attempts to add polytypes to lambda

attempts to add polytypes to lambda.hs

-- original from <https://www.cs.ox.ac.uk/projects/gip/school/tc.hs> but URLs don't last long in academia
--
{-# LANGUAGE ExistentialQuantification,GADTs,TypeOperators #-}

-- This typechecker, written by Stephanie Weirich at Dagstuhl (Sept 04)
-- demonstrates that it's possible to write functions of type
--    tc :: String -> Term a
-- where Term a is our strongly-typed GADT.
-- That is, generate a typed term from an untyped source; Lennart
-- Augustsson set this as a challenge.
--
-- In fact the main function goes
--    tc :: UTerm -> exists ty. (Ty ty, Term ty)
-- so the type checker returns a pair of an expression and its type,
-- wrapped, of course, in an existential.

module Main where

import Data.Typeable

-- Untyped world --------------------------------------------
data UTerm = UVar String
       | ULam String UType UTerm
       | UApp UTerm UTerm
       | UConBool Bool
       | UIf UTerm UTerm UTerm
       | UTLam String UTerm
       | UTApp UTerm UType

data UType = UBool | UArr UType UType | UVarTy String

-- Type to term functions
data t :-> a
data Type

-- Typed world -----------------------------------------------
data Ty t where
  Bool :: Ty Bool
  Arr  :: Ty a -> Ty b -> Ty (a -> b)
  Type :: String -> Ty Type
  TypeVar :: String -> Var g t -> Ty t

instance Show (Ty t) where show = showType

data Term g t where
  Var :: Var g t -> Term g t
  Lam :: Ty a -> Term (g,a) b -> Term g (a->b)
  App :: Term g (s -> t) -> Term g s -> Term g t
  ConBool :: Bool -> Term g Bool
  If :: Term g Bool -> Term g a -> Term g a -> Term g a
  TypeTerm :: Ty a -> Term g Type

data Var g t where
  ZVar :: Var (h,t) t
  SVar :: Var h t -> Var (h,s) t
deriving instance Eq (Var g t)

instance Show (Var g t) where
  show v = "v" ++ show (go v) where
    go :: forall g t. Var g t -> Int
    go ZVar = 0
    go (SVar v) = 1 + go v

data Typed thing = forall ty. Typed (Ty ty) (thing ty)

-- Typechecking types
data ExType = forall t. ExType (Ty t)

tcType :: TyEnv g -> UType -> ExType
tcType e (UVarTy s) = case lookupVar s e of Typed t v -> ExType $ TypeVar s v
tcType _ UBool = ExType Bool
tcType e (UArr t1 t2) = case tcType e t1 of { ExType t1' ->
              case tcType e t2 of { ExType t2' ->
              ExType (Arr t1' t2') }}

-- The type environment and lookup
data TyEnv g where
  Nil  :: TyEnv g
  Cons :: String -> Ty t -> TyEnv h -> TyEnv (h,t)

lookupVar :: String -> TyEnv g -> Typed (Var g)
lookupVar _ Nil = error "Variable not found"
lookupVar v (Cons s ty e)
  | v==s      = Typed ty ZVar
  | otherwise = case lookupVar v e of
           Typed ty v -> Typed ty (SVar v)

data Equal a b where
  Equal :: Equal c c

cmpTy :: Ty a -> Ty b -> Maybe (Equal a b)
cmpTy Bool Bool = Just Equal
cmpTy (Type s) (Type t) | s == t = Just Equal | otherwise = Nothing
cmpTy (TypeVar s v) (TypeVar t v') = do
  undefined
cmpTy (Arr a1 a2) (Arr b1 b2)
  = do    { Equal <- cmpTy a1 b1
    ; Equal <- cmpTy a2 b2
    ; return Equal }
cmpTy a b = error $ "cmpTy: " ++ show (a,b)

-- Typechecking terms
tc :: UTerm -> TyEnv g -> Typed (Term g)
tc (UVar v) env = case lookupVar v env of
            Typed ty v -> Typed ty (Var v)
tc (UConBool b) env
  = Typed Bool (ConBool b)
tc (ULam s ty body) env
  = case tcType env ty of { ExType bndr_ty' ->
    case tc body (Cons s bndr_ty' env) of { Typed body_ty' body' ->
    Typed (Arr bndr_ty' body_ty')
      (Lam bndr_ty' body') }}
tc (UApp e1 e2) env
  = case tc e1 env of { Typed (Arr bndr_ty body_ty) e1' ->
    case tc e2 env of { Typed arg_ty e2' ->
    case cmpTy arg_ty bndr_ty of
    Nothing -> error "Type error"
    Just Equal -> Typed body_ty (App e1' e2') }}
tc (UIf e1 e2 e3) env
  = case tc e1 env of { Typed Bool e1' ->
    case tc e2 env of { Typed t2   e2' ->
    case tc e3 env of { Typed t3   e3' ->
    case cmpTy t2 t3 of
    Nothing -> error "Type error"
    Just Equal -> Typed t2 (If e1' e2' e3') }}}
tc (UTLam s body) env
  = let bndr_ty' = Type s in
    case tc body (Cons s bndr_ty' env) of { Typed body_ty' body' ->
    Typed (Arr bndr_ty' body_ty')
      (Lam bndr_ty' body') }
tc (UTApp e1 arg_ty) env
  = case tcType env arg_ty of
     ExType arg_ty' ->
       case tc e1 env of { Typed t1'@(Arr (Type s) _) e1' ->
         instantiate t1' e1' s arg_ty'
         }

instantiate :: forall a b g. Ty (Type -> a) -> Term g (Type -> a) -> String -> Ty b -> Typed (Term g)
instantiate (Arr Type{} ty') e arg_name arg = go ty' (App e (TypeTerm arg)) where
  go :: forall x. Ty x -> Term g x -> Typed (Term g)
  go t (Lam ty e) = Typed (subst arg_name arg t) (Lam (subst arg_name arg ty) (substE arg_name arg ty e)
  go t ex = Typed t ex

subst :: String -> Ty a -> Ty b -> ExType
subst = ...

showType :: Ty a -> String
showType (TypeVar s v) = s
showType Bool = "Bool"
showType (Type v) = "(" ++ show v ++ " :: Type)"
showType (Arr t1 t2) = "(" ++ showType t1 ++ ") -> (" ++ showType t2 ++ ")"

uNot = ULam "x" UBool (UIf (UVar "x") (UConBool False) (UConBool True))

uId = UTLam "t" (ULam "x" (UVarTy "t") (UVar "x"))

testId = UApp (UTApp uId UBool) (UConBool True)

test :: UTerm
test = UApp uNot (UConBool True)

main = putStrLn (case tc testId Nil of
            Typed ty _ -> showType ty
        )