LambdaCalculusAndTypeSoundness: remove the other use of evaluation contexts

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Adam Chlipala 2021-03-27 18:57:03 -04:00
parent 008c45351a
commit f5aed26c77

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@ -306,68 +306,53 @@ Module Ulc.
Qed.
(** * Small-step semantics with evaluation contexts *)
(** * Small-step semantics *)
(* We can also port to this setting our small-step semantics style based on
* evaluation contexts. *)
(* We can also port to this setting our small-step semantics style. *)
Inductive context : Set :=
| Hole : context
| App1 : context -> exp -> context
| App2 : exp -> context -> context.
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e,
plug Hole e e
| PlugApp1 : forall c e1 e2 e,
plug c e1 e
-> plug (App1 c e2) e1 (App e e2)
| PlugApp2 : forall c e1 e2 e,
value e1
-> plug c e2 e
-> plug (App2 e1 c) e2 (App e1 e).
(* Subtle point: the [value] hypothesis right above enforces a well-formedness
* condition on contexts that may actually be plugged. We don't allow
* skipping over a lefthand subterm of an application when that term has
* evaluation work left to do. This condition is the essence of
* *call-by-value* instead of other evaluation strategies. Details are
* largely beyond our scope here. *)
(* Compared to the small-step contextual semantics from two chapters back, we
* skip a [step0] relation, since function application (called "beta
* reduction") is the only option here. *)
(* Function application (called "beta reduction") is the big rule here. *)
Inductive step : exp -> exp -> Prop :=
| ContextBeta : forall c x e v e1 e2,
| ContextBeta : forall x e v,
value v
-> plug c (App (Abs x e) v) e1
-> plug c (subst v x e) e2
-> step e1 e2.
-> step (App (Abs x e) v) (subst v x e)
Local Hint Constructors plug step : core.
(* However, we also need bureaucractic rules for pushing evaluation inside
* applications. *)
| App1 : forall e1 e1' e2,
step e1 e1'
-> step (App e1 e2) (App e1' e2)
| App2 : forall v e2 e2',
value v
-> step e2 e2'
-> step (App v e2) (App v e2').
(* Note how that last rule enforces a deterministic evaluation order!
* We call it *call-by-value*. *)
Local Hint Constructors step : core.
(* Here we now go through a proof of equivalence between big- and small-step
* semantics, though we won't spend any further commentary on it. *)
Lemma step_eval'' : forall v c x e e1 e2 v0,
value v
-> plug c (App (Abs x e) v) e1
-> plug c (subst v x e) e2
-> eval e2 v0
-> eval e1 v0.
Proof.
induct c; invert 2; invert 1; simplify; eauto.
invert H0; eauto.
invert H0; eauto.
Qed.
Local Hint Resolve step_eval'' : core.
Lemma step_eval' : forall e1 e2,
step e1 e2
-> forall v, eval e2 v
-> eval e1 v.
Proof.
invert 1; simplify; eauto.
induct 1; simplify; eauto.
invert H0.
econstructor.
apply IHstep.
eassumption.
eassumption.
assumption.
invert H1.
econstructor.
eassumption.
apply IHstep.
eassumption.
assumption.
Qed.
Local Hint Resolve step_eval' : core.
@ -380,76 +365,23 @@ Module Ulc.
induct 1; eauto.
Qed.
Lemma plug_functional : forall C e e1,
plug C e e1
-> forall e2, plug C e e2
-> e1 = e2.
Local Hint Resolve eval_value : core.
Theorem step_app1 : forall e1 e1' e2,
step^* e1 e1'
-> step^* (App e1 e2) (App e1' e2).
Proof.
induct 1; invert 1; simplify; try f_equal; eauto.
induct 1; eauto.
Qed.
Lemma plug_mirror : forall C e e', plug C e e'
-> forall e1, exists e1', plug C e1 e1'.
Theorem step_app2 : forall e2 e2' v,
value v
-> step^* e2 e2'
-> step^* (App v e2) (App v e2').
Proof.
induct 1; simplify; eauto.
specialize (IHplug e0); first_order; eauto.
specialize (IHplug e0); first_order; eauto.
induct 2; eauto.
Qed.
Fixpoint compose (C1 C2 : context) : context :=
match C2 with
| Hole => C1
| App1 C2' e => App1 (compose C1 C2') e
| App2 v C2' => App2 v (compose C1 C2')
end.
Lemma compose_ok : forall C1 C2 e1 e2 e3,
plug C1 e1 e2
-> plug C2 e2 e3
-> plug (compose C1 C2) e1 e3.
Proof.
induct 2; simplify; eauto.
Qed.
Local Hint Resolve compose_ok : core.
Lemma step_plug : forall e1 e2,
step e1 e2
-> forall C e1' e2', plug C e1 e1'
-> plug C e2 e2'
-> step e1' e2'.
Proof.
invert 1; simplify; eauto.
Qed.
Lemma stepStar_plug : forall e1 e2,
step^* e1 e2
-> forall C e1' e2', plug C e1 e1'
-> plug C e2 e2'
-> step^* e1' e2'.
Proof.
induct 1; simplify.
assert (e1' = e2') by (eapply plug_functional; eassumption).
subst.
constructor.
assert (exists y', plug C y y') by eauto using plug_mirror.
invert H3.
eapply step_plug in H.
econstructor.
eassumption.
eapply IHtrc.
eassumption.
assumption.
eassumption.
assumption.
Qed.
Local Hint Resolve stepStar_plug eval_value : core.
Theorem eval_step : forall e v,
eval e v
-> step^* e v.
@ -457,10 +389,16 @@ Module Ulc.
induct 1; eauto.
eapply trc_trans.
eapply stepStar_plug with (e1 := e1) (e2 := Abs x e1') (C := App1 Hole e2); eauto.
apply step_app1.
eassumption.
eapply trc_trans.
eapply stepStar_plug with (e1 := e2) (e2 := v2) (C := App2 (Abs x e1') Hole); eauto.
eapply step_app2.
constructor.
eassumption.
econstructor.
constructor.
eauto.
assumption.
Qed.
End Ulc.