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LambdaCalculusAndTypeSoundness: Church numerals
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Adam Chlipala
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1 changed files with 226 additions and 11 deletions
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@ -14,7 +14,7 @@ Module Ulc.
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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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| Abs y e1 => Abs y (if y ==v 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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@ -34,7 +34,6 @@ Module Ulc.
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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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@ -42,6 +41,15 @@ Module Ulc.
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Hint Constructors eval value.
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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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(** 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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@ -50,6 +58,192 @@ Module Ulc.
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induct 1; eauto.
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Qed.
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Hint Resolve eval_value.
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Coercion Var : var >-> exp.
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Notation "\ x , e" := (Abs x e) (at level 50).
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Infix "@" := App (at level 49, left associativity).
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Definition zero := \"f", \"x", "x".
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Definition plus1 := \"n", \"f", \"x", "f" @ ("n" @ "f" @ "x").
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Fixpoint canonical' (n : nat) : exp :=
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match n with
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| O => "x"
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| S n' => "f" @ ((\"f", \"x", canonical' n') @ "f" @ "x")
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end.
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Definition canonical n := \"f", \"x", canonical' n.
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Definition represents (e : exp) (n : nat) :=
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eval e (canonical n).
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Theorem zero_ok : represents zero 0.
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Proof.
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unfold zero, represents, canonical.
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simplify.
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econstructor.
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Qed.
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Theorem plus1_ok : forall e n, represents e n
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-> represents (plus1 @ e) (S n).
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Proof.
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unfold plus1, represents, canonical; simplify.
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econstructor.
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econstructor.
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eassumption.
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simplify.
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econstructor.
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Qed.
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Definition add := \"n", \"m", "n" @ plus1 @ "m".
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Example add_1_2 : exists v,
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eval (add @ (plus1 @ zero) @ (plus1 @ (plus1 @ zero))) v
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/\ eval (plus1 @ (plus1 @ (plus1 @ zero))) v.
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Proof.
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eexists; propositional.
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repeat (econstructor; simplify).
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repeat econstructor.
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Qed.
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Lemma subst_m_canonical' : forall m n,
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subst m "m" (canonical' n) = canonical' n.
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Proof.
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induct n; simplify; equality.
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Qed.
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Lemma add_ok' : forall m n,
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eval
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(subst (\ "f", (\ "x", canonical' m)) "x"
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(subst (\ "n", (\ "f", (\ "x", "f" @ (("n" @ "f") @ "x")))) "f"
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(canonical' n))) (canonical (n + m)).
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Proof.
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induct n; simplify.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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simplify.
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econstructor.
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econstructor.
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simplify.
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eassumption.
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simplify.
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econstructor.
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Qed.
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Theorem add_ok : forall n ne m me,
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represents ne n
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-> represents me m
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-> represents (add @ ne @ me) (n + m).
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Proof.
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unfold represents; simplify.
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econstructor.
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econstructor.
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econstructor.
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eassumption.
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simplify.
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econstructor.
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eassumption.
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simplify.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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simplify.
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econstructor.
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econstructor.
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rewrite subst_m_canonical'.
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apply add_ok'.
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Qed.
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Definition mult := \"n", \"m", "n" @ (add @ "m") @ zero.
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Example mult_1_2 : exists v,
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eval (mult @ (plus1 @ zero) @ (plus1 @ (plus1 @ zero))) v
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/\ eval (plus1 @ (plus1 @ zero)) v.
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Proof.
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eexists; propositional.
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repeat (econstructor; simplify).
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repeat econstructor.
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Qed.
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Lemma mult_ok' : forall m n,
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eval
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(subst (\ "f", (\ "x", "x")) "x"
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(subst
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(\ "m",
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((\ "f", (\ "x", canonical' m)) @
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(\ "n", (\ "f", (\ "x", "f" @ (("n" @ "f") @ "x"))))) @ "m")
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"f" (canonical' n))) (canonical (n * m)).
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Proof.
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induct n; simplify.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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simplify.
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econstructor.
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econstructor.
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simplify.
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eassumption.
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simplify.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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simplify.
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econstructor.
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econstructor.
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rewrite subst_m_canonical'.
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apply add_ok'.
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Qed.
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Theorem mult_ok : forall n ne m me,
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represents ne n
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-> represents me m
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-> represents (mult @ ne @ me) (n * m).
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Proof.
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unfold represents; simplify.
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econstructor.
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econstructor.
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econstructor.
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eassumption.
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simplify.
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econstructor.
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eassumption.
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simplify.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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simplify.
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econstructor.
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simplify.
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econstructor.
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econstructor.
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simplify.
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rewrite subst_m_canonical'.
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apply mult_ok'.
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Qed.
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(** * Small-step semantics with evaluation contexts *)
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@ -86,15 +280,6 @@ Module Ulc.
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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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@ -314,6 +499,36 @@ Module Stlc.
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Hint Constructors value plug step0 step hasty.
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Infix "-->" := Fun (at level 60, right associativity).
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Coercion Const : nat >-> exp.
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Infix "^+^" := Plus (at level 50).
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Coercion Var : var >-> exp.
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Notation "\ x , e" := (Abs x e) (at level 51).
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Infix "@" := App (at level 49, left associativity).
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Example one_plus_one : hasty $0 (1 ^+^ 1) Nat.
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Proof.
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repeat (econstructor; simplify).
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Qed.
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Example add : hasty $0 (\"n", \"m", "n" ^+^ "m") (Nat --> Nat --> Nat).
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Proof.
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repeat (econstructor; simplify).
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Qed.
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Example eleven : hasty $0 ((\"n", \"m", "n" ^+^ "m") @ 7 @ 4) Nat.
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Proof.
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repeat (econstructor; simplify).
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Qed.
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Example seven_the_long_way : hasty $0 ((\"x", "x") @ (\"x", "x") @ 7) Nat.
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Proof.
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repeat (econstructor; simplify).
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Qed.
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(** * Some examples of typed programs *)
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(** * Let's prove type soundness first without much automation. *)
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