module interp3a where

-- No scoping errors for this interpreter
-- No typing errors 

open import Data.Unit
open import Data.Product
open import Data.Nat

data Typ : Set where
 Nat  : Typ
 _~>_ : Typ -> Typ -> Typ

Ctx : Set


data Var : Ctx -> Typ -> Set where
  VZ   : forall {g t} -> Var (t × g) t
  VS   : forall {g t' t} -> Var g t -> Var (t' × g) t 

data Exp : Set -> Set -> Set where 
  EVar : forall {g t} -> Var g t -> Exp g t
  ELit : forall {g} -> ℕ -> Exp g ℕ 
  ELam : forall {g a b} -> Exp (a × g) b -> Exp g (a -> b)
  EApp : forall {g a b} -> Exp g (a -> b) -> Exp g a -> Exp g b

sLookup : forall {g t} -> Var g t -> g -> t
sLookup VZ     (v , g) = v
sLookup (VS x) (v , g) = sLookup x g

interp : forall {g t} -> g -> Exp g t -> t
interp e (EVar x)     = sLookup x e
interp e (ELit i)     = i
interp e (ELam t)     = \x -> interp (x , e) t
interp e (EApp t1 t2) = (interp e t1) (interp e t2)

t0 = interp tt (EApp (ELam (EVar VZ)) (ELit 2))

-- doesn't type check
-- t1 = interp () (Var (S Z))

-- doesn't type check
-- t2 = interp () (App (Lit 2) (Lit 3))