# Typechecking

(* $Date: 2011-04-12 22:39:07 -0400 (Tue, 12 Apr 2011) $ *)

Require Export Stlc.

Require Import Relations.

The has_type relation of the STLC defines what it means for a
term to belong to a type (in some context). But it doesn't, by
itself, tell us how to
Fortunately, the rules defining has_type are

*check*whether or not a term is well typed.*syntax directed*— they exactly follow the shape of the term. This makes it straightforward to translate the typing rules into clauses of a typechecking*function*that takes a term and a context and either returns the term's type or else signals that the term is not typable.Fixpoint beq_ty (T1 T2:ty) : bool :=

match T1,T2 with

| ty_Bool, ty_Bool =>

true

| ty_arrow T11 T12, ty_arrow T21 T22 =>

andb (beq_ty T11 T21) (beq_ty T12 T22)

| _,_ =>

false

end.

... and we need to establish the usual two-way connection between
the boolean result returned by beq_ty and the logical
proposition that its inputs are equal.

Lemma beq_ty_refl : ∀ T1,

beq_ty T1 T1 = true.

Proof.

intros T1. induction T1; simpl.

reflexivity.

rewrite IHT1_1. rewrite IHT1_2. reflexivity. Qed.

Lemma beq_ty__eq : ∀ T1 T2,

beq_ty T1 T2 = true → T1 = T2.

Proof with auto.

intros T1. induction T1; intros T2 Hbeq; destruct T2; inversion Hbeq.

Case "T1=ty_Bool".

reflexivity.

Case "T1=ty_arrow T1_1 T1_2".

apply andb_true in H0. destruct H0 as [Hbeq1 Hbeq2].

apply IHT1_1 in Hbeq1. apply IHT1_2 in Hbeq2. subst... Qed.

## The Typechecker

Fixpoint type_check (Γ:context) (t:tm) : option ty :=

match t with

| tm_var x => Γ x

| tm_abs x T11 t12 => match type_check (extend Γ x T11) t12 with

| Some T12 => Some (ty_arrow T11 T12)

| _ => None

end

| tm_app t1 t2 => match type_check Γ t1, type_check Γ t2 with

| Some (ty_arrow T11 T12),Some T2 =>

if beq_ty T11 T2 then Some T12 else None

| _,_ => None

end

| tm_true => Some ty_Bool

| tm_false => Some ty_Bool

| tm_if x t f => match type_check Γ x with

| Some ty_Bool =>

match type_check Γ t, type_check Γ f with

| Some T1, Some T2 =>

if beq_ty T1 T2 then Some T1 else None

| _,_ => None

end

| _ => None

end

end.

## Properties

*sound*and

*complete*for the original has_type relation — that is, type_check and has_type define the same partial function.

Theorem type_checking_sound : ∀ Γ t T,

type_check Γ t = Some T → has_type Γ t T.

Proof with eauto.

intros Γ t. generalize dependent Γ.

tm_cases (induction t) Case; intros Γ T Htc; inversion Htc.

Case "tm_var"...

Case "tm_app".

remember (type_check Γ t1) as TO1.

remember (type_check Γ t2) as TO2.

destruct TO1 as [T1|]; try solve by inversion;

destruct T1 as [|T11 T12]; try solve by inversion.

destruct TO2 as [T2|]; try solve by inversion.

remember (beq_ty T11 T2) as b.

destruct b; try solve by inversion.

symmetry in Heqb. apply beq_ty__eq in Heqb.

inversion H0; subst...

Case "tm_abs".

rename i into y. rename t into T1.

remember (extend Γ y T1) as G'.

remember (type_check G' t0) as TO2.

destruct TO2; try solve by inversion.

inversion H0; subst...

Case "tm_true"...

Case "tm_false"...

Case "tm_if".

remember (type_check Γ t1) as TOc.

remember (type_check Γ t2) as TO1.

remember (type_check Γ t3) as TO2.

destruct TOc as [Tc|]; try solve by inversion.

destruct Tc; try solve by inversion.

destruct TO1 as [T1|]; try solve by inversion.

destruct TO2 as [T2|]; try solve by inversion.

remember (beq_ty T1 T2) as b.

destruct b; try solve by inversion.

symmetry in Heqb. apply beq_ty__eq in Heqb.

inversion H0. subst. subst...

Qed.

Theorem type_checking_complete : ∀ Γ t T,

has_type Γ t T → type_check Γ t = Some T.

Proof with auto.

intros Γ t T Hty.

has_type_cases (induction Hty) Case; simpl.

Case "T_Var"...

Case "T_Abs". rewrite IHHty...

Case "T_App".

rewrite IHHty1. rewrite IHHty2.

rewrite (beq_ty_refl T11)...

Case "T_True"...

Case "T_False"...

Case "T_If". rewrite IHHty1. rewrite IHHty2.

rewrite IHHty3. rewrite (beq_ty_refl T)...

Qed.

End STLCChecker.