diff --git a/lib/Quox/Syntax/Term/Split.idr b/lib/Quox/Syntax/Term/Split.idr
index 091b227..621ecd8 100644
--- a/lib/Quox/Syntax/Term/Split.idr
+++ b/lib/Quox/Syntax/Term/Split.idr
@@ -3,84 +3,50 @@ module Quox.Syntax.Term.Split
 import Quox.Syntax.Term.Base
 import Quox.Syntax.Term.Subst
 
-import Data.So
+import Quox.No
 import Data.Vect
 
 %default total
 
 
-public export
-data IsLam : Pred $ Term {} where ItIsLam : IsLam $ Lam x body
-
 public export %inline
-isLam : Dec1 IsLam
-isLam (TYPE _)   = No $ \case _ impossible
-isLam (Pi {})    = No $ \case _ impossible
-isLam (Lam {})   = Yes ItIsLam
-isLam (Eq {})    = No $ \case _ impossible
-isLam (DLam {})  = No $ \case _ impossible
-isLam (E _)      = No $ \case _ impossible
-isLam (CloT {})  = No $ \case _ impossible
-isLam (DCloT {}) = No $ \case _ impossible
+isLam : Term {} -> Bool
+isLam (Lam {}) = True
+isLam _        = False
 
 public export
 0 NotLam : Pred $ Term {}
-NotLam = Not . IsLam
+NotLam = No . isLam
 
 
-public export
-data IsDLam : Pred $ Term {} where ItIsDLam : IsDLam $ DLam i body
-
 public export %inline
-isDLam : Dec1 IsDLam
-isDLam (TYPE _)   = No $ \case _ impossible
-isDLam (Pi {})    = No $ \case _ impossible
-isDLam (Eq {})    = No $ \case _ impossible
-isDLam (Lam {})   = No $ \case _ impossible
-isDLam (DLam {})  = Yes ItIsDLam
-isDLam (E _)      = No $ \case _ impossible
-isDLam (CloT {})  = No $ \case _ impossible
-isDLam (DCloT {}) = No $ \case _ impossible
+isDLam : Term {} -> Bool
+isDLam (DLam {}) = True
+isDLam _         = False
 
 public export
 0 NotDLam : Pred $ Term {}
-NotDLam = Not . IsDLam
+NotDLam = No . isDLam
 
 
-public export
-data IsApp : Pred $ Elim {} where ItIsApp : IsApp $ f :@ s
-
 public export %inline
-isApp : Dec1 IsApp
-isApp (F _)      = No $ \case _ impossible
-isApp (B _)      = No $ \case _ impossible
-isApp (_ :@ _)   = Yes ItIsApp
-isApp (_ :% _)   = No $ \case _ impossible
-isApp (_ :# _)   = No $ \case _ impossible
-isApp (CloE {})  = No $ \case _ impossible
-isApp (DCloE {}) = No $ \case _ impossible
+isApp : Elim {} -> Bool
+isApp (_ :@ _) = True
+isApp _        = False
 
 public export
 0 NotApp : Pred $ Elim {}
-NotApp = Not . IsApp
+NotApp = No . isApp
 
 
-public export
-data IsDApp : Pred $ Elim {} where ItIsDApp : IsDApp $ f :% d
-
 public export %inline
-isDApp : Dec1 IsDApp
-isDApp (F _)      = No $ \case _ impossible
-isDApp (B _)      = No $ \case _ impossible
-isDApp (_ :@ _)   = No $ \case _ impossible
-isDApp (_ :% _)   = Yes ItIsDApp
-isDApp (_ :# _)   = No $ \case _ impossible
-isDApp (CloE {})  = No $ \case _ impossible
-isDApp (DCloE {}) = No $ \case _ impossible
+isDApp : Elim {} -> Bool
+isDApp (_ :% _) = True
+isDApp _        = False
 
 public export
 0 NotDApp : Pred $ Elim {}
-NotDApp = Not . IsDApp
+NotDApp = No . isDApp
 
 
 infixl 9 :@@
@@ -98,9 +64,9 @@ record GetArgs q d n where
 
 export
 getArgs' : Elim q d n -> List (Term q d n) -> GetArgs q d n
-getArgs' fun args with (isApp fun)
-  getArgs' (f :@ a) args | Yes _  = getArgs' f (a :: args)
-  getArgs' fun args      | No  no = GotArgs {fun, args, notApp = no}
+getArgs' fun args = case nchoose $ isApp fun of
+  Left  y => let f :@ a = fun in getArgs' f (a :: args)
+  Right n => GotArgs {fun, args, notApp = n}
 
 ||| splits an application into its head and arguments. if it's not an
 ||| application then the list is just empty
@@ -124,9 +90,9 @@ record GetDArgs q d n where
 
 export
 getDArgs' : Elim q d n -> List (Dim d) -> GetDArgs q d n
-getDArgs' fun args with (isDApp fun)
-  getDArgs' (f :% a) args | Yes yes = getDArgs' f (a :: args)
-  getDArgs' fun args      | No  no  = GotDArgs {fun, args, notDApp = no}
+getDArgs' fun args = case nchoose $ isDApp fun of
+  Left  y => let f :% d = fun in getDArgs' f (d :: args)
+  Right n => GotDArgs {fun, args, notDApp = n}
 
 ||| splits a dimension application into its head and arguments. if it's not an
 ||| d application then the list is just empty
@@ -172,11 +138,10 @@ export
 getLams' : forall lams, rest.
            Vect lams Name -> Term q d rest -> (0 eq : lams + n = rest) ->
            GetLams q d n
-getLams' xs s eq with (isLam s)
-  getLams' xs (Lam x sbody) Refl | Yes _ =
-    let body = assert_smaller (Lam x sbody) sbody.term in
-    getLams' (x :: xs) body Refl
-  getLams' xs s eq | No no = GotLams xs s eq no
+getLams' xs s Refl = case nchoose $ isLam s of
+  Left  y => let Lam x body = s in
+             getLams' (x :: xs) (assert_smaller s body.term) Refl
+  Right n => GotLams xs s Refl n
 
 export %inline
 getLams : Term q d n -> GetLams q d n
@@ -196,11 +161,10 @@ export
 getDLams' : forall lams, rest.
            Vect lams Name -> Term q rest n -> (0 eq : lams + d = rest) ->
            GetDLams q d n
-getDLams' is s eq with (isDLam s)
-  getDLams' is (DLam i sbody) Refl | Yes _ =
-    let body = assert_smaller (DLam i sbody) sbody.term in
-    getDLams' (i :: is) body Refl
-  getDLams' is s eq | No no = GotDLams is s eq no
+getDLams' is s Refl = case nchoose $ isDLam s of
+  Left  y => let DLam i body = s in
+             getDLams' (i :: is) (assert_smaller s body.term) Refl
+  Right n => GotDLams is s Refl n
 
 export %inline
 getDLams : Term q d n -> GetDLams q d n