves us a lot of effort
* adapting old definitions. So far, variables have been read-only, but we can
* easily enough make them mutable. The types of some of our relations will
* change, but surprisingly much will remain intact. *)
Module StlcMutable.
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)
(* New cases: *)
| GetVar (x : var)
| SetVar (x : var) (e : exp).
(* Note that we are distinguishing between mutable and immutable variables,
* keeping the latter to bind in [Abs] and [Match]. *)
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)
| GetVar y => GetVar y
| SetVar y e2' => SetVar y (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 case: *)
| SetVar1 : var -> context -> context.
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e, plug Hole e e
| PlugPlus1 : forall e e' C e2,
plug C e e'
-> plug (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall e e' C e2,
plug C e e'
-> plug (App1 C e2) e (App e' e2)
| PlugApp2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (App2 v1 C) e (App v1 e')
| PlugPair1 : forall e e' C e2,
plug C e e'
-> plug (Pair1 C e2) e (Pair e' e2)
| PlugPair2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Pair2 v1 C) e (Pair v1 e')
| PlugFst1 : forall e e' C,
plug C e e'
-> plug (Fst1 C) e (Fst e')
| PlugSnd1 : forall e e' C,
plug C e e'
-> plug (Snd1 C) e (Snd e')
| PlugInl1 : forall e e' C,
plug C e e'
-> plug (Inl1 C) e (Inl e')
| PlugInr1 : forall e e' C,
plug C e e'
-> plug (Inr1 C) e (Inr e')
| PluMatch1 : forall e e' C x1 e1 x2 e2,
plug C e e'
-> plug (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2)
(* Our new plugging rules *)
| PluSetVar1 : forall x e e' C,
plug C e e'
-> plug (SetVar1 x C) e (SetVar x e').
Definition valuation := fmap var exp.
Inductive step0 : valuation * exp -> valuation * exp -> Prop :=
| Beta : forall env x e v,
value v
-> step0 (env, App (Abs x e) v) (env, subst v x e)
| Add : forall env n1 n2,
step0 (env, Plus (Const n1) (Const n2)) (env, Const (n1 + n2))
| FstPair : forall env v1 v2,
value v1
-> value v2
-> step0 (env, Fst (Pair v1 v2)) (env, v1)
| SndPair : forall env v1 v2,
value v1
-> value v2
-> step0 (env, Snd (Pair v1 v2)) (env, v2)
| MatchInl : forall env v x1 e1 x2 e2,
value v
-> step0 (env, Match (Inl v) x1 e1 x2 e2) (env, subst v x1 e1)
| MatchInr : forall env v x1 e1 x2 e2,
value v
-> step0 (env, Match (Inr v) x1 e1 x2 e2) (env, subst v x2 e2)
| Read : forall env x v,
env $? x = Some v
-> step0 (env, GetVar x) (env, v)
| Overwrite : forall env x v,
value v
-> step0 (env, SetVar x v) (env $+ (x, v), v).
Inductive step : valuation * exp -> valuation * exp -> Prop :=
| StepRule : forall C env1 e1 env2 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step0 (env1, e1) (env2, e2)
-> step (env1, e1') (env2, e2').
Definition trsys_of (env : valuation) (e : exp) := {|
Initial := {(env, e)};
Step := step
|}.
Inductive type :=
| Nat
| Fun (dom ran : type)
| Prod (t1 t2 : type)
| Sum (t1 t2 : type).
(* Now there will be two typing contexts, one for mutable variables and one
* for immutable. *)
Inductive hasty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> hasty M G (Var x) t
| HtConst : forall G n,
hasty M G (Const n) Nat
| HtPlus : forall G e1 e2,
hasty M G e1 Nat
-> hasty M G e2 Nat
-> hasty M G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
hasty M (G $+ (x, t1)) e1 t2
-> hasty M G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
hasty M G e1 (Fun t1 t2)
-> hasty M G e2 t1
-> hasty M G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
hasty M G e1 t1
-> hasty M G e2 t2
-> hasty M G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
hasty M G e1 (Prod t1 t2)
-> hasty M G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
hasty M G e1 (Prod t1 t2)
-> hasty M G (Snd e1) t2
| HtInl : forall G e1 t1 t2,
hasty M G e1 t1
-> hasty M G (Inl e1) (Sum t1 t2)
| HtInr : forall G e1 t1 t2,
hasty M G e1 t2
-> hasty M G (Inr e1) (Sum t1 t2)
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
hasty M G e (Sum t1 t2)
-> hasty M (G $+ (x1, t1)) e1 t
-> hasty M (G $+ (x2, t2)) e2 t
-> hasty M G (Match e x1 e1 x2 e2) t
(* New cases: *)
| HtGetVar : forall G x t,
M $? x = Some t
-> hasty M G (GetVar x) t
| HtSetVar : forall G x e t,
M $? x = Some t
-> hasty M G e t
-> hasty M G (SetVar x e) t.
Local Hint Constructors value plug step0 step hasty : core.
(* Let's characterize what makes a typing context and a variable valuation
* compatible. *)
Definition compatible (M : fmap var type) (env : valuation) :=
forall x t, M $? x = Some t
-> exists v, env $? x = Some v
/\ hasty M $0 v t.
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
(* Add a few extra underscores below, for new [hasty] argument. *)
| [ 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
(* New cases: *)
| [ H1 : compatible ?M ?env, H2 : ?M $? ?x = Some _ |- _ ] =>
(assert (exists v, env $? x = Some v) by eauto; fail 1)
|| (pose proof (H1 _ _ H2); first_order)
| [ H1 : forall env, compatible ?M env -> _, H2 : compatible ?M _ |- _ ] =>
specialize (H1 _ H2); first_order
end; subst.
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
Lemma progress : forall M e t,
hasty M $0 e t
-> forall env, compatible M env
-> value e
\/ exists st', step (env, e) st'.
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 M G e t, hasty M G e t
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
-> hasty M G' e t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve weakening : core.
Lemma hasty_change : forall M G e t,
hasty M G e t
-> forall G', G' = G
-> hasty M G' e t.
Proof.
t.
Qed.
Local Hint Resolve hasty_change : core.
Lemma substitution : forall M G x t' e t e',
hasty M (G $+ (x, t')) e t
-> hasty M $0 e' t'
-> hasty M G (subst e' x e) t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve substitution : core.
Lemma compatible_bind : forall M env x v t,
compatible M env
-> M $? x = Some t
-> hasty M $0 v t
-> compatible M (env $+ (x, v)).
Proof.
unfold compatible; first_order.
cases (x ==v x0); subst; simplify.
rewrite H0 in H2.
invert H2.
eauto.
apply H in H0; first_order.
Qed.
Local Hint Resolve compatible_bind : core.
Lemma compatible_hasty : forall M env x v t,
compatible M env
-> env $? x = Some v
-> M $? x = Some t
-> hasty M $0 v t.
Proof.
t.
specialize (H _ _ H1); first_order.
rewrite H0 in H.
invert H.
assumption.
Qed.
Local Hint Immediate compatible_hasty : core.
Lemma preservation0_exp : forall M env1 e1 env2 e2,
step0 (env1, e1) (env2, e2)
-> forall t, hasty M $0 e1 t
-> compatible M env1
-> hasty M $0 e2 t.
Proof.
invert 1; t.
Qed.
Lemma preservation0_env : forall M env1 e1 env2 e2,
step0 (env1, e1) (env2, e2)
-> forall t, hasty M $0 e1 t
-> compatible M env1
-> compatible M env2.
Proof.
invert 1; t.
Qed.
Local Hint Resolve preservation0_exp preservation0_env : core.
Lemma preservation'_exp : forall M C e1 e1',
plug C e1 e1'
-> forall e2 e2' t env1 env2, plug C e2 e2'
-> step0 (env1, e1) (env2, e2)
-> hasty M $0 e1' t
-> compatible M env1
-> hasty M $0 e2' t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve preservation'_exp : core.
Lemma preservation'_env' : forall M C e1 e1',
plug C e1 e1'
-> forall t e2 env1 env2, step0 (env1, e1) (env2, e2)
-> hasty M $0 e1' t
-> compatible M env1
-> compatible M env2.
Proof.
induct 1; t.
Qed.
Lemma preservation'_env : forall M C e1 e1' t e2 env1 env2,
step0 (env1, e1) (env2, e2)
-> plug C e1 e1'
-> hasty M $0 e1' t
-> compatible M env1
-> compatible M env2.
Proof.
simplify; eapply preservation'_env' with (e2 := e2); eauto.
Qed.
Local Hint Immediate preservation'_env : core.
Lemma preservation : forall M env1 e1 env2 e2,
step (env1, e1) (env2, e2)
-> forall t, hasty M $0 e1 t
-> compatible M env1
-> hasty M $0 e2 t /\ compatible M env2.
Proof.
invert 1; t.
Qed.
Local Hint Resolve progress preservation : core.
Theorem safety : forall M env e t, hasty M $0 e t
-> compatible M env
-> invariantFor (trsys_of env e)
(fun st => value (snd st)
\/ exists st', step st st').
Proof.
simplify.
apply invariant_weaken with (invariant1 := fun st => hasty M $0 (snd st) t /\ compatible M (fst st)); eauto.
apply invariant_induction; simplify.
propositional; subst; auto.
invert H1.
cases s; cases s'; simplify.
eauto.
propositional.
cases s; simplify.
eauto.
Qed.
End StlcMutable.