diff --git a/example/fib.disco b/example/fib.disco
new file mode 100644
index 00000000..b808e726
--- /dev/null
+++ b/example/fib.disco
@@ -0,0 +1,25 @@
+-- Normal, should be memoized
+fib : N -> N 
+fib 0 = 0 
+fib 1 = 1
+fib x = fib (x .- 1) + fib (x .- 2)
+
+-- Multiple arrows, should be memoized 
+fibA : N -> N -> N -> N 
+fibA 0 _ _ = 0
+fibA 1 _ _ = 1
+fibA x n1 n2 = fibA (x .- 1) n1 n2 + fibA (x .- 2) n1 n2
+
+-- Higher-order, should not be memoized 
+fibH : N -> N -> (N -> N) -> N 
+fibH 0 _ _ = 0
+fibH 1 _ _ = 1
+fibH x n f = fibH (x .- 1) n f + fibH (x .- 2) n f
+
+fibHH : N -> N -> (N -> N -> (N -> N) -> N) -> N 
+fibHH 0 _ _ = 0 
+fibHH 1 _ _ = 1
+fibHH x n f = fibHH (x .- 1) n f + fibHH (x .- 2) n f
+
+help : N -> N 
+help _ = 30
diff --git a/example/rsa.disco b/example/rsa.disco
index f84eddd9..0d2c8a19 100644
--- a/example/rsa.disco
+++ b/example/rsa.disco
@@ -19,17 +19,6 @@ import list
 -- and `decrypt` functions can be used to encrypt and decrypt 
 -- lists of natural numbers.
 
-fib : N -> N 
-fib 0 = 0
-fib 1 = 1
-fib x = fib (x .- 1) + fib (x .- 2)
-
-foo : (N -> Bool) -> N -> Bool 
-foo f n = f n 
-
-tru : N -> Bool 
-tru _ = True
-
 encrypt : N * N -> List(N) -> List(N)
 encrypt key xs = each (encrypt1 key, xs)
 
diff --git a/src/Disco/AST/Core.hs b/src/Disco/AST/Core.hs
index 1c240397..2c7322c5 100644
--- a/src/Disco/AST/Core.hs
+++ b/src/Disco/AST/Core.hs
@@ -16,6 +16,7 @@
 module Disco.AST.Core (
   -- * Core AST
   RationalDisplay (..),
+  ShouldMemo (..),
   Core (..),
   Op (..),
   opArity,
@@ -62,6 +63,8 @@ instance Monoid RationalDisplay where
   mempty = Fraction
   mappend = (P.<>)
 
+data ShouldMemo = Memo | NoMemo deriving (Show, Generic, Data, Alpha)
+
 -- | AST for the desugared, untyped core language.
 data Core where
   -- | A variable.
@@ -87,7 +90,7 @@ data Core where
   -- | A projection from a product type, i.e. @fst@ or @snd@.
   CProj :: Side -> Core -> Core
   -- | An anonymous function.
-  CAbs :: Bool -> Bind [Name Core] Core -> Core
+  CAbs :: ShouldMemo -> Bind [Name Core] Core -> Core
   -- | Function application.
   CApp :: Core -> Core -> Core
   -- | A "test frame" under which a test case is run. Records the
diff --git a/src/Disco/Compile.hs b/src/Disco/Compile.hs
index 70e055a2..e38a8b34 100644
--- a/src/Disco/Compile.hs
+++ b/src/Disco/Compile.hs
@@ -170,10 +170,10 @@ compileDTerm term@(DTAbs q _ _) = do
   cbody <- compileDTerm body
   case q of
     Lam -> if canMemo tys
-      then return $ abstractMemo xs cbody
-      else return $ abstract xs cbody
-    Ex -> return $ quantify (OExists tys) (abstract xs cbody)
-    All -> return $ quantify (OForall tys) (abstract xs cbody)
+      then return $ abstract Memo xs cbody
+      else return $ abstract NoMemo xs cbody
+    Ex -> return $ quantify (OExists tys) (abstract NoMemo xs cbody)
+    All -> return $ quantify (OForall tys) (abstract NoMemo xs cbody)
  where
   -- Gather nested abstractions with the same quantifier.
   unbindDeep :: Member Fresh r => DTerm -> Sem r ([Name DTerm], [Type], DTerm)
@@ -183,46 +183,44 @@ compileDTerm term@(DTAbs q _ _) = do
     return (name : ns, ty : tys, body)
   unbindDeep t = return ([], [], t)
 
-  abstract :: [Name DTerm] -> Core -> Core
-  abstract xs body = CAbs False (bind (map coerce xs) body)
-
-  abstractMemo :: [Name DTerm] -> Core -> Core
-  abstractMemo xs body = CAbs True (bind (map coerce xs) body)
+  abstract :: ShouldMemo -> [Name DTerm] -> Core -> Core
+  abstract m xs body = CAbs m (bind (map coerce xs) body)
 
   quantify :: Op -> Core -> Core
   quantify op = CApp (CConst op)
 
-  -- Intend on changing this, but functional for now
-  canMemo :: [Type] -> Bool
-  canMemo [] = True
-  canMemo (x : xs) = case x of
-   TyAtom a -> checkAtom a && canMemo xs
-   TyCon CArr tys -> arrMemo tys && canMemo tys && canMemo xs
-   TyCon c tys -> checkCon c && canMemo tys && canMemo xs
-
-  arrMemo :: [Type] -> Bool
-  arrMemo [] = True
-  arrMemo (x : xs) = case x of
-   TyCon CArr _ -> False
-   _ -> arrMemo xs
-
-  checkCon :: Con -> Bool
-  checkCon (CUser _) = False
-  checkCon CGraph = False
-  checkCon CMap = False
-  checkCon (CContainer a) = checkAtom a
-  checkCon _ = True
+  canMemo :: [Type] -> Bool 
+  canMemo = all canMemoTy
 
-  checkAtom :: Atom -> Bool
-  checkAtom (AVar _) = False
-  checkAtom (ABase b) = checkBase b
-
-  checkBase :: BaseTy -> Bool
-  checkBase CtrList = False
-  checkBase CtrBag = False
-  checkBase CtrSet = False
-  checkBase Gen = False
-  checkBase _ = True
+  canMemoTy :: Type -> Bool 
+  canMemoTy (TyAtom a) = canMemoAtom a 
+  -- Anti-higher-order while permitting multiple arrows
+  canMemoTy (TyCon CArr tys@(t:_)) = case t of 
+   TyCon CArr _ -> False
+   _ -> all canMemoTy tys
+  canMemoTy (TyCon c tys) = canMemoCon c && all canMemoTy tys
+
+  canMemoCon :: Con -> Bool 
+  canMemoCon = \case 
+   CArr -> False
+   CUser _ -> False 
+   CGraph -> False 
+   CMap -> False 
+   CContainer a -> canMemoAtom a 
+   _ -> True
+
+  canMemoAtom :: Atom -> Bool 
+  canMemoAtom (AVar _) = False 
+  canMemoAtom (ABase b) = canMemoBase b 
+
+  canMemoBase :: BaseTy -> Bool 
+  canMemoBase = \case 
+   Gen -> False 
+   CtrSet -> False 
+   CtrBag -> False 
+   CtrList -> False 
+   P -> False 
+   _ -> True
 
 
 -- Special case for Cons, which compiles to a constructor application
@@ -272,13 +270,13 @@ compilePrim ty p@(PrimUOp _) = compilePrimErr p ty
 compilePrim _ (PrimBOp Cons) = do
   hd <- fresh (string2Name "hd")
   tl <- fresh (string2Name "tl")
-  return $ CAbs False $ bind [hd, tl] $ CInj R (CPair (CVar (localName hd)) (CVar (localName tl)))
+  return $ CAbs NoMemo $ bind [hd, tl] $ CInj R (CPair (CVar (localName hd)) (CVar (localName tl)))
 compilePrim _ PrimLeft = do
   a <- fresh (string2Name "a")
-  return $ CAbs False $ bind [a] $ CInj L (CVar (localName a))
+  return $ CAbs NoMemo $ bind [a] $ CInj L (CVar (localName a))
 compilePrim _ PrimRight = do
   a <- fresh (string2Name "a")
-  return $ CAbs False $ bind [a] $ CInj R (CVar (localName a))
+  return $ CAbs NoMemo $ bind [a] $ CInj R (CVar (localName a))
 compilePrim (ty1 :*: ty2 :->: resTy) (PrimBOp bop) = return $ compileBOp ty1 ty2 resTy bop
 compilePrim ty p@(PrimBOp _) = compilePrimErr p ty
 compilePrim _ PrimSqrt = return $ CConst OSqrt
@@ -380,13 +378,13 @@ compilePrimErr p ty = error $ "Impossible! compilePrim " ++ show p ++ " on bad t
 --   of type (Unit → τ), in order to delay evaluation until explicitly
 --   applying it to the unit value.
 compileCase :: Member Fresh r => [DBranch] -> Sem r Core
-compileCase [] = return $ CAbs False (bind [string2Name "_"] (CConst OMatchErr))
+compileCase [] = return $ CAbs NoMemo (bind [string2Name "_"] (CConst OMatchErr))
 -- empty case ==>  λ _ . error
 
 compileCase (b : bs) = do
   c1 <- compileBranch b
   c2 <- compileCase bs
-  return $ CAbs False (bind [string2Name "_"] (CApp c1 c2))
+  return $ CAbs NoMemo (bind [string2Name "_"] (CApp c1 c2))
 
 -- | Compile a branch of a case expression of type τ to a core
 --   language expression of type (Unit → τ) → τ.  The idea is that it
@@ -400,7 +398,7 @@ compileBranch b = do
   c <- compileDTerm e
   k <- fresh (string2Name "k") -- Fresh name for the failure continuation
   bc <- compileGuards (fromTelescope gs) k c
-  return $ CAbs False (bind [k] bc)
+  return $ CAbs NoMemo (bind [k] bc)
 
 -- | 'compileGuards' takes a list of guards, the name of the failure
 --   continuation of type (Unit → τ), and a Core term of type τ to
@@ -422,25 +420,25 @@ compileGuards (DGPat (unembed -> s) p : gs) k e = do
 --   calls the failure continuation in the case of failure, or the
 --   rest of the guards in the case of success.
 compileMatch :: Member Fresh r => DPattern -> Core -> Name Core -> Core -> Sem r Core
-compileMatch (DPVar _ x) s _ e = return $ CApp (CAbs False (bind [coerce x] e)) s
+compileMatch (DPVar _ x) s _ e = return $ CApp (CAbs NoMemo (bind [coerce x] e)) s
 -- Note in the below two cases that we can't just discard s since
 -- that would result in a lazy semantics.  With an eager/strict
 -- semantics, we have to make sure s gets evaluated even if its
 -- value is then discarded.
-compileMatch (DPWild _) s _ e = return $ CApp (CAbs False (bind [string2Name "_"] e)) s
-compileMatch DPUnit s _ e = return $ CApp (CAbs False (bind [string2Name "_"] e)) s
+compileMatch (DPWild _) s _ e = return $ CApp (CAbs NoMemo (bind [string2Name "_"] e)) s
+compileMatch DPUnit s _ e = return $ CApp (CAbs NoMemo (bind [string2Name "_"] e)) s
 compileMatch (DPPair _ x1 x2) s _ e = do
   y <- fresh (string2Name "y")
 
   -- {? e when s is (x1,x2) ?}   ==>   (\y. (\x1.\x2. e) (fst y) (snd y)) s
   return $
     CApp
-      ( CAbs False
+      ( CAbs NoMemo
           ( bind
               [y]
               ( CApp
                   ( CApp
-                      (CAbs False (bind [coerce x1, coerce x2] e))
+                      (CAbs NoMemo (bind [coerce x1, coerce x2] e))
                       (CProj L (CVar (localName y)))
                   )
                   (CProj R (CVar (localName y)))
diff --git a/src/Disco/Interpret/CESK.hs b/src/Disco/Interpret/CESK.hs
index be2f8431..ddd31db4 100644
--- a/src/Disco/Interpret/CESK.hs
+++ b/src/Disco/Interpret/CESK.hs
@@ -179,11 +179,11 @@ step cesk = case cesk of
   (In (CAbs mem b) e k) -> do
     (xs, body) <- unbind b
     case mem of
-      True -> do 
+      Memo -> do 
          cell <- allocateValue (VTrie T.empty)
          -- cell <- allocateValue (VMap M.empty)
          return $ Out (VClo (Just (cell,[])) e xs body) k
-      False -> return $ Out (VClo Nothing e xs body) k
+      NoMemo -> return $ Out (VClo Nothing e xs body) k
 
   (In (CApp c1 c2) e k) -> return $ In c1 e (FArg e c2 : k)
   (In (CType ty) _ k) -> return $ Out (VType ty) k