{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE BlockArguments #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE DeriveAnyClass #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE DerivingStrategies #-}
module Elab where

import Control.Arrow (Arrow(first))
import Control.Monad.Reader
import Control.Exception

import qualified Data.Map.Strict as Map
import qualified Data.Sequence as Seq
import qualified Data.Set as Set
import qualified Data.Text as T
import Data.Traversable
import Data.Text (Text)
import Data.Map (Map)
import Data.Typeable
import Data.Foldable

import Elab.Eval.Formula (possible, truthAssignments)
import Elab.WiredIn
import Elab.Monad
import Elab.Eval

import qualified Presyntax.Presyntax as P

import Prettyprinter

import Syntax.Pretty
import Syntax
import Syntax.Subst

infer :: P.Expr -> ElabM (Term, NFType)
infer (P.Span ex a b) = withSpan a b $ infer ex

infer (P.Var t) = do
  name <- getNameFor t
  nft <- getNfType name
  pure (Ref name, nft)

infer (P.App p f x) = do
  (f, f_ty) <- infer f
  porp <- isPiType p f_ty
  case porp of
    It'sProd d r w -> do
      x <- check x d
      x_nf <- eval x
      pure (App p (w f) x, r x_nf)
    It'sPath li le ri wp -> do
      x <- check x VI
      x_nf <- eval x
      pure (IElim (quote (fun li)) (quote le) (quote ri) (wp f) x, li x_nf)
    It'sPartial phi a w -> do
      x <- check x (VIsOne phi)
      pure (App P.Ex (w f) x, a)
    It'sPartialP phi a w -> do
      x <- check x (VIsOne phi)
      x_nf <- eval x
      pure (App P.Ex (w f) x, a @@ x_nf)

infer (P.Proj1 x) = do
  (tm, ty) <- infer x
  (d, _, wp) <- isSigmaType ty
  pure (Proj1 (wp tm), d)

infer (P.Proj2 x) = do
  (tm, ty) <- infer x
  tm_nf <- eval tm
  (_, r, wp) <- isSigmaType ty
  pure (Proj2 (wp tm), r (vProj1 tm_nf))

infer exp = do
  t <- newMeta VType
  tm <- switch $ check exp t
  pure (tm, t)

check :: P.Expr -> NFType -> ElabM Term
check (P.Span ex a b) ty = withSpan a b $ check ex ty

check (P.Lam p var body) (VPi p' dom (Closure _ rng)) | p == p' =
  assume (Bound var 0) dom $ \name ->
    Lam p name <$> check body (rng (VVar name))

check tm (VPi P.Im dom (Closure var rng)) =
  assume var dom $ \name ->
    Lam P.Im name <$> check tm (rng (VVar name))

check (P.Lam p v b) ty = do
  porp <- isPiType p =<< forceIO ty
  case porp of
    It'sProd d r wp ->
      assume (Bound v 0) d $ \name ->
        wp . Lam p name <$> check b (r (VVar name))

    It'sPath li le ri wp -> do
      tm <- assume (Bound v 0) VI $ \var ->
        Lam P.Ex var <$> check b (force (li (VVar var)))

      tm_nf <- eval tm

      unify (tm_nf @@ VI0) le
        `catchElab` (throwElab . WhenCheckingEndpoint le ri VI0)

      unify (tm_nf @@ VI1) ri
        `catchElab` (throwElab . WhenCheckingEndpoint le ri VI1)

      pure (wp (PathIntro (quote (fun li)) (quote le) (quote ri) tm))

    It'sPartial phi a wp ->
      assume (Bound v 0) (VIsOne phi) $ \var ->
        wp . Lam p var <$> check b a

    It'sPartialP phi a wp ->
      assume (Bound v 0) (VIsOne phi) $ \var ->
        wp . Lam p var <$> check b (a @@ VVar var)

check (P.Pair a b) ty = do
  (d, r, wp) <- isSigmaType =<< forceIO ty
  a <- check a d
  a_nf <- eval a
  b <- check b (r a_nf)
  pure (wp (Pair a b))

check (P.Pi p s d r) ty = do
  isSort ty
  d <- check d ty
  d_nf <- eval d
  assume (Bound s 0) d_nf \var -> do
    r <- check r ty
    pure (Pi p var d r)

check (P.Sigma s d r) ty = do
  isSort ty
  d <- check d ty
  d_nf <- eval d
  assume (Bound s 0) d_nf \var -> do
    r <- check r ty
    pure (Sigma var d r)

check (P.Let items body) ty = do
  checkLetItems mempty items \decs -> do
    body <- check body ty
    pure (Let decs body)

check (P.LamSystem bs) ty = do
  (extent, dom) <- isPartialType ty
  let dom_q = quote dom
  eqns <- for (zip [(0 :: Int)..] bs) $ \(n, (formula, rhs)) -> do
    phi <- checkFormula (P.condF formula)
    rhses <-
        case P.condV formula of
          Just t -> assume (Bound t 0) (VIsOne phi) $ \var -> do
            env <- ask
            for (truthAssignments phi (getEnv env)) $ \e -> do
              let env' = env{ getEnv = e }
              (Just var,) <$> check rhs (eval' env' dom_q)
          Nothing -> do
            env <- ask
            for (truthAssignments phi (getEnv env)) $ \e -> do
              let env' = env{ getEnv = e }
              (Nothing,) <$> check rhs (eval' env' dom_q)
    pure (n, (phi, head rhses))

  unify extent (foldl ior VI0 (map (fst . snd) eqns))

  for_ eqns $ \(n, (formula, (binding, rhs))) -> do
    let
      k = case binding of
        Just v -> assume v (VIsOne formula) . const
        Nothing -> id
    k $ for_ eqns $ \(n', (formula', (binding, rhs'))) -> do
      let
        k = case binding of
          Just v -> assume v (VIsOne formula) . const
          Nothing -> id
        truth = possible mempty (iand formula formula')
        add [] = id
        add ((~(HVar x), True):xs) = redefine x VI VI1 . add xs
        add ((~(HVar x), False):xs) = redefine x VI VI0 . add xs
      k $ when ((n /= n') && fst truth) . add (Map.toList (snd truth)) $ do
        vl <- eval rhs
        vl' <- eval rhs'
        unify vl vl'
          `withNote` vsep [ pretty "These two cases must agree because they are both possible:"
                          , indent 2 $ pretty '*' <+> prettyTm (quote formula) <+> operator (pretty "=>") <+> prettyTm rhs
                          , indent 2 $ pretty '*' <+> prettyTm (quote formula') <+> operator (pretty "=>") <+> prettyTm rhs'
                          ]
          `withNote` (pretty "Consider this face, where both are true:" <+> showFace (snd truth))

  name <- newName
  let
    mkB name (Just v, b) = App P.Ex (Lam P.Ex v b) (Ref name)
    mkB _ (Nothing, b) = b
  pure (Lam P.Ex name (System (Map.fromList (map (\(_, (x, y)) -> (quote x, mkB name y)) eqns))))

check (P.LamCase pats) ty =
  do
    porp <- isPiType P.Ex ty
    case porp of
      It'sProd dom rng wp -> do
        name <- newName
        let range = Lam P.Ex name (quote (rng (VVar name)))

        cases <- checkPatterns range [] pats \partialPats (pat, rhs) -> do
          checkPattern pat dom \pat wp boundary pat_nf -> do
            rhs <- check rhs (rng pat_nf)
            case boundary of
              -- If we're checking a higher constructor then we need to
              -- compute what the case expression computed so far does
              -- with all the faces

              -- and make sure that the current case agrees with that
              -- boundary
              Just boundary -> do
                rhs_nf <- eval (wp rhs)
                cases <- partialPats

                let
                  (ty, a, b) = case pat_nf of
                    VNe (HCon ty (ConName _ _ a b)) _ -> (ty, a, b)
                    VNe (HPCon _ ty (ConName _ _ a b)) _ -> (ty, a, b)
                    _ -> undefined
                dummies <- replicateM ((a + b) - length (getBoundaryNames boundary)) newName
                let
                  base = appDummies (VVar <$> dummies) ty rhs_nf
                  sys = boundaryFormulas (drop a dummies ++ getBoundaryNames boundary) (getBoundaryMap boundary)
                
                for_ (Map.toList sys) \(formula, side) -> do
                  let rhs = cases @@ side
                  for_ (truthAssignments formula mempty) $ \i -> do
                    let vl = foldl (\v n -> vApp P.Ex v (snd (i Map.! n))) base (getBoundaryNames boundary)
                    unify vl rhs
                      `withNote` vcat [ pretty "These must be the same because of the face"
                                      , indent 2 $ prettyTm (quote formula) <+> operator (pretty "=>") <+> prettyTm (quote (zonk side))
                                      ]
                      `withNote` (pretty "Mandated by the constructor" <+> prettyTm (quote pat_nf))
              _ -> pure ()

            pure (pat, wp rhs)
        let x = wp (Lam P.Ex name (Case range (Ref name) cases))
        pure x
      _ -> do
        dom <- newMeta VTypeω
        n <- newName' (Bound (T.singleton 'x') 0)
        assume n dom \_ -> do
          rng <- newMeta VTypeω
          throwElab $ NotEqual (VPi P.Ex dom (Closure n (const rng))) ty
  where
    checkPatterns _ acc [] _ = pure (reverse acc)
    checkPatterns rng acc (x:xs) k = do
      n <- newName
      (p, t) <- k (eval (Lam P.Ex n (Case rng (Ref n) acc))) x
      checkPatterns rng ((p, t):acc) xs k
    
    appDummies (v:vl) (VPi p _ (Closure _ r)) x = appDummies vl (r v) (vApp p x v)
    appDummies [] _ x = x
    appDummies vs t _ = error (show (vs, t))

    boundaryFormulas [] (VSystem fs) = fs
    boundaryFormulas (x:xs) k = boundaryFormulas xs $ k @@ VVar x
    boundaryFormulas a b = error (show (a, b))

check exp ty = do
  (tm, has) <- switch $ infer exp
  wp <- isConvertibleTo has ty
  pure (wp tm)

checkPattern :: P.Pattern -> NFSort -> (Term -> (Term -> Term) -> Maybe Boundary -> Value -> ElabM a) -> ElabM a
checkPattern (P.PCap var) dom cont = do
  name <- asks (Map.lookup var . nameMap)
  case name of
    Just name@(ConName _ _ skip arity) -> do
      unless (arity == 0) $ throwElab $ UnsaturatedCon name
      (ty, wp, _) <- instantiate =<< getNfType name
      unify ty dom
      wrap <- skipLams skip
      cont (Con name) wrap Nothing =<< eval (wp (Con name))
    Just name -> throwElab $ NotACon name
    Nothing -> assume (Bound var 0) dom \name -> cont (Ref name) (Lam P.Ex name) Nothing (VVar name)

checkPattern (P.PCon var args) dom cont =
  do
    name <- asks (Map.lookup var . nameMap)
    case name of
      Just name@(ConName _ _ nskip arity) -> do
        unless (arity == length args) $ throwElab $ UnsaturatedCon name
        (ty, wp, xs) <- instantiate =<< getNfType name
        _ <- isConvertibleTo (skipBinders arity ty) dom

        skip <- skipLams nskip
        t <- asks (Map.lookup name . boundaries)

        con <- quote <$> getValue name

        bindNames args ty $ \names wrap ->
          cont (Con name) (skip . wrap) (instBoundary xs <$> t) =<< eval (foldl (\x y -> App P.Ex x (Ref y)) (wp con) names)

      Just name -> throwElab $ NotACon name
      _ -> throwElab $ NotInScope (Bound var 0)
  where
    skipBinders :: Int -> NFType -> NFType
    skipBinders 0 t = t
    skipBinders n (VPi _ _ (Closure v r)) = skipBinders (n - 1) (r (VVar v))
    skipBinders _ _ = error $ "constructor type is wrong?"

    bindNames (n:ns) (VPi p d (Closure _ r)) k =
      assume (Bound n 0) d \n -> bindNames ns (r (VVar n)) \ns w ->
        k (n:ns) (Lam p n . w)
    bindNames [] _ k = k [] id
    bindNames xs t _ = error $ show (xs, t)

instBoundary :: [Value] -> Boundary -> Boundary
instBoundary metas (Boundary x y) = Boundary x (foldl (vApp P.Ex) y metas)

instantiate :: NFType -> ElabM (NFType, Term -> Term, [Value])
instantiate (VPi P.Im d (Closure _ k)) = do
  t <- newMeta d
  (ty, w, xs) <- instantiate (k t)
  pure (ty, \inner -> w (App P.Im inner (quote t)), t:xs)
instantiate x = pure (x, id, [])

skipLams :: Int -> ElabM (Term -> Term)
skipLams 0 = pure id
skipLams k = do
  n <- newName
  (Lam P.Im n . ) <$> skipLams (k - 1)

checkLetItems :: Map Text (Maybe NFType) -> [P.LetItem] -> ([(Name, Term, Term)] -> ElabM a) -> ElabM a
checkLetItems _   [] cont = cont []
checkLetItems map (P.LetDecl v t:xs) cont = do
  t <- check t VTypeω
  t_nf <- eval t
  assume (Defined v 0) t_nf \_ ->
    checkLetItems (Map.insert v (Just t_nf) map) xs cont

checkLetItems map (P.LetBind name rhs:xs) cont = do
  case Map.lookup name map of
    Nothing -> do
      (tm, ty) <- infer rhs
      tm_nf <- eval tm
      makeLetDef (Defined name 0) ty tm_nf \name' ->
        checkLetItems map xs \xs ->
          cont ((name', quote ty, tm):xs)
    Just Nothing -> throwElab $ Redefinition (Defined name 0)
    Just (Just ty_nf) -> do
      rhs <- check rhs ty_nf
      rhs_nf <- eval rhs
      makeLetDef (Defined name 0) ty_nf rhs_nf \name' ->
        checkLetItems (Map.insert name Nothing map) xs \xs ->
          cont ((name', quote ty_nf, rhs):xs)

checkFormula :: P.Formula -> ElabM Value
checkFormula P.FTop = pure VI1
checkFormula P.FBot = pure VI0
checkFormula (P.FAnd x y) = iand <$> checkFormula x <*> checkFormula y
checkFormula (P.FOr x y) = ior <$> checkFormula x <*> checkFormula y
checkFormula (P.FIs0 x) = do
  nm <- getNameFor x
  ty <- getNfType nm
  unify ty VI
  pure (inot (VVar nm))
checkFormula (P.FIs1 x) = do
  nm <- getNameFor x
  ty <- getNfType nm
  unify ty VI
  pure (VVar nm)

isSort :: NFType -> ElabM ()
isSort t = isSort (force t) where
  isSort VType              = pure ()
  isSort VTypeω             = pure ()
  isSort vt@(VNe HMeta{} _) = unify vt VType
  isSort vt = throwElab $ NotEqual vt VType

data ProdOrPath
  = It'sProd { prodDmn  :: NFType
             , prodRng  :: NFType -> NFType
             , prodWrap :: Term -> Term
             }
  | It'sPath { pathLine  :: NFType -> NFType
             , pathLeft  :: Value
             , pathRight :: Value
             , pathWrap  :: Term -> Term
             }
  | It'sPartial { partialExtent :: NFEndp
                , partialDomain :: Value
                , partialWrap   :: Term -> Term
                }
  | It'sPartialP { partialExtent :: NFEndp
                 , partialDomain :: Value
                 , partialWrap   :: Term -> Term
                 }

isPiType :: P.Plicity -> NFType -> ElabM ProdOrPath
isPiType p x = isPiType p (force x) where
  isPiType p (VPi p' d (Closure _ k)) | p == p' = pure (It'sProd d k id)
  isPiType P.Ex (VPath li le ri) = pure (It'sPath (li @@) le ri id)
  isPiType P.Ex (VPartial phi a) = pure (It'sPartial phi a id)
  isPiType P.Ex (VPartialP phi a) = pure (It'sPartialP phi a id)

  isPiType P.Ex (VPi P.Im d (Closure _ k)) = do
    meta <- newMeta d
    porp <- isPiType P.Ex (k meta)
    pure $ case porp of
      It'sProd d r w -> It'sProd d r (\f -> w (App P.Im f (quote meta)))
      It'sPath l x y w -> It'sPath l x y (\f -> w (App P.Im f (quote meta)))
      It'sPartial phi a w -> It'sPartial phi a (\f -> w (App P.Im f (quote meta)))
      It'sPartialP phi a w -> It'sPartialP phi a (\f -> w (App P.Im f (quote meta)))
  isPiType p t = do
    dom <- newMeta VType
    name <- newName
    assume name dom $ \name -> do
      rng <- newMeta VType
      wp <- isConvertibleTo t (VPi p dom (Closure name (const rng)))
      pure (It'sProd dom (const rng) wp)

isSigmaType :: NFType -> ElabM (Value, NFType -> NFType, Term -> Term)
isSigmaType t = isSigmaType (force t) where
  isSigmaType (VSigma d (Closure _ k)) = pure (d, k, id)
  isSigmaType t = do
    dom <- newMeta VType
    name <- newName
    assume name dom $ \name -> do
      rng <- newMeta VType
      wp <- isConvertibleTo t (VSigma dom (Closure name (const rng)))
      pure (dom, const rng, wp)

isPartialType :: NFType -> ElabM (NFEndp, Value)
isPartialType t = isPartialType (force t) where
  isPartialType (VPartial phi a) = pure (phi, a)
  isPartialType (VPartialP phi a) = pure (phi, a)
  isPartialType t = do
    phi <- newMeta VI
    dom <- newMeta (VPartial phi VType)
    unify t (VPartial phi dom)
    pure (phi, dom)

checkStatement :: P.Statement -> ElabM a -> ElabM a
checkStatement (P.SpanSt s a b) k = withSpan a b $ checkStatement s k

checkStatement (P.Decl name ty) k = do
  ty <- check ty VTypeω
  ty_nf <- eval ty
  assumes (flip Defined 0 <$> name) ty_nf (const k)

checkStatement (P.Postulate []) k = k
checkStatement (P.Postulate ((name, ty):xs)) k = do
  ty <- check ty VTypeω
  ty_nf <- eval ty
  assume (Defined name 0) ty_nf \name ->
    local (\e -> e { definedNames = Set.insert name (definedNames e) }) (checkStatement (P.Postulate xs) k)

checkStatement (P.Defn name rhs) k = do
  ty <- asks (Map.lookup name . nameMap)
  case ty of
    Nothing -> do
      (tm, ty) <- infer rhs
      tm_nf <- eval tm
      makeLetDef (Defined name 0) ty tm_nf (const k)
    Just nm -> do
      ty_nf <- getNfType nm
      t <- asks (Set.member nm . definedNames)
      when t $ throwElab (Redefinition (Defined name 0))

      rhs <- check rhs ty_nf
      rhs_nf <- evalFix (Defined name 0) ty_nf rhs
      makeLetDef (Defined name 0) ty_nf rhs_nf $ \name ->
        local (\e -> e { definedNames = Set.insert name (definedNames e) }) k

checkStatement (P.Builtin winame var) k = do
  wi <-
    case Map.lookup winame wiredInNames of
      Just wi -> pure wi
      _ -> throwElab $ NoSuchPrimitive winame

  let
    check = do
      nm <- getNameFor var
      ty <- getNfType nm
      unify ty (wiType wi)
        `withNote` hsep [ pretty "Previous definition of", pretty nm, pretty "here" ]
        `seeAlso` nm

  env <- ask
  liftIO $
    runElab check env `catch` \(_ :: NotInScope) -> pure ()

  define (Defined var 0) (wiType wi) (wiValue wi) $ \name ->
    local (\e -> e { definedNames = Set.insert name (definedNames e) }) k

checkStatement (P.ReplNf e) k = do
  (e, _) <- infer e
  e_nf <- eval e
  h <- asks commHook
  liftIO (h e_nf)
  k

checkStatement (P.ReplTy e) k = do
  (_, ty) <- infer e
  h <- asks commHook
  liftIO (h ty)
  k

checkStatement (P.Data name tele retk constrs) k =
  do
    checkTeleRetk True tele retk \kind tele undef -> do
      kind_nf <- eval kind
      defineInternal (Defined name 0) kind_nf (\name' -> VNe (mkHead name') mempty) \name' -> 
        checkCons tele (VNe (mkHead name') (Seq.fromList (map makeProj tele))) constrs (local (markAsDef name' . undef) k)
  where
    makeProj (x, p, _) = PApp p (VVar x)

    markAsDef x e = e { definedNames = Set.insert x (definedNames e) }

    mkHead name
      | any (\case { (_, _, P.Path{}) -> True; _ -> False}) constrs = HData True name
      | otherwise = HData False name

    checkTeleRetk allKan [] retk cont = do
      t <- check retk VTypeω
      t_nf <- eval t
      when allKan $ unify t_nf VType
      cont t [] id
    checkTeleRetk allKan ((x, p, t):xs) retk cont = do
      (t, ty) <- infer t
      _ <- isConvertibleTo ty VTypeω
      let
        allKan' = case ty of
          VType -> allKan
          _ -> False
      t_nf <- eval t
      let rm nm e = e{ nameMap = Map.delete (getNameText nm) (nameMap e), getEnv = Map.delete nm (getEnv e) }
      assume (Bound x 0) t_nf $ \nm -> checkTeleRetk allKan' xs retk \k xs w -> cont (Pi p nm t k) ((nm, p, t_nf):xs) (rm nm . w)

    checkCons _ _et [] k = k

    checkCons n ret ((s, e, P.Point x ty):xs) k = withSpan s e $ do
      t <- check ty VTypeω
      ty_nf <- eval t
      let
        (args, ret') = splitPi ty_nf
        closed = close n t
        n' = map (\(x, _, y) -> (x, P.Im, y)) n
      unify ret' ret
      closed_nf <- eval closed
      defineInternal (ConName x 0 (length n') (length args)) closed_nf (makeCon closed_nf mempty n' args) \_ -> checkCons n ret xs k
    
    checkCons n ret ((s, e, P.Path name indices return faces):xs) k = withSpan s e $ do
      (con, closed_nf, value, boundary) <- assumes (flip Bound 0 <$> indices) VI \indices -> do
        t <- check return VTypeω
        ty_nf <- eval t
        let
          (args, ret') = splitPi ty_nf
          closed = close n (addArgs args (addInterval indices (quote ret')))
          n' = map (\(x, _, y) -> (x, P.Im, y)) n

          addArgs = flip $ foldr (\(x, p, t) -> Pi p x (quote t))
          addInterval = flip $ foldr (\n -> Pi P.Ex n I)

          envArgs ((x, _, y):xs) = assume x y . const . envArgs xs
          envArgs [] = id

        closed_nf <- eval closed
        unify ret' ret

        faces <- envArgs args $ for faces \(f, t) -> do
          phi <- checkFormula f
          t <- check t ret
          pure (phi, (quote phi, t))
        
        system <- eval $ foldr (\x -> Lam P.Ex x) (System (Map.fromList (map snd faces))) (map (\(x, _, _) -> x) n' ++ map (\(x, _, _) -> x) args ++ indices)

        unify (foldr ior VI0 (map fst faces)) (totalProp indices)
          `withNote` pretty "The formula determining the endpoints of a higher constructor must be a classical tautology"

        pure (ConName name 0 (length n') (length args + length indices), closed_nf, makePCon closed_nf mempty n' args indices system, Boundary indices system)
      defineInternal con closed_nf value \name -> addBoundary name boundary $ checkCons n ret xs k

    close []             t = t
    close ((x, _, y):xs) t = Pi P.Im x (quote y) (close xs t)

    splitPi (VPi p y (Closure x k)) = first ((x, p, y):) $ splitPi (k (VVar x))
    splitPi t = ([], t)

    makeCon cty sp []              [] con = VNe (HCon cty con) sp
    makeCon cty sp ((nm, p, _):xs) ys con = VLam p $ Closure nm \a -> makeCon cty (sp Seq.:|> PApp p a) xs ys con
    makeCon cty sp [] ((nm, p, _):ys) con = VLam p $ Closure nm \a -> makeCon cty (sp Seq.:|> PApp p a) [] ys con

    makePCon cty sp []              [] [] sys con = VNe (HPCon sys cty con) sp
    makePCon cty sp ((nm, p, _):xs) ys zs sys con = VLam p $ Closure nm \a -> makePCon cty (sp Seq.:|> PApp p a) xs ys zs (sys @@ a) con
    makePCon cty sp [] ((nm, p, _):ys) zs sys con = VLam p $ Closure nm \a -> makePCon cty (sp Seq.:|> PApp p a) [] ys zs (sys @@ a) con
    makePCon cty sp [] []         (nm:zs) sys con = VLam P.Ex $ Closure nm \a -> makePCon cty (sp Seq.:|> PApp P.Ex a) [] [] zs (sys @@ a) con

    totalProp (x:xs) = ior (inot (VVar x)) (VVar x) `ior` totalProp xs
    totalProp [] = VI0

evalFix :: Name -> NFType -> Term -> ElabM Value
evalFix name nft term = do
  env <- ask
  pure . fix $ \val -> eval' env{ getEnv = Map.insert name (nft, val) (getEnv env) } term

checkProgram :: [P.Statement] -> ElabM a -> ElabM a
checkProgram [] k = k
checkProgram (st:sts) k = checkStatement st $ checkProgram sts k

newtype Redefinition = Redefinition { getRedefName :: Name }
  deriving (Show, Typeable, Exception)

data WhenCheckingEndpoint = WhenCheckingEndpoint { leftEndp :: Value, rightEndp :: Value, whichIsWrong :: NFEndp, exc :: SomeException }
  deriving (Show, Typeable, Exception)

data UnsaturatedCon = UnsaturatedCon { theConstr :: Name }
  deriving          (Show, Typeable)
  deriving anyclass (Exception)

data NotACon = NotACon { theNotConstr :: Name }
  deriving          (Show, Typeable)
  deriving anyclass (Exception)