diff --git a/src/lib/Builder.hs b/src/lib/Builder.hs
index 8e7563511..5f1c372de 100644
--- a/src/lib/Builder.hs
+++ b/src/lib/Builder.hs
@@ -786,8 +786,8 @@ buildMap :: (Emits n, ScopableBuilder r m)
          -> (forall l. (Emits l, DExt n l) => Atom r l -> m l (Atom r l))
          -> m n (Atom r n)
 buildMap xs f = do
-  TabTy d (_:>t) _ <- return $ getType xs
-  buildFor noHint Fwd (IxType t d) \i ->
+  TabPi t <- return $ getType xs
+  buildFor noHint Fwd (tabIxType t) \i ->
     tabApp (sink xs) (Var i) >>= f
 
 unzipTab :: (Emits n, Builder r m) => Atom r n -> m n (Atom r n, Atom r n)
@@ -857,8 +857,8 @@ zeroAt ty = liftEmitBuilder $ go ty where
   go = \case
    BaseTy bt  -> return $ Con $ Lit $ zeroLit bt
    ProdTy tys -> ProdVal <$> mapM go tys
-   TabTy d (b:>t) bodyTy -> buildFor (getNameHint b) Fwd (IxType t d) \i ->
-     go =<< applySubst (b @> SubstVal (Var i)) bodyTy
+   TabPi tabPi -> buildFor (getNameHint tabPi) Fwd (tabIxType tabPi) \i ->
+     go =<< instantiateTabPiTy (sink tabPi) (Var i)
    _ -> unreachable
   zeroLit bt = case bt of
     Scalar Float64Type -> Float64Lit 0.0
@@ -902,8 +902,8 @@ tangentBaseMonoidFor ty = do
 addTangent :: (Emits n, SBuilder m) => SAtom n -> SAtom n -> m n (SAtom n)
 addTangent x y = do
   case getType x of
-    TabTy d (b:>t) _  ->
-      liftEmitBuilder $ buildFor (getNameHint b) Fwd (IxType t d) \i -> do
+    TabPi t ->
+      liftEmitBuilder $ buildFor (getNameHint t) Fwd (tabIxType t) \i -> do
         bindM2 addTangent (tabApp (sink x) (Var i)) (tabApp (sink y) (Var i))
     TC con -> case con of
       BaseType (Scalar _) -> emitOp $ BinOp FAdd x y
diff --git a/src/lib/CheapReduction.hs b/src/lib/CheapReduction.hs
index 9a35c2ed5..4c42bbda2 100644
--- a/src/lib/CheapReduction.hs
+++ b/src/lib/CheapReduction.hs
@@ -15,7 +15,7 @@ module CheapReduction
   , unwrapLeadingNewtypesType, wrapNewtypesData, liftSimpAtom, liftSimpType
   , liftSimpFun, makeStructRepVal, NonAtomRenamer (..), Visitor (..), VisitGeneric (..)
   , visitAtomPartial, visitTypePartial, visitAtomDefault, visitTypeDefault, Visitor2
-  , visitBinders, visitPiDefault, visitAlt, toAtomVar, instantiatePiTy
+  , visitBinders, visitPiDefault, visitAlt, toAtomVar, instantiatePiTy, instantiateTabPiTy
   , bindersToVars, bindersToAtoms)
   where
 
@@ -474,6 +474,10 @@ instantiatePiTy :: (EnvReader m, IRRep r) => PiType r n -> [Atom r n] -> m n (Ef
 instantiatePiTy (PiType bs effTy) xs = do
   applySubst (bs @@> (SubstVal <$> xs)) effTy
 
+instantiateTabPiTy :: (EnvReader m, IRRep r) => TabPiType r n -> Atom r n -> m n (Type r n)
+instantiateTabPiTy (TabPiType _ b resultTy) x = do
+  applySubst (b @> SubstVal x) resultTy
+
 -- Returns a representation type (type of an TypeCon-typed Newtype payload)
 -- given a list of instantiated DataConDefs.
 dataDefRep :: DataConDefs n -> CType n
diff --git a/src/lib/CheckType.hs b/src/lib/CheckType.hs
index 671067d04..47cf2df19 100644
--- a/src/lib/CheckType.hs
+++ b/src/lib/CheckType.hs
@@ -788,11 +788,7 @@ checkTabApp ty (i:rest) = do
   resultTy' <- applySubst (b@>SubstVal i') resultTy
   checkTabApp resultTy' rest
 
-checkArgTys
-  :: (Typer m r, SubstB AtomSubstVal b, BindsNames b, BindsOneAtomName r b, IRRep r)
-  => Nest b o o'
-  -> [Atom r o]
-  -> m i o ()
+checkArgTys :: (Typer m r, IRRep r) => Nest (Binder r) o o' -> [Atom r o] -> m i o ()
 checkArgTys Empty [] = return ()
 checkArgTys (Nest b bs) (x:xs) = do
   dropSubst $ x |: binderType b
@@ -930,15 +926,14 @@ checkedInstantiateTyConDef (TyConDef _ _ bs cons) (TyConParams _ xs) = do
   checkedApplyNaryAbs (Abs bs cons) xs
 
 checkedApplyNaryAbs
-  :: forall b r e o m
-  .  ( BindsOneAtomName r b, EnvReader m, Fallible1 m, SinkableE e
-     , SubstE AtomSubstVal e, IRRep r, SubstB AtomSubstVal b)
-  => Abs (Nest b) e o -> [Atom r o] -> m o (e o)
+  :: forall r e o m
+  .  ( EnvReader m, Fallible1 m, SinkableE e , SubstE AtomSubstVal e, IRRep r)
+  => Abs (Nest (Binder r)) e o -> [Atom r o] -> m o (e o)
 checkedApplyNaryAbs (Abs bsTop e) xsTop = do
   go (EmptyAbs bsTop) xsTop
   applySubst (bsTop@@>(SubstVal<$>xsTop)) e
  where
-   go :: EmptyAbs (Nest b) o -> [Atom r o] -> m o ()
+   go :: EmptyAbs (Nest (Binder r)) o -> [Atom r o] -> m o ()
    go (Abs Empty UnitE) [] = return ()
    go (Abs (Nest b bs) UnitE) (x:xs) = do
      checkAlphaEq (binderType b) (getType x)
diff --git a/src/lib/Export.hs b/src/lib/Export.hs
index 42dcc7ba1..f7ab3184d 100644
--- a/src/lib/Export.hs
+++ b/src/lib/Export.hs
@@ -175,8 +175,8 @@ parseTabTy = go []
       NewtypeTyCon Nat    -> return $ Just $ RectContArrayPtr IdxRepScalarBaseTy shape
       TabTy d (b:>ixty) a -> do
         maybeN <- case IxType ixty d of
-          (IxType (NewtypeTyCon (Fin n)) _) -> return $ Just n
-          (IxType _ (IxDictRawFin n)) -> return $ Just n
+          IxType (NewtypeTyCon (Fin n)) _ -> return $ Just n
+          IxType _ (IxDictRawFin n) -> return $ Just n
           _ -> return Nothing
         maybeDim <- case maybeN of
           Just (Var v)    -> do
diff --git a/src/lib/Imp.hs b/src/lib/Imp.hs
index 1a2ab54dd..bfb73537c 100644
--- a/src/lib/Imp.hs
+++ b/src/lib/Imp.hs
@@ -366,16 +366,16 @@ toImpRefOp refDest' m = do
           ans <- liftBuilderImp $ emitBlock (sink body')
           storeAtom accDest ans
         False -> case accTy of
-          TabTy d (b:>t) eltTy -> do
-            let ixTy = IxType t d
+          TabPi t -> do
+            let ixTy = tabIxType t
             n <- indexSetSizeImp ixTy
             emitLoop noHint Fwd n \i -> do
               idx <- unsafeFromOrdinalImp (sink ixTy) i
               xElt <- liftBuilderImp $ tabApp (sink x) (sink idx)
               yElt <- liftBuilderImp $ tabApp (sink y) (sink idx)
-              eltTy' <- applySubst (b@>SubstVal idx) eltTy
+              eltTy <- instantiateTabPiTy (sink t) idx
               ithDest <- indexDest (sink accDest) idx
-              liftMonoidCombine ithDest eltTy' (sink bc) xElt yElt
+              liftMonoidCombine ithDest eltTy (sink bc) xElt yElt
           _ -> error $ "Base monoid type mismatch: can't lift " ++
                  pprint baseTy ++ " to " ++ pprint accTy
 
@@ -578,15 +578,15 @@ toImpTypedHof (TypedHof (EffTy _ resultTy') hof) = do
         alphaEq xTy accTy >>= \case
           True -> storeAtom accDest x
           False -> case accTy of
-            TabTy d (b:>t) eltTy -> do
-              let ixTy = IxType t d
+            TabPi t -> do
+              let ixTy = tabIxType t
               n <- indexSetSizeImp ixTy
               emitLoop noHint Fwd n \i -> do
                 idx <- unsafeFromOrdinalImp (sink ixTy) i
                 x' <- sinkM x
-                eltTy' <- applySubst (b@>SubstVal idx) eltTy
+                eltTy <- instantiateTabPiTy (sink t) idx
                 ithDest <- indexDest (sink accDest) idx
-                liftMonoidEmpty ithDest eltTy' x'
+                liftMonoidEmpty ithDest eltTy x'
             _ -> error $ "Base monoid type mismatch: can't lift " ++
                   pprint xTy ++ " to " ++ pprint accTy
 
@@ -1002,11 +1002,11 @@ buildGarbageVal ty =
 -- === Operations on dests ===
 
 indexDest :: Emits n => Dest n -> SAtom n -> SubstImpM i n (Dest n)
-indexDest (Dest destValTy@(TabTy d (b:>t) eltTy) tree) i = do
-  eltTy' <- applySubst (b@>SubstVal i) eltTy
-  ord <- ordinalImp (IxType t d) i
-  leafTys <- typeToTree destValTy
-  Dest eltTy' <$> forM (zipTrees leafTys tree) \(leafTy, ptr) -> do
+indexDest (Dest (TabPi tabTy) tree) i = do
+  eltTy <- instantiateTabPiTy tabTy i
+  ord <- ordinalImp (tabIxType tabTy) i
+  leafTys <- typeToTree $ TabPi tabTy
+  Dest eltTy <$> forM (zipTrees leafTys tree) \(leafTy, ptr) -> do
     BufferType ixStruct _ <- return $ getRefBufferType leafTy
     offset <- computeOffsetImp ixStruct ord
     impOffset ptr offset
@@ -1026,10 +1026,10 @@ indexRepValParam :: Emits n
   => SRepVal n -> SAtom n -> (SType n -> SType n)
   -> (IExpr n -> SubstImpM i n (IExpr n))
   -> SubstImpM i n (SRepVal n)
-indexRepValParam (RepVal tabTy@(TabPi (TabPiType d (b:>t) eltTy)) vals) i tyFunc func = do
-  eltTy' <- applySubst (b@>SubstVal i) eltTy
-  ord <- ordinalImp (IxType t d) i
-  leafTys <- typeToTree tabTy
+indexRepValParam (RepVal (TabPi tabTy) vals) i tyFunc func = do
+  eltTy <- instantiateTabPiTy tabTy i
+  ord <- ordinalImp (tabIxType tabTy) i
+  leafTys <- typeToTree (TabPi tabTy)
   vals' <- forM (zipTrees leafTys vals) \(leafTy, ptr) -> do
     BufferPtr (BufferType ixStruct _) <- return $ getIExprInterpretation leafTy
     offset <- computeOffsetImp ixStruct ord
@@ -1041,7 +1041,7 @@ indexRepValParam (RepVal tabTy@(TabPi (TabPiType d (b:>t) eltTy)) vals) i tyFunc
       _                       -> func ptr'
   -- `func` may have changed the types of the `vals'`.  The caller must also
   -- supply `tyFunc` to reflect that change in the SType.
-  return $ RepVal (tyFunc eltTy') vals'
+  return $ RepVal (tyFunc eltTy) vals'
 indexRepValParam _ _ _ _ = error "expected table type"
 {-# INLINE indexRepValParam #-}
 
diff --git a/src/lib/Lower.hs b/src/lib/Lower.hs
index 8eaf8cbeb..bce5b8050 100644
--- a/src/lib/Lower.hs
+++ b/src/lib/Lower.hs
@@ -153,12 +153,12 @@ lowerFor _ _ _ _ _ = error "expected a unary lambda expression"
 lowerTabCon :: forall i o. Emits o
   => Maybe (Dest SimpIR o) -> SType i -> [SAtom i] -> LowerM i o (SExpr o)
 lowerTabCon maybeDest tabTy elems = do
-  tabTy'@(TabPi (TabPiType dict (_:>t) _)) <- substM tabTy
+  TabPi tabTy' <- substM tabTy
   dest <- case maybeDest of
     Just  d -> return d
-    Nothing -> emitExpr $ PrimOp $ DAMOp $ AllocDest tabTy'
+    Nothing -> emitExpr $ PrimOp $ DAMOp $ AllocDest $ TabPi tabTy'
   Abs bord ufoBlock <- buildAbs noHint IdxRepTy \ord -> do
-    buildBlock $ unsafeFromOrdinal (sink $ IxType t dict) $ Var $ sink ord
+    buildBlock $ unsafeFromOrdinal (sink $ tabIxType tabTy') $ Var $ sink ord
   -- This is emitting a chain of RememberDest ops to force `dest` to be used
   -- linearly, and to force reads of the `Freeze dest'` result not to be
   -- reordered in front of the writes.
diff --git a/src/lib/QueryTypePure.hs b/src/lib/QueryTypePure.hs
index 2501cbf8f..9be267241 100644
--- a/src/lib/QueryTypePure.hs
+++ b/src/lib/QueryTypePure.hs
@@ -119,8 +119,8 @@ instance IRRep r => HasType r (Con r) where
 
 getSuperclassType :: RNest CBinder n l -> Nest CBinder l l' -> Int -> CType n
 getSuperclassType _ Empty = error "bad index"
-getSuperclassType bsAbove (Nest b bs) = \case
-  0 -> ignoreHoistFailure $ hoist bsAbove $ binderType b
+getSuperclassType bsAbove (Nest b@(_:>t) bs) = \case
+  0 -> ignoreHoistFailure $ hoist bsAbove t
   i -> getSuperclassType (RNest bsAbove b) bs (i-1)
 
 instance IRRep r => HasType r (Expr r) where
@@ -213,6 +213,9 @@ rawStrType = case newName "n" of
 rawFinTabType :: IRRep r => Atom r n -> Type r n -> Type r n
 rawFinTabType n eltTy = IxType IdxRepTy (IxDictRawFin n) ==> eltTy
 
+tabIxType :: TabPiType r n -> IxType r n
+tabIxType (TabPiType d (_:>t) _) = IxType t d
+
 typesAsBinderNest
   :: (SinkableE e, HoistableE e, IRRep r)
   => [Type r n] -> e n -> Abs (Nest (Binder r)) e n
diff --git a/src/lib/RuntimePrint.hs b/src/lib/RuntimePrint.hs
index 3255773ad..4a4c2c6a5 100644
--- a/src/lib/RuntimePrint.hs
+++ b/src/lib/RuntimePrint.hs
@@ -185,8 +185,8 @@ bufferTy h = do
 extendBuffer :: (Emits n, CBuilder m) => CAtom n -> CAtom n -> m n ()
 extendBuffer buf tab = do
   RefTy h _ <- return $ getType buf
-  TabTy d (_:>t) _ <- return $ getType tab
-  n <- applyIxMethodCore Size (IxType t d) []
+  TabPi t <- return $ getType tab
+  n <- applyIxMethodCore Size (tabIxType t) []
   void $ applyPreludeFunction "stack_extend_internal" [n, h, buf, tab]
 
 -- argument has type `Word8`
@@ -237,8 +237,8 @@ forEachTabElt
   -> (forall l. (Emits l, DExt n l) => CAtom l -> CAtom l -> m l ())
   -> m n ()
 forEachTabElt tab cont = do
-  TabTy d (_:>t) _ <- return $ getType tab
-  let ixTy = IxType t d
+  TabPi t <- return $ getType tab
+  let ixTy = tabIxType t
   void $ buildFor "i" Fwd ixTy \i -> do
     x <- tabApp (sink tab) (Var i)
     i' <- applyIxMethodCore Ordinal (sink ixTy) [Var i]
diff --git a/src/lib/Simplify.hs b/src/lib/Simplify.hs
index c151937b6..a40b01fa5 100644
--- a/src/lib/Simplify.hs
+++ b/src/lib/Simplify.hs
@@ -153,12 +153,12 @@ getRepType ty = go ty where
         x <- liftSimpAtom (sink l) (Var $ binderVar b')
         r' <- go =<< applySubst (b@>SubstVal x) r
         return $ DepPairTy $ DepPairType expl b' r'
-    TabPi (TabPiType d (b:>t) bodyTy) -> do
-      let ixTy = IxType t d
+    TabPi tabTy -> do
+      let ixTy = tabIxType tabTy
       IxType t' d' <- simplifyIxType ixTy
-      withFreshBinder (getNameHint b) t' \b' -> do
+      withFreshBinder (getNameHint tabTy) t' \b' -> do
         x <- liftSimpAtom (sink $ ixTypeType ixTy) (Var $ binderVar b')
-        bodyTy' <- go =<< applySubst (b@>SubstVal x) bodyTy
+        bodyTy' <- go =<< instantiateTabPiTy (sink tabTy) x
         return $ TabPi $ TabPiType d' b' bodyTy'
     NewtypeTyCon con -> do
       (_, ty') <- unwrapNewtypeType con
@@ -1025,7 +1025,7 @@ simplifyCustomLinearization (Abs runtimeBs staticArgs) actives rule = do
       return $ activeArg':rest
     buildTangentArgs _ _ _ = error "zip error"
 
-    fromNonDepNest :: (HoistableB b, BindsOneAtomName CoreIR b) => Nest b n l -> [CType n]
+    fromNonDepNest :: Nest CBinder n l -> [CType n]
     fromNonDepNest Empty = []
     fromNonDepNest (Nest b bs) =
       case ignoreHoistFailure $ hoist b (Abs bs UnitE) of
diff --git a/src/lib/Types/Core.hs b/src/lib/Types/Core.hs
index 9e9062d3f..d09f7c69f 100644
--- a/src/lib/Types/Core.hs
+++ b/src/lib/Types/Core.hs
@@ -897,17 +897,13 @@ data LinearizationSpec (n::S) =
   LinearizationSpec (TopFunName n) [Active]
   deriving (Show, Generic)
 
--- === BindsOneAtomName ===
+-- === Binder utils ===
 
-class BindsOneName b (AtomNameC r) => BindsOneAtomName (r::IR) (b::B) | b -> r where
-  binderType :: b n l -> Type r n
-  binderVar  :: DExt n l => b n l -> AtomVar r l
+binderType :: Binder r n l -> Type r n
+binderType (_:>ty) = ty
 
-bindersTypes :: (IRRep r, Distinct l, ProvesExt b, BindsNames b, BindsOneAtomName r b)
-             => Nest b n l -> [Type r l]
-bindersTypes Empty = []
-bindersTypes n@(Nest b bs) = ty : bindersTypes bs
-  where ty = withExtEvidence n $ sink (binderType b)
+binderVar  :: (IRRep r, DExt n l) => Binder r n l -> AtomVar r l
+binderVar (b:>ty) = AtomVar (binderName b) (sink ty)
 
 nestToAtomVars :: (Distinct l, Ext n l, IRRep r)
                => Nest (Binder r) n l -> [AtomVar r l]
@@ -916,14 +912,6 @@ nestToAtomVars = \case
   Nest b bs -> withExtEvidence b $ withSubscopeDistinct bs $
     sink (binderVar b) : nestToAtomVars bs
 
-instance IRRep r => BindsOneAtomName r (BinderP (AtomNameC r) (Type r)) where
-  binderType (_ :> ty) = ty
-  binderVar (b:>t) = AtomVar (binderName b) (sink t)
-
-toBinderNest :: BindsOneAtomName r b => Nest b n l -> Nest (Binder r) n l
-toBinderNest Empty = Empty
-toBinderNest (Nest b bs) = Nest (asNameBinder b :> binderType b) (toBinderNest bs)
-
 -- === ToBinding ===
 
 atomBindingToBinding :: AtomBinding r n -> Binding (AtomNameC r) n
@@ -957,14 +945,6 @@ instance (ToBinding e1 c, ToBinding e2 c) => ToBinding (EitherE e1 e2) c where
   toBinding (LeftE  e) = toBinding e
   toBinding (RightE e) = toBinding e
 
--- === HasArgType ===
-
-class HasArgType (e::E) (r::IR) | e -> r where
-  argType :: e n -> Type r n
-
-instance HasArgType (TabPiType r) r where
-  argType (TabPiType _ (_:>ty) _) = ty
-
 -- === Pattern synonyms ===
 
 -- XXX: only use this pattern when you're actually expecting a type. If it's
@@ -2055,6 +2035,9 @@ instance IRRep r => AlphaEqE (TabPiType r) where
 instance IRRep r => AlphaHashableE (TabPiType r) where
   hashWithSaltE env salt (TabPiType _ b t) = hashWithSaltE env salt $ Abs b t
 
+instance HasNameHint (TabPiType r n) where
+  getNameHint (TabPiType _ b _) = getNameHint b
+
 instance IRRep r => SinkableE      (TabPiType r)
 instance IRRep r => HoistableE     (TabPiType r)
 instance IRRep r => RenameE        (TabPiType r)