LambdaCalculusAndTypeSoundness: Church numerals

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Adam Chlipala 2016-03-13 14:44:41 -04:00
parent 55257f669d
commit a36ebc7802

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@ -14,7 +14,7 @@ Module Ulc.
Fixpoint subst (rep : exp) (x : string) (e : exp) : exp := Fixpoint subst (rep : exp) (x : string) (e : exp) : exp :=
match e with match e with
| Var y => if string_dec y x then rep else Var y | Var y => if string_dec y x then rep else Var y
| Abs y e1 => Abs y (if string_dec y x then e1 else subst rep x e1) | Abs y e1 => Abs y (if y ==v x then e1 else subst rep x e1)
| App e1 e2 => App (subst rep x e1) (subst rep x e2) | App e1 e2 => App (subst rep x e1) (subst rep x e2)
end. end.
@ -34,7 +34,6 @@ Module Ulc.
(** Note that we omit a [Var] case, since variable terms can't be *closed*. *) (** Note that we omit a [Var] case, since variable terms can't be *closed*. *)
(** Which terms are values, that is, final results of execution? *) (** Which terms are values, that is, final results of execution? *)
Inductive value : exp -> Prop := Inductive value : exp -> Prop :=
| Value : forall x e, value (Abs x e). | Value : forall x e, value (Abs x e).
@ -42,6 +41,15 @@ Module Ulc.
Hint Constructors eval value. Hint Constructors eval value.
Theorem value_eval : forall v,
value v
-> eval v v.
Proof.
invert 1; eauto.
Qed.
Hint Resolve value_eval.
(** Actually, let's prove that [eval] only produces values. *) (** Actually, let's prove that [eval] only produces values. *)
Theorem eval_value : forall e v, Theorem eval_value : forall e v,
eval e v eval e v
@ -50,6 +58,192 @@ Module Ulc.
induct 1; eauto. induct 1; eauto.
Qed. Qed.
Hint Resolve eval_value.
Coercion Var : var >-> exp.
Notation "\ x , e" := (Abs x e) (at level 50).
Infix "@" := App (at level 49, left associativity).
Definition zero := \"f", \"x", "x".
Definition plus1 := \"n", \"f", \"x", "f" @ ("n" @ "f" @ "x").
Fixpoint canonical' (n : nat) : exp :=
match n with
| O => "x"
| S n' => "f" @ ((\"f", \"x", canonical' n') @ "f" @ "x")
end.
Definition canonical n := \"f", \"x", canonical' n.
Definition represents (e : exp) (n : nat) :=
eval e (canonical n).
Theorem zero_ok : represents zero 0.
Proof.
unfold zero, represents, canonical.
simplify.
econstructor.
Qed.
Theorem plus1_ok : forall e n, represents e n
-> represents (plus1 @ e) (S n).
Proof.
unfold plus1, represents, canonical; simplify.
econstructor.
econstructor.
eassumption.
simplify.
econstructor.
Qed.
Definition add := \"n", \"m", "n" @ plus1 @ "m".
Example add_1_2 : exists v,
eval (add @ (plus1 @ zero) @ (plus1 @ (plus1 @ zero))) v
/\ eval (plus1 @ (plus1 @ (plus1 @ zero))) v.
Proof.
eexists; propositional.
repeat (econstructor; simplify).
repeat econstructor.
Qed.
Lemma subst_m_canonical' : forall m n,
subst m "m" (canonical' n) = canonical' n.
Proof.
induct n; simplify; equality.
Qed.
Lemma add_ok' : forall m n,
eval
(subst (\ "f", (\ "x", canonical' m)) "x"
(subst (\ "n", (\ "f", (\ "x", "f" @ (("n" @ "f") @ "x")))) "f"
(canonical' n))) (canonical (n + m)).
Proof.
induct n; simplify.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
simplify.
econstructor.
econstructor.
simplify.
eassumption.
simplify.
econstructor.
Qed.
Theorem add_ok : forall n ne m me,
represents ne n
-> represents me m
-> represents (add @ ne @ me) (n + m).
Proof.
unfold represents; simplify.
econstructor.
econstructor.
econstructor.
eassumption.
simplify.
econstructor.
eassumption.
simplify.
econstructor.
econstructor.
econstructor.
econstructor.
simplify.
econstructor.
econstructor.
rewrite subst_m_canonical'.
apply add_ok'.
Qed.
Definition mult := \"n", \"m", "n" @ (add @ "m") @ zero.
Example mult_1_2 : exists v,
eval (mult @ (plus1 @ zero) @ (plus1 @ (plus1 @ zero))) v
/\ eval (plus1 @ (plus1 @ zero)) v.
Proof.
eexists; propositional.
repeat (econstructor; simplify).
repeat econstructor.
Qed.
Lemma mult_ok' : forall m n,
eval
(subst (\ "f", (\ "x", "x")) "x"
(subst
(\ "m",
((\ "f", (\ "x", canonical' m)) @
(\ "n", (\ "f", (\ "x", "f" @ (("n" @ "f") @ "x"))))) @ "m")
"f" (canonical' n))) (canonical (n * m)).
Proof.
induct n; simplify.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
simplify.
econstructor.
econstructor.
simplify.
eassumption.
simplify.
econstructor.
econstructor.
econstructor.
econstructor.
simplify.
econstructor.
econstructor.
rewrite subst_m_canonical'.
apply add_ok'.
Qed.
Theorem mult_ok : forall n ne m me,
represents ne n
-> represents me m
-> represents (mult @ ne @ me) (n * m).
Proof.
unfold represents; simplify.
econstructor.
econstructor.
econstructor.
eassumption.
simplify.
econstructor.
eassumption.
simplify.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
simplify.
econstructor.
simplify.
econstructor.
econstructor.
simplify.
rewrite subst_m_canonical'.
apply mult_ok'.
Qed.
(** * Small-step semantics with evaluation contexts *) (** * Small-step semantics with evaluation contexts *)
@ -86,15 +280,6 @@ Module Ulc.
(** We can move directly to establishing inclusion from basic small steps to contextual small steps. *) (** We can move directly to establishing inclusion from basic small steps to contextual small steps. *)
Theorem value_eval : forall v,
value v
-> eval v v.
Proof.
invert 1; eauto.
Qed.
Hint Resolve value_eval.
Lemma step_eval'' : forall v c x e e1 e2 v0, Lemma step_eval'' : forall v c x e e1 e2 v0,
value v value v
-> plug c (App (Abs x e) v) e1 -> plug c (App (Abs x e) v) e1
@ -314,6 +499,36 @@ Module Stlc.
Hint Constructors value plug step0 step hasty. Hint Constructors value plug step0 step hasty.
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).
Example one_plus_one : hasty $0 (1 ^+^ 1) Nat.
Proof.
repeat (econstructor; simplify).
Qed.
Example add : hasty $0 (\"n", \"m", "n" ^+^ "m") (Nat --> Nat --> Nat).
Proof.
repeat (econstructor; simplify).
Qed.
Example eleven : hasty $0 ((\"n", \"m", "n" ^+^ "m") @ 7 @ 4) Nat.
Proof.
repeat (econstructor; simplify).
Qed.
Example seven_the_long_way : hasty $0 ((\"x", "x") @ (\"x", "x") @ 7) Nat.
Proof.
repeat (econstructor; simplify).
Qed.
(** * Some examples of typed programs *)
(** * Let's prove type soundness first without much automation. *) (** * Let's prove type soundness first without much automation. *)