typecheck/TcGenFunctor.hs

    {-
    (c) The University of Glasgow 2011

    The deriving code for the Functor, Foldable, and Traversable classes
    (equivalent to the code in TcGenDeriv, for other classes)
    -}

    {-# LANGUAGE ScopedTypeVariables #-}
    {-# LANGUAGE TypeFamilies #-}
    {-# LANGUAGE FlexibleContexts #-}
    {-# LANGUAGE LambdaCase #-}

    module TcGenFunctor (
            FFoldType(..), functorLikeTraverse,
            deepSubtypesContaining, foldDataConArgs,

            gen_Functor_binds, gen_Foldable_binds, gen_Traversable_binds
        ) where

    import GhcPrelude

    import Bag
    import DataCon
    import FastString
    import GHC.Hs
    import Panic
    import PrelNames
    import RdrName
    import SrcLoc
    import State
    import TcGenDeriv
    import TcType
    import TyCon
    import TyCoRep
    import Type
    import Util
    import Var
    import VarSet
    import MkId (coerceId)
    import TysWiredIn (true_RDR, false_RDR)

    import Data.Maybe (catMaybes, isJust)

    {-
    ************************************************************************
    *                                                                      *
                            Functor instances

     see http://www.mail-archive.com/haskell-prime@haskell.org/msg02116.html

    *                                                                      *
    ************************************************************************

    For the data type:

      data T a = T1 Int a | T2 (T a)

    We generate the instance:

      instance Functor T where
          fmap f (T1 b1 a) = T1 b1 (f a)
          fmap f (T2 ta)   = T2 (fmap f ta)

    Notice that we don't simply apply 'fmap' to the constructor arguments.
    Rather
      - Do nothing to an argument whose type doesn't mention 'a'
      - Apply 'f' to an argument of type 'a'
      - Apply 'fmap f' to other arguments
    That's why we have to recurse deeply into the constructor argument types,
    rather than just one level, as we typically do.

    What about types with more than one type parameter?  In general, we only
    derive Functor for the last position:

      data S a b = S1 [b] | S2 (a, T a b)
      instance Functor (S a) where
        fmap f (S1 bs)    = S1 (fmap f bs)
        fmap f (S2 (p,q)) = S2 (a, fmap f q)

    However, we have special cases for
             - tuples
             - functions

    More formally, we write the derivation of fmap code over type variable
    'a for type 'b as ($fmap 'a 'b).  In this general notation the derived
    instance for T is:

      instance Functor T where
          fmap f (T1 x1 x2) = T1 ($(fmap 'a 'b1) x1) ($(fmap 'a 'a) x2)
          fmap f (T2 x1)    = T2 ($(fmap 'a '(T a)) x1)

      $(fmap 'a 'b)          =  \x -> x     -- when b does not contain a
      $(fmap 'a 'a)          =  f
      $(fmap 'a '(b1,b2))    =  \x -> case x of (x1,x2) -> ($(fmap 'a 'b1) x1, $(fmap 'a 'b2) x2)
      $(fmap 'a '(T b1 b2))  =  fmap $(fmap 'a 'b2)   -- when a only occurs in the last parameter, b2
      $(fmap 'a '(b -> c))   =  \x b -> $(fmap 'a' 'c) (x ($(cofmap 'a 'b) b))

    For functions, the type parameter 'a can occur in a contravariant position,
    which means we need to derive a function like:

      cofmap :: (a -> b) -> (f b -> f a)

    This is pretty much the same as $fmap, only without the $(cofmap 'a 'a) case:

      $(cofmap 'a 'b)          =  \x -> x     -- when b does not contain a
      $(cofmap 'a 'a)          =  error "type variable in contravariant position"
      $(cofmap 'a '(b1,b2))    =  \x -> case x of (x1,x2) -> ($(cofmap 'a 'b1) x1, $(cofmap 'a 'b2) x2)
      $(cofmap 'a '[b])        =  map $(cofmap 'a 'b)
      $(cofmap 'a '(T b1 b2))  =  fmap $(cofmap 'a 'b2)   -- when a only occurs in the last parameter, b2
      $(cofmap 'a '(b -> c))   =  \x b -> $(cofmap 'a' 'c) (x ($(fmap 'a 'c) b))

    Note that the code produced by $(fmap _ _) is always a higher order function,
    with type `(a -> b) -> (g a -> g b)` for some g. When we need to do pattern
    matching on the type, this means create a lambda function (see the (,) case above).
    The resulting code for fmap can look a bit weird, for example:

      data X a = X (a,Int)
      -- generated instance
      instance Functor X where
          fmap f (X x) = (\y -> case y of (x1,x2) -> X (f x1, (\z -> z) x2)) x

    The optimizer should be able to simplify this code by simple inlining.

    An older version of the deriving code tried to avoid these applied
    lambda functions by producing a meta level function. But the function to
    be mapped, `f`, is a function on the code level, not on the meta level,
    so it was eta expanded to `\x -> [| f $x |]`. This resulted in too much eta expansion.
    It is better to produce too many lambdas than to eta expand, see ticket #7436.
    -}

    gen_Functor_binds :: SrcSpan -> TyCon -> (LHsBinds GhcPs, BagDerivStuff)
    -- When the argument is phantom, we can use  fmap _ = coerce
    -- See Note [Phantom types with Functor, Foldable, and Traversable]
    gen_Functor_binds loc tycon
      | Phantom <- last (tyConRoles tycon)
      = (unitBag fmap_bind, emptyBag)
      where
        fmap_name = L loc fmap_RDR
        fmap_bind = mkRdrFunBind fmap_name fmap_eqns
        fmap_eqns = [mkSimpleMatch fmap_match_ctxt
                                   [nlWildPat]
                                   coerce_Expr]
        fmap_match_ctxt = mkPrefixFunRhs fmap_name

    gen_Functor_binds loc tycon
      = (listToBag [fmap_bind, replace_bind], emptyBag)
      where
        data_cons = tyConDataCons tycon
        fmap_name = L loc fmap_RDR

        -- See Note [EmptyDataDecls with Functor, Foldable, and Traversable]
        fmap_bind = mkRdrFunBindEC 2 id fmap_name fmap_eqns
        fmap_match_ctxt = mkPrefixFunRhs fmap_name

        fmap_eqn con = flip evalState bs_RDRs $
                         match_for_con fmap_match_ctxt [f_Pat] con =<< parts
          where
            parts = sequence $ foldDataConArgs ft_fmap con

        fmap_eqns = map fmap_eqn data_cons

        ft_fmap :: FFoldType (State [RdrName] (LHsExpr GhcPs))
        ft_fmap = FT { ft_triv = mkSimpleLam $ \x -> return x
                       -- fmap f = \x -> x
                     , ft_var  = return f_Expr
                       -- fmap f = f
                     , ft_fun  = \g h -> do
                         gg <- g
                         hh <- h
                         mkSimpleLam2 $ \x b -> return $
                           nlHsApp hh (nlHsApp x (nlHsApp gg b))
                       -- fmap f = \x b -> h (x (g b))
                     , ft_tup = \t gs -> do
                         gg <- sequence gs
                         mkSimpleLam $ mkSimpleTupleCase (match_for_con CaseAlt) t gg
                       -- fmap f = \x -> case x of (a1,a2,..) -> (g1 a1,g2 a2,..)
                     , ft_ty_app = \_ g -> nlHsApp fmap_Expr <$> g
                       -- fmap f = fmap g
                     , ft_forall = \_ g -> g
                     , ft_bad_app = panic "in other argument in ft_fmap"
                     , ft_co_var = panic "contravariant in ft_fmap" }

        -- See Note [Deriving <$]
        replace_name = L loc replace_RDR

        -- See Note [EmptyDataDecls with Functor, Foldable, and Traversable]
        replace_bind = mkRdrFunBindEC 2 id replace_name replace_eqns
        replace_match_ctxt = mkPrefixFunRhs replace_name

        replace_eqn con = flip evalState bs_RDRs $
            match_for_con replace_match_ctxt [z_Pat] con =<< parts
          where
            parts = traverse (fmap replace) $ foldDataConArgs ft_replace con

        replace_eqns = map replace_eqn data_cons

        ft_replace :: FFoldType (State [RdrName] Replacer)
        ft_replace = FT { ft_triv = fmap Nested $ mkSimpleLam $ \x -> return x
                       -- (p <$) = \x -> x
                     , ft_var  = fmap Immediate $ mkSimpleLam $ \_ -> return z_Expr
                       -- (p <$) = const p
                     , ft_fun  = \g h -> do
                         gg <- replace <$> g
                         hh <- replace <$> h
                         fmap Nested $ mkSimpleLam2 $ \x b -> return $
                           nlHsApp hh (nlHsApp x (nlHsApp gg b))
                       -- (<$) p = \x b -> h (x (g b))
                     , ft_tup = \t gs -> do
                         gg <- traverse (fmap replace) gs
                         fmap Nested . mkSimpleLam $
                              mkSimpleTupleCase (match_for_con CaseAlt) t gg
                       -- (p <$) = \x -> case x of (a1,a2,..) -> (g1 a1,g2 a2,..)
                     , ft_ty_app = \_ gm -> do
                           g <- gm
                           case g of
                             Nested g' -> pure . Nested $
                                              nlHsApp fmap_Expr $ g'
                             Immediate _ -> pure . Nested $
                                              nlHsApp replace_Expr z_Expr
                       -- (p <$) = fmap (p <$)
                     , ft_forall = \_ g -> g
                     , ft_bad_app = panic "in other argument in ft_replace"
                     , ft_co_var = panic "contravariant in ft_replace" }

        -- Con a1 a2 ... -> Con (f1 a1) (f2 a2) ...
        match_for_con :: HsMatchContext RdrName
                      -> [LPat GhcPs] -> DataCon -> [LHsExpr GhcPs]
                      -> State [RdrName] (LMatch GhcPs (LHsExpr GhcPs))
        match_for_con ctxt = mkSimpleConMatch ctxt $
            \con_name xs -> return $ nlHsApps con_name xs  -- Con x1 x2 ..

    -- See Note [Deriving <$]
    data Replacer = Immediate {replace :: LHsExpr GhcPs}
                  | Nested {replace :: LHsExpr GhcPs}

    {- Note [Deriving <$]
       ~~~~~~~~~~~~~~~~~~

    We derive the definition of <$. Allowing this to take the default definition
    can lead to memory leaks: mapping over a structure with a constant function can
    fill the result structure with trivial thunks that retain the values from the
    original structure. The simplifier seems to handle this all right for simple
    types, but not for recursive ones. Consider

    data Tree a = Bin !(Tree a) a !(Tree a) | Tip deriving Functor

    -- fmap _ Tip = Tip
    -- fmap f (Bin l v r) = Bin (fmap f l) (f v) (fmap f r)

    Using the default definition of <$, we get (<$) x = fmap (\_ -> x) and that
    simplifies no further. Why is that? `fmap` is defined recursively, so GHC
    cannot inline it. The static argument transformation would turn the definition
    into a non-recursive one

    -- fmap f = go where
    --   go Tip = Tip
    --   go (Bin l v r) = Bin (go l) (f v) (go r)

    which GHC could inline, producing an efficient definion of `<$`. But there are
    several problems. First, GHC does not perform the static argument transformation
    by default, even with -O2. Second, even when it does perform the static argument
    transformation, it does so only when there are at least two static arguments,
    which is not the case for fmap. Finally, when the type in question is
    non-regular, such as

    data Nesty a = Z a | S (Nesty a) (Nest (a, a))

    the function argument is no longer (entirely) static, so the static argument
    transformation will do nothing for us.

    Applying the default definition of `<$` will produce a tree full of thunks that
    look like ((\_ -> x) x0), which represents unnecessary thunk allocation and
    also retention of the previous value, potentially leaking memory. Instead, we
    derive <$ separately. Two aspects are different from fmap: the case of the
    sought type variable (ft_var) and the case of a type application (ft_ty_app).
    The interesting one is ft_ty_app. We have to distinguish two cases: the
    "immediate" case where the type argument *is* the sought type variable, and
    the "nested" case where the type argument *contains* the sought type variable.

    The immediate case:

    Suppose we have

    data Imm a = Imm (F ... a)

    Then we want to define

    x <$ Imm q = Imm (x <$ q)

    The nested case:

    Suppose we have

    data Nes a = Nes (F ... (G a))

    Then we want to define

    x <$ Nes q = Nes (fmap (x <$) q)

    We use the Replacer type to tag whether the expression derived for applying
    <$ to the last type variable was the ft_var case (immediate) or one of the
    others (letting ft_forall pass through as usual).

    We could, but do not, give tuples special treatment to improve efficiency
    in some cases. Suppose we have

    data Nest a = Z a | S (Nest (a,a))

    The optimal definition would be

    x <$ Z _ = Z x
    x <$ S t = S ((x, x) <$ t)

    which produces a result with maximal internal sharing. The reason we do not
    attempt to treat this case specially is that we have no way to give
    user-provided tuple-like types similar treatment. If the user changed the
    definition to

    data Pair a = Pair a a
    data Nest a = Z a | S (Nest (Pair a))

    they would experience a surprising degradation in performance. -}

    {-
    Utility functions related to Functor deriving.

    Since several things use the same pattern of traversal, this is abstracted into functorLikeTraverse.
    This function works like a fold: it makes a value of type 'a' in a bottom up way.
    -}

    -- Generic traversal for Functor deriving
    -- See Note [FFoldType and functorLikeTraverse]
    data FFoldType a      -- Describes how to fold over a Type in a functor like way
       = FT { ft_triv    :: a
              -- ^ Does not contain variable
            , ft_var     :: a
              -- ^ The variable itself
            , ft_co_var  :: a
              -- ^ The variable itself, contravariantly
            , ft_fun     :: a -> a -> a
              -- ^ Function type
            , ft_tup     :: TyCon -> [a] -> a
              -- ^ Tuple type
            , ft_ty_app  :: Type -> a -> a
              -- ^ Type app, variable only in last argument
            , ft_bad_app :: a
              -- ^ Type app, variable other than in last argument
            , ft_forall  :: TcTyVar -> a -> a
              -- ^ Forall type
         }

    functorLikeTraverse :: forall a.
                           TyVar         -- ^ Variable to look for
                        -> FFoldType a   -- ^ How to fold
                        -> Type          -- ^ Type to process
                        -> a
    functorLikeTraverse var (FT { ft_triv = caseTrivial,     ft_var = caseVar
                                , ft_co_var = caseCoVar,     ft_fun = caseFun
                                , ft_tup = caseTuple,        ft_ty_app = caseTyApp
                                , ft_bad_app = caseWrongArg, ft_forall = caseForAll })
                        ty
      = fst (go False ty)
      where
        go :: Bool        -- Covariant or contravariant context
           -> Type
           -> (a, Bool)   -- (result of type a, does type contain var)

        go co ty | Just ty' <- tcView ty = go co ty'
        go co (TyVarTy    v) | v == var = (if co then caseCoVar else caseVar,True)
        go co (FunTy { ft_arg = x, ft_res = y, ft_af = af })
           | InvisArg <- af = go co y
           | xc || yc       = (caseFun xr yr,True)
           where (xr,xc) = go (not co) x
                 (yr,yc) = go co       y
        go co (AppTy    x y) | xc = (caseWrongArg,   True)
                             | yc = (caseTyApp x yr, True)
            where (_, xc) = go co x
                  (yr,yc) = go co y
        go co ty@(TyConApp con args)
           | not (or xcs)     = (caseTrivial, False)   -- Variable does not occur
           -- At this point we know that xrs, xcs is not empty,
           -- and at least one xr is True
           | isTupleTyCon con = (caseTuple con xrs, True)
           | or (init xcs)    = (caseWrongArg, True)         -- T (..var..)    ty
           | Just (fun_ty, _) <- splitAppTy_maybe ty         -- T (..no var..) ty
                              = (caseTyApp fun_ty (last xrs), True)
           | otherwise        = (caseWrongArg, True)   -- Non-decomposable (eg type function)
           where
             -- When folding over an unboxed tuple, we must explicitly drop the
             -- runtime rep arguments, or else GHC will generate twice as many
             -- variables in a unboxed tuple pattern match and expression as it
             -- actually needs. See #12399
             (xrs,xcs) = unzip (map (go co) (dropRuntimeRepArgs args))
        go co (ForAllTy (Bndr v vis) x)
           | isVisibleArgFlag vis = panic "unexpected visible binder"
           | v /= var && xc       = (caseForAll v xr,True)
           where (xr,xc) = go co x

        go _ _ = (caseTrivial,False)

    -- Return all syntactic subterms of ty that contain var somewhere
    -- These are the things that should appear in instance constraints
    deepSubtypesContaining :: TyVar -> Type -> [TcType]
    deepSubtypesContaining tv
      = functorLikeTraverse tv
            (FT { ft_triv = []
                , ft_var = []
                , ft_fun = (++)
                , ft_tup = \_ xs -> concat xs
                , ft_ty_app = (:)
                , ft_bad_app = panic "in other argument in deepSubtypesContaining"
                , ft_co_var = panic "contravariant in deepSubtypesContaining"
                , ft_forall = \v xs -> filterOut ((v `elemVarSet`) . tyCoVarsOfType) xs })

    foldDataConArgs :: FFoldType a -> DataCon -> [a]
    -- Fold over the arguments of the datacon
    foldDataConArgs ft con
      = map foldArg (dataConOrigArgTys con)
      where
        foldArg
          = case getTyVar_maybe (last (tyConAppArgs (dataConOrigResTy con))) of
                 Just tv -> functorLikeTraverse tv ft
                 Nothing -> const (ft_triv ft)
        -- If we are deriving Foldable for a GADT, there is a chance that the last
        -- type variable in the data type isn't actually a type variable at all.
        -- (for example, this can happen if the last type variable is refined to
        -- be a concrete type such as Int). If the last type variable is refined
        -- to be a specific type, then getTyVar_maybe will return Nothing.
        -- See Note [DeriveFoldable with ExistentialQuantification]
        --
        -- The kind checks have ensured the last type parameter is of kind *.

    -- Make a HsLam using a fresh variable from a State monad
    mkSimpleLam :: (LHsExpr GhcPs -> State [RdrName] (LHsExpr GhcPs))
                -> State [RdrName] (LHsExpr GhcPs)
    -- (mkSimpleLam fn) returns (\x. fn(x))
    mkSimpleLam lam =
        get >>= \case
          n:names -> do
            put names
            body <- lam (nlHsVar n)
            return (mkHsLam [nlVarPat n] body)
          _ -> panic "mkSimpleLam"

    mkSimpleLam2 :: (LHsExpr GhcPs -> LHsExpr GhcPs
                 -> State [RdrName] (LHsExpr GhcPs))
                 -> State [RdrName] (LHsExpr GhcPs)
    mkSimpleLam2 lam =
        get >>= \case
          n1:n2:names -> do
            put names
            body <- lam (nlHsVar n1) (nlHsVar n2)
            return (mkHsLam [nlVarPat n1,nlVarPat n2] body)
          _ -> panic "mkSimpleLam2"

    -- "Con a1 a2 a3 -> fold [x1 a1, x2 a2, x3 a3]"
    --
    -- @mkSimpleConMatch fold extra_pats con insides@ produces a match clause in
    -- which the LHS pattern-matches on @extra_pats@, followed by a match on the
    -- constructor @con@ and its arguments. The RHS folds (with @fold@) over @con@
    -- and its arguments, applying an expression (from @insides@) to each of the
    -- respective arguments of @con@.
    mkSimpleConMatch :: Monad m => HsMatchContext RdrName
                     -> (RdrName -> [LHsExpr GhcPs] -> m (LHsExpr GhcPs))
                     -> [LPat GhcPs]
                     -> DataCon
                     -> [LHsExpr GhcPs]
                     -> m (LMatch GhcPs (LHsExpr GhcPs))
    mkSimpleConMatch ctxt fold extra_pats con insides = do
        let con_name = getRdrName con
        let vars_needed = takeList insides as_RDRs
        let bare_pat = nlConVarPat con_name vars_needed
        let pat = if null vars_needed
              then bare_pat
              else nlParPat bare_pat
        rhs <- fold con_name
                    (zipWith (\i v -> i `nlHsApp` nlHsVar v) insides vars_needed)
        return $ mkMatch ctxt (extra_pats ++ [pat]) rhs
                         (noLoc emptyLocalBinds)

    -- "Con a1 a2 a3 -> fmap (\b2 -> Con a1 b2 a3) (traverse f a2)"
    --
    -- @mkSimpleConMatch2 fold extra_pats con insides@ behaves very similarly to
    -- 'mkSimpleConMatch', with two key differences:
    --
    -- 1. @insides@ is a @[Maybe (LHsExpr RdrName)]@ instead of a
    --    @[LHsExpr RdrName]@. This is because it filters out the expressions
    --    corresponding to arguments whose types do not mention the last type
    --    variable in a derived 'Foldable' or 'Traversable' instance (i.e., the
    --    'Nothing' elements of @insides@).
    --
    -- 2. @fold@ takes an expression as its first argument instead of a
    --    constructor name. This is because it uses a specialized
    --    constructor function expression that only takes as many parameters as
    --    there are argument types that mention the last type variable.
    --
    -- See Note [Generated code for DeriveFoldable and DeriveTraversable]
    mkSimpleConMatch2 :: Monad m
                      => HsMatchContext RdrName
                      -> (LHsExpr GhcPs -> [LHsExpr GhcPs]
                                          -> m (LHsExpr GhcPs))
                      -> [LPat GhcPs]
                      -> DataCon
                      -> [Maybe (LHsExpr GhcPs)]
                      -> m (LMatch GhcPs (LHsExpr GhcPs))
    mkSimpleConMatch2 ctxt fold extra_pats con insides = do
        let con_name = getRdrName con
            vars_needed = takeList insides as_RDRs
            pat = nlConVarPat con_name vars_needed
            -- Make sure to zip BEFORE invoking catMaybes. We want the variable
            -- indicies in each expression to match up with the argument indices
            -- in con_expr (defined below).
            exps = catMaybes $ zipWith (\i v -> (`nlHsApp` nlHsVar v) <$> i)
                                       insides vars_needed
            -- An element of argTysTyVarInfo is True if the constructor argument
            -- with the same index has a type which mentions the last type
            -- variable.
            argTysTyVarInfo = map isJust insides
            (asWithTyVar, asWithoutTyVar) = partitionByList argTysTyVarInfo as_Vars

            con_expr
              | null asWithTyVar = nlHsApps con_name asWithoutTyVar
              | otherwise =
                  let bs   = filterByList  argTysTyVarInfo bs_RDRs
                      vars = filterByLists argTysTyVarInfo bs_Vars as_Vars
                  in mkHsLam (map nlVarPat bs) (nlHsApps con_name vars)

        rhs <- fold con_expr exps
        return $ mkMatch ctxt (extra_pats ++ [pat]) rhs
                         (noLoc emptyLocalBinds)

    -- "case x of (a1,a2,a3) -> fold [x1 a1, x2 a2, x3 a3]"
    mkSimpleTupleCase :: Monad m => ([LPat GhcPs] -> DataCon -> [a]
                                     -> m (LMatch GhcPs (LHsExpr GhcPs)))
                      -> TyCon -> [a] -> LHsExpr GhcPs -> m (LHsExpr GhcPs)
    mkSimpleTupleCase match_for_con tc insides x
      = do { let data_con = tyConSingleDataCon tc
           ; match <- match_for_con [] data_con insides
           ; return $ nlHsCase x [match] }

    {-
    ************************************************************************
    *                                                                      *
                            Foldable instances

     see http://www.mail-archive.com/haskell-prime@haskell.org/msg02116.html

    *                                                                      *
    ************************************************************************

    Deriving Foldable instances works the same way as Functor instances,
    only Foldable instances are not possible for function types at all.
    Given (data T a = T a a (T a) deriving Foldable), we get:

      instance Foldable T where
          foldr f z (T x1 x2 x3) =
            $(foldr 'a 'a) x1 ( $(foldr 'a 'a) x2 ( $(foldr 'a '(T a)) x3 z ) )

    -XDeriveFoldable is different from -XDeriveFunctor in that it filters out
    arguments to the constructor that would produce useless code in a Foldable
    instance. For example, the following datatype:

      data Foo a = Foo Int a Int deriving Foldable

    would have the following generated Foldable instance:

      instance Foldable Foo where
        foldr f z (Foo x1 x2 x3) = $(foldr 'a 'a) x2

    since neither of the two Int arguments are folded over.

    The cases are:

      $(foldr 'a 'a)         =  f
      $(foldr 'a '(b1,b2))   =  \x z -> case x of (x1,x2) -> $(foldr 'a 'b1) x1 ( $(foldr 'a 'b2) x2 z )
      $(foldr 'a '(T b1 b2)) =  \x z -> foldr $(foldr 'a 'b2) z x  -- when a only occurs in the last parameter, b2

    Note that the arguments to the real foldr function are the wrong way around,
    since (f :: a -> b -> b), while (foldr f :: b -> t a -> b).

    One can envision a case for types that don't contain the last type variable:

      $(foldr 'a 'b)         =  \x z -> z     -- when b does not contain a

    But this case will never materialize, since the aforementioned filtering
    removes all such types from consideration.
    See Note [Generated code for DeriveFoldable and DeriveTraversable].

    Foldable instances differ from Functor and Traversable instances in that
    Foldable instances can be derived for data types in which the last type
    variable is existentially quantified. In particular, if the last type variable
    is refined to a more specific type in a GADT:

      data GADT a where
          G :: a ~ Int => a -> G Int

    then the deriving machinery does not attempt to check that the type a contains
    Int, since it is not syntactically equal to a type variable. That is, the
    derived Foldable instance for GADT is:

      instance Foldable GADT where
          foldr _ z (GADT _) = z

    See Note [DeriveFoldable with ExistentialQuantification].

    Note [Deriving null]
    ~~~~~~~~~~~~~~~~~~~~

    In some cases, deriving the definition of 'null' can produce much better
    results than the default definition. For example, with

      data SnocList a = Nil | Snoc (SnocList a) a

    the default definition of 'null' would walk the entire spine of a
    nonempty snoc-list before concluding that it is not null. But looking at
    the Snoc constructor, we can immediately see that it contains an 'a', and
    so 'null' can return False immediately if it matches on Snoc. When we
    derive 'null', we keep track of things that cannot be null. The interesting
    case is type application. Given

      data Wrap a = Wrap (Foo (Bar a))

    we use

      null (Wrap fba) = all null fba

    but if we see

      data Wrap a = Wrap (Foo a)

    we can just use

      null (Wrap fa) = null fa

    Indeed, we allow this to happen even for tuples:

      data Wrap a = Wrap (Foo (a, Int))

    produces

      null (Wrap fa) = null fa

    As explained in Note [Deriving <$], giving tuples special performance treatment
    could surprise users if they switch to other types, but Ryan Scott seems to
    think it's okay to do it for now.
    -}

    gen_Foldable_binds :: SrcSpan -> TyCon -> (LHsBinds GhcPs, BagDerivStuff)
    -- When the parameter is phantom, we can use foldMap _ _ = mempty
    -- See Note [Phantom types with Functor, Foldable, and Traversable]
    gen_Foldable_binds loc tycon
      | Phantom <- last (tyConRoles tycon)
      = (unitBag foldMap_bind, emptyBag)
      where
        foldMap_name = L loc foldMap_RDR
        foldMap_bind = mkRdrFunBind foldMap_name foldMap_eqns
        foldMap_eqns = [mkSimpleMatch foldMap_match_ctxt
                                      [nlWildPat, nlWildPat]
                                      mempty_Expr]
        foldMap_match_ctxt = mkPrefixFunRhs foldMap_name

    gen_Foldable_binds loc tycon
      | null data_cons  -- There's no real point producing anything but
                        -- foldMap for a type with no constructors.
      = (unitBag foldMap_bind, emptyBag)

      | otherwise
      = (listToBag [foldr_bind, foldMap_bind, null_bind], emptyBag)
      where
        data_cons = tyConDataCons tycon

        foldr_bind = mkRdrFunBind (L loc foldable_foldr_RDR) eqns
        eqns = map foldr_eqn data_cons
        foldr_eqn con
          = evalState (match_foldr z_Expr [f_Pat,z_Pat] con =<< parts) bs_RDRs
          where
            parts = sequence $ foldDataConArgs ft_foldr con

        foldMap_name = L loc foldMap_RDR

        -- See Note [EmptyDataDecls with Functor, Foldable, and Traversable]
        foldMap_bind = mkRdrFunBindEC 2 (const mempty_Expr)
                          foldMap_name foldMap_eqns

        foldMap_eqns = map foldMap_eqn data_cons

        foldMap_eqn con
          = evalState (match_foldMap [f_Pat] con =<< parts) bs_RDRs
          where
            parts = sequence $ foldDataConArgs ft_foldMap con

        -- Given a list of NullM results, produce Nothing if any of
        -- them is NotNull, and otherwise produce a list of Maybes
        -- with Justs representing unknowns and Nothings representing
        -- things that are definitely null.
        convert :: [NullM a] -> Maybe [Maybe a]
        convert = traverse go where
          go IsNull = Just Nothing
          go NotNull = Nothing
          go (NullM a) = Just (Just a)

        null_name = L loc null_RDR
        null_match_ctxt = mkPrefixFunRhs null_name
        null_bind = mkRdrFunBind null_name null_eqns
        null_eqns = map null_eqn data_cons
        null_eqn con
          = flip evalState bs_RDRs $ do
              parts <- sequence $ foldDataConArgs ft_null con
              case convert parts of
                Nothing -> return $
                  mkMatch null_match_ctxt [nlParPat (nlWildConPat con)]
                    false_Expr (noLoc emptyLocalBinds)
                Just cp -> match_null [] con cp

        -- Yields 'Just' an expression if we're folding over a type that mentions
        -- the last type parameter of the datatype. Otherwise, yields 'Nothing'.
        -- See Note [FFoldType and functorLikeTraverse]
        ft_foldr :: FFoldType (State [RdrName] (Maybe (LHsExpr GhcPs)))
        ft_foldr
          = FT { ft_triv    = return Nothing
                 -- foldr f = \x z -> z
               , ft_var     = return $ Just f_Expr
                 -- foldr f = f
               , ft_tup     = \t g -> do
                   gg  <- sequence g
                   lam <- mkSimpleLam2 $ \x z ->
                     mkSimpleTupleCase (match_foldr z) t gg x
                   return (Just lam)
                 -- foldr f = (\x z -> case x of ...)
               , ft_ty_app  = \_ g -> do
                   gg <- g
                   mapM (\gg' -> mkSimpleLam2 $ \x z -> return $
                     nlHsApps foldable_foldr_RDR [gg',z,x]) gg
                 -- foldr f = (\x z -> foldr g z x)
               , ft_forall  = \_ g -> g
               , ft_co_var  = panic "contravariant in ft_foldr"
               , ft_fun     = panic "function in ft_foldr"
               , ft_bad_app = panic "in other argument in ft_foldr" }

        match_foldr :: LHsExpr GhcPs
                    -> [LPat GhcPs]
                    -> DataCon
                    -> [Maybe (LHsExpr GhcPs)]
                    -> State [RdrName] (LMatch GhcPs (LHsExpr GhcPs))
        match_foldr z = mkSimpleConMatch2 LambdaExpr $ \_ xs -> return (mkFoldr xs)
          where
            -- g1 v1 (g2 v2 (.. z))
            mkFoldr :: [LHsExpr GhcPs] -> LHsExpr GhcPs
            mkFoldr = foldr nlHsApp z

        -- See Note [FFoldType and functorLikeTraverse]
        ft_foldMap :: FFoldType (State [RdrName] (Maybe (LHsExpr GhcPs)))
        ft_foldMap
          = FT { ft_triv = return Nothing
                 -- foldMap f = \x -> mempty
               , ft_var  = return (Just f_Expr)
                 -- foldMap f = f
               , ft_tup  = \t g -> do
                   gg  <- sequence g
                   lam <- mkSimpleLam $ mkSimpleTupleCase match_foldMap t gg
                   return (Just lam)
                 -- foldMap f = \x -> case x of (..,)
               , ft_ty_app = \_ g -> fmap (nlHsApp foldMap_Expr) <$> g
                 -- foldMap f = foldMap g
               , ft_forall = \_ g -> g
               , ft_co_var = panic "contravariant in ft_foldMap"
               , ft_fun = panic "function in ft_foldMap"
               , ft_bad_app = panic "in other argument in ft_foldMap" }

        match_foldMap :: [LPat GhcPs]
                      -> DataCon
                      -> [Maybe (LHsExpr GhcPs)]
                      -> State [RdrName] (LMatch GhcPs (LHsExpr GhcPs))
        match_foldMap = mkSimpleConMatch2 CaseAlt $ \_ xs -> return (mkFoldMap xs)
          where
            -- mappend v1 (mappend v2 ..)
            mkFoldMap :: [LHsExpr GhcPs] -> LHsExpr GhcPs
            mkFoldMap [] = mempty_Expr
            mkFoldMap xs = foldr1 (\x y -> nlHsApps mappend_RDR [x,y]) xs

        -- See Note [FFoldType and functorLikeTraverse]
        -- Yields NullM an expression if we're folding over an expression
        -- that may or may not be null. Yields IsNull if it's certainly
        -- null, and yields NotNull if it's certainly not null.
        -- See Note [Deriving null]
        ft_null :: FFoldType (State [RdrName] (NullM (LHsExpr GhcPs)))
        ft_null
          = FT { ft_triv = return IsNull
                 -- null = \_ -> True
               , ft_var  = return NotNull
                 -- null = \_ -> False
               , ft_tup  = \t g -> do
                   gg  <- sequence g
                   case convert gg of
                     Nothing -> pure NotNull
                     Just ggg ->
                       NullM <$> (mkSimpleLam $ mkSimpleTupleCase match_null t ggg)
                 -- null = \x -> case x of (..,)
               , ft_ty_app = \_ g -> flip fmap g $ \nestedResult ->
                                  case nestedResult of
                                    -- If e definitely contains the parameter,
                                    -- then we can test if (G e) contains it by
                                    -- simply checking if (G e) is null
                                    NotNull -> NullM null_Expr
                                    -- This case is unreachable--it will actually be
                                    -- caught by ft_triv
                                    IsNull -> IsNull
                                    -- The general case uses (all null),
                                    -- (all (all null)), etc.
                                    NullM nestedTest -> NullM $
                                                        nlHsApp all_Expr nestedTest
                 -- null fa = null fa, or null fa = all null fa, or null fa = True
               , ft_forall = \_ g -> g
               , ft_co_var = panic "contravariant in ft_null"
               , ft_fun = panic "function in ft_null"
               , ft_bad_app = panic "in other argument in ft_null" }

        match_null :: [LPat GhcPs]
                      -> DataCon
                      -> [Maybe (LHsExpr GhcPs)]
                      -> State [RdrName] (LMatch GhcPs (LHsExpr GhcPs))
        match_null = mkSimpleConMatch2 CaseAlt $ \_ xs -> return (mkNull xs)
          where
            -- v1 && v2 && ..
            mkNull :: [LHsExpr GhcPs] -> LHsExpr GhcPs
            mkNull [] = true_Expr
            mkNull xs = foldr1 (\x y -> nlHsApps and_RDR [x,y]) xs

    data NullM a =
        IsNull   -- Definitely null
      | NotNull  -- Definitely not null
      | NullM a  -- Unknown

    {-
    ************************************************************************
    *                                                                      *
                            Traversable instances

     see http://www.mail-archive.com/haskell-prime@haskell.org/msg02116.html
    *                                                                      *
    ************************************************************************

    Again, Traversable is much like Functor and Foldable.

    The cases are:

      $(traverse 'a 'a)          =  f
      $(traverse 'a '(b1,b2))    =  \x -> case x of (x1,x2) ->
         liftA2 (,) ($(traverse 'a 'b1) x1) ($(traverse 'a 'b2) x2)
      $(traverse 'a '(T b1 b2))  =  traverse $(traverse 'a 'b2)  -- when a only occurs in the last parameter, b2

    Like -XDeriveFoldable, -XDeriveTraversable filters out arguments whose types
    do not mention the last type parameter. Therefore, the following datatype:

      data Foo a = Foo Int a Int

    would have the following derived Traversable instance:

      instance Traversable Foo where
        traverse f (Foo x1 x2 x3) =
          fmap (\b2 -> Foo x1 b2 x3) ( $(traverse 'a 'a) x2 )

    since the two Int arguments do not produce any effects in a traversal.

    One can envision a case for types that do not mention the last type parameter:

      $(traverse 'a 'b)          =  pure     -- when b does not contain a

    But this case will never materialize, since the aforementioned filtering
    removes all such types from consideration.
    See Note [Generated code for DeriveFoldable and DeriveTraversable].
    -}

    gen_Traversable_binds :: SrcSpan -> TyCon -> (LHsBinds GhcPs, BagDerivStuff)
    -- When the argument is phantom, we can use traverse = pure . coerce
    -- See Note [Phantom types with Functor, Foldable, and Traversable]
    gen_Traversable_binds loc tycon
      | Phantom <- last (tyConRoles tycon)
      = (unitBag traverse_bind, emptyBag)
      where
        traverse_name = L loc traverse_RDR
        traverse_bind = mkRdrFunBind traverse_name traverse_eqns
        traverse_eqns =
            [mkSimpleMatch traverse_match_ctxt
                           [nlWildPat, z_Pat]
                           (nlHsApps pure_RDR [nlHsApp coerce_Expr z_Expr])]
        traverse_match_ctxt = mkPrefixFunRhs traverse_name

    gen_Traversable_binds loc tycon
      = (unitBag traverse_bind, emptyBag)
      where
        data_cons = tyConDataCons tycon

        traverse_name = L loc traverse_RDR

        -- See Note [EmptyDataDecls with Functor, Foldable, and Traversable]
        traverse_bind = mkRdrFunBindEC 2 (nlHsApp pure_Expr)
                                       traverse_name traverse_eqns
        traverse_eqns = map traverse_eqn data_cons
        traverse_eqn con
          = evalState (match_for_con [f_Pat] con =<< parts) bs_RDRs
          where
            parts = sequence $ foldDataConArgs ft_trav con

        -- Yields 'Just' an expression if we're folding over a type that mentions
        -- the last type parameter of the datatype. Otherwise, yields 'Nothing'.
        -- See Note [FFoldType and functorLikeTraverse]
        ft_trav :: FFoldType (State [RdrName] (Maybe (LHsExpr GhcPs)))
        ft_trav
          = FT { ft_triv    = return Nothing
                 -- traverse f = pure x
               , ft_var     = return (Just f_Expr)
                 -- traverse f = f x
               , ft_tup     = \t gs -> do
                   gg  <- sequence gs
                   lam <- mkSimpleLam $ mkSimpleTupleCase match_for_con t gg
                   return (Just lam)
                 -- traverse f = \x -> case x of (a1,a2,..) ->
                 --                           liftA2 (,,) (g1 a1) (g2 a2) <*> ..
               , ft_ty_app  = \_ g -> fmap (nlHsApp traverse_Expr) <$> g
                 -- traverse f = traverse g
               , ft_forall  = \_ g -> g
               , ft_co_var  = panic "contravariant in ft_trav"
               , ft_fun     = panic "function in ft_trav"
               , ft_bad_app = panic "in other argument in ft_trav" }

        -- Con a1 a2 ... -> liftA2 (\b1 b2 ... -> Con b1 b2 ...) (g1 a1)
        --                    (g2 a2) <*> ...
        match_for_con :: [LPat GhcPs]
                      -> DataCon
                      -> [Maybe (LHsExpr GhcPs)]
                      -> State [RdrName] (LMatch GhcPs (LHsExpr GhcPs))
        match_for_con = mkSimpleConMatch2 CaseAlt $
                                                 \con xs -> return (mkApCon con xs)
          where
            -- liftA2 (\b1 b2 ... -> Con b1 b2 ...) x1 x2 <*> ..
            mkApCon :: LHsExpr GhcPs -> [LHsExpr GhcPs] -> LHsExpr GhcPs
            mkApCon con [] = nlHsApps pure_RDR [con]
            mkApCon con [x] = nlHsApps fmap_RDR [con,x]
            mkApCon con (x1:x2:xs) =
                foldl' appAp (nlHsApps liftA2_RDR [con,x1,x2]) xs
              where appAp x y = nlHsApps ap_RDR [x,y]

    -----------------------------------------------------------------------

    f_Expr, z_Expr, fmap_Expr, replace_Expr, mempty_Expr, foldMap_Expr,
        traverse_Expr, coerce_Expr, pure_Expr, true_Expr, false_Expr,
        all_Expr, null_Expr :: LHsExpr GhcPs
    f_Expr        = nlHsVar f_RDR
    z_Expr        = nlHsVar z_RDR
    fmap_Expr     = nlHsVar fmap_RDR
    replace_Expr  = nlHsVar replace_RDR
    mempty_Expr   = nlHsVar mempty_RDR
    foldMap_Expr  = nlHsVar foldMap_RDR
    traverse_Expr = nlHsVar traverse_RDR
    coerce_Expr   = nlHsVar (getRdrName coerceId)
    pure_Expr     = nlHsVar pure_RDR
    true_Expr     = nlHsVar true_RDR
    false_Expr    = nlHsVar false_RDR
    all_Expr      = nlHsVar all_RDR
    null_Expr     = nlHsVar null_RDR

    f_RDR, z_RDR :: RdrName
    f_RDR = mkVarUnqual (fsLit "f")
    z_RDR = mkVarUnqual (fsLit "z")

    as_RDRs, bs_RDRs :: [RdrName]
    as_RDRs = [ mkVarUnqual (mkFastString ("a"++show i)) | i <- [(1::Int) .. ] ]
    bs_RDRs = [ mkVarUnqual (mkFastString ("b"++show i)) | i <- [(1::Int) .. ] ]

    as_Vars, bs_Vars :: [LHsExpr GhcPs]
    as_Vars = map nlHsVar as_RDRs
    bs_Vars = map nlHsVar bs_RDRs

    f_Pat, z_Pat :: LPat GhcPs
    f_Pat = nlVarPat f_RDR
    z_Pat = nlVarPat z_RDR

    {-
    Note [DeriveFoldable with ExistentialQuantification]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Functor and Traversable instances can only be derived for data types whose
    last type parameter is truly universally polymorphic. For example:

      data T a b where
        T1 ::                 b   -> T a b   -- YES, b is unconstrained
        T2 :: Ord b   =>      b   -> T a b   -- NO, b is constrained by (Ord b)
        T3 :: b ~ Int =>      b   -> T a b   -- NO, b is constrained by (b ~ Int)
        T4 ::                 Int -> T a Int -- NO, this is just like T3
        T5 :: Ord a   => a -> b   -> T a b   -- YES, b is unconstrained, even
                                             -- though a is existential
        T6 ::                 Int -> T Int b -- YES, b is unconstrained

    For Foldable instances, however, we can completely lift the constraint that
    the last type parameter be truly universally polymorphic. This means that T
    (as defined above) can have a derived Foldable instance:

      instance Foldable (T a) where
        foldr f z (T1 b)   = f b z
        foldr f z (T2 b)   = f b z
        foldr f z (T3 b)   = f b z
        foldr f z (T4 b)   = z
        foldr f z (T5 a b) = f b z
        foldr f z (T6 a)   = z

        foldMap f (T1 b)   = f b
        foldMap f (T2 b)   = f b
        foldMap f (T3 b)   = f b
        foldMap f (T4 b)   = mempty
        foldMap f (T5 a b) = f b
        foldMap f (T6 a)   = mempty

    In a Foldable instance, it is safe to fold over an occurrence of the last type
    parameter that is not truly universally polymorphic. However, there is a bit
    of subtlety in determining what is actually an occurrence of a type parameter.
    T3 and T4, as defined above, provide one example:

      data T a b where
        ...
        T3 :: b ~ Int => b   -> T a b
        T4 ::            Int -> T a Int
        ...

      instance Foldable (T a) where
        ...
        foldr f z (T3 b) = f b z
        foldr f z (T4 b) = z
        ...
        foldMap f (T3 b) = f b
        foldMap f (T4 b) = mempty
        ...

    Notice that the argument of T3 is folded over, whereas the argument of T4 is
    not. This is because we only fold over constructor arguments that
    syntactically mention the universally quantified type parameter of that
    particular data constructor. See foldDataConArgs for how this is implemented.

    As another example, consider the following data type. The argument of each
    constructor has the same type as the last type parameter:

      data E a where
        E1 :: (a ~ Int) => a   -> E a
        E2 ::              Int -> E Int
        E3 :: (a ~ Int) => a   -> E Int
        E4 :: (a ~ Int) => Int -> E a

    Only E1's argument is an occurrence of a universally quantified type variable
    that is syntactically equivalent to the last type parameter, so only E1's
    argument will be folded over in a derived Foldable instance.

    See #10447 for the original discussion on this feature. Also see
    https://gitlab.haskell.org/ghc/ghc/wikis/commentary/compiler/derive-functor
    for a more in-depth explanation.

    Note [FFoldType and functorLikeTraverse]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Deriving Functor, Foldable, and Traversable all require generating expressions
    which perform an operation on each argument of a data constructor depending
    on the argument's type. In particular, a generated operation can be different
    depending on whether the type mentions the last type variable of the datatype
    (e.g., if you have data T a = MkT a Int, then a generated foldr expression would
    fold over the first argument of MkT, but not the second).

    This pattern is abstracted with the FFoldType datatype, which provides hooks
    for the user to specify how a constructor argument should be folded when it
    has a type with a particular "shape". The shapes are as follows (assume that
    a is the last type variable in a given datatype):

    * ft_triv:    The type does not mention the last type variable at all.
                  Examples: Int, b

    * ft_var:     The type is syntactically equal to the last type variable.
                  Moreover, the type appears in a covariant position (see
                  the Deriving Functor instances section of the user's guide
                  for an in-depth explanation of covariance vs. contravariance).
                  Example: a (covariantly)

    * ft_co_var:  The type is syntactically equal to the last type variable.
                  Moreover, the type appears in a contravariant position.
                  Example: a (contravariantly)

    * ft_fun:     A function type which mentions the last type variable in
                  the argument position, result position or both.
                  Examples: a -> Int, Int -> a, Maybe a -> [a]

    * ft_tup:     A tuple type which mentions the last type variable in at least
                  one of its fields. The TyCon argument of ft_tup represents the
                  particular tuple's type constructor.
                  Examples: (a, Int), (Maybe a, [a], Either a Int), (# Int, a #)

    * ft_ty_app:  A type is being applied to the last type parameter, where the
                  applied type does not mention the last type parameter (if it
                  did, it would fall under ft_bad_app). The Type argument to
                  ft_ty_app represents the applied type.

                  Note that functions, tuples, and foralls are distinct cases
                  and take precedence of ft_ty_app. (For example, (Int -> a) would
                  fall under (ft_fun Int a), not (ft_ty_app ((->) Int) a).
                  Examples: Maybe a, Either b a

    * ft_bad_app: A type application uses the last type parameter in a position
                  other than the last argument. This case is singled out because
                  Functor, Foldable, and Traversable instances cannot be derived
                  for datatypes containing arguments with such types.
                  Examples: Either a Int, Const a b

    * ft_forall:  A forall'd type mentions the last type parameter on its right-
                  hand side (and is not quantified on the left-hand side). This
                  case is present mostly for plumbing purposes.
                  Example: forall b. Either b a

    If FFoldType describes a strategy for folding subcomponents of a Type, then
    functorLikeTraverse is the function that applies that strategy to the entirety
    of a Type, returning the final folded-up result.

    foldDataConArgs applies functorLikeTraverse to every argument type of a
    constructor, returning a list of the fold results. This makes foldDataConArgs
    a natural way to generate the subexpressions in a generated fmap, foldr,
    foldMap, or traverse definition (the subexpressions must then be combined in
    a method-specific fashion to form the final generated expression).

    Deriving Generic1 also does validity checking by looking for the last type
    variable in certain positions of a constructor's argument types, so it also
    uses foldDataConArgs. See Note [degenerate use of FFoldType] in TcGenGenerics.

    Note [Generated code for DeriveFoldable and DeriveTraversable]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We adapt the algorithms for -XDeriveFoldable and -XDeriveTraversable based on
    that of -XDeriveFunctor. However, there an important difference between deriving
    the former two typeclasses and the latter one, which is best illustrated by the
    following scenario:

      data WithInt a = WithInt a Int# deriving (Functor, Foldable, Traversable)

    The generated code for the Functor instance is straightforward:

      instance Functor WithInt where
        fmap f (WithInt a i) = WithInt (f a) i

    But if we use too similar of a strategy for deriving the Foldable and
    Traversable instances, we end up with this code:

      instance Foldable WithInt where
        foldMap f (WithInt a i) = f a <> mempty

      instance Traversable WithInt where
        traverse f (WithInt a i) = fmap WithInt (f a) <*> pure i

    This is unsatisfying for two reasons:

    1. The Traversable instance doesn't typecheck! Int# is of kind #, but pure
       expects an argument whose type is of kind *. This effectively prevents
       Traversable from being derived for any datatype with an unlifted argument
       type (#11174).

    2. The generated code contains superfluous expressions. By the Monoid laws,
       we can reduce (f a <> mempty) to (f a), and by the Applicative laws, we can
       reduce (fmap WithInt (f a) <*> pure i) to (fmap (\b -> WithInt b i) (f a)).

    We can fix both of these issues by incorporating a slight twist to the usual
    algorithm that we use for -XDeriveFunctor. The differences can be summarized
    as follows:

    1. In the generated expression, we only fold over arguments whose types
       mention the last type parameter. Any other argument types will simply
       produce useless 'mempty's or 'pure's, so they can be safely ignored.

    2. In the case of -XDeriveTraversable, instead of applying ConName,
       we apply (\b_i ... b_k -> ConName a_1 ... a_n), where

       * ConName has n arguments
       * {b_i, ..., b_k} is a subset of {a_1, ..., a_n} whose indices correspond
         to the arguments whose types mention the last type parameter. As a
         consequence, taking the difference of {a_1, ..., a_n} and
         {b_i, ..., b_k} yields the all the argument values of ConName whose types
         do not mention the last type parameter. Note that [i, ..., k] is a
         strictly increasing—but not necessarily consecutive—integer sequence.

         For example, the datatype

           data Foo a = Foo Int a Int a

         would generate the following Traversable instance:

           instance Traversable Foo where
             traverse f (Foo a1 a2 a3 a4) =
               fmap (\b2 b4 -> Foo a1 b2 a3 b4) (f a2) <*> f a4

    Technically, this approach would also work for -XDeriveFunctor as well, but we
    decide not to do so because:

    1. There's not much benefit to generating, e.g., ((\b -> WithInt b i) (f a))
       instead of (WithInt (f a) i).

    2. There would be certain datatypes for which the above strategy would
       generate Functor code that would fail to typecheck. For example:

         data Bar f a = Bar (forall f. Functor f => f a) deriving Functor

       With the conventional algorithm, it would generate something like:

         fmap f (Bar a) = Bar (fmap f a)

       which typechecks. But with the strategy mentioned above, it would generate:

         fmap f (Bar a) = (\b -> Bar b) (fmap f a)

       which does not typecheck, since GHC cannot unify the rank-2 type variables
       in the types of b and (fmap f a).

    Note [Phantom types with Functor, Foldable, and Traversable]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Given a type F :: * -> * whose type argument has a phantom role, we can always
    produce lawful Functor and Traversable instances using

        fmap _ = coerce
        traverse _ = pure . coerce

    Indeed, these are equivalent to any *strictly lawful* instances one could
    write, except that this definition of 'traverse' may be lazier.  That is, if
    instances obey the laws under true equality (rather than up to some equivalence
    relation), then they will be essentially equivalent to these. These definitions
    are incredibly cheap, so we want to use them even if it means ignoring some
    non-strictly-lawful instance in an embedded type.

    Foldable has far fewer laws to work with, which leaves us unwelcome
    freedom in implementing it. At a minimum, we would like to ensure that
    a derived foldMap is always at least as good as foldMapDefault with a
    derived traverse. To accomplish that, we must define

       foldMap _ _ = mempty

    in these cases.

    This may have different strictness properties from a standard derivation.
    Consider

       data NotAList a = Nil | Cons (NotAList a) deriving Foldable

    The usual deriving mechanism would produce

       foldMap _ Nil = mempty
       foldMap f (Cons x) = foldMap f x

    which is strict in the entire spine of the NotAList.

    Final point: why do we even care about such types? Users will rarely if ever
    map, fold, or traverse over such things themselves, but other derived
    instances may:

       data Hasn'tAList a = NotHere a (NotAList a) deriving Foldable

    Note [EmptyDataDecls with Functor, Foldable, and Traversable]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    There are some slightly tricky decisions to make about how to handle
    Functor, Foldable, and Traversable instances for types with no constructors.
    For fmap, the two basic options are

       fmap _ _ = error "Sorry, no constructors"

    or

       fmap _ z = case z of

    In most cases, the latter is more helpful: if the thunk passed to fmap
    throws an exception, we're generally going to be much more interested in
    that exception than in the fact that there aren't any constructors.

    In order to match the semantics for phantoms (see note above), we need to
    be a bit careful about 'traverse'. The obvious definition would be

       traverse _ z = case z of

    but this is stricter than the one for phantoms. We instead use

       traverse _ z = pure $ case z of

    For foldMap, the obvious choices are

       foldMap _ _ = mempty

    or

       foldMap _ z = case z of

    We choose the first one to be consistent with what foldMapDefault does for
    a derived Traversable instance.
    -}