LambdaCalculusAndTypeSoundness_template update

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Adam Chlipala 2021-03-27 19:15:05 -04:00
parent 8c2c0f5cfa
commit bcbb2181be

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@ -37,7 +37,7 @@ Module Ulc.
Inductive value : exp -> Prop :=
| Value : forall x e, value (Abs x e).
Hint Constructors eval value.
Local Hint Constructors eval value : core.
Theorem value_eval : forall v,
value v
@ -46,7 +46,7 @@ Module Ulc.
invert 1; eauto.
Qed.
Hint Resolve value_eval.
Local Hint Resolve value_eval : core.
Theorem eval_value : forall e v,
eval e v
@ -55,7 +55,7 @@ Module Ulc.
induct 1; eauto.
Qed.
Hint Resolve eval_value.
Local Hint Resolve eval_value : core.
(* Some notations, to let us write more normal-looking lambda terms *)
Coercion Var : var >-> exp.
@ -276,59 +276,51 @@ Module Ulc.
Qed.
(** * Small-step semantics with evaluation contexts *)
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).
(** * Small-step semantics *)
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)
Hint Constructors plug step.
(* 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').
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.
Hint Resolve step_eval''.
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.
Hint Resolve step_eval'.
Local Hint Resolve step_eval' : core.
Theorem step_eval : forall e v,
step^* e v
@ -338,76 +330,23 @@ 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.
Hint Resolve compose_ok.
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.
Hint Resolve stepStar_plug eval_value.
Theorem eval_step : forall e v,
eval e v
-> step^* e v.
@ -415,10 +354,16 @@ 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.
@ -444,50 +389,32 @@ Module Stlc.
| App e2' e2'' => App (subst e1 x e2') (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.
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').
Inductive step0 : exp -> exp -> Prop :=
Inductive step : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
-> step (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2)).
Inductive step : exp -> exp -> Prop :=
| StepRule : forall C e1 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step0 e1 e2
-> step e1' e2'.
step (Plus (Const n1) (Const n2)) (Const (n1 + n2))
| 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')
| Plus1 : forall e1 e1' e2,
step e1 e1'
-> step (Plus e1 e2) (Plus e1' e2)
| Plus2 : forall v e2 e2',
value v
-> step e2 e2'
-> step (Plus v e2) (Plus v e2').
Definition trsys_of (e : exp) := {|
Initial := {e};
Step := step
|}.
Inductive type :=
| Nat (* Numbers *)
| Fun (dom ran : type) (* Functions *).
@ -510,7 +437,7 @@ Module Stlc.
-> hasty G e2 t1
-> hasty G (App e1 e2) t2.
Hint Constructors value plug step0 step hasty.
Local Hint Constructors value step hasty : core.
(* Some notation to make it more pleasant to write programs *)
Infix "-->" := Fun (at level 60, right associativity).
@ -564,7 +491,7 @@ Module Stlc.
Proof.
Admitted.
Hint Resolve hasty_change.
Local Hint Resolve hasty_change : core.
Lemma preservation : forall e1 e2,
step e1 e2