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LambdaCalculusAndTypeSoundness: untyped lambda calculus semantics, two ways
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Require Import Frap.
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Require Import Frap.
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Module Ulc.
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Inductive exp : Set :=
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| Var : var -> exp
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| Abs : var -> exp -> exp
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| App : exp -> exp -> exp.
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Fixpoint subst (rep : exp) (x : string) (e : exp) : exp :=
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match e with
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| Var y => if string_dec y x then rep else Var y
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| Abs y e1 => Abs y (if string_dec y x then e1 else subst rep x e1)
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| App e1 e2 => App (subst rep x e1) (subst rep x e2)
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end.
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(** * Big-step semantics *)
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(** This is the most straightforward way to give semantics to lambda terms:
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* We evaluate any closed term into a value (that is, an [Abs]). *)
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Inductive eval : exp -> exp -> Prop :=
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| BigAbs : forall x e,
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eval (Abs x e) (Abs x e)
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| BigApp : forall e1 x e1' e2 v2 v,
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eval e1 (Abs x e1')
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-> eval e2 v2
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-> eval (subst v2 x e1') v
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-> eval (App e1 e2) v.
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(** Note that we omit a [Var] case, since variable terms can't be *closed*. *)
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(** Which terms are values, that is, final results of execution? *)
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Inductive value : exp -> Prop :=
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| Value : forall x e, value (Abs x e).
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(** We're cheating a bit here, *assuming* that the term is also closed. *)
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Hint Constructors eval value.
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(** Actually, let's prove that [eval] only produces values. *)
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Theorem eval_value : forall e v,
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eval e v
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-> value v.
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Proof.
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induct 1; eauto.
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Qed.
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(** * Small-step semantics with evaluation contexts *)
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Inductive context : Set :=
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| Hole : context
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| App1 : context -> exp -> context
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| App2 : exp -> context -> context.
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Inductive plug : context -> exp -> exp -> Prop :=
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| PlugHole : forall e,
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plug Hole e e
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| PlugApp1 : forall c e1 e2 e,
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plug c e1 e
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-> plug (App1 c e2) e1 (App e e2)
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| PlugApp2 : forall c e1 e2 e,
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value e1
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-> plug c e2 e
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-> plug (App2 e1 c) e2 (App e1 e).
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(* Subtle point: the [value] hypothesis right above enforces a well-formedness
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* condition on contexts that may actually be plugged. We don't allow
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* skipping over a lefthand subterm of an application when that term has
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* evaluation work left to do. This condition is the essence of
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* *call-by-value* instead of other evaluation strategies. Details are
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* largely beyond our scope here. *)
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Inductive step : exp -> exp -> Prop :=
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| ContextBeta : forall c x e v e1 e2,
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value v
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-> plug c (App (Abs x e) v) e1
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-> plug c (subst v x e) e2
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-> step e1 e2.
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Hint Constructors plug step.
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(** We can move directly to establishing inclusion from basic small steps to contextual small steps. *)
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Theorem value_eval : forall v,
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value v
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-> eval v v.
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Proof.
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invert 1; eauto.
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Qed.
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Hint Resolve value_eval.
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Lemma step_eval'' : forall v c x e e1 e2 v0,
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value v
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-> plug c (App (Abs x e) v) e1
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-> plug c (subst v x e) e2
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-> eval e2 v0
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-> eval e1 v0.
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Proof.
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induct c; invert 2; invert 1; simplify; eauto.
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invert H0; eauto.
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invert H0; eauto.
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Qed.
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Hint Resolve step_eval''.
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Lemma step_eval' : forall e1 e2,
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step e1 e2
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-> forall v, eval e2 v
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-> eval e1 v.
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Proof.
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invert 1; simplify; eauto.
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Qed.
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Hint Resolve step_eval'.
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Theorem step_eval : forall e v,
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step^* e v
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-> value v
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-> eval e v.
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Proof.
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induct 1; eauto.
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Qed.
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Lemma plug_functional : forall C e e1,
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plug C e e1
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-> forall e2, plug C e e2
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-> e1 = e2.
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Proof.
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induct 1; invert 1; simplify; try f_equal; eauto.
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Qed.
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Lemma plug_mirror : forall C e e', plug C e e'
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-> forall e1, exists e1', plug C e1 e1'.
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Proof.
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induct 1; simplify; eauto.
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specialize (IHplug e0); first_order; eauto.
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specialize (IHplug e0); first_order; eauto.
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Qed.
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Fixpoint compose (C1 C2 : context) : context :=
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match C2 with
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| Hole => C1
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| App1 C2' e => App1 (compose C1 C2') e
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| App2 v C2' => App2 v (compose C1 C2')
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end.
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Lemma compose_ok : forall C1 C2 e1 e2 e3,
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plug C1 e1 e2
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-> plug C2 e2 e3
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-> plug (compose C1 C2) e1 e3.
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Proof.
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induct 2; simplify; eauto.
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Qed.
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Hint Resolve compose_ok.
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Lemma step_plug : forall e1 e2,
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step e1 e2
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-> forall C e1' e2', plug C e1 e1'
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-> plug C e2 e2'
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-> step e1' e2'.
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Proof.
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invert 1; simplify; eauto.
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Qed.
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Lemma stepStar_plug : forall e1 e2,
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step^* e1 e2
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-> forall C e1' e2', plug C e1 e1'
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-> plug C e2 e2'
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-> step^* e1' e2'.
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Proof.
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induct 1; simplify.
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assert (e1' = e2') by (eapply plug_functional; eassumption).
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subst.
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constructor.
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assert (exists y', plug C y y') by eauto using plug_mirror.
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invert H3.
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eapply step_plug in H.
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econstructor.
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eassumption.
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eapply IHtrc.
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eassumption.
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assumption.
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eassumption.
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assumption.
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Qed.
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Hint Resolve stepStar_plug eval_value.
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Theorem eval_step : forall e v,
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eval e v
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-> step^* e v.
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Proof.
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induct 1; eauto.
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eapply trc_trans.
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eapply stepStar_plug with (e1 := e1) (e2 := Abs x e1') (C := App1 Hole e2); eauto.
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eapply trc_trans.
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eapply stepStar_plug with (e1 := e2) (e2 := v2) (C := App2 (Abs x e1') Hole); eauto.
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eauto.
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Qed.
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End Ulc.
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Module Stlc.
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(* Expression syntax *)
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(* Expression syntax *)
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Inductive exp : Set :=
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Inductive exp : Set :=
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| Var (x : var)
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| Var (x : var)
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@ -519,3 +726,4 @@ Proof.
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apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t); eauto.
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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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apply invariant_induction; simplify; eauto; equality.
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Qed.
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Qed.
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End Stlc.
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