e ::= zero     -- Cero
    | succ e   -- Sucesor
    | pred e   -- Predecesor
    | isZero e
    | true 
    | false
    | if e then e else e

data Expr
  = Zero
  | Succ Expr
  | Pred Expr
  | IsZero Expr
  | Tru
  | Fals
  | If Expr Expr Expr
  deriving (Show, Eq)

T ::= Natural
    | Boolean

data Type
  = Natural
  | Boolean

check :: Expr -> Maybe Type

check :: Expr -> Maybe Type

e ::= v           -- Variable
    | 0, 1, 2..   -- Naturals
    | True, False -- Bools
    | λ x : T. e  -- Lambda
    | e1 e2       -- Aplicación

data Expr
  = Var Name
  | Lit Ground
  | App Expr Expr
  | Lam Name Type Expr

data Ground
  = LInt Int
  | LBool Bool

T ::= Natural
    | Boolean
    | T1 -> T2

data Type
  = TInt
  | TBool
  | TArr Type Type

type Context =
    [(Name, Type)]

check :: Expr -> Either TypeError Type

check :: Expr -> Context -> Either TypeError Type

check :: Expr -> Context -> Either TypeError Type
check (Var n) ctx = do
  case lookup n ctx of
    Just ty -> pure ty
    Nothing -> Left $ "Variable " <> n <> " not in scope!"

check :: Expr -> Context -> Either TypeError Type
check (Lam ty var body) ctx = do
  rangeT <- check body ((var, ty) : ctx)
  pure (TArr ty rangeT)

check :: Expr -> Context -> Either TypeError Type
check (App term1 term2) ctx = do
  ty1 <- check term1 ctx
  ty2 <- check term2 ctx
  case ty1 of
    Func domain range ->
      if domain == ty2
        then pure range
        else Left ("Wrong domain, expected " <> show domain <> " but got " <> show ty2)
 
    _ -> Left ("Expected " <> show term1 <> " to be a function")

check :: Expr -> Context -> Either TypeError Type
check (Lit l) _ =
  case l of
    LInt _ -> TInt
    LBool _ -> TBool