GHC/Core/TyCo/FVs.hs

    {-# LANGUAGE CPP #-}

    module GHC.Core.TyCo.FVs
      (     shallowTyCoVarsOfType, shallowTyCoVarsOfTypes,
            tyCoVarsOfType,        tyCoVarsOfTypes,
            tyCoVarsOfTypeDSet, tyCoVarsOfTypesDSet,

            tyCoFVsBndr, tyCoFVsVarBndr, tyCoFVsVarBndrs,
            tyCoFVsOfType, tyCoVarsOfTypeList,
            tyCoFVsOfTypes, tyCoVarsOfTypesList,
            deepTcvFolder,

            shallowTyCoVarsOfTyVarEnv, shallowTyCoVarsOfCoVarEnv,

            shallowTyCoVarsOfCo, shallowTyCoVarsOfCos,
            tyCoVarsOfCo, tyCoVarsOfCos, tyCoVarsOfMCo,
            coVarsOfType, coVarsOfTypes,
            coVarsOfCo, coVarsOfCos,
            tyCoVarsOfCoDSet,
            tyCoFVsOfCo, tyCoFVsOfCos,
            tyCoVarsOfCoList,

            almostDevoidCoVarOfCo,

            -- Injective free vars
            injectiveVarsOfType, injectiveVarsOfTypes,
            invisibleVarsOfType, invisibleVarsOfTypes,

            -- Any and No Free vars
            anyFreeVarsOfType, anyFreeVarsOfTypes, anyFreeVarsOfCo,
            noFreeVarsOfType, noFreeVarsOfTypes, noFreeVarsOfCo,

            -- * Well-scoped free variables
            scopedSort, tyCoVarsOfTypeWellScoped,
            tyCoVarsOfTypesWellScoped,

            -- * Closing over kinds
            closeOverKindsDSet, closeOverKindsList,
            closeOverKinds,

            -- * Raw materials
            Endo(..), runTyCoVars
      ) where

    #include "HsVersions.h"

    import GHC.Prelude

    import {-# SOURCE #-} GHC.Core.Type (coreView, partitionInvisibleTypes)

    import Data.Monoid as DM ( Endo(..), Any(..) )
    import GHC.Core.TyCo.Rep
    import GHC.Core.TyCon
    import GHC.Types.Var
    import GHC.Utils.FV

    import GHC.Types.Unique.FM
    import GHC.Types.Var.Set
    import GHC.Types.Var.Env
    import GHC.Utils.Misc
    import GHC.Utils.Panic

    {-
    %************************************************************************
    %*                                                                      *
                     Free variables of types and coercions
    %*                                                                      *
    %************************************************************************
    -}

    {- Note [Shallow and deep free variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Definitions

    * Shallow free variables of a type: the variables
      affected by substitution. Specifically, the (TyVarTy tv)
      and (CoVar cv) that appear
        - In the type and coercions appearing in the type
        - In shallow free variables of the kind of a Forall binder
      but NOT in the kind of the /occurrences/ of a type variable.

    * Deep free variables of a type: shallow free variables, plus
      the deep free variables of the kinds of those variables.
      That is,  deepFVs( t ) = closeOverKinds( shallowFVs( t ) )

    Examples:

      Type                     Shallow     Deep
      ---------------------------------
      (a : (k:Type))           {a}        {a,k}
      forall (a:(k:Type)). a   {k}        {k}
      (a:k->Type) (b:k)        {a,b}      {a,b,k}
    -}

    {- Note [Free variables of types]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The family of functions tyCoVarsOfType, tyCoVarsOfTypes etc, returns
    a VarSet that is closed over the types of its variables.  More precisely,
      if    S = tyCoVarsOfType( t )
      and   (a:k) is in S
      then  tyCoVarsOftype( k ) is a subset of S

    Example: The tyCoVars of this ((a:* -> k) Int) is {a, k}.

    We could /not/ close over the kinds of the variable occurrences, and
    instead do so at call sites, but it seems that we always want to do
    so, so it's easiest to do it here.

    It turns out that getting the free variables of types is performance critical,
    so we profiled several versions, exploring different implementation strategies.

    1. Baseline version: uses FV naively. Essentially:

       tyCoVarsOfType ty = fvVarSet $ tyCoFVsOfType ty

       This is not nice, because FV introduces some overhead to implement
       determinism, and through its "interesting var" function, neither of which
       we need here, so they are a complete waste.

    2. UnionVarSet version: instead of reusing the FV-based code, we simply used
       VarSets directly, trying to avoid the overhead of FV. E.g.:

       -- FV version:
       tyCoFVsOfType (AppTy fun arg)    a b c = (tyCoFVsOfType fun `unionFV` tyCoFVsOfType arg) a b c

       -- UnionVarSet version:
       tyCoVarsOfType (AppTy fun arg)    = (tyCoVarsOfType fun `unionVarSet` tyCoVarsOfType arg)

       This looks deceptively similar, but while FV internally builds a list- and
       set-generating function, the VarSet functions manipulate sets directly, and
       the latter performs a lot worse than the naive FV version.

    3. Accumulator-style VarSet version: this is what we use now. We do use VarSet
       as our data structure, but delegate the actual work to a new
       ty_co_vars_of_...  family of functions, which use accumulator style and the
       "in-scope set" filter found in the internals of FV, but without the
       determinism overhead.

    See #14880.

    Note [Closing over free variable kinds]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    tyCoVarsOfType and tyCoFVsOfType, while traversing a type, will also close over
    free variable kinds. In previous GHC versions, this happened naively: whenever
    we would encounter an occurrence of a free type variable, we would close over
    its kind. This, however is wrong for two reasons (see #14880):

    1. Efficiency. If we have Proxy (a::k) -> Proxy (a::k) -> Proxy (a::k), then
       we don't want to have to traverse k more than once.

    2. Correctness. Imagine we have forall k. b -> k, where b has
       kind k, for some k bound in an outer scope. If we look at b's kind inside
       the forall, we'll collect that k is free and then remove k from the set of
       free variables. This is plain wrong. We must instead compute that b is free
       and then conclude that b's kind is free.

    An obvious first approach is to move the closing-over-kinds from the
    occurrences of a type variable to after finding the free vars - however, this
    turns out to introduce performance regressions, and isn't even entirely
    correct.

    In fact, it isn't even important *when* we close over kinds; what matters is
    that we handle each type var exactly once, and that we do it in the right
    context.

    So the next approach we tried was to use the "in-scope set" part of FV or the
    equivalent argument in the accumulator-style `ty_co_vars_of_type` function, to
    say "don't bother with variables we have already closed over". This should work
    fine in theory, but the code is complicated and doesn't perform well.

    But there is a simpler way, which is implemented here. Consider the two points
    above:

    1. Efficiency: we now have an accumulator, so the second time we encounter 'a',
       we'll ignore it, certainly not looking at its kind - this is why
       pre-checking set membership before inserting ends up not only being faster,
       but also being correct.

    2. Correctness: we have an "in-scope set" (I think we should call it it a
      "bound-var set"), specifying variables that are bound by a forall in the type
      we are traversing; we simply ignore these variables, certainly not looking at
      their kind.

    So now consider:

        forall k. b -> k

    where b :: k->Type is free; but of course, it's a different k! When looking at
    b -> k we'll have k in the bound-var set. So we'll ignore the k. But suppose
    this is our first encounter with b; we want the free vars of its kind. But we
    want to behave as if we took the free vars of its kind at the end; that is,
    with no bound vars in scope.

    So the solution is easy. The old code was this:

      ty_co_vars_of_type (TyVarTy v) is acc
        | v `elemVarSet` is  = acc
        | v `elemVarSet` acc = acc
        | otherwise          = ty_co_vars_of_type (tyVarKind v) is (extendVarSet acc v)

    Now all we need to do is take the free vars of tyVarKind v *with an empty
    bound-var set*, thus:

    ty_co_vars_of_type (TyVarTy v) is acc
      | v `elemVarSet` is  = acc
      | v `elemVarSet` acc = acc
      | otherwise          = ty_co_vars_of_type (tyVarKind v) emptyVarSet (extendVarSet acc v)
                                                              ^^^^^^^^^^^

    And that's it. This works because a variable is either bound or free. If it is bound,
    then we won't look at it at all. If it is free, then all the variables free in its
    kind are free -- regardless of whether some local variable has the same Unique.
    So if we're looking at a variable occurrence at all, then all variables in its
    kind are free.
    -}

    {- *********************************************************************
    *                                                                      *
              Endo for free variables
    *                                                                      *
    ********************************************************************* -}

    {- Note [Acumulating parameter free variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We can use foldType to build an accumulating-parameter version of a
    free-var finder, thus:

        fvs :: Type -> TyCoVarSet
        fvs ty = appEndo (foldType folder ty) emptyVarSet

    Recall that
        foldType :: TyCoFolder env a -> env -> Type -> a

        newtype Endo a = Endo (a -> a)   -- In Data.Monoid
        instance Monoid a => Monoid (Endo a) where
           (Endo f) `mappend` (Endo g) = Endo (f.g)

        appEndo :: Endo a -> a -> a
        appEndo (Endo f) x = f x

    So `mappend` for Endos is just function composition.

    It's very important that, after optimisation, we end up with
    * an arity-three function
    * that is strict in the accumulator

       fvs env (TyVarTy v) acc
          | v `elemVarSet` env = acc
          | v `elemVarSet` acc = acc
          | otherwise          = acc `extendVarSet` v
       fvs env (AppTy t1 t2)   = fvs env t1 (fvs env t2 acc)
       ...

    The "strict in the accumulator" part is to ensure that in the
    AppTy equation we don't build a thunk for (fvs env t2 acc).

    The optimiser does do all this, but not very robustly. It depends
    critially on the basic arity-2 function not being exported, so that
    all its calls are visibly to three arguments. This analysis is
    done by the Call Arity pass.

    TL;DR: check this regularly!
    -}

    runTyCoVars :: Endo TyCoVarSet -> TyCoVarSet
    {-# INLINE runTyCoVars #-}
    runTyCoVars f = appEndo f emptyVarSet

    noView :: Type -> Maybe Type
    noView _ = Nothing

    {- *********************************************************************
    *                                                                      *
              Deep free variables
              See Note [Shallow and deep free variables]
    *                                                                      *
    ********************************************************************* -}

    tyCoVarsOfType :: Type -> TyCoVarSet
    tyCoVarsOfType ty = runTyCoVars (deep_ty ty)
    -- Alternative:
    --   tyCoVarsOfType ty = closeOverKinds (shallowTyCoVarsOfType ty)

    tyCoVarsOfTypes :: [Type] -> TyCoVarSet
    tyCoVarsOfTypes tys = runTyCoVars (deep_tys tys)
    -- Alternative:
    --   tyCoVarsOfTypes tys = closeOverKinds (shallowTyCoVarsOfTypes tys)

    tyCoVarsOfCo :: Coercion -> TyCoVarSet
    -- See Note [Free variables of Coercions]
    tyCoVarsOfCo co = runTyCoVars (deep_co co)

    tyCoVarsOfMCo :: MCoercion -> TyCoVarSet
    tyCoVarsOfMCo MRefl    = emptyVarSet
    tyCoVarsOfMCo (MCo co) = tyCoVarsOfCo co

    tyCoVarsOfCos :: [Coercion] -> TyCoVarSet
    tyCoVarsOfCos cos = runTyCoVars (deep_cos cos)

    deep_ty  :: Type       -> Endo TyCoVarSet
    deep_tys :: [Type]     -> Endo TyCoVarSet
    deep_co  :: Coercion   -> Endo TyCoVarSet
    deep_cos :: [Coercion] -> Endo TyCoVarSet
    (deep_ty, deep_tys, deep_co, deep_cos) = foldTyCo deepTcvFolder emptyVarSet

    deepTcvFolder :: TyCoFolder TyCoVarSet (Endo TyCoVarSet)
    deepTcvFolder = TyCoFolder { tcf_view = noView
                               , tcf_tyvar = do_tcv, tcf_covar = do_tcv
                               , tcf_hole  = do_hole, tcf_tycobinder = do_bndr }
      where
        do_tcv is v = Endo do_it
          where
            do_it acc | v `elemVarSet` is  = acc
                      | v `elemVarSet` acc = acc
                      | otherwise          = appEndo (deep_ty (varType v)) $
                                             acc `extendVarSet` v

        do_bndr is tcv _ = extendVarSet is tcv
        do_hole is hole  = do_tcv is (coHoleCoVar hole)
                           -- See Note [CoercionHoles and coercion free variables]
                           -- in GHC.Core.TyCo.Rep

    {- *********************************************************************
    *                                                                      *
              Shallow free variables
              See Note [Shallow and deep free variables]
    *                                                                      *
    ********************************************************************* -}

    shallowTyCoVarsOfType :: Type -> TyCoVarSet
    -- See Note [Free variables of types]
    shallowTyCoVarsOfType ty = runTyCoVars (shallow_ty ty)

    shallowTyCoVarsOfTypes :: [Type] -> TyCoVarSet
    shallowTyCoVarsOfTypes tys = runTyCoVars (shallow_tys tys)

    shallowTyCoVarsOfCo :: Coercion -> TyCoVarSet
    shallowTyCoVarsOfCo co = runTyCoVars (shallow_co co)

    shallowTyCoVarsOfCos :: [Coercion] -> TyCoVarSet
    shallowTyCoVarsOfCos cos = runTyCoVars (shallow_cos cos)

    -- | Returns free variables of types, including kind variables as
    -- a non-deterministic set. For type synonyms it does /not/ expand the
    -- synonym.
    shallowTyCoVarsOfTyVarEnv :: TyVarEnv Type -> TyCoVarSet
    -- See Note [Free variables of types]
    shallowTyCoVarsOfTyVarEnv tys = shallowTyCoVarsOfTypes (nonDetEltsUFM tys)
      -- It's OK to use nonDetEltsUFM here because we immediately
      -- forget the ordering by returning a set

    shallowTyCoVarsOfCoVarEnv :: CoVarEnv Coercion -> TyCoVarSet
    shallowTyCoVarsOfCoVarEnv cos = shallowTyCoVarsOfCos (nonDetEltsUFM cos)
      -- It's OK to use nonDetEltsUFM here because we immediately
      -- forget the ordering by returning a set

    shallow_ty  :: Type       -> Endo TyCoVarSet
    shallow_tys :: [Type]     -> Endo TyCoVarSet
    shallow_co  :: Coercion   -> Endo TyCoVarSet
    shallow_cos :: [Coercion] -> Endo TyCoVarSet
    (shallow_ty, shallow_tys, shallow_co, shallow_cos) = foldTyCo shallowTcvFolder emptyVarSet

    shallowTcvFolder :: TyCoFolder TyCoVarSet (Endo TyCoVarSet)
    shallowTcvFolder = TyCoFolder { tcf_view = noView
                                  , tcf_tyvar = do_tcv, tcf_covar = do_tcv
                                  , tcf_hole  = do_hole, tcf_tycobinder = do_bndr }
      where
        do_tcv is v = Endo do_it
          where
            do_it acc | v `elemVarSet` is  = acc
                      | v `elemVarSet` acc = acc
                      | otherwise          = acc `extendVarSet` v

        do_bndr is tcv _ = extendVarSet is tcv
        do_hole _ _  = mempty   -- Ignore coercion holes

    {- *********************************************************************
    *                                                                      *
              Free coercion variables
    *                                                                      *
    ********************************************************************* -}

    {- Note [Finding free coercion varibles]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Here we are only interested in the free /coercion/ variables.
    We can achieve this through a slightly different TyCo folder.

    Notice that we look deeply, into kinds.

    See #14880.
    -}

    coVarsOfType  :: Type       -> CoVarSet
    coVarsOfTypes :: [Type]     -> CoVarSet
    coVarsOfCo    :: Coercion   -> CoVarSet
    coVarsOfCos   :: [Coercion] -> CoVarSet

    coVarsOfType  ty  = runTyCoVars (deep_cv_ty ty)
    coVarsOfTypes tys = runTyCoVars (deep_cv_tys tys)
    coVarsOfCo    co  = runTyCoVars (deep_cv_co co)
    coVarsOfCos   cos = runTyCoVars (deep_cv_cos cos)

    deep_cv_ty  :: Type       -> Endo CoVarSet
    deep_cv_tys :: [Type]     -> Endo CoVarSet
    deep_cv_co  :: Coercion   -> Endo CoVarSet
    deep_cv_cos :: [Coercion] -> Endo CoVarSet
    (deep_cv_ty, deep_cv_tys, deep_cv_co, deep_cv_cos) = foldTyCo deepCoVarFolder emptyVarSet

    deepCoVarFolder :: TyCoFolder TyCoVarSet (Endo CoVarSet)
    deepCoVarFolder = TyCoFolder { tcf_view = noView
                                 , tcf_tyvar = do_tyvar, tcf_covar = do_covar
                                 , tcf_hole  = do_hole, tcf_tycobinder = do_bndr }
      where
        do_tyvar _ _  = mempty
          -- This do_tyvar means we won't see any CoVars in this
          -- TyVar's kind.   This may be wrong; but it's the way it's
          -- always been.  And its awkward to change, because
          -- the tyvar won't end up in the accumulator, so
          -- we'd look repeatedly.  Blargh.

        do_covar is v = Endo do_it
          where
            do_it acc | v `elemVarSet` is  = acc
                      | v `elemVarSet` acc = acc
                      | otherwise          = appEndo (deep_cv_ty (varType v)) $
                                             acc `extendVarSet` v

        do_bndr is tcv _ = extendVarSet is tcv
        do_hole is hole  = do_covar is (coHoleCoVar hole)
                           -- See Note [CoercionHoles and coercion free variables]
                           -- in GHC.Core.TyCo.Rep

    {- *********************************************************************
    *                                                                      *
              Closing over kinds
    *                                                                      *
    ********************************************************************* -}

    ------------- Closing over kinds -----------------

    closeOverKinds :: TyCoVarSet -> TyCoVarSet
    -- For each element of the input set,
    -- add the deep free variables of its kind
    closeOverKinds vs = nonDetStrictFoldVarSet do_one vs vs
      where
        do_one v acc = appEndo (deep_ty (varType v)) acc

    {- --------------- Alternative version 1 (using FV) ------------
    closeOverKinds = fvVarSet . closeOverKindsFV . nonDetEltsUniqSet
    -}

    {- ---------------- Alternative version 2 -------------

    -- | Add the kind variables free in the kinds of the tyvars in the given set.
    -- Returns a non-deterministic set.
    closeOverKinds :: TyCoVarSet -> TyCoVarSet
    closeOverKinds vs
       = go vs vs
      where
        go :: VarSet   -- Work list
           -> VarSet   -- Accumulator, always a superset of wl
           -> VarSet
        go wl acc
          | isEmptyVarSet wl = acc
          | otherwise        = go wl_kvs (acc `unionVarSet` wl_kvs)
          where
            k v inner_acc = ty_co_vars_of_type (varType v) acc inner_acc
            wl_kvs = nonDetFoldVarSet k emptyVarSet wl
            -- wl_kvs = union of shallow free vars of the kinds of wl
            --          but don't bother to collect vars in acc

    -}

    {- ---------------- Alternative version 3 -------------
    -- | Add the kind variables free in the kinds of the tyvars in the given set.
    -- Returns a non-deterministic set.
    closeOverKinds :: TyVarSet -> TyVarSet
    closeOverKinds vs = close_over_kinds vs emptyVarSet

    close_over_kinds :: TyVarSet  -- Work list
                     -> TyVarSet  -- Accumulator
                     -> TyVarSet
    -- Precondition: in any call (close_over_kinds wl acc)
    --  for every tv in acc, the shallow kind-vars of tv
    --  are either in the work list wl, or in acc
    -- Postcondition: result is the deep free vars of (wl `union` acc)
    close_over_kinds wl acc
      = nonDetFoldVarSet do_one acc wl
      where
        do_one :: Var -> TyVarSet -> TyVarSet
        -- (do_one v acc) adds v and its deep free-vars to acc
        do_one v acc | v `elemVarSet` acc
                     = acc
                     | otherwise
                     = close_over_kinds (shallowTyCoVarsOfType (varType v)) $
                       acc `extendVarSet` v
    -}

    {- *********************************************************************
    *                                                                      *
              The FV versions return deterministic results
    *                                                                      *
    ********************************************************************* -}

    -- | Given a list of tyvars returns a deterministic FV computation that
    -- returns the given tyvars with the kind variables free in the kinds of the
    -- given tyvars.
    closeOverKindsFV :: [TyVar] -> FV
    closeOverKindsFV tvs =
      mapUnionFV (tyCoFVsOfType . tyVarKind) tvs `unionFV` mkFVs tvs

    -- | Add the kind variables free in the kinds of the tyvars in the given set.
    -- Returns a deterministically ordered list.
    closeOverKindsList :: [TyVar] -> [TyVar]
    closeOverKindsList tvs = fvVarList $ closeOverKindsFV tvs

    -- | Add the kind variables free in the kinds of the tyvars in the given set.
    -- Returns a deterministic set.
    closeOverKindsDSet :: DTyVarSet -> DTyVarSet
    closeOverKindsDSet = fvDVarSet . closeOverKindsFV . dVarSetElems

    -- | `tyCoFVsOfType` that returns free variables of a type in a deterministic
    -- set. For explanation of why using `VarSet` is not deterministic see
    -- Note [Deterministic FV] in "GHC.Utils.FV".
    tyCoVarsOfTypeDSet :: Type -> DTyCoVarSet
    -- See Note [Free variables of types]
    tyCoVarsOfTypeDSet ty = fvDVarSet $ tyCoFVsOfType ty

    -- | `tyCoFVsOfType` that returns free variables of a type in deterministic
    -- order. For explanation of why using `VarSet` is not deterministic see
    -- Note [Deterministic FV] in "GHC.Utils.FV".
    tyCoVarsOfTypeList :: Type -> [TyCoVar]
    -- See Note [Free variables of types]
    tyCoVarsOfTypeList ty = fvVarList $ tyCoFVsOfType ty

    -- | Returns free variables of types, including kind variables as
    -- a deterministic set. For type synonyms it does /not/ expand the
    -- synonym.
    tyCoVarsOfTypesDSet :: [Type] -> DTyCoVarSet
    -- See Note [Free variables of types]
    tyCoVarsOfTypesDSet tys = fvDVarSet $ tyCoFVsOfTypes tys

    -- | Returns free variables of types, including kind variables as
    -- a deterministically ordered list. For type synonyms it does /not/ expand the
    -- synonym.
    tyCoVarsOfTypesList :: [Type] -> [TyCoVar]
    -- See Note [Free variables of types]
    tyCoVarsOfTypesList tys = fvVarList $ tyCoFVsOfTypes tys

    -- | The worker for `tyCoFVsOfType` and `tyCoFVsOfTypeList`.
    -- The previous implementation used `unionVarSet` which is O(n+m) and can
    -- make the function quadratic.
    -- It's exported, so that it can be composed with
    -- other functions that compute free variables.
    -- See Note [FV naming conventions] in "GHC.Utils.FV".
    --
    -- Eta-expanded because that makes it run faster (apparently)
    -- See Note [FV eta expansion] in "GHC.Utils.FV" for explanation.
    tyCoFVsOfType :: Type -> FV
    -- See Note [Free variables of types]
    tyCoFVsOfType (TyVarTy v)        f bound_vars (acc_list, acc_set)
      | not (f v) = (acc_list, acc_set)
      | v `elemVarSet` bound_vars = (acc_list, acc_set)
      | v `elemVarSet` acc_set = (acc_list, acc_set)
      | otherwise = tyCoFVsOfType (tyVarKind v) f
                                   emptyVarSet   -- See Note [Closing over free variable kinds]
                                   (v:acc_list, extendVarSet acc_set v)
    tyCoFVsOfType (TyConApp _ tys)   f bound_vars acc = tyCoFVsOfTypes tys f bound_vars acc
    tyCoFVsOfType (LitTy {})         f bound_vars acc = emptyFV f bound_vars acc
    tyCoFVsOfType (AppTy fun arg)    f bound_vars acc = (tyCoFVsOfType fun `unionFV` tyCoFVsOfType arg) f bound_vars acc
    tyCoFVsOfType (FunTy _ w arg res)  f bound_vars acc = (tyCoFVsOfType w `unionFV` tyCoFVsOfType arg `unionFV` tyCoFVsOfType res) f bound_vars acc
    tyCoFVsOfType (ForAllTy bndr ty) f bound_vars acc = tyCoFVsBndr bndr (tyCoFVsOfType ty)  f bound_vars acc
    tyCoFVsOfType (CastTy ty co)     f bound_vars acc = (tyCoFVsOfType ty `unionFV` tyCoFVsOfCo co) f bound_vars acc
    tyCoFVsOfType (CoercionTy co)    f bound_vars acc = tyCoFVsOfCo co f bound_vars acc

    tyCoFVsBndr :: TyCoVarBinder -> FV -> FV
    -- Free vars of (forall b. <thing with fvs>)
    tyCoFVsBndr (Bndr tv _) fvs = tyCoFVsVarBndr tv fvs

    tyCoFVsVarBndrs :: [Var] -> FV -> FV
    tyCoFVsVarBndrs vars fvs = foldr tyCoFVsVarBndr fvs vars

    tyCoFVsVarBndr :: Var -> FV -> FV
    tyCoFVsVarBndr var fvs
      = tyCoFVsOfType (varType var)   -- Free vars of its type/kind
        `unionFV` delFV var fvs       -- Delete it from the thing-inside

    tyCoFVsOfTypes :: [Type] -> FV
    -- See Note [Free variables of types]
    tyCoFVsOfTypes (ty:tys) fv_cand in_scope acc = (tyCoFVsOfType ty `unionFV` tyCoFVsOfTypes tys) fv_cand in_scope acc
    tyCoFVsOfTypes []       fv_cand in_scope acc = emptyFV fv_cand in_scope acc

    -- | Get a deterministic set of the vars free in a coercion
    tyCoVarsOfCoDSet :: Coercion -> DTyCoVarSet
    -- See Note [Free variables of types]
    tyCoVarsOfCoDSet co = fvDVarSet $ tyCoFVsOfCo co

    tyCoVarsOfCoList :: Coercion -> [TyCoVar]
    -- See Note [Free variables of types]
    tyCoVarsOfCoList co = fvVarList $ tyCoFVsOfCo co

    tyCoFVsOfMCo :: MCoercion -> FV
    tyCoFVsOfMCo MRefl    = emptyFV
    tyCoFVsOfMCo (MCo co) = tyCoFVsOfCo co

    tyCoFVsOfCo :: Coercion -> FV
    -- Extracts type and coercion variables from a coercion
    -- See Note [Free variables of types]
    tyCoFVsOfCo (Refl ty) fv_cand in_scope acc
      = tyCoFVsOfType ty fv_cand in_scope acc
    tyCoFVsOfCo (GRefl _ ty mco) fv_cand in_scope acc
      = (tyCoFVsOfType ty `unionFV` tyCoFVsOfMCo mco) fv_cand in_scope acc
    tyCoFVsOfCo (TyConAppCo _ _ cos) fv_cand in_scope acc = tyCoFVsOfCos cos fv_cand in_scope acc
    tyCoFVsOfCo (AppCo co arg) fv_cand in_scope acc
      = (tyCoFVsOfCo co `unionFV` tyCoFVsOfCo arg) fv_cand in_scope acc
    tyCoFVsOfCo (ForAllCo tv kind_co co) fv_cand in_scope acc
      = (tyCoFVsVarBndr tv (tyCoFVsOfCo co) `unionFV` tyCoFVsOfCo kind_co) fv_cand in_scope acc
    tyCoFVsOfCo (FunCo _ w co1 co2)    fv_cand in_scope acc
      = (tyCoFVsOfCo co1 `unionFV` tyCoFVsOfCo co2 `unionFV` tyCoFVsOfCo w) fv_cand in_scope acc
    tyCoFVsOfCo (CoVarCo v) fv_cand in_scope acc
      = tyCoFVsOfCoVar v fv_cand in_scope acc
    tyCoFVsOfCo (HoleCo h) fv_cand in_scope acc
      = tyCoFVsOfCoVar (coHoleCoVar h) fv_cand in_scope acc
        -- See Note [CoercionHoles and coercion free variables]
    tyCoFVsOfCo (AxiomInstCo _ _ cos) fv_cand in_scope acc = tyCoFVsOfCos cos fv_cand in_scope acc
    tyCoFVsOfCo (UnivCo p _ t1 t2) fv_cand in_scope acc
      = (tyCoFVsOfProv p `unionFV` tyCoFVsOfType t1
                         `unionFV` tyCoFVsOfType t2) fv_cand in_scope acc
    tyCoFVsOfCo (SymCo co)          fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfCo (TransCo co1 co2)   fv_cand in_scope acc = (tyCoFVsOfCo co1 `unionFV` tyCoFVsOfCo co2) fv_cand in_scope acc
    tyCoFVsOfCo (NthCo _ _ co)      fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfCo (LRCo _ co)         fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfCo (InstCo co arg)     fv_cand in_scope acc = (tyCoFVsOfCo co `unionFV` tyCoFVsOfCo arg) fv_cand in_scope acc
    tyCoFVsOfCo (KindCo co)         fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfCo (SubCo co)          fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfCo (AxiomRuleCo _ cs)  fv_cand in_scope acc = tyCoFVsOfCos cs fv_cand in_scope acc

    tyCoFVsOfCoVar :: CoVar -> FV
    tyCoFVsOfCoVar v fv_cand in_scope acc
      = (unitFV v `unionFV` tyCoFVsOfType (varType v)) fv_cand in_scope acc

    tyCoFVsOfProv :: UnivCoProvenance -> FV
    tyCoFVsOfProv (PhantomProv co)    fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfProv (ProofIrrelProv co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
    tyCoFVsOfProv (PluginProv _)      fv_cand in_scope acc = emptyFV fv_cand in_scope acc

    tyCoFVsOfCos :: [Coercion] -> FV
    tyCoFVsOfCos []       fv_cand in_scope acc = emptyFV fv_cand in_scope acc
    tyCoFVsOfCos (co:cos) fv_cand in_scope acc = (tyCoFVsOfCo co `unionFV` tyCoFVsOfCos cos) fv_cand in_scope acc

    ----- Whether a covar is /Almost Devoid/ in a type or coercion ----

    -- | Given a covar and a coercion, returns True if covar is almost devoid in
    -- the coercion. That is, covar can only appear in Refl and GRefl.
    -- See last wrinkle in Note [Unused coercion variable in ForAllCo] in "GHC.Core.Coercion"
    almostDevoidCoVarOfCo :: CoVar -> Coercion -> Bool
    almostDevoidCoVarOfCo cv co =
      almost_devoid_co_var_of_co co cv

    almost_devoid_co_var_of_co :: Coercion -> CoVar -> Bool
    almost_devoid_co_var_of_co (Refl {}) _ = True   -- covar is allowed in Refl and
    almost_devoid_co_var_of_co (GRefl {}) _ = True  -- GRefl, so we don't look into
                                                    -- the coercions
    almost_devoid_co_var_of_co (TyConAppCo _ _ cos) cv
      = almost_devoid_co_var_of_cos cos cv
    almost_devoid_co_var_of_co (AppCo co arg) cv
      = almost_devoid_co_var_of_co co cv
      && almost_devoid_co_var_of_co arg cv
    almost_devoid_co_var_of_co (ForAllCo v kind_co co) cv
      = almost_devoid_co_var_of_co kind_co cv
      && (v == cv || almost_devoid_co_var_of_co co cv)
    almost_devoid_co_var_of_co (FunCo _ w co1 co2) cv
      = almost_devoid_co_var_of_co w cv
      && almost_devoid_co_var_of_co co1 cv
      && almost_devoid_co_var_of_co co2 cv
    almost_devoid_co_var_of_co (CoVarCo v) cv = v /= cv
    almost_devoid_co_var_of_co (HoleCo h)  cv = (coHoleCoVar h) /= cv
    almost_devoid_co_var_of_co (AxiomInstCo _ _ cos) cv
      = almost_devoid_co_var_of_cos cos cv
    almost_devoid_co_var_of_co (UnivCo p _ t1 t2) cv
      = almost_devoid_co_var_of_prov p cv
      && almost_devoid_co_var_of_type t1 cv
      && almost_devoid_co_var_of_type t2 cv
    almost_devoid_co_var_of_co (SymCo co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_co (TransCo co1 co2) cv
      = almost_devoid_co_var_of_co co1 cv
      && almost_devoid_co_var_of_co co2 cv
    almost_devoid_co_var_of_co (NthCo _ _ co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_co (LRCo _ co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_co (InstCo co arg) cv
      = almost_devoid_co_var_of_co co cv
      && almost_devoid_co_var_of_co arg cv
    almost_devoid_co_var_of_co (KindCo co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_co (SubCo co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_co (AxiomRuleCo _ cs) cv
      = almost_devoid_co_var_of_cos cs cv

    almost_devoid_co_var_of_cos :: [Coercion] -> CoVar -> Bool
    almost_devoid_co_var_of_cos [] _ = True
    almost_devoid_co_var_of_cos (co:cos) cv
      = almost_devoid_co_var_of_co co cv
      && almost_devoid_co_var_of_cos cos cv

    almost_devoid_co_var_of_prov :: UnivCoProvenance -> CoVar -> Bool
    almost_devoid_co_var_of_prov (PhantomProv co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_prov (ProofIrrelProv co) cv
      = almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_prov (PluginProv _) _ = True

    almost_devoid_co_var_of_type :: Type -> CoVar -> Bool
    almost_devoid_co_var_of_type (TyVarTy _) _ = True
    almost_devoid_co_var_of_type (TyConApp _ tys) cv
      = almost_devoid_co_var_of_types tys cv
    almost_devoid_co_var_of_type (LitTy {}) _ = True
    almost_devoid_co_var_of_type (AppTy fun arg) cv
      = almost_devoid_co_var_of_type fun cv
      && almost_devoid_co_var_of_type arg cv
    almost_devoid_co_var_of_type (FunTy _ w arg res) cv
      = almost_devoid_co_var_of_type w cv
      && almost_devoid_co_var_of_type arg cv
      && almost_devoid_co_var_of_type res cv
    almost_devoid_co_var_of_type (ForAllTy (Bndr v _) ty) cv
      = almost_devoid_co_var_of_type (varType v) cv
      && (v == cv || almost_devoid_co_var_of_type ty cv)
    almost_devoid_co_var_of_type (CastTy ty co) cv
      = almost_devoid_co_var_of_type ty cv
      && almost_devoid_co_var_of_co co cv
    almost_devoid_co_var_of_type (CoercionTy co) cv
      = almost_devoid_co_var_of_co co cv

    almost_devoid_co_var_of_types :: [Type] -> CoVar -> Bool
    almost_devoid_co_var_of_types [] _ = True
    almost_devoid_co_var_of_types (ty:tys) cv
      = almost_devoid_co_var_of_type ty cv
      && almost_devoid_co_var_of_types tys cv

    {- *********************************************************************
    *                                                                      *
                     Injective free vars
    *                                                                      *
    ********************************************************************* -}

    -- | Returns the free variables of a 'Type' that are in injective positions.
    -- Specifically, it finds the free variables while:
    --
    -- * Expanding type synonyms
    --
    -- * Ignoring the coercion in @(ty |> co)@
    --
    -- * Ignoring the non-injective fields of a 'TyConApp'
    --
    --
    -- For example, if @F@ is a non-injective type family, then:
    --
    -- @
    -- injectiveTyVarsOf( Either c (Maybe (a, F b c)) ) = {a,c}
    -- @
    --
    -- If @'injectiveVarsOfType' ty = itvs@, then knowing @ty@ fixes @itvs@.
    -- More formally, if
    -- @a@ is in @'injectiveVarsOfType' ty@
    -- and  @S1(ty) ~ S2(ty)@,
    -- then @S1(a)  ~ S2(a)@,
    -- where @S1@ and @S2@ are arbitrary substitutions.
    --
    -- See @Note [When does a tycon application need an explicit kind signature?]@.
    injectiveVarsOfType :: Bool   -- ^ Should we look under injective type families?
                                  -- See Note [Coverage condition for injective type families]
                                  -- in "GHC.Tc.Instance.Family".
                        -> Type -> FV
    injectiveVarsOfType look_under_tfs = go
      where
        go ty                  | Just ty' <- coreView ty
                               = go ty'
        go (TyVarTy v)         = unitFV v `unionFV` go (tyVarKind v)
        go (AppTy f a)         = go f `unionFV` go a
        go (FunTy _ w ty1 ty2) = go w `unionFV` go ty1 `unionFV` go ty2
        go (TyConApp tc tys)   =
          case tyConInjectivityInfo tc of
            Injective inj
              |  look_under_tfs || not (isTypeFamilyTyCon tc)
              -> mapUnionFV go $
                 filterByList (inj ++ repeat True) tys
                             -- Oversaturated arguments to a tycon are
                             -- always injective, hence the repeat True
            _ -> emptyFV
        go (ForAllTy (Bndr tv _) ty) = go (tyVarKind tv) `unionFV` delFV tv (go ty)
        go LitTy{}                   = emptyFV
        go (CastTy ty _)             = go ty
        go CoercionTy{}              = emptyFV

    -- | Returns the free variables of a 'Type' that are in injective positions.
    -- Specifically, it finds the free variables while:
    --
    -- * Expanding type synonyms
    --
    -- * Ignoring the coercion in @(ty |> co)@
    --
    -- * Ignoring the non-injective fields of a 'TyConApp'
    --
    -- See @Note [When does a tycon application need an explicit kind signature?]@.
    injectiveVarsOfTypes :: Bool -- ^ look under injective type families?
                                 -- See Note [Coverage condition for injective type families]
                                 -- in "GHC.Tc.Instance.Family".
                         -> [Type] -> FV
    injectiveVarsOfTypes look_under_tfs = mapUnionFV (injectiveVarsOfType look_under_tfs)

    {- *********************************************************************
    *                                                                      *
                     Invisible vars
    *                                                                      *
    ********************************************************************* -}

    -- | Returns the set of variables that are used invisibly anywhere within
    -- the given type. A variable will be included even if it is used both visibly
    -- and invisibly. An invisible use site includes:
    --   * In the kind of a variable
    --   * In the kind of a bound variable in a forall
    --   * In a coercion
    --   * In a Specified or Inferred argument to a function
    -- See Note [VarBndrs, TyCoVarBinders, TyConBinders, and visibility] in "GHC.Core.TyCo.Rep"
    invisibleVarsOfType :: Type -> FV
    invisibleVarsOfType = go
      where
        go ty                 | Just ty' <- coreView ty
                              = go ty'
        go (TyVarTy v)        = go (tyVarKind v)
        go (AppTy f a)        = go f `unionFV` go a
        go (FunTy _ w ty1 ty2) = go w `unionFV` go ty1 `unionFV` go ty2
        go (TyConApp tc tys)  = tyCoFVsOfTypes invisibles `unionFV`
                                invisibleVarsOfTypes visibles
          where (invisibles, visibles) = partitionInvisibleTypes tc tys
        go (ForAllTy tvb ty)  = tyCoFVsBndr tvb $ go ty
        go LitTy{}            = emptyFV
        go (CastTy ty co)     = tyCoFVsOfCo co `unionFV` go ty
        go (CoercionTy co)    = tyCoFVsOfCo co

    -- | Like 'invisibleVarsOfType', but for many types.
    invisibleVarsOfTypes :: [Type] -> FV
    invisibleVarsOfTypes = mapUnionFV invisibleVarsOfType

    {- *********************************************************************
    *                                                                      *
                     Any free vars
    *                                                                      *
    ********************************************************************* -}

    {-# INLINE afvFolder #-}   -- so that specialization to (const True) works
    afvFolder :: (TyCoVar -> Bool) -> TyCoFolder TyCoVarSet DM.Any
    afvFolder check_fv = TyCoFolder { tcf_view = noView
                                    , tcf_tyvar = do_tcv, tcf_covar = do_tcv
                                    , tcf_hole = do_hole, tcf_tycobinder = do_bndr }
      where
        do_tcv is tv = Any (not (tv `elemVarSet` is) && check_fv tv)
        do_hole _ _  = Any False    -- I'm unsure; probably never happens
        do_bndr is tv _ = is `extendVarSet` tv

    anyFreeVarsOfType :: (TyCoVar -> Bool) -> Type -> Bool
    anyFreeVarsOfType check_fv ty = DM.getAny (f ty)
      where (f, _, _, _) = foldTyCo (afvFolder check_fv) emptyVarSet

    anyFreeVarsOfTypes :: (TyCoVar -> Bool) -> [Type] -> Bool
    anyFreeVarsOfTypes check_fv tys = DM.getAny (f tys)
      where (_, f, _, _) = foldTyCo (afvFolder check_fv) emptyVarSet

    anyFreeVarsOfCo :: (TyCoVar -> Bool) -> Coercion -> Bool
    anyFreeVarsOfCo check_fv co = DM.getAny (f co)
      where (_, _, f, _) = foldTyCo (afvFolder check_fv) emptyVarSet

    noFreeVarsOfType :: Type -> Bool
    noFreeVarsOfType ty = not $ DM.getAny (f ty)
      where (f, _, _, _) = foldTyCo (afvFolder (const True)) emptyVarSet

    noFreeVarsOfTypes :: [Type] -> Bool
    noFreeVarsOfTypes tys = not $ DM.getAny (f tys)
      where (_, f, _, _) = foldTyCo (afvFolder (const True)) emptyVarSet

    noFreeVarsOfCo :: Coercion -> Bool
    noFreeVarsOfCo co = not $ DM.getAny (f co)
      where (_, _, f, _) = foldTyCo (afvFolder (const True)) emptyVarSet

    {- *********************************************************************
    *                                                                      *
                     scopedSort
    *                                                                      *
    ********************************************************************* -}

    {- Note [ScopedSort]
    ~~~~~~~~~~~~~~~~~~~~
    Consider

      foo :: Proxy a -> Proxy (b :: k) -> Proxy (a :: k2) -> ()

    This function type is implicitly generalised over [a, b, k, k2]. These
    variables will be Specified; that is, they will be available for visible
    type application. This is because they are written in the type signature
    by the user.

    However, we must ask: what order will they appear in? In cases without
    dependency, this is easy: we just use the lexical left-to-right ordering
    of first occurrence. With dependency, we cannot get off the hook so
    easily.

    We thus state:

     * These variables appear in the order as given by ScopedSort, where
       the input to ScopedSort is the left-to-right order of first occurrence.

    Note that this applies only to *implicit* quantification, without a
    `forall`. If the user writes a `forall`, then we just use the order given.

    ScopedSort is defined thusly (as proposed in #15743):
      * Work left-to-right through the input list, with a cursor.
      * If variable v at the cursor is depended on by any earlier variable w,
        move v immediately before the leftmost such w.

    INVARIANT: The prefix of variables before the cursor form a valid telescope.

    Note that ScopedSort makes sense only after type inference is done and all
    types/kinds are fully settled and zonked.

    -}

    -- | Do a topological sort on a list of tyvars,
    --   so that binders occur before occurrences
    -- E.g. given  [ a::k, k::*, b::k ]
    -- it'll return a well-scoped list [ k::*, a::k, b::k ]
    --
    -- This is a deterministic sorting operation
    -- (that is, doesn't depend on Uniques).
    --
    -- It is also meant to be stable: that is, variables should not
    -- be reordered unnecessarily. This is specified in Note [ScopedSort]
    -- See also Note [Ordering of implicit variables] in "GHC.Rename.HsType"

    scopedSort :: [TyCoVar] -> [TyCoVar]
    scopedSort = go [] []
      where
        go :: [TyCoVar] -- already sorted, in reverse order
           -> [TyCoVarSet] -- each set contains all the variables which must be placed
                           -- before the tv corresponding to the set; they are accumulations
                           -- of the fvs in the sorted tvs' kinds

                           -- This list is in 1-to-1 correspondence with the sorted tyvars
                           -- INVARIANT:
                           --   all (\tl -> all (`subVarSet` head tl) (tail tl)) (tails fv_list)
                           -- That is, each set in the list is a superset of all later sets.

           -> [TyCoVar] -- yet to be sorted
           -> [TyCoVar]
        go acc _fv_list [] = reverse acc
        go acc  fv_list (tv:tvs)
          = go acc' fv_list' tvs
          where
            (acc', fv_list') = insert tv acc fv_list

        insert :: TyCoVar       -- var to insert
               -> [TyCoVar]     -- sorted list, in reverse order
               -> [TyCoVarSet]  -- list of fvs, as above
               -> ([TyCoVar], [TyCoVarSet])   -- augmented lists
        insert tv []     []         = ([tv], [tyCoVarsOfType (tyVarKind tv)])
        insert tv (a:as) (fvs:fvss)
          | tv `elemVarSet` fvs
          , (as', fvss') <- insert tv as fvss
          = (a:as', fvs `unionVarSet` fv_tv : fvss')

          | otherwise
          = (tv:a:as, fvs `unionVarSet` fv_tv : fvs : fvss)
          where
            fv_tv = tyCoVarsOfType (tyVarKind tv)

           -- lists not in correspondence
        insert _ _ _ = panic "scopedSort"

    -- | Get the free vars of a type in scoped order
    tyCoVarsOfTypeWellScoped :: Type -> [TyVar]
    tyCoVarsOfTypeWellScoped = scopedSort . tyCoVarsOfTypeList

    -- | Get the free vars of types in scoped order
    tyCoVarsOfTypesWellScoped :: [Type] -> [TyVar]
    tyCoVarsOfTypesWellScoped = scopedSort . tyCoVarsOfTypesList