diff --git a/Setup.hs b/Setup.hs
index e8ef27d..09e11ce 100644
--- a/Setup.hs
+++ b/Setup.hs
@@ -1,3 +1,4 @@
+
 import Distribution.Simple
 
 main = defaultMain
diff --git a/cubical.cabal b/cubical.cabal
index deeb799..4072bef 100644
--- a/cubical.cabal
+++ b/cubical.cabal
@@ -46,6 +46,8 @@ executable cubical
                      , Elab.WiredIn
                      , Elab.Eval.Formula
 
+                     , Debug
+
   build-tool-depends:  alex:alex >= 3.2.4 && < 4.0
                     ,  happy:happy >= 1.19.12 && < 2.0
 
diff --git a/intro.tt b/intro.tt
index 5f1e436..9bcefaf 100644
--- a/intro.tt
+++ b/intro.tt
@@ -292,6 +292,19 @@ hfill {A} {phi} u a0 i = fill (\i -> A) {phi} u a0 i
 transRefl : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans p refl) p
 transRefl p j i = fill (\i -> A) {ior i (inot i)} (\k [ (i = i0) -> x, (i = i1) -> y ]) (inS (p i)) (inot j)
 
+rightCancel : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans p (sym p)) refl
+rightCancel p j i = cube p i1 j i where
+  cube : {A : Type} {x : A} {y : A} (p : Path x y) -> I -> I -> I -> A
+  cube {A} {x} p k j i =
+    hfill {A} (\ k [ (i = i0) -> x
+                  , (i = i1) -> p (iand (inot k) (inot j))
+                  , (j = i1) -> x
+                  ])
+              (inS (p (iand i (inot j)))) k
+
+leftCancel : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans (sym p) p) refl
+leftCancel p = rightCancel (sym p)
+
 transpFill : {A : I -> Type} (x : A i0) -> PathP A x (transp (\i -> A i) x)
 transpFill {A} x i = fill (\i -> A i) (\k []) (inS x) i
 
@@ -313,7 +326,8 @@ transpFun p q f = refl
 transpDFun : {A : I -> Type} {B : (i : I) -> A i -> Type}
           -> (f : (x : A i0) -> B i0 x) 
           -> Path (transp (\i -> (x : A i) -> B i x) f)
-                  (\x -> transp (\i -> B i (fill (\j -> A (inot j)) (\k []) (inS x) (inot i))) (f (fill (\j -> A (inot j)) (\k []) (inS x) i1)))
+                  (\x -> transp (\i -> B i (fill (\j -> A (inot j)) (\k []) (inS x) (inot i)))
+                          (f (fill (\j -> A (inot j)) (\k []) (inS x) i1)))
 transpDFun f = refl
 
 -- When considering the more general case of a composition respecing sides,
@@ -512,6 +526,13 @@ J : {A : Type} {x : A}
  -> P y p
 J P d p = transp (\i -> P (p i) (\j -> p (iand i j))) d
 
+Jay : {A : Type} {x : A}
+      (P : ((y : A) * Path x y) -> Type)
+      (d : P (x, refl))
+      (s : (y : A) * Path x y)
+   -> P s
+Jay P d s = transp (\i -> P ((singContr {A} {x}).2 s i)) d
+
 
 -- Isomorphisms
 ---------------
@@ -598,6 +619,9 @@ IsoToEquiv {A} {B} iso = (f, \y -> (fCenter y, fIsCenter y)) where
   fIsCenter : (y : B) (w : fiber f y) -> Path (fCenter y) w
   fIsCenter y w = lemIso y (fCenter y).1 w.1 (fCenter y).2 w.2
 
+IsoToId : {A : Type} {B : Type} -> Iso A B -> Path A B
+IsoToId i = univalence (IsoToEquiv i)
+
 -- We can prove that any involutive function is an isomorphism, since
 -- such a function is its own inverse.
 
@@ -899,14 +923,18 @@ windingBase = refl
 goAround : Int -> Path base base
 goAround =
   \case
-    pos n -> goAround_nat n
-    neg n -> sym (goAround_nat (succ n))
+    pos n -> forwards n
+    neg n -> backwards n
   where
-    goAround_nat : Nat -> Path base base
-    goAround_nat = \case
+    forwards : Nat -> Path base base
+    forwards = \case
       zero -> refl
-      succ n -> trans (\i -> loop i) (goAround_nat n)
-
+      succ n -> trans (forwards n) (\i -> loop i)
+    backwards : Nat -> Path base base
+    backwards = \case
+      zero -> \i -> loop (inot i)
+      succ n -> trans (backwards n) (\i -> loop (inot i))
+    
 -- One particularly general higher inductive type is the homotopy pushout,
 -- which can be seen as a kind of sum B + C with the extra condition that
 -- whenever x and y are in the image of f (resp. g), inl x ≡ inr y.
@@ -1139,7 +1167,7 @@ S2IsSuspS1 = univalence (IsoToEquiv iso) where
   fromS2 = \case { base2 -> north; surf2 i j -> suspSurf i j i1 }
 
   toFromS2 : (x : S2) -> Path (toS2 (fromS2 x)) x
-  toFromS2 = \case { base2 -> refl; surf2 i j -> refl }
+  toFromS2 = \case { base2 -> refl; surf2 i j -> \k -> toS2 (suspSurf i j (inot k)) }
 
   fromToS2 : (x : Susp S1) -> Path (fromS2 (toS2 x)) x
   fromToS2 = \case { north -> refl; south -> \i -> merid base i; merid x i -> meridCase i x } where
@@ -1177,7 +1205,7 @@ S3IsSuspS2 = univalence (IsoToEquiv iso) where
   fromS3 = \case { base3 -> north; surf3 i j k -> suspSurf i j k i1 }
 
   toFromS3 : (x : S3) -> Path (toS3 (fromS3 x)) x
-  toFromS3 = \case { base3 -> refl; surf3 i j k -> refl }
+  toFromS3 = \case { base3 -> refl; surf3 i j k -> \l -> toS3 (suspSurf i j k (inot l)) }
 
   fromToS3 : (x : Susp S2) -> Path (fromS3 (toS3 x)) x
   fromToS3 = \case { north -> refl; south -> \i -> merid base2 i; merid x i -> meridCase i x } where
@@ -1264,4 +1292,45 @@ rightIsOne : {a : I} {b : I} -> Eq_s b i1 -> Eq_s (ior a b) i1
 rightIsOne {a} {b} p = J_s {I} {i1} (\i p -> IsOne (ior a i)) refl_s (sym_s p)
 
 bothAreOne : {a : I} {b : I} -> Eq_s a i1 -> Eq_s b i1 -> Eq_s (iand a b) i1
-bothAreOne {a} {b} p q = J_s {I} {i1} (\i p -> IsOne (iand i b)) q (sym_s p)
\ No newline at end of file
+bothAreOne {a} {b} p q = J_s {I} {i1} (\i p -> IsOne (iand i b)) q (sym_s p)
+
+test : {X : Type} -> (S1 -> X) -> (a : X) * Path a a
+test {X} f = (f base, \i -> f (loop i))
+
+test' : {X : Type} -> ((a : X) * Path a a) -> S1 -> X
+test' {X} p = \case
+  base -> p.1
+  loop i -> p.2 i
+
+test_test' : {X : Type} -> (f : S1 -> X) -> Path (test' (test f)) f
+test_test' {X} f = funext {S1} {\s -> X} {\x -> test' (test f) x} {f} h where
+  h : (x : S1) -> Path (test' (test f) x) (f x)
+  h = \case
+    base -> refl
+    loop i -> refl
+
+test'_test : {X : Type} -> (x : (a : X) * Path a a) -> Path (test (test' x)) x
+test'_test x = refl
+
+test'' : {X : Type} -> Path (S1 -> X) ((a : X) * Path a a)
+test'' = IsoToId {S1 -> X} {(a : X) * Path a a} (test, test', test'_test, test_test')
+
+-- HoTT book lemma 8.9.1
+encodeDecode : {A : Type} {a0 : A}
+            -> (code : A -> Type)
+            -> (c0 : code a0)
+            -> (decode : (x : A) -> code x -> (Path a0 x))
+            -> ((c : code a0) -> Path (transp (\i -> code (decode a0 c i)) c0) c)
+            -> Path (decode a0 c0) refl
+            -> Path (Path a0 a0) (code a0)
+encodeDecode code c0 decode encDec based = IsoToId (encode {a0}, decode a0, encDec, decEnc) where
+  encode : {x : A} -> Path a0 x -> code x
+  encode alpha = transp (\i -> code (alpha i)) c0
+
+  encodeRefl : Path (encode refl) c0
+  encodeRefl = sym (transpFill {\i -> code a0} c0)
+
+  decEnc : {x : A} (p : Path a0 x) -> Path (decode x (encode p)) p
+  decEnc p = J (\x p -> Path (decode x (encode p)) p) q p where
+    q : Path (decode a0 (encode refl)) refl
+    q = transp (\i -> Path (decode a0 (encodeRefl (inot i))) refl) based
\ No newline at end of file
diff --git a/src/Debug.hs b/src/Debug.hs
new file mode 100644
index 0000000..18f011b
--- /dev/null
+++ b/src/Debug.hs
@@ -0,0 +1,38 @@
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE KindSignatures #-}
+module Debug where
+
+import qualified Debug.Trace as D
+
+#if defined(RELEASE)
+import GHC.Exts
+#else
+import GHC.Stack
+import Prettyprinter
+#endif
+
+traceDoc :: Doc a -> b -> b
+
+#if defined(RELEASE)
+type DebugCallStack = (() :: Constraint)
+traceDoc !_ v = v
+#else
+type DebugCallStack = HasCallStack
+traceDoc x = D.trace (show x)
+#endif
+
+trace :: Pretty a => a -> b -> b
+trace x = traceDoc (pretty x)
+
+traceWith :: Pretty a => String -> a -> b -> b
+traceWith s x = traceDoc (pretty s <> pretty ": " <> pretty x)
+
+traceId :: Pretty a => a -> a
+traceId x = traceDoc (pretty x) x
+
+traceDocM :: (Applicative m) => Doc a -> m ()
+traceDocM x = traceDoc x (pure ())
+
+traceM :: (Applicative m, Pretty a) => a -> m ()
+traceM = traceDocM . pretty
diff --git a/src/Elab.hs b/src/Elab.hs
index b1cea92..1219446 100644
--- a/src/Elab.hs
+++ b/src/Elab.hs
@@ -15,6 +15,7 @@ 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.Maybe (fromMaybe)
 import Data.Traversable
 import Data.Text (Text)
 import Data.Map (Map)
@@ -32,7 +33,6 @@ import Prettyprinter
 
 import Syntax.Pretty
 import Syntax
-import Data.Maybe (fromMaybe)
 
 infer :: P.Expr -> ElabM (Term, NFType)
 infer (P.Span ex a b) = withSpan a b $ infer ex
@@ -188,7 +188,7 @@ check (P.LamSystem bs) ty = do
                           , 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))
+          `withNote` (pretty "Consider this face, where both are true:" <+> showFace False (snd truth))
 
   name <- newName
   let
@@ -232,14 +232,14 @@ check (P.LamCase pats) ty =
                   let rhs = cases @@ side
                   for_ (truthAssignments formula mempty) $ \i -> do
                     let vl = foldl (\v n -> vApp P.Ex v (lookup n)) base (getBoundaryNames boundary)
-                        lookup n = fromMaybe VI0 (snd <$> (Map.lookup n i))
+                        lookup n = fromMaybe (VVar n) (snd <$> (Map.lookup n i))
                     unify vl rhs
+                      `withNote` (pretty "From the boundary conditions of the constructor" <+> prettyTm (quote pat_nf) <> pretty ":")
                       `withNote` vcat [ pretty "These must be the same because of the face"
-                                      , indent 2 $ prettyTm (quote formula) <+> operator (pretty "=>") <+> prettyTm (quote (zonk side))
-                                      , pretty "which evaluates into"
-                                      , indent 2 $ prettyTm (quote formula) <+> operator (pretty "=>") <+> prettyVl rhs
+                                      , indent 2 $ prettyVl (zonk formula) <+> operator (pretty "=>") <+> prettyVl (zonk side)
+                                      , pretty "which is mapped to"
+                                      , indent 2 $ prettyVl (zonk formula) <+> operator (pretty "=>") <+> prettyVl (zonk rhs)
                                       ]
-                      `withNote` (pretty "Mandated by the constructor" <+> prettyTm (quote pat_nf))
               _ -> pure ()
 
             pure (pat, n_lams, wp rhs)
@@ -502,7 +502,7 @@ checkStatement (P.ReplNf e) k = do
   (e, _) <- infer e
   e_nf <- eval e
   h <- asks commHook
-  liftIO (h (prettyVl e_nf))
+  liftIO $ h . prettyVl =<< zonkIO e_nf
   k
 
 checkStatement (P.ReplTy e) k = do
@@ -624,4 +624,4 @@ data NotACon = NotACon { theNotConstr :: Name }
   deriving anyclass (Exception)
 
 newtype DeclaredUndefined = DeclaredUndefined { declaredUndefName :: Name }
-  deriving (Eq, Show, Exception)
\ No newline at end of file
+  deriving (Eq, Show, Exception)
diff --git a/src/Elab/Eval.hs b/src/Elab/Eval.hs
index 6c57dda..2534b0e 100644
--- a/src/Elab/Eval.hs
+++ b/src/Elab/Eval.hs
@@ -82,17 +82,14 @@ zonkIO (VSystem fs) = do
   t <- for (Map.toList fs) $ \(a, b) -> (,) <$> zonkIO a <*> zonkIO b
   pure (mkVSystem (Map.fromList t))
 zonkIO (VSub a b c) = VSub <$> zonkIO a <*> zonkIO b <*> zonkIO c
-zonkIO (VInc a b c) = VInc <$> zonkIO a <*> zonkIO b <*> zonkIO c
+zonkIO (VInc a b c) = incS <$> zonkIO a <*> zonkIO b <*> zonkIO c
 zonkIO (VComp a b c d) = comp <$> zonkIO a <*> zonkIO b <*> zonkIO c <*> zonkIO d
 zonkIO (VHComp a b c d) = hComp <$> zonkIO a <*> zonkIO b <*> zonkIO c <*> zonkIO d
 
 zonkIO (VGlueTy a phi ty e)   = glueType <$> zonkIO a <*> zonkIO phi <*> zonkIO ty <*> zonkIO e
 zonkIO (VGlue a phi ty e t x) = glueElem <$> zonkIO a <*> zonkIO phi <*> zonkIO ty <*> zonkIO e <*> zonkIO t <*> zonkIO x
 zonkIO (VUnglue a phi ty e x) = unglue   <$> zonkIO a <*> zonkIO phi <*> zonkIO ty <*> zonkIO e <*> zonkIO x
-zonkIO (VCase env t x xs) = do
-  env' <- (\x -> x {getEnv = env}) <$> emptyEnv
-  let xs' = map (\(a, i, n) -> (a, i, quote (eval' env' n))) xs
-  evalCase env' . (@@) <$> zonkIO t <*> zonkIO x <*> pure xs'
+zonkIO (VCase env t x xs) = pure $ VCase env t x xs
 
 zonkIO (VEqStrict a x y) = VEqStrict <$> zonkIO a <*> zonkIO x <*> zonkIO y
 zonkIO (VReflStrict a x) = VReflStrict <$> zonkIO a <*> zonkIO x
@@ -130,7 +127,7 @@ eval' env (App p f x) = vApp p (eval' env f) (eval' env x)
 
 eval' env (Lam p s t) =
   VLam p $ Closure s $ \a ->
-    eval' env { getEnv = Map.insert s (error "type of abs", a) (getEnv env) } t
+    eval' env { getEnv = Map.insert s (error ("type of abs " ++ show (pretty (Lam p s t))), a) (getEnv env) } t
 
 eval' env (Pi p s d t) =
   VPi p (eval' env d) $ Closure s $ \a ->
@@ -166,7 +163,7 @@ eval' e (PartialP x y) = VPartialP (eval' e x) (eval' e y)
 eval' e (System fs) = mkVSystem (Map.fromList $ map (\(x, y) -> (eval' e x, eval' e y)) $ Map.toList $ fs)
 
 eval' e (Sub a phi u)   = VSub (eval' e a) (eval' e phi) (eval' e u)
-eval' e (Inc a phi u)   = VInc (eval' e a) (eval' e phi) (eval' e u)
+eval' e (Inc a phi u)   = incS (eval' e a) (eval' e phi) (eval' e u)
 eval' e (Ouc a phi u x) = outS (eval' e a) (eval' e phi) (eval' e u) (eval' e x)
 
 eval' e (Comp a phi u a0) = comp (eval' e a) (eval' e phi) (eval' e u) (eval' e a0)
@@ -195,11 +192,14 @@ evalCase env rng sc [] = VCase (getEnv env) (fun rng) sc []
 
 evalCase env rng (VSystem fs) cases = VSystem (fmap (flip (evalCase env rng) cases) fs)
 
-evalCase env rng (VHComp a phi u a0) cases =
-  comp (fun \i -> rng (v i)) phi (system \i is1 -> evalCase env rng (u @@ i @@ is1) cases)
-         (VInc (rng a) phi (evalCase env rng (outS a0 phi (u @@ VI0) a0) cases))
+evalCase env rng (VHComp a φ u u0) cases =
+  comp (fun \i -> rng (v i))
+       φ
+       (system \i is1 -> α (u @@ i @@ is1))
+       (VInc (rng a) φ (α (outS a φ (u @@ VI0) u0)))
   where
-    v = Elab.WiredIn.fill (fun (const a)) phi u a0
+    v = Elab.WiredIn.fill (fun (const a)) φ u u0
+    α x = evalCase env rng x cases
 
 evalCase env _ sc ((Ref _, _, k):_) = eval' env k @@ sc
 
@@ -213,10 +213,10 @@ evalCase env rng (val@(VNe (HPCon _ _ x) sp)) ((Con x', _, k):xs)
 
 evalCase env rng sc xs = VCase (getEnv env) (fun rng) sc xs
 
-evalFix' :: ElabEnv -> Name -> NFType -> Term -> Value
+evalFix' :: HasCallStack => ElabEnv -> Name -> NFType -> Term -> Value
 evalFix' env name nft term = fix $ \val -> eval' env{ getEnv = Map.insert name (nft, val) (getEnv env) } term
 
-evalFix :: Name -> NFType -> Term -> ElabM Value
+evalFix :: HasCallStack => Name -> NFType -> Term -> ElabM Value
 evalFix name nft term = do
   t <- ask
   pure (evalFix' t name nft term)
@@ -231,8 +231,10 @@ unify' topa topb = join $ go <$> forceIO topa <*> forceIO topb where
   go (VNe (HMeta mv) sp) rhs = solveMeta mv sp rhs
   go rhs (VNe (HMeta mv) sp) = solveMeta mv sp rhs
 
-  go (VNe (HPCon s _ _) _) rhs = go (force s) rhs
-  go lhs (VNe (HPCon s _ _) _) = go lhs (force s)
+  go (VNe (HPCon _ _ x) sp) (VNe (HPCon _ _ y) sp')
+    | x == y = traverse_ (uncurry unify'Spine) (Seq.zip sp sp')
+  go (VNe (HPCon s _ _) _) rhs | Just v <- trivialSystem s = go v rhs
+  go lhs (VNe (HPCon s _ _) _) | Just v <- trivialSystem s = go lhs v
 
   go (VCase e _ a b) (VCase e' _ a' b') = do
     env <- ask
@@ -343,6 +345,11 @@ unify' topa topb = join $ go <$> forceIO topa <*> forceIO topb where
     | compareDNFs x y = pure ()
     | otherwise = fail
 
+trivialSystem :: Value -> Maybe Value
+trivialSystem = go . force where
+  go VSystem{} = Nothing
+  go x = Just x
+
 unify'Spine :: Projection -> Projection -> ElabM ()
 unify'Spine (PApp a v) (PApp a' v')
   | a == a' = unify' v v'
@@ -568,7 +575,7 @@ substituteIO sub = substituteIO . force where
     t <- for (Map.toList fs) $ \(a, b) -> (,) <$> substituteIO a <*> substituteIO b
     pure (mkVSystem (Map.fromList t))
   substituteIO (VSub a b c) = VSub <$> substituteIO a <*> substituteIO b <*> substituteIO c
-  substituteIO (VInc a b c) = VInc <$> substituteIO a <*> substituteIO b <*> substituteIO c
+  substituteIO (VInc a b c) = incS <$> substituteIO a <*> substituteIO b <*> substituteIO c
   substituteIO (VComp a b c d) = comp <$> substituteIO a <*> substituteIO b <*> substituteIO c <*> substituteIO d
   substituteIO (VHComp a b c d) = hComp <$> substituteIO a <*> substituteIO b <*> substituteIO c <*> substituteIO d
 
@@ -576,8 +583,8 @@ substituteIO sub = substituteIO . force where
   substituteIO (VGlue a phi ty e t x) = glueElem <$> substituteIO a <*> substituteIO phi <*> substituteIO ty <*> substituteIO e <*> substituteIO t <*> substituteIO x
   substituteIO (VUnglue a phi ty e x) = unglue   <$> substituteIO a <*> substituteIO phi <*> substituteIO ty <*> substituteIO e <*> substituteIO x
   substituteIO (VCase env t x xs) = VCase env <$> substituteIO t <*> substituteIO x <*> pure xs
-  substituteIO (VEqStrict a x y) = VEqStrict <$> zonkIO a <*> zonkIO x <*> zonkIO y
-  substituteIO (VReflStrict a x) = VReflStrict <$> zonkIO a <*> zonkIO x
+  substituteIO (VEqStrict a x y) = VEqStrict <$> substituteIO a <*> substituteIO x <*> substituteIO y
+  substituteIO (VReflStrict a x) = VReflStrict <$> substituteIO a <*> substituteIO x
 
 substitute :: Map Name Value -> Value -> Value
 substitute sub = unsafePerformIO . substituteIO sub
@@ -594,7 +601,7 @@ substituteSp _ PProj2 = pure PProj2
 mkVSystem :: Map.Map Value Value -> Value
 mkVSystem vals =
   let map' = Map.fromList (Map.toList vals >>= go)
-      go (x, y) = 
+      go (x, y) =
         case (force x, y) of
           (VI0, _) -> []
           (VIOr _ _, VSystem y) -> Map.toList y >>= go
@@ -653,7 +660,7 @@ vProj1 (VPair a _) = a
 vProj1 (VNe h sp) = VNe h (sp Seq.:|> PProj1)
 vProj1 (GluedVl h sp vl) = GluedVl h (sp Seq.:|> PProj1) (vProj1 vl)
 vProj1 (VSystem fs) = VSystem (fmap vProj1 fs)
-vProj1 (VInc (VSigma a _) b c) = VInc a b (vProj1 c)
+vProj1 (VInc (VSigma a _) b c) = incS a b (vProj1 c)
 vProj1 (VCase env rng sc branches) =
   VCase env rng sc (map (projIntoCase Proj1) branches)
 vProj1 x = error $ "can't proj1 " ++ show (prettyTm (quote x))
@@ -663,7 +670,7 @@ vProj2 (VPair _ b) = b
 vProj2 (VNe h sp)  = VNe h (sp Seq.:|> PProj2)
 vProj2 (GluedVl h sp vl) = GluedVl h (sp Seq.:|> PProj2) (vProj2 vl)
 vProj2 (VSystem fs) = VSystem (fmap vProj2 fs)
-vProj2 (VInc (VSigma _ (Closure _ r)) b c) = VInc (r (vProj1 c)) b (vProj2 c)
+vProj2 (VInc (VSigma _ (Closure _ r)) b c) = incS (r (vProj1 c)) b (vProj2 c)
 vProj2 (VCase env rng sc branches) =
   VCase env rng sc (map (projIntoCase Proj2) branches)
 vProj2 x = error $ "can't proj2 " ++ show (prettyTm (quote x))
diff --git a/src/Elab/Eval/Formula.hs b/src/Elab/Eval/Formula.hs
index 70fc9c3..ae313e8 100644
--- a/src/Elab/Eval/Formula.hs
+++ b/src/Elab/Eval/Formula.hs
@@ -2,13 +2,13 @@ module Elab.Eval.Formula where
 
 import qualified Data.Map.Strict as Map
 import qualified Data.Sequence as Seq
+import qualified Data.Set as Set
 import Data.Map.Strict (Map)
+import Data.Set (Set)
 
 import Syntax
 
 import {-# SOURCE #-} Elab.WiredIn (inot, ior, iand)
-import Data.Set (Set)
-import qualified Data.Set as Set
 
 toDnf :: Value -> Maybe Value
 toDnf = fmap (dnf2Val . normalise) . val2Dnf where
diff --git a/src/Elab/WiredIn.hs b/src/Elab/WiredIn.hs
index 93da268..c034ec9 100644
--- a/src/Elab/WiredIn.hs
+++ b/src/Elab/WiredIn.hs
@@ -4,6 +4,9 @@
 {-# LANGUAGE DerivingStrategies #-}
 {-# LANGUAGE DeriveAnyClass #-}
 {-# LANGUAGE ViewPatterns #-}
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE KindSignatures #-}
 module Elab.WiredIn where
 
 import Control.Exception
@@ -15,10 +18,13 @@ import Data.Map.Strict (Map)
 import Data.Text (Text)
 import Data.Typeable
 
+import Debug
+
 import Elab.Eval
 
 import GHC.Stack (HasCallStack)
 
+import Presyntax.Presyntax (Plicity(Im, Ex))
 import qualified Presyntax.Presyntax as P
 
 import Syntax.Pretty (prettyTm, prettyVl)
@@ -71,17 +77,17 @@ wiValue WiInterval = VI
 wiValue WiI0       = VI0
 wiValue WiI1       = VI1
 
-wiValue WiIAnd      = fun \x -> fun \y -> iand x y
-wiValue WiIOr       = fun \x -> fun \y -> ior x y
-wiValue WiINot      = fun inot
-wiValue WiPathP     = fun \a -> fun \x -> fun \y -> VPath a x y
+wiValue WiIAnd      = functions [(Ex, "i"), (Ex, "j")] \[i, j] -> iand i j
+wiValue WiIOr       = functions [(Ex, "i"), (Ex, "j")] \[i, j] -> ior i j
+wiValue WiINot      = fun' "x" inot
+wiValue WiPathP     = functions [(Ex, "A"), (Ex, "x"), (Ex, "y")] \[a, x, y] -> VPath a x y
 
-wiValue WiPartial  = fun \phi -> fun \r -> VPartial phi r
-wiValue WiPartialP = fun \phi -> fun \r -> VPartialP phi r
-wiValue WiSub      = fun \a -> fun \phi -> fun \u -> VSub a phi u
-wiValue WiInS      = forallI \a -> forallI \phi -> fun \u -> VInc a phi u
-wiValue WiOutS     = forallI \a -> forallI \phi -> forallI \u -> fun \x -> outS a phi u x
-wiValue WiComp     = fun \a -> forallI \phi -> fun \u -> fun \x -> VInc (a @@ VI1) phi (comp a phi u x)
+wiValue WiPartial  = functions [(Ex, "phi"), (Ex, "A")] \[phi, a] -> VPartial phi a
+wiValue WiPartialP = functions [(Ex, "phi"), (Ex, "A")] \[phi, a] -> VPartialP phi a
+wiValue WiSub      = functions [(Ex, "A"), (Ex, "phi"), (Ex, "u")] \[a, phi, u] -> VSub a phi u
+wiValue WiInS      = functions [(Im, "A"), (Im, "phi"), (Ex, "u")] \[a, phi, u] -> incS a phi u
+wiValue WiOutS     = functions [(Im, "A"), (Im, "phi"), (Im, "u"), (Ex, "u0")] \[a, phi, u, x] -> outS a phi u x
+wiValue WiComp     = fun' "A" \a -> forallI \phi -> fun' "u" \u -> fun' "u0" \x -> incS (a @@ VI1) phi (comp a phi u x)
 
 wiValue WiGlue     = fun \a -> forallI \phi -> fun \t -> fun \e -> glueType a phi t e
 wiValue WiGlueElem = forallI \a -> forallI \phi -> forallI \ty -> forallI \eqv -> fun \x -> fun \y -> glueElem a phi ty eqv x y
@@ -109,6 +115,11 @@ line k = VLam P.Ex $ Closure (Bound "i" 0) (k . force)
 fun' :: String -> (Value -> Value) -> Value
 fun' x k = VLam P.Ex $ Closure (Bound (T.pack x) 0) (k . force)
 
+functions :: [(P.Plicity, String)] -> ([Value] -> Value) -> Value
+functions args cont = go args [] where
+  go [] acc = cont (reverse acc)
+  go ((p, x):xs) acc = VLam p $ Closure (Bound (T.pack x) 0) \arg -> go xs (arg:acc)
+
 forallI :: (Value -> Value) -> Value
 forallI k = VLam P.Im $ Closure (Bound "x" 0) (k . force)
 
@@ -130,7 +141,6 @@ forAll' n a b = VPi P.Im a (Closure (Bound (T.pack n) 0) b)
 forAll :: Value -> (Value -> Value) -> Value
 forAll = forAll' "x"
 
-
 wiredInNames :: Map Text WiredIn
 wiredInNames = Map.fromList
   [ ("Pretype", WiPretype)
@@ -153,7 +163,7 @@ wiredInNames = Map.fromList
   , ("Glue", WiGlue)
   , ("glue", WiGlueElem)
   , ("unglue", WiUnglue)
-  
+
   , ("Eq_s", WiSEq)
   , ("refl_s", WiSRefl)
   , ("K_s", WiSK)
@@ -221,21 +231,26 @@ ielim line left right fn i =
 
         _ -> error $ "can't ielim " ++ show (prettyTm (quote fn))
 
-outS :: HasCallStack => NFSort -> NFEndp -> Value -> Value -> Value
+incS :: DebugCallStack => NFSort -> NFEndp -> Value -> Value
+incS _ _ (force -> VNe h (sp Seq.:|> POuc _ _ _))
+  = VNe h sp
+incS a phi u = VInc a phi u
+
+outS :: DebugCallStack => NFSort -> NFEndp -> Value -> Value -> Value
 outS _ (force -> VI1) u _  = u @@ VReflStrict VI VI1
 
 outS _ _phi _ (VInc _ _ x) = x
 outS _ VI0 _  x            = x
 
-outS a phi u  (GluedVl x sp vl) = GluedVl x (sp Seq.:|> POuc a phi u) (outS a phi u vl)
-outS a phi u  (VNe x sp) = VNe x (sp Seq.:|> POuc a phi u)
+outS a phi u (GluedVl x sp vl) = GluedVl x (sp Seq.:|> POuc a phi u) (outS a phi u vl)
+outS a phi u (VNe x sp) = VNe x (sp Seq.:|> POuc a phi u)
 outS a phi u (VSystem fs) = mkVSystem (fmap (outS a phi u) fs)
 outS _ _ _ v             = error $ "can't outS " ++ show (prettyTm (quote v))
 
 -- Composition
-comp :: HasCallStack => NFLine -> NFEndp -> Value -> Value -> Value
+comp :: DebugCallStack => NFLine -> NFEndp -> Value -> Value -> Value
 comp _a VI1 u _a0 =  u @@ VI1 @@ VReflStrict VI VI1
-comp a psi@phi u incA0@(compOutS (a @@ VI0) phi (u @@ VI0) -> a0) =
+comp a psi@phi u incA0@(outS (a @@ VI0) phi (u @@ VI0) -> a0) =
   case force (a @@ VVar name) of
     VPi{} ->
       let
@@ -243,20 +258,21 @@ comp a psi@phi u incA0@(compOutS (a @@ VI0) phi (u @@ VI0) -> a0) =
         dom  i = let VPi _ d _ = force (a @@ i)              in d
         rng  i = let VPi _ _ (Closure _ r) =  force (a @@ i) in r
 
-        y' i y = fill (fun (dom . inot)) VI0 (fun \_ -> fun \_ -> VSystem mempty) (VInc (dom VI0) phi y) i
+        y' i y = fill (fun (dom . inot)) VI0 (fun \_ -> fun \_ -> VSystem mempty) (incS (dom VI0) phi y) i
         ybar i y = y' (inot i) y
        in VLam (plic VI1) . Closure (Bound "x" 0) $ \arg ->
               comp (line \i -> rng i (ybar i arg))
                    phi
                    (system \i isone -> vApp (plic i) (u @@ i @@ isone) (ybar i arg))
-                   (VInc (rng VI0 (ybar VI0 arg)) phi (vApp (plic VI0) a0 (ybar VI0 arg)))
+                   (incS (rng VI0 (ybar VI0 arg)) phi (vApp (plic VI0) a0 (ybar VI0 arg)))
+
     VSigma{} ->
       let
         dom i = let VSigma d _             = force (a @@ i) in d
         rng i = let VSigma _ (Closure _ r) = force (a @@ i) in r
 
-        w i = fill (fun dom) phi (system \i isone -> vProj1 (u @@ i @@ isone)) (VInc (dom VI0) phi (vProj1 a0)) i
-        c2 = comp (fun \x -> rng x (w x)) phi (system \i isone -> vProj2 (u @@ i @@ isone)) (VInc (rng VI0 (w VI0)) phi (vProj2 a0))
+        w i = fill (fun dom) phi (system \i isone -> vProj1 (u @@ i @@ isone)) (incS (dom VI0) phi (vProj1 a0)) i
+        c2 = comp (fun \x -> rng x (w x)) phi (system \i isone -> vProj2 (u @@ i @@ isone)) (incS (rng VI0 (w VI0)) phi (vProj2 a0))
        in
          VPair (w VI1) c2
 
@@ -272,7 +288,7 @@ comp a psi@phi u incA0@(compOutS (a @@ VI0) phi (u @@ VI0) -> a0) =
             (system \i isone -> mkVSystem (Map.fromList [ (phi, ielim (a' VI0) (u' VI0) (v' VI0) (u @@ i @@ isone) j)
                                                         , (j, v' i)
                                                         , (inot j, u' i)]))
-            (VInc (a' VI0 @@ VI0 @@ j) phi (ielim (a' VI0 @@ VI0) (u' VI0) (v' VI0) a0 j))
+            (incS (a' VI0 @@ VI0 @@ j) phi (ielim (a' VI0 @@ VI0) (u' VI0) (v' VI0) a0 j))
 
     VGlueTy _ thePhi theTypes theEquivs ->
       let
@@ -290,8 +306,8 @@ comp a psi@phi u incA0@(compOutS (a @@ VI0) phi (u @@ VI0) -> a0) =
         a0 = unglue (base VI0) (phi VI0) (types VI0) (equivs VI0) b0
 
         del = faceForall phi
-        a1' = comp (line base)  psi (system a) (VInc (base VI0) psi a0)
-        t1' = comp (line (const (types VI0))) psi (line (b @@)) (VInc (base VI0) psi b0)
+        a1' = comp (line base)  psi (system a) (incS (base VI0) psi a0)
+        t1' = comp (line (const (types VI0))) psi (line (b @@)) (incS (base VI0) psi b0)
 
         (omega_st, omega_t, omega_rep) = pres types base equivs psi (b @@) b0
         omega = outS omega_t psi omega_rep omega_st
@@ -309,7 +325,7 @@ comp a psi@phi u incA0@(compOutS (a @@ VI0) phi (u @@ VI0) -> a0) =
           (phi VI1 `ior` psi)
           (system \j _u -> mkVSystem (Map.fromList [ (phi VI1, ielim (base VI1) a1' (vProj1 (equivs VI1 @@ VReflStrict VI VI1)) alpha j)
                                                    , (psi, a VI1 (VReflStrict VI VI1))]))
-          (VInc (base VI1) (phi VI1 `ior` psi) a1')
+          (incS (base VI1) (phi VI1 `ior` psi) a1')
         b1 = glueElem (base VI1) (phi VI1) (types VI1) (equivs VI1) (fun (const t1)) a1
        in b1
 
@@ -321,14 +337,11 @@ comp a psi@phi u incA0@(compOutS (a @@ VI0) phi (u @@ VI0) -> a0) =
       case force a0 of
         VNe (HCon con_type con_name) con_args ->
           VNe (HCon con_type con_name) $ compConArgs makeSetFiller (length args) (a @@) con_type con_args phi u
-        _ -> VComp a phi u (VInc (a @@ VI0) phi a0)
+        _ -> VComp a phi u (incS (a @@ VI0) phi a0)
 
     VNe (HData True name) args -> compHIT name (length args) (a @@) phi u incA0
 
-    VLam{} -> error $ "comp VLam " ++ show (prettyTm (quote a))
-    VSystem map -> mkVSystem (fmap (\x -> comp x psi u incA0) map)
-
-    _ -> VComp a phi u (VInc (a @@ VI0) phi a0)
+    _ -> VComp a phi u (incS (a @@ VI0) phi a0)
   where
     {-# NOINLINE name #-}
     name = unsafePerformIO newName
@@ -350,12 +363,12 @@ compHIT :: HasCallStack => Name -> Int -> (NFEndp -> NFSort) -> NFEndp -> Value
 compHIT name n a phi u a0 =
   case force phi of
     VI1 -> u @@ VI1 @@ VReflStrict VI VI1
-    VI0 | n == 0 -> compOutS (a VI0) phi u a0
-        | otherwise -> transHit name a VI0 (compOutS (a VI0) phi u a0)
+    VI0 | n == 0 -> outS (a VI0) phi u a0
+        | otherwise -> transHit name a VI0 (outS (a VI0) phi u a0)
     x -> go n a x u a0
   where
     go 0 a phi u a0 = VHComp (a VI0) phi u a0
-    go _ a phi u a0 = VHComp (a VI1) phi (system \i n -> transSqueeze name a VI0 (\i -> u @@ i @@ n) i) (transHit name a VI0 (compOutS (a VI1) phi (u @@ VI1 @@ VReflStrict VI VI1) a0))
+    go _ a phi u a0 = VHComp (a VI1) phi (system \i n -> transSqueeze name a VI0 (\i -> u @@ i @@ n) i) (transHit name a VI0 (outS (a VI0) phi (u @@ VI1 @@ VReflStrict VI VI1) a0))
 
 compConArgs :: (Name -> Int -> Value -> t1 -> t2 -> Value -> Value)
             -> Int
@@ -396,13 +409,6 @@ nthArg i (force -> VNe hd s) =
 nthArg i (force -> VSystem vs) = VSystem (fmap (nthArg i) vs)
 nthArg i xs = error $ "can't get " ++ show i ++ "th argument of " ++ show (prettyTm (quote xs))
 
-compOutS :: HasCallStack => NFSort -> NFEndp -> Value -> Value -> Value
-compOutS a b c d = compOutS a b c (force d) where
-  compOutS _ _hi _0 vl@VComp{}   = error $ "unwrapped composition given as input to composition operation is fuckign illegal " ++ show (prettyTm (quote (zonk vl)))
-  compOutS _ _hi _0 vl@VHComp{}  = error $ "unwrapped composition (gay) given as input to composition operation is fuckign illegal " ++ show (prettyTm (quote (zonk vl)))
-  compOutS _ _hi _0 (VInc _ _ x) = x
-  compOutS a phi a0 v = outS a phi a0 v
-
 system :: (Value -> Value -> Value) -> Value
 system k = VLam P.Ex $ Closure (Bound "i" 0) \i -> VLam P.Ex $ Closure (Bound "[i]" 0) \isone -> k i isone
 
@@ -414,18 +420,18 @@ fill a phi u a0 j =
                                                    , (inot j, outS a phi (u @@ VI0) a0)]))
        a0
 
-hComp :: NFSort -> NFEndp -> Value -> Value -> Value
+hComp :: DebugCallStack => NFSort -> NFEndp -> Value -> Value -> Value
 hComp _ (force -> VI1) u _ = u @@ VI1 @@ VReflStrict VI VI1
 hComp a phi u a0 = VHComp a phi u a0
 
-glueType :: NFSort -> NFEndp -> NFPartial -> NFPartial -> Value
+glueType :: DebugCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> Value
 glueType a phi tys eqvs = VGlueTy a phi tys eqvs
 
-glueElem :: NFSort -> NFEndp -> NFPartial -> NFPartial -> NFPartial -> Value -> Value
+glueElem :: DebugCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> NFPartial -> Value -> Value
 glueElem _a (force -> VI1) _tys _eqvs t _vl = t @@ VReflStrict VI VI1
 glueElem a phi tys eqvs t vl = VGlue a phi tys eqvs t vl
 
-unglue   :: HasCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> Value -> Value
+unglue   :: DebugCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> Value -> Value
 unglue _a (force -> VI1) _tys eqvs x = vProj1 (eqvs @@ VReflStrict VI VI1) @@ x
 unglue _a _phi _tys _eqvs (force -> VGlue _ _ _ _ t vl) = outS _a _phi (t @@ VReflStrict VI VI1) vl
 unglue a phi tys eqvs (force -> VSystem fs) = VSystem (fmap (unglue a phi tys eqvs) fs)
@@ -462,18 +468,18 @@ equiv a b = GluedVl (HCon VType (Defined (T.pack "Equiv") (-1))) sp $ exists' "f
   sp = Seq.fromList [ PApp P.Ex a, PApp P.Ex b ]
 
 pres :: (NFEndp -> NFSort) -> (NFEndp -> NFSort) -> (NFEndp -> Value) -> NFEndp -> (NFEndp -> Value) -> Value -> (Value, NFSort, Value)
-pres tyT tyA f phi t t0 = (VInc pathT phi (VLine (tyA VI1) c1 c2 (line path)), pathT, fun $ \u -> VLine (fun (const (tyA VI1))) c1 c2 (fun (const (f VI1 @@ (t VI1 @@ u))))) where
+pres tyT tyA f phi t t0 = (incS pathT phi (VLine (tyA VI1) c1 c2 (line path)), pathT, fun $ \u -> VLine (fun (const (tyA VI1))) c1 c2 (fun (const (f VI1 @@ (t VI1 @@ u))))) where
   pathT = VPath (fun (const (tyA VI1))) c1 c2
-  c1 = comp (line tyA) phi (system \i u -> f i @@ (t i @@ u)) (VInc (tyA VI0) phi (f VI0 @@ t0))
+  c1 = comp (line tyA) phi (system \i u -> f i @@ (t i @@ u)) (incS (tyA VI0) phi (f VI0 @@ t0))
   c2 = f VI1 @@ comp (line tyT) phi (system \i u -> t i @@ u) t0
 
   a0 = f VI0 @@ t0
   v = fill (fun tyT) phi (system \i u -> t i @@ u) t0
 
-  path j = comp (fun tyA) (phi `ior` j) (system \i _ -> f i @@ (v i)) (VInc (tyA VI0) phi a0)
+  path j = comp (fun tyA) (phi `ior` j) (system \i _ -> f i @@ (v i)) (incS (tyA VI0) phi a0)
 
 opEquiv :: HasCallStack => Value -> Value -> Value -> NFEndp -> Value -> Value -> Value -> (Value, NFSort, Value)
-opEquiv aT tT f phi t p a = (VInc ty phi v, ty, fun \u -> VPair (t @@ u) (p @@ u)) where
+opEquiv aT tT f phi t p a = (incS ty phi v, ty, fun \u -> VPair (t @@ u) (p @@ u)) where
   fn = vProj1 f
   ty  = exists' "f" tT \x -> VPath (line (const aT)) a (fn @@ x)
   v   = contr ty (vProj2 f @@ a) phi (\u -> VPair (t @@ u) (p @@ u))
@@ -483,17 +489,17 @@ contr a aC phi u =
   comp (line (const a))
     phi
     (system \i is1 -> ielim (line (const a)) (vProj1 aC) (u is1) (vProj2 aC @@ u is1) i)
-    (VInc a phi (vProj1 aC))
+    (incS a phi (vProj1 aC))
 
 transp :: (NFEndp -> Value) -> Value -> Value
-transp line a0 = comp (fun line) VI0 (system \_ _ -> VSystem mempty) (VInc (line VI0) VI0 a0)
+transp line a0 = comp (fun line) VI0 (system \_ _ -> VSystem mempty) (incS (line VI0) VI0 a0)
 
 gtrans :: (NFEndp -> Value) -> NFEndp -> Value -> Value
-gtrans line phi a0 = comp (fun line) phi (system \_ _ -> mkVSystem (Map.singleton phi a0)) (VInc (line VI0) VI0 a0)
+gtrans line phi a0 = comp (fun line) phi (system \_ _ -> mkVSystem (Map.singleton phi a0)) (incS (line VI0) VI0 a0)
 
 transHit :: Name -> (NFEndp -> Value) -> NFEndp -> Value -> Value
 transHit name line phi x = transHit name line phi (force x) where
-  transHit name line phi (VHComp _ psi u u0) = VHComp (line VI1) psi (system \i j -> transHit name line phi (u @@ i @@ j)) (transHit name line phi (compOutS (line VI0) phi u u0))
+  transHit name line phi (VHComp _ psi u u0) = VHComp (line VI1) psi (system \i j -> transHit name line phi (u @@ i @@ j)) (transHit name line phi (outS (line VI0) phi u u0))
   transHit name line phi (VNe (HCon con_type con_name) spine) | ourType = x' where
     x' = VNe (HCon con_type con_name) $ compConArgs (makeTransFiller name) nargs line con_type spine phi ()
     (_, VNe hd (length -> nargs)) = unPi con_type
@@ -538,17 +544,17 @@ makeEquiv' line' = VPair f $ fun \y -> VPair (fib y) (fun \u -> p (vProj1 u) (vP
 
     f = fun \x -> transp (line @@) x
     g = fun \x -> transp ((line @@) . inot) x
-    u i = fun \x -> fill line VI0 (system \_ _ -> mkVSystem mempty) (VInc a VI0 x) i
-    v i = fun \x -> fill (fun ((line @@) . inot)) VI0 (system \_ _ -> mkVSystem mempty) (VInc a VI1 x) (inot i)
+    u i = fun \x -> fill line VI0 (system \_ _ -> mkVSystem mempty) (incS a VI0 x) i
+    v i = fun \x -> fill (fun ((line @@) . inot)) VI0 (system \_ _ -> mkVSystem mempty) (incS a VI1 x) (inot i)
 
     fib y = VPair (g @@ y) (VLine b y (f @@ (g @@ y)) (fun (theta0 y VI1)))
-    theta0 y i j = fill line (ior j (inot j)) (system \i _ -> mkVSystem (Map.fromList [(j, v i @@ y), (inot j, u i @@ (g @@ y))])) (VInc a (ior j (inot j)) (g @@ y)) i
+    theta0 y i j = fill line (ior j (inot j)) (system \i _ -> mkVSystem (Map.fromList [(j, v i @@ y), (inot j, u i @@ (g @@ y))])) (incS a (ior j (inot j)) (g @@ y)) i
     theta1 x beta y i j =
       fill (fun ((line @@) . inot))
            (ior j (inot j))
            (system \i _ -> mkVSystem (Map.fromList [ (inot j, v (inot i) @@ y)
                                                    , (j, u (inot i) @@ x)]))
-           (VInc b (ior j (inot j)) (ielim b y (f @@ x) beta y))
+           (incS b (ior j (inot j)) (ielim b y (f @@ x) beta y))
            (inot i)
     omega x beta y = theta1 x beta y VI0
     delta x beta y j k = comp line (ior k (ior (inot k) (ior j (inot j))))
@@ -556,7 +562,7 @@ makeEquiv' line' = VPair f $ fun \y -> VPair (fib y) (fun \u -> p (vProj1 u) (vP
                                                                            , (k, theta1 x beta y i j)
                                                                            , (inot j, v i @@ y)
                                                                            , (j, u i @@ omega x beta y k)]))
-                                   (VInc a (ior k (ior (inot k) (ior j (inot j)))) (omega x beta y (iand j k)))
+                                   (incS a (ior k (ior (inot k) (ior j (inot j)))) (omega x beta y (iand j k)))
     p x beta y = VLine (exists a \x -> VPath b y (f @@ x)) (fib y) (VPair x beta) $ fun \k ->
       VPair (omega x beta y k) (VLine (VPath b y (f @@ x)) (vProj2 (fib y)) beta $ fun \j -> delta x beta y j k)
 
@@ -571,7 +577,7 @@ idEquiv a = VPair idfun idisequiv where
 
   id_fiber y = VPair y (VLine a y y (fun (const y)))
 
-strictK :: Value -> Value -> Value -> Value -> Value -> Value
+strictK :: DebugCallStack => Value -> Value -> Value -> Value -> Value -> Value
 strictK _ _ _ pr (VReflStrict _ _) = pr
 strictK a x bigp pr (VNe h sp) = VNe h (sp Seq.:|> PK a x bigp pr)
 strictK a x bigp pr (VCase env rng sc cases) = VCase env rng sc (map (projIntoCase func) cases) where
@@ -579,7 +585,7 @@ strictK a x bigp pr (VCase env rng sc cases) = VCase env rng sc (map (projIntoCa
 strictK a x bigp pr (GluedVl h sp vl) = GluedVl h (sp Seq.:|> PK a x bigp pr) (strictK a x bigp pr vl)
 strictK _ _ _ _r eq = error $ "can't K " ++ show (prettyVl eq)
 
-strictJ :: Value -> Value -> Value -> Value -> Value -> Value -> Value
+strictJ :: DebugCallStack => Value -> Value -> Value -> Value -> Value -> Value -> Value
 strictJ _a _x _bigp pr _ (VReflStrict _ _) = pr
 strictJ a x bigp pr y (VNe h sp) = VNe h (sp Seq.:|> PJ a x bigp pr y)
 strictJ a x bigp pr y (VCase env rng sc cases) = VCase env rng sc (map (projIntoCase func) cases) where
@@ -592,4 +598,4 @@ projIntoCase fun (pat, nLams, term) = (pat, nLams, go nLams term) where
   go 0 x = fun x
   go n (Lam p x r) = Lam p x (go (n - 1) r)
   go n (PathIntro l a b r) = PathIntro l a b (go (n - 1) r)
-  go _ x = x
\ No newline at end of file
+  go _ x = x
diff --git a/src/Elab/WiredIn.hs-boot b/src/Elab/WiredIn.hs-boot
index bfca174..9c0890d 100644
--- a/src/Elab/WiredIn.hs-boot
+++ b/src/Elab/WiredIn.hs-boot
@@ -1,8 +1,9 @@
-module Elab.WiredIn where
-
-import GHC.Stack.Types
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE KindSignatures #-}
+module Elab.WiredIn (wiType, wiValue, iand, ior, inot, ielim, incS, outS, comp, fill, hComp, glueType, glueElem, unglue, fun, system, strictK, strictJ, projIntoCase) where
 
 import Syntax
+import Debug (DebugCallStack)
 
 wiType  :: WiredIn -> NFType
 wiValue :: WiredIn -> NFType
@@ -11,19 +12,20 @@ iand, ior :: NFEndp -> NFEndp -> NFEndp
 inot      :: NFEndp -> NFEndp
 ielim     :: NFSort -> Value -> Value -> Value -> NFEndp -> Value
 
-outS :: HasCallStack => NFSort -> NFEndp -> Value -> Value -> Value
-comp :: HasCallStack => NFLine -> NFEndp -> Value -> Value -> Value
-fill :: HasCallStack => NFLine -> NFEndp -> Value -> Value -> Value -> Value
-hComp :: NFSort -> NFEndp -> Value -> Value -> Value
+incS   :: DebugCallStack => NFSort -> NFEndp -> Value -> Value
+outS  :: DebugCallStack => NFSort -> NFEndp -> Value -> Value -> Value
+comp  :: DebugCallStack => NFLine -> NFEndp -> Value -> Value -> Value
+fill  :: DebugCallStack => NFLine -> NFEndp -> Value -> Value -> Value -> Value
+hComp :: DebugCallStack => NFSort -> NFEndp -> Value -> Value -> Value
 
-glueType :: NFSort -> NFEndp -> NFPartial -> NFPartial -> Value
-glueElem :: NFSort -> NFEndp -> NFPartial -> NFPartial -> NFPartial -> Value -> Value
-unglue   :: HasCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> Value -> Value
+glueType :: DebugCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> Value
+glueElem :: DebugCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> NFPartial -> Value -> Value
+unglue   :: DebugCallStack => NFSort -> NFEndp -> NFPartial -> NFPartial -> Value -> Value
 
 fun :: (Value -> Value) -> Value
 system :: (Value -> Value -> Value) -> Value
 
-strictK :: NFSort -> Value -> NFSort -> Value -> Value -> Value
-strictJ :: NFSort -> Value -> NFSort -> Value -> Value -> Value -> Value
+strictK :: DebugCallStack => NFSort -> Value -> NFSort -> Value -> Value -> Value
+strictJ :: DebugCallStack => NFSort -> Value -> NFSort -> Value -> Value -> Value -> Value
 
 projIntoCase :: (Term -> Term) -> (Term, Int, Term) -> (Term, Int, Term)
\ No newline at end of file
diff --git a/src/Main.hs b/src/Main.hs
index 42f05a6..45eb248 100644
--- a/src/Main.hs
+++ b/src/Main.hs
@@ -22,7 +22,7 @@ import Data.Foldable
 import Data.Maybe
 import Data.IORef
 
-import Debug.Trace
+import Debug (traceDocM)
 
 import Elab.Monad hiding (switch)
 import Elab.WiredIn
@@ -64,7 +64,7 @@ evalArgExpr env str =
     Right e ->
       flip runElab env (do
         (e, _) <- infer e
-        liftIO . putStrLn . show . prettyTm . quote . zonk =<< Elab.Eval.eval e)
+        liftIO . T.putStrLn . render . prettyTm . quote . zonk =<< Elab.Eval.eval e)
       `catch` \e -> do
           displayExceptions' inp (e :: SomeException)
     Left e -> liftIO $ print e
@@ -267,5 +267,5 @@ dumpTokens = do
   case tokenClass t of
     TokEof -> pure ()
     _ -> do
-      traceM (show t)
+      traceDocM (viaShow t)
       dumpTokens
diff --git a/src/Syntax.hs b/src/Syntax.hs
index 04fbcd0..9233de2 100644
--- a/src/Syntax.hs
+++ b/src/Syntax.hs
@@ -201,7 +201,8 @@ quoteWith names (VNe h sp) = foldl goSpine (goHead h) sp where
         case Map.lookup VI1 f of
           Just vl -> constantly (length sp) vl
           _ -> PCon (quote sys) v
-      _ -> PCon (quote sys) v
+      VLam{} -> PCon (quote sys) v
+      s -> constantly (length sp) s
   goHead (HData x v) = Data x v
 
   goSpine t (PApp p v) = App p t (quoteWith names v)
@@ -220,6 +221,9 @@ quoteWith names (GluedVl _ Seq.Empty x) = quoteWith names x
 quoteWith names (GluedVl h sp (VLam p (Closure n k))) =
   quoteWith names (VLam p (Closure n (\a -> GluedVl h (sp Seq.:|> PApp p a) (k a))))
 
+quoteWith names (GluedVl h sp (VLine ty x y (VLam p (Closure n k)))) =
+  quoteWith names (VLine ty x y (VLam p (Closure n (\a -> GluedVl h (sp Seq.:|> PIElim ty x y a) (k a)))))
+
 quoteWith names (GluedVl h sp vl)
   | GluedVl _ _ inner <- vl = quoteWith names (GluedVl h sp inner)
   | alwaysShort vl = quoteWith names vl
@@ -328,4 +332,4 @@ unPi :: Value -> ([(Plicity, Value)], Value)
 unPi (VPi p d (Closure n k)) =
   let (a, x) = unPi (k (VVar n))
    in ((p, d):a, x)
-unPi x = ([], x)
\ No newline at end of file
+unPi x = ([], x)
diff --git a/src/Syntax/Pretty.hs b/src/Syntax/Pretty.hs
index 0b26feb..5a40f55 100644
--- a/src/Syntax/Pretty.hs
+++ b/src/Syntax/Pretty.hs
@@ -2,9 +2,9 @@
 {-# OPTIONS_GHC -Wno-orphans #-}
 {-# LANGUAGE ViewPatterns #-}
 {-# LANGUAGE NamedFieldPuns #-}
+{-# LANGUAGE CPP #-}
 module Syntax.Pretty where
 
-
 import qualified Data.Map.Strict as Map
 import qualified Data.Text.Lazy as L
 import qualified Data.Text as T
@@ -21,8 +21,8 @@ import Syntax
 instance Pretty Name where
   pretty = pretty . getNameText
 
-prettyTm :: Term -> Doc AnsiStyle
-prettyTm = go True 0 where
+prettyTm' :: Bool -> Term -> Doc AnsiStyle
+prettyTm' implicits = go True 0 where
   go t p =
     \case
       Ref v    -> pretty v
@@ -30,12 +30,16 @@ prettyTm = go True 0 where
       PCon _ v -> keyword $ pretty v
       Data _ v -> keyword $ pretty v
 
-      App Im f _ -> go t p f
+      App Im f x
+        | implicits -> parenIf (p >= arg_prec) $
+            go False fun_prec f
+        <+> braces (go False 0 x)
+        | otherwise -> go t p f
       App Ex f x ->
         parenIf (p >= arg_prec) $
               go False fun_prec f
           <+> group (go False arg_prec x)
-      
+
       Lam Ex v (App Ex f (Ref v')) | v == v' -> instead f
       Lam i v t ->
         let
@@ -50,8 +54,8 @@ prettyTm = go True 0 where
           parenIf (p >= fun_prec) . group $
             pretty '\\' <> hsep (map (\(i, v) -> braceIf (i == Im) (pretty v)) as)
               <+> arrow
-              <+> nest 2 (go False 0 b)
-      
+              <+> nest 2 (align (go False 0 b))
+
       Pi _ (T.unpack . getNameText -> "_") d r ->
         parenIf (p >= fun_prec) $
           group (go False dom_prec d)
@@ -67,12 +71,12 @@ prettyTm = go True 0 where
           parenIf (p >= fun_prec) $
             plic i (pretty x <+> colon <+> go False 0 d)
               <> c <> go t 0 r
-      
+
       Let binds body ->
         parenIf (p >= fun_prec) $
           align $ keyword (pretty "let")
               <> line
-              <> indent 2 (prettyBinds binds)
+              <> indent 2 (prettyBinds False binds)
               <> keyword (pretty "in")
               <+> go False 0 body
 
@@ -104,7 +108,7 @@ prettyTm = go True 0 where
 
       PathP _ x y -> parenIf (p >= arg_prec) $
         go False 0 x <+> operator (pretty "≡") <+> go False 0 y
-      
+
       IElim _a _x _y f i -> instead (App Ex f i)
       PathIntro _a _x _y f -> instead f
 
@@ -116,18 +120,18 @@ prettyTm = go True 0 where
         let
           face (f, t) = go False 0 f <+> operator (pretty "=>") <+> align (go False 0 t)
         in
-          braces (line <> indent 2 (vsep (map face (Map.toList fs))) <> line)
+          braces (line <> indent 2 (align (vsep (punctuate comma (map face (Map.toList fs))))) <> line)
 
       Sub a phi u -> apps (con "Sub") [(Ex, a), (Ex, phi), (Ex, u)]
-      Inc a phi u -> apps (con "inS") [(Ex, a), (Ex, phi), (Ex, u)]
-      Ouc a phi u a0 -> apps (con "outS") [(Ex, a), (Ex, phi), (Ex, u), (Ex, a0)]
+      Inc a phi u -> apps (con "inS") [(Im, a), (Im, phi), (Ex, u)]
+      Ouc a phi u a0 -> apps (con "outS") [(Im, a), (Im, phi), (Im, u), (Ex, a0)]
 
       GlueTy a phi t e -> apps (con "primGlue") [(Ex, a), (Ex, phi), (Ex, t), (Ex, e)]
       Glue _a _phi _ty _e t im -> apps (con "glue") [(Ex, t), (Ex, im)]
       Unglue _a _phi _ty _e t -> apps (con "unglue") [(Ex, t)]
 
-      Comp a phi u a0 -> apps (con "comp") [(Ex, a), (Ex, phi), (Ex, u), (Ex, a0)]
-      HComp a phi u a0 -> apps (con "hcomp") [(Ex, a), (Ex, phi), (Ex, u), (Ex, a0)]
+      Comp a phi u a0 -> apps (con "comp") [(Ex, a), (Im, phi), (Ex, u), (Ex, a0)]
+      HComp a phi u a0 -> apps (con "hcomp") [(Im, a), (Im, phi), (Ex, u), (Ex, a0)]
 
       Case _ t cs ->
         let
@@ -162,7 +166,7 @@ prettyTm = go True 0 where
                   | otherwise = x
 
       con x = Con (Bound (T.pack x) 0)
-      
+
       plic = \case
         Ex -> parens
         Im -> braces
@@ -170,24 +174,43 @@ prettyTm = go True 0 where
       arrow = operator (pretty "->")
 
       instead = go t p
-      apps :: Term -> [(Plicity, Term)] -> Doc AnsiStyle 
+      apps :: Term -> [(Plicity, Term)] -> Doc AnsiStyle
       apps f xs = instead (foldl (\f (p, x) -> App p f x) f xs)
 
-prettyBinds :: [(Name, Term, Term)] -> Doc AnsiStyle
-prettyBinds [] = mempty
-prettyBinds ((x, ty, tm):bs) =
-     pretty x <+> colon <+> align (prettyTm ty)
+prettyBinds :: Bool -> [(Name, Term, Term)] -> Doc AnsiStyle
+prettyBinds _ [] = mempty
+prettyBinds imp ((x, ty, tm):bs) =
+     pretty x <+> colon <+> align (prettyTm' imp ty)
   <> line
-  <> pretty x <+> equals <+> align (prettyTm tm)
+  <> pretty x <+> equals <+> align (prettyTm' imp tm)
   <> line
-  <> prettyBinds bs
+  <> prettyBinds imp bs
 
-showFace :: Map Head Bool -> Doc AnsiStyle
-showFace = hsep . map go . Map.toList where
-  go (h, b) = parens $ prettyTm (quote (VNe h mempty)) <+> operator (pretty "=") <+> pretty (if b then "i1" else "i0")
+showFace :: Bool -> Map Head Bool -> Doc AnsiStyle
+showFace imp = hsep . map go . Map.toList where
+  go (h, b) = parens $ prettyTm' imp (quote (VNe h mempty)) <+> operator (pretty "=") <+> pretty (if b then "i1" else "i0")
+
+prettyVl' :: Bool -> Value -> Doc AnsiStyle
+prettyVl' b = prettyTm' b . quote
+
+instance Pretty Term where
+  pretty = unAnnotate . prettyTm
+
+prettyTm :: Term -> Doc AnsiStyle
+prettyTm = prettyTm' printImplicits
+
+instance Pretty Value where
+  pretty = unAnnotate . prettyVl
 
 prettyVl :: Value -> Doc AnsiStyle
-prettyVl = prettyTm . quote
+prettyVl = prettyVl' printImplicits
+
+printImplicits :: Bool
+#if defined(RELEASE)
+printImplicits = False
+#else
+printImplicits = True
+#endif
 
 render :: Doc AnsiStyle -> Text
 render = L.toStrict . renderLazy . layoutSmart defaultLayoutOptions
@@ -203,4 +226,4 @@ keyword :: Doc AnsiStyle -> Doc AnsiStyle
 keyword = annotate (color Magenta)
 
 operator :: Doc AnsiStyle -> Doc AnsiStyle
-operator = annotate (color Yellow)
\ No newline at end of file
+operator = annotate (color Yellow)