diff --git a/src/Elab.hs b/src/Elab.hs
index d9332b4..6949c13 100644
--- a/src/Elab.hs
+++ b/src/Elab.hs
@@ -64,7 +64,7 @@ check env (P.Lam s b) expected = do
       pure (PathI (quote a) (quote x) (quote y) s bd)
 
 check env (P.Let v t d b) expected = do
-  ty <- check env t VType
+  ty <- check env t VTypeω
   let ty' = eval env ty
   d <- check env d ty'
   let d' = eval env d
@@ -84,7 +84,7 @@ check env (P.Partial fs) ty = do
   let ourFaces    = Systems.faces formula
       extentFaces = Systems.faces extent
 
-  unless (formula == extent) $
+  unless (toDNF formula == toDNF extent) $
     throwIO $ IncompleteSystem formula extent
 
   let range = formulaToTm $ toDNF formula
@@ -164,21 +164,35 @@ infer env (P.App f x) = do
       pure (PathP (quote a) (quote ai0) (quote ai1) fun arg, a @@ arg')
 
 infer env (P.Pi s d r) = do
-  dom <- check env d VType
+  (dom, ty) <- infer env d
+  case ty of
+    VType -> pure VType
+    VTypeω -> pure VTypeω
+    _ -> throwIO . UnifyError $ NotSort ty
   let d' = eval env dom
-  rng <- check env { names = Map.insert s (d', VVar s) (names env) } r VType
-  pure (Pi s dom rng, VType)
+  (rng, rng_t) <- infer env { names = Map.insert s (d', VVar s) (names env) } r
+  case ty of
+    VType -> pure VType
+    VTypeω -> pure VTypeω
+    _ -> throwIO . UnifyError $ NotSort ty
+  pure (Pi s dom rng, rng_t)
 
 infer env (P.Sigma s d r) = do
-  dom <- check env d VType
+  (dom, ty) <- infer env d
+  rng_t <-
+    case ty of
+      VType -> pure VType
+      VTypeω -> pure VTypeω
+      _ -> throwIO . UnifyError $ NotSort ty
   let d' = eval env dom
-  rng <- check env { names = Map.insert s (d', VVar s) (names env) } r VType
-  pure (Sigma s dom rng, VType)
+  rng <- check env { names = Map.insert s (d', VVar s) (names env) } r rng_t
+  pure (Sigma s dom rng, rng_t)
 
-infer env P.Type = pure (Type, VType)
-infer env P.I    = pure (I,    VType)
-infer env P.I0   = pure (I0,   VI)
-infer env P.I1   = pure (I1,   VI)
+infer env P.Type  = pure (Type, VType)
+infer env P.Typeω = pure (Typeω, VTypeω)
+infer env P.I     = pure (I, VTypeω)
+infer env P.I0    = pure (I0, VI)
+infer env P.I1    = pure (I1, VI)
 
 infer env (P.Cut e t) = do
   t <- check env t VType
@@ -211,7 +225,7 @@ infer env P.PartialT = do
         Lam "A" Type $
           Partial (Var "r") (Var "A")
     , VPi "I" VI \i ->
-        VPi "A" VType (const VType))
+        VPi "A" VType (const VTypeω))
 
 infer env P.Comp = do
   let u_t a r = VPi "i" VI \i -> VPartial r (a @@ i)
@@ -244,7 +258,7 @@ infer env (P.INot x) = (, VI) . INot <$> check env x VI
 infer env P.Lam{}    = error "can't infer type for lambda"
 
 infer env (P.Let v t d b) = do
-  ty <- check env t VType
+  ty <- check env t VTypeω
   let ty' = eval env ty
   d <- check env d ty'
   let d' = eval env d
@@ -285,4 +299,4 @@ checkFormula env (P.Is1 x) =
     Nothing -> throwIO (NotInScope x)
 
 index_t :: Value
-index_t = VPi "_" VI (const VType)
+index_t = VPi "_" VI (const VTypeω)
diff --git a/src/Eval.hs b/src/Eval.hs
index 29b8049..40410d9 100644
--- a/src/Eval.hs
+++ b/src/Eval.hs
@@ -49,6 +49,7 @@ VNe hd xs @@ vl = VNe hd (PApp vl:xs)
 VLam _ _ k        @@ vl = k vl
 VEqGlued a b      @@ vl = VEqGlued (a @@ vl) (b @@ vl)
 VOfSub a phi u0 x @@ vl = x @@ vl
+VSystem fs        @@ vl = mapVSystem (VSystem fs) (@@ vl)
 f @@ _  = error $ "can't apply argument to " ++ show f
 
 proj1 :: Value -> Value
@@ -81,7 +82,7 @@ pathp env p x y (VNe hd sp) i =
 pathp env p x y (VOfSub _ _ _ v) i = pathp env p x y v i
 
 comp :: Env -> Value -> Formula -> Value -> Value -> Value
-comp env a@(VLam ivar VI fam) phi u a0 = go (fam undefined) phi u a0 where
+comp env a@(VLam ivar VI fam) phi u a0 = go (fam (VVar "woopsie")) phi u a0 where
   i = VVar ivar
   stuck :: Value
   stuck = maybeAddEq $ VComp a (toValue phi) u a0
@@ -96,14 +97,14 @@ comp env a@(VLam ivar VI fam) phi u a0 = go (fam undefined) phi u a0 where
   go VPi{} phi u a0 =
     let
       dom x = let VPi _ d _ = fam x in d
-      rng x = let VPi _ d _ = fam x in d
+      rng x = let VPi _ _ r = fam x in r
 
       ai1 = dom VI0
-      y' i y   = fill env i (dom . inot . fam) P.Bot (VSystem emptySystem) y
+      y' i y   = fill env i (dom . inot) P.Bot (VSystem emptySystem) y
       ybar i y = y' (inot i) y
      in VLam "x" ai1 \arg ->
        comp env
-        (VLam ivar VI (\i -> rng (ybar i arg)))
+        (VLam ivar VI (\i -> rng i (ybar i arg)))
         phi
         (VLam "i" VI \i -> mapVSystem (u @@ i) (@@ ybar i arg)) 
         (a0 @@ ybar VI0 arg)
@@ -138,8 +139,10 @@ comp env a@(VLam ivar VI fam) phi u a0 = go (fam undefined) phi u a0 where
                 let
                   VSystem (FMap sys) = p @@ v
                   sys' = fmap (flip (pathp env ai0 u0 v0) j) sys
-               in mkVSystem $ Map.fromList [(phi, mapVSystem (p @@ v) (flip (pathp env ai0 u0 v0) j))
-                                           , (notFormula jc, u' v), (jc, v' v)])
+               in mkVSystem $ Map.fromList [ (phi, mapVSystem (p @@ v) (flip (pathp env ai0 u0 v0) j))
+                                           , (notFormula jc, u' v)
+                                           , (jc, v' v)
+                                           ])
               (pathp env (ai0 @@ VI0) u0 v0 p0 j)
 
   go a P.Top u a0 = u @@ VI1
@@ -250,6 +253,7 @@ data UnifyError
   | NotPiType      Value
   | NotPartialType Formula Value
   | NotSigmaType   Value
+  | NotSort        Value
   deriving (Show, Typeable, Exception)
 
 unify :: Env -> Value -> Value -> IO ()
@@ -326,6 +330,9 @@ unify env vl (VSystem fs)
   | ((_, vl'):_) <- Map.toList (Map.filterWithKey (\f _ -> isTrue (toValue' env f)) (getSystem fs))
   = unify env vl' vl
 
+unify env VType VTypeω = pure ()
+unify env VTypeω VTypeω = pure ()
+
 unify env x y =
   case sameCube env x y of
     Just True -> pure ()
@@ -418,7 +425,7 @@ fill :: Env
 fill env i a phi u a0 =
   comp env
     (VLam "j" VI \j -> a (i `iand` j))
-    (phi `P.Or` ifc)
+    (phi `P.Or` notFormula ifc)
     (VLam "j" VI \j ->
       mkVSystem (Map.fromList [ (phi, uiand j)
                               , (notFormula ifc, a0) ]))
diff --git a/src/Main.hs b/src/Main.hs
index 3ba2e80..aa45f62 100644
--- a/src/Main.hs
+++ b/src/Main.hs
@@ -67,7 +67,7 @@ main = do
 
 repl :: IO ()
 repl = runInputT defaultSettings (go (Env mempty)) where
-  go env = getInputLine "λ " >>= \case
+  go env = getInputLine "> " >>= \case
     Just i | ":sq " `isPrefixOf` i -> do
       case parseString body (replicate 4 ' ' ++ drop 4 i) of
         Right exp -> 
@@ -84,7 +84,7 @@ repl = runInputT defaultSettings (go (Env mempty)) where
         Right (Assume vs) ->
           let
             loop env ((v, t):vs) = do
-              tm <- liftIO $ check env t VType
+              tm <- liftIO $ check env t VTypeω
               let ty = eval env tm
               loop env{ names = Map.insert v (ty, VVar v) (names env) } vs
             loop env [] = go env
diff --git a/src/Presyntax.hs b/src/Presyntax.hs
index 3a2bedb..6a2b032 100644
--- a/src/Presyntax.hs
+++ b/src/Presyntax.hs
@@ -14,6 +14,7 @@ data Exp
 
   | Pi  String Exp Exp
   | Type
+  | Typeω
 
   | I
   | I0 | I1
diff --git a/src/Presyntax/Lexer.hs b/src/Presyntax/Lexer.hs
index a9f8897..878cc95 100644
--- a/src/Presyntax/Lexer.hs
+++ b/src/Presyntax/Lexer.hs
@@ -12,6 +12,7 @@ data TokenClass
   | Tok_lambda
 
   | Tok_type
+  | Tok_typeω
   | Tok_path
   | Tok_phi
   | Tok_sub
@@ -178,6 +179,7 @@ lexString = go 0 0 0 where
 
   finishIdent c
     | T.pack "Type" == c    = Tok_type
+    | T.pack "Typeω" == c   = Tok_typeω
     | T.pack "Path" == c    = Tok_path
     | T.pack "Partial" == c = Tok_phi
     | T.pack "Sub" == c     = Tok_sub
diff --git a/src/Syntax.hs b/src/Syntax.hs
index 0922267..40b9b5d 100644
--- a/src/Syntax.hs
+++ b/src/Syntax.hs
@@ -1,3 +1,4 @@
+{-# LANGUAGE ViewPatterns #-}
 {-# LANGUAGE LambdaCase #-}
 {-# LANGUAGE PatternSynonyms #-}
 module Syntax where
@@ -19,7 +20,7 @@ data Term
   | Let String Term Term Term
 
   | Pi  String Term Term
-  | Type
+  | Type | Typeω
 
   | I
   | I0 | I1
@@ -110,6 +111,7 @@ instance Show Term where
          in showBinder (Pi v d r)
 
       Type -> showString "Type"
+      Typeω -> showString "Typeω"
       I    -> showChar 'I'
       I0   -> showString "i0"
       I1   -> showString "i1"
@@ -129,9 +131,11 @@ instance Show Term where
       Path a x y -> showsPrec p (App (App (App (Var "Path") a) x) y)
       Sub a x y -> showsPrec p (App (App (App (Var "Sub") a) x) y)
       Partial r a -> showsPrec p (App (App (Var "Partial") r) a)
-      Comp a phi u a0 -> showsPrec p (foldl App (Var "comp") [a, phi, u, a0])
       InclSub _a _phi _u0 a0 -> showsPrec p a0
 
+      Comp a I0 (Lam _ I (System (FMap (Map.null -> True)))) a0 -> showsPrec p (foldl App (Var "transp") [a, a0])
+      Comp a phi u a0 -> showsPrec p (foldl App (Var "comp") [a, phi, u, a0])
+
       PathI a x y s b -> showParen (p >= fun_prec) $
         showString ("λ " ++ s ++ " -> ")
         . shows b
@@ -163,7 +167,7 @@ data Value
   = VNe   String [Proj]
   | VLam  String Value (Value -> Value)
   | VPi   String Value (Value -> Value)
-  | VType
+  | VType | VTypeω
   | VI | VI0 | VI1
   | VEqGlued Value Value -- e which is def. eq. to e'
   | VPair Value Value
@@ -207,10 +211,11 @@ quote = go mempty where
     let n = rename s scope
      in Sigma n (go scope d) (go (Set.insert n scope) (r (VVar n)))
 
-  go scope VType = Type
-  go scope VI0   = I0
-  go scope VI1   = I1
-  go scope VI    = I
+  go _ VType = Type
+  go _ VI0   = I0
+  go _ VI1   = I1
+  go _ VI    = I
+  go _ VTypeω = Typeω
 
   go scope (VIAnd x y) = IAnd (go scope x) (go scope y)
   go scope (VIOr x y) = IOr (go scope x) (go scope y)